-This is ../../gmp/doc/gmp.info, produced by makeinfo version 4.8 from
-../../gmp/doc/gmp.texi.
+This is gmp.info, produced by makeinfo version 6.7 from gmp.texi.
- This manual describes how to install and use the GNU multiple
-precision arithmetic library, version 5.0.1.
+This manual describes how to install and use the GNU multiple precision
+arithmetic library, version 6.2.1.
- Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
-2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free
-Software Foundation, Inc.
-
- Permission is granted to copy, distribute and/or modify this
-document under the terms of the GNU Free Documentation License, Version
-1.3 or any later version published by the Free Software Foundation;
-with no Invariant Sections, with the Front-Cover Texts being "A GNU
-Manual", and with the Back-Cover Texts being "You have freedom to copy
-and modify this GNU Manual, like GNU software". A copy of the license
-is included in *Note GNU Free Documentation License::.
+ Copyright 1991, 1993-2016, 2018-2020 Free Software Foundation, Inc.
+ Permission is granted to copy, distribute and/or modify this document
+under the terms of the GNU Free Documentation License, Version 1.3 or
+any later version published by the Free Software Foundation; with no
+Invariant Sections, with the Front-Cover Texts being "A GNU Manual", and
+with the Back-Cover Texts being "You have freedom to copy and modify
+this GNU Manual, like GNU software". A copy of the license is included
+in *note GNU Free Documentation License::.
INFO-DIR-SECTION GNU libraries
START-INFO-DIR-ENTRY
* gmp: (gmp). GNU Multiple Precision Arithmetic Library.
END-INFO-DIR-ENTRY
+\1f
+File: gmp.info, Node: Exact Remainder, Next: Small Quotient Division, Prev: Exact Division, Up: Division Algorithms
+
+15.2.6 Exact Remainder
+----------------------
+
+If the exact division algorithm is done with a full subtraction at each
+stage and the dividend isn't a multiple of the divisor, then low zero
+limbs are produced but with a remainder in the high limbs. For dividend
+a, divisor d, quotient q, and b = 2^mp_bits_per_limb, this remainder r
+is of the form
+
+ a = q*d + r*b^n
+
+ n represents the number of zero limbs produced by the subtractions,
+that being the number of limbs produced for q. r will be in the range
+0<=r<d and can be viewed as a remainder, but one shifted up by a factor
+of b^n.
+
+ Carrying out full subtractions at each stage means the same number of
+cross products must be done as a normal division, but there's still some
+single limb divisions saved. When d is a single limb some
+simplifications arise, providing good speedups on a number of
+processors.
+
+ The functions 'mpn_divexact_by3', 'mpn_modexact_1_odd' and the
+internal 'mpn_redc_X' functions differ subtly in how they return r,
+leading to some negations in the above formula, but all are essentially
+the same.
+
+ Clearly r is zero when a is a multiple of d, and this leads to
+divisibility or congruence tests which are potentially more efficient
+than a normal division.
+
+ The factor of b^n on r can be ignored in a GCD when d is odd, hence
+the use of 'mpn_modexact_1_odd' by 'mpn_gcd_1' and 'mpz_kronecker_ui'
+etc (*note Greatest Common Divisor Algorithms::).
+
+ Montgomery's REDC method for modular multiplications uses operands of
+the form of x*b^-n and y*b^-n and on calculating (x*b^-n)*(y*b^-n) uses
+the factor of b^n in the exact remainder to reach a product in the same
+form (x*y)*b^-n (*note Modular Powering Algorithm::).
+
+ Notice that r generally gives no useful information about the
+ordinary remainder a mod d since b^n mod d could be anything. If
+however b^n == 1 mod d, then r is the negative of the ordinary
+remainder. This occurs whenever d is a factor of b^n-1, as for example
+with 3 in 'mpn_divexact_by3'. For a 32 or 64 bit limb other such
+factors include 5, 17 and 257, but no particular use has been found for
+this.
+
+\1f
+File: gmp.info, Node: Small Quotient Division, Prev: Exact Remainder, Up: Division Algorithms
+
+15.2.7 Small Quotient Division
+------------------------------
+
+An NxM division where the number of quotient limbs Q=N-M is small can be
+optimized somewhat.
+
+ An ordinary basecase division normalizes the divisor by shifting it
+to make the high bit set, shifting the dividend accordingly, and
+shifting the remainder back down at the end of the calculation. This is
+wasteful if only a few quotient limbs are to be formed. Instead a
+division of just the top 2*Q limbs of the dividend by the top Q limbs of
+the divisor can be used to form a trial quotient. This requires only
+those limbs normalized, not the whole of the divisor and dividend.
+
+ A multiply and subtract then applies the trial quotient to the M-Q
+unused limbs of the divisor and N-Q dividend limbs (which includes Q
+limbs remaining from the trial quotient division). The starting trial
+quotient can be 1 or 2 too big, but all cases of 2 too big and most
+cases of 1 too big are detected by first comparing the most significant
+limbs that will arise from the subtraction. An addback is done if the
+quotient still turns out to be 1 too big.
+
+ This whole procedure is essentially the same as one step of the
+basecase algorithm done in a Q limb base, though with the trial quotient
+test done only with the high limbs, not an entire Q limb "digit"
+product. The correctness of this weaker test can be established by
+following the argument of Knuth section 4.3.1 exercise 20 but with the
+v2*q>b*r+u2 condition appropriately relaxed.
+
+\1f
+File: gmp.info, Node: Greatest Common Divisor Algorithms, Next: Powering Algorithms, Prev: Division Algorithms, Up: Algorithms
+
+15.3 Greatest Common Divisor
+============================
+
+* Menu:
+
+* Binary GCD::
+* Lehmer's Algorithm::
+* Subquadratic GCD::
+* Extended GCD::
+* Jacobi Symbol::
+
+\1f
+File: gmp.info, Node: Binary GCD, Next: Lehmer's Algorithm, Prev: Greatest Common Divisor Algorithms, Up: Greatest Common Divisor Algorithms
+
+15.3.1 Binary GCD
+-----------------
+
+At small sizes GMP uses an O(N^2) binary style GCD. This is described
+in many textbooks, for example Knuth section 4.5.2 algorithm B. It
+simply consists of successively reducing odd operands a and b using
+
+ a,b = abs(a-b),min(a,b)
+ strip factors of 2 from a
+
+ The Euclidean GCD algorithm, as per Knuth algorithms E and A,
+repeatedly computes the quotient q = floor(a/b) and replaces a,b by v, u
+- q v. The binary algorithm has so far been found to be faster than the
+Euclidean algorithm everywhere. One reason the binary method does well
+is that the implied quotient at each step is usually small, so often
+only one or two subtractions are needed to get the same effect as a
+division. Quotients 1, 2 and 3 for example occur 67.7% of the time, see
+Knuth section 4.5.3 Theorem E.
+
+ When the implied quotient is large, meaning b is much smaller than a,
+then a division is worthwhile. This is the basis for the initial a mod
+b reductions in 'mpn_gcd' and 'mpn_gcd_1' (the latter for both Nx1 and
+1x1 cases). But after that initial reduction, big quotients occur too
+rarely to make it worth checking for them.
+
+
+ The final 1x1 GCD in 'mpn_gcd_1' is done in the generic C code as
+described above. For two N-bit operands, the algorithm takes about 0.68
+iterations per bit. For optimum performance some attention needs to be
+paid to the way the factors of 2 are stripped from a.
+
+ Firstly it may be noted that in twos complement the number of low
+zero bits on a-b is the same as b-a, so counting or testing can begin on
+a-b without waiting for abs(a-b) to be determined.
+
+ A loop stripping low zero bits tends not to branch predict well,
+since the condition is data dependent. But on average there's only a
+few low zeros, so an option is to strip one or two bits arithmetically
+then loop for more (as done for AMD K6). Or use a lookup table to get a
+count for several bits then loop for more (as done for AMD K7). An
+alternative approach is to keep just one of a or b odd and iterate
+
+ a,b = abs(a-b), min(a,b)
+ a = a/2 if even
+ b = b/2 if even
+
+ This requires about 1.25 iterations per bit, but stripping of a
+single bit at each step avoids any branching. Repeating the bit strip
+reduces to about 0.9 iterations per bit, which may be a worthwhile
+tradeoff.
+
+ Generally with the above approaches a speed of perhaps 6 cycles per
+bit can be achieved, which is still not terribly fast with for instance
+a 64-bit GCD taking nearly 400 cycles. It's this sort of time which
+means it's not usually advantageous to combine a set of divisibility
+tests into a GCD.
+
+ Currently, the binary algorithm is used for GCD only when N < 3.
+
+\1f
+File: gmp.info, Node: Lehmer's Algorithm, Next: Subquadratic GCD, Prev: Binary GCD, Up: Greatest Common Divisor Algorithms
+
+15.3.2 Lehmer's algorithm
+-------------------------
+
+Lehmer's improvement of the Euclidean algorithms is based on the
+observation that the initial part of the quotient sequence depends only
+on the most significant parts of the inputs. The variant of Lehmer's
+algorithm used in GMP splits off the most significant two limbs, as
+suggested, e.g., in "A Double-Digit Lehmer-Euclid Algorithm" by Jebelean
+(*note References::). The quotients of two double-limb inputs are
+collected as a 2 by 2 matrix with single-limb elements. This is done by
+the function 'mpn_hgcd2'. The resulting matrix is applied to the inputs
+using 'mpn_mul_1' and 'mpn_submul_1'. Each iteration usually reduces
+the inputs by almost one limb. In the rare case of a large quotient, no
+progress can be made by examining just the most significant two limbs,
+and the quotient is computed using plain division.
+
+ The resulting algorithm is asymptotically O(N^2), just as the
+Euclidean algorithm and the binary algorithm. The quadratic part of the
+work are the calls to 'mpn_mul_1' and 'mpn_submul_1'. For small sizes,
+the linear work is also significant. There are roughly N calls to the
+'mpn_hgcd2' function. This function uses a couple of important
+optimizations:
+
+ * It uses the same relaxed notion of correctness as 'mpn_hgcd' (see
+ next section). This means that when called with the most
+ significant two limbs of two large numbers, the returned matrix
+ does not always correspond exactly to the initial quotient sequence
+ for the two large numbers; the final quotient may sometimes be one
+ off.
+
+ * It takes advantage of the fact the quotients are usually small.
+ The division operator is not used, since the corresponding
+ assembler instruction is very slow on most architectures. (This
+ code could probably be improved further, it uses many branches that
+ are unfriendly to prediction).
+
+ * It switches from double-limb calculations to single-limb
+ calculations half-way through, when the input numbers have been
+ reduced in size from two limbs to one and a half.
+
+\1f
+File: gmp.info, Node: Subquadratic GCD, Next: Extended GCD, Prev: Lehmer's Algorithm, Up: Greatest Common Divisor Algorithms
+
+15.3.3 Subquadratic GCD
+-----------------------
+
+For inputs larger than 'GCD_DC_THRESHOLD', GCD is computed via the HGCD
+(Half GCD) function, as a generalization to Lehmer's algorithm.
+
+ Let the inputs a,b be of size N limbs each. Put S = floor(N/2) + 1.
+Then HGCD(a,b) returns a transformation matrix T with non-negative
+elements, and reduced numbers (c;d) = T^{-1} (a;b). The reduced numbers
+c,d must be larger than S limbs, while their difference abs(c-d) must
+fit in S limbs. The matrix elements will also be of size roughly N/2.
+
+ The HGCD base case uses Lehmer's algorithm, but with the above stop
+condition that returns reduced numbers and the corresponding
+transformation matrix half-way through. For inputs larger than
+'HGCD_THRESHOLD', HGCD is computed recursively, using the divide and
+conquer algorithm in "On Schönhage's algorithm and subquadratic integer
+GCD computation" by Möller (*note References::). The recursive
+algorithm consists of these main steps.
+
+ * Call HGCD recursively, on the most significant N/2 limbs. Apply
+ the resulting matrix T_1 to the full numbers, reducing them to a
+ size just above 3N/2.
+
+ * Perform a small number of division or subtraction steps to reduce
+ the numbers to size below 3N/2. This is essential mainly for the
+ unlikely case of large quotients.
+
+ * Call HGCD recursively, on the most significant N/2 limbs of the
+ reduced numbers. Apply the resulting matrix T_2 to the full
+ numbers, reducing them to a size just above N/2.
+
+ * Compute T = T_1 T_2.
+
+ * Perform a small number of division and subtraction steps to satisfy
+ the requirements, and return.
+
+ GCD is then implemented as a loop around HGCD, similarly to Lehmer's
+algorithm. Where Lehmer repeatedly chops off the top two limbs, calls
+'mpn_hgcd2', and applies the resulting matrix to the full numbers, the
+sub-quadratic GCD chops off the most significant third of the limbs (the
+proportion is a tuning parameter, and 1/3 seems to be more efficient
+than, e.g, 1/2), calls 'mpn_hgcd', and applies the resulting matrix.
+Once the input numbers are reduced to size below 'GCD_DC_THRESHOLD',
+Lehmer's algorithm is used for the rest of the work.
+
+ The asymptotic running time of both HGCD and GCD is O(M(N)*log(N)),
+where M(N) is the time for multiplying two N-limb numbers.
+
+\1f
+File: gmp.info, Node: Extended GCD, Next: Jacobi Symbol, Prev: Subquadratic GCD, Up: Greatest Common Divisor Algorithms
+
+15.3.4 Extended GCD
+-------------------
+
+The extended GCD function, or GCDEXT, calculates gcd(a,b) and also
+cofactors x and y satisfying a*x+b*y=gcd(a,b). All the algorithms used
+for plain GCD are extended to handle this case. The binary algorithm is
+used only for single-limb GCDEXT. Lehmer's algorithm is used for sizes
+up to 'GCDEXT_DC_THRESHOLD'. Above this threshold, GCDEXT is
+implemented as a loop around HGCD, but with more book-keeping to keep
+track of the cofactors. This gives the same asymptotic running time as
+for GCD and HGCD, O(M(N)*log(N))
+
+ One difference to plain GCD is that while the inputs a and b are
+reduced as the algorithm proceeds, the cofactors x and y grow in size.
+This makes the tuning of the chopping-point more difficult. The current
+code chops off the most significant half of the inputs for the call to
+HGCD in the first iteration, and the most significant two thirds for the
+remaining calls. This strategy could surely be improved. Also the stop
+condition for the loop, where Lehmer's algorithm is invoked once the
+inputs are reduced below 'GCDEXT_DC_THRESHOLD', could maybe be improved
+by taking into account the current size of the cofactors.
+
+\1f
+File: gmp.info, Node: Jacobi Symbol, Prev: Extended GCD, Up: Greatest Common Divisor Algorithms
+
+15.3.5 Jacobi Symbol
+--------------------
+
+Jacobi symbol (A/B)
+
+ Initially if either operand fits in a single limb, a reduction is
+done with either 'mpn_mod_1' or 'mpn_modexact_1_odd', followed by the
+binary algorithm on a single limb. The binary algorithm is well suited
+to a single limb, and the whole calculation in this case is quite
+efficient.
+
+ For inputs larger than 'GCD_DC_THRESHOLD', 'mpz_jacobi',
+'mpz_legendre' and 'mpz_kronecker' are computed via the HGCD (Half GCD)
+function, as a generalization to Lehmer's algorithm.
+
+ Most GCD algorithms reduce a and b by repeatatily computing the
+quotient q = floor(a/b) and iteratively replacing
+
+ a, b = b, a - q * b
+
+ Different algorithms use different methods for calculating q, but the
+core algorithm is the same if we use *note Lehmer's Algorithm:: or *note
+HGCD: Subquadratic GCD.
+
+ At each step it is possible to compute if the reduction inverts the
+Jacobi symbol based on the two least significant bits of A and B. For
+more details see "Efficient computation of the Jacobi symbol" by Möller
+(*note References::).
+
+ A small set of bits is thus used to track state
+ * current sign of result (1 bit)
+
+ * two least significant bits of A and B (4 bits)
+
+ * a pointer to which input is currently the denominator (1 bit)
+
+ In all the routines sign changes for the result are accumulated using
+fast bit twiddling which avoids conditional jumps.
+
+ The final result is calculated after verifying the inputs are coprime
+(GCD = 1) by raising (-1)^e
+
+ Much of the HGCD code is shared directly with the HGCD
+implementations, such as the 2x2 matrix calculation, *Note Lehmer's
+Algorithm:: basecase and 'GCD_DC_THRESHOLD'.
+
+ The asymptotic running time is O(M(N)*log(N)), where M(N) is the time
+for multiplying two N-limb numbers.
+
\1f
File: gmp.info, Node: Powering Algorithms, Next: Root Extraction Algorithms, Prev: Greatest Common Divisor Algorithms, Up: Algorithms
-16.4 Powering Algorithms
+15.4 Powering Algorithms
========================
* Menu:
\1f
File: gmp.info, Node: Normal Powering Algorithm, Next: Modular Powering Algorithm, Prev: Powering Algorithms, Up: Powering Algorithms
-16.4.1 Normal Powering
+15.4.1 Normal Powering
----------------------
-Normal `mpz' or `mpf' powering uses a simple binary algorithm,
+Normal 'mpz' or 'mpf' powering uses a simple binary algorithm,
successively squaring and then multiplying by the base when a 1 bit is
seen in the exponent, as per Knuth section 4.6.3. The "left to right"
-variant described there is used rather than algorithm A, since it's
-just as easy and can be done with somewhat less temporary memory.
+variant described there is used rather than algorithm A, since it's just
+as easy and can be done with somewhat less temporary memory.
\1f
File: gmp.info, Node: Modular Powering Algorithm, Prev: Normal Powering Algorithm, Up: Powering Algorithms
-16.4.2 Modular Powering
+15.4.2 Modular Powering
-----------------------
Modular powering is implemented using a 2^k-ary sliding window
made to minimize the average number of multiplications that must
supplement the squaring.
- The modular multiplies and squares use either a simple division or
+ The modular multiplies and squarings use either a simple division or
the REDC method by Montgomery (*note References::). REDC is a little
faster, essentially saving N single limb divisions in a fashion similar
to an exact remainder (*note Exact Remainder::).
\1f
File: gmp.info, Node: Root Extraction Algorithms, Next: Radix Conversion Algorithms, Prev: Powering Algorithms, Up: Algorithms
-16.5 Root Extraction Algorithms
+15.5 Root Extraction Algorithms
===============================
* Menu:
\1f
File: gmp.info, Node: Square Root Algorithm, Next: Nth Root Algorithm, Prev: Root Extraction Algorithms, Up: Root Extraction Algorithms
-16.5.1 Square Root
+15.5.1 Square Root
------------------
Square roots are taken using the "Karatsuba Square Root" algorithm by
An input n is split into four parts of k bits each, so with b=2^k we
have n = a3*b^3 + a2*b^2 + a1*b + a0. Part a3 must be "normalized" so
-that either the high or second highest bit is set. In GMP, k is kept
-on a limb boundary and the input is left shifted (by an even number of
+that either the high or second highest bit is set. In GMP, k is kept on
+a limb boundary and the input is left shifted (by an even number of
bits) to normalize.
The square root of the high two parts is taken, by recursive
r = r + 2*s - 1
s = s - 1
- The algorithm is expressed in a divide and conquer form, but as
-noted in the paper it can also be viewed as a discrete variant of
-Newton's method, or as a variation on the schoolboy method (no longer
-taught) for square roots two digits at a time.
+ The algorithm is expressed in a divide and conquer form, but as noted
+in the paper it can also be viewed as a discrete variant of Newton's
+method, or as a variation on the schoolboy method (no longer taught) for
+square roots two digits at a time.
If the remainder r is not required then usually only a few high limbs
-of r and u need to be calculated to determine whether an adjustment to
-s is required. This optimization is not currently implemented.
+of r and u need to be calculated to determine whether an adjustment to s
+is required. This optimization is not currently implemented.
In the Karatsuba multiplication range this algorithm is
O(1.5*M(N/2)), where M(n) is the time to multiply two numbers of n
Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range.
The algorithm does all its calculations in integers and the resulting
-`mpn_sqrtrem' is used for both `mpz_sqrt' and `mpf_sqrt'. The extended
-precision given by `mpf_sqrt_ui' is obtained by padding with zero limbs.
+'mpn_sqrtrem' is used for both 'mpz_sqrt' and 'mpf_sqrt'. The extended
+precision given by 'mpf_sqrt_ui' is obtained by padding with zero limbs.
\1f
File: gmp.info, Node: Nth Root Algorithm, Next: Perfect Square Algorithm, Prev: Square Root Algorithm, Up: Root Extraction Algorithms
-16.5.2 Nth Root
+15.5.2 Nth Root
---------------
Integer Nth roots are taken using Newton's method with the following
The initial approximation a[1] is generated bitwise by successively
powering a trial root with or without new 1 bits, aiming to be just
-above the true root. The iteration converges quadratically when
-started from a good approximation. When n is large more initial bits
-are needed to get good convergence. The current implementation is not
-particularly well optimized.
+above the true root. The iteration converges quadratically when started
+from a good approximation. When n is large more initial bits are needed
+to get good convergence. The current implementation is not particularly
+well optimized.
\1f
File: gmp.info, Node: Perfect Square Algorithm, Next: Perfect Power Algorithm, Prev: Nth Root Algorithm, Up: Root Extraction Algorithms
-16.5.3 Perfect Square
+15.5.3 Perfect Square
---------------------
A significant fraction of non-squares can be quickly identified by
checking whether the input is a quadratic residue modulo small integers.
- `mpz_perfect_square_p' first tests the input mod 256, which means
-just examining the low byte. Only 44 different values occur for
-squares mod 256, so 82.8% of inputs can be immediately identified as
+ 'mpz_perfect_square_p' first tests the input mod 256, which means
+just examining the low byte. Only 44 different values occur for squares
+mod 256, so 82.8% of inputs can be immediately identified as
non-squares.
- On a 32-bit system similar tests are done mod 9, 5, 7, 13 and 17,
-for a total 99.25% of inputs identified as non-squares. On a 64-bit
-system 97 is tested too, for a total 99.62%.
+ On a 32-bit system similar tests are done mod 9, 5, 7, 13 and 17, for
+a total 99.25% of inputs identified as non-squares. On a 64-bit system
+97 is tested too, for a total 99.62%.
These moduli are chosen because they're factors of 2^24-1 (or 2^48-1
for 64-bits), and such a remainder can be quickly taken just using
-additions (see `mpn_mod_34lsub1').
+additions (see 'mpn_mod_34lsub1').
- When nails are in use moduli are instead selected by the `gen-psqr.c'
-program and applied with an `mpn_mod_1'. The same 2^24-1 or 2^48-1
+ When nails are in use moduli are instead selected by the 'gen-psqr.c'
+program and applied with an 'mpn_mod_1'. The same 2^24-1 or 2^48-1
could be done with nails using some extra bit shifts, but this is not
currently implemented.
- In any case each modulus is applied to the `mpn_mod_34lsub1' or
-`mpn_mod_1' remainder and a table lookup identifies non-squares. By
-using a "modexact" style calculation, and suitably permuted tables,
-just one multiply each is required, see the code for details. Moduli
-are also combined to save operations, so long as the lookup tables
-don't become too big. `gen-psqr.c' does all the pre-calculations.
+ In any case each modulus is applied to the 'mpn_mod_34lsub1' or
+'mpn_mod_1' remainder and a table lookup identifies non-squares. By
+using a "modexact" style calculation, and suitably permuted tables, just
+one multiply each is required, see the code for details. Moduli are
+also combined to save operations, so long as the lookup tables don't
+become too big. 'gen-psqr.c' does all the pre-calculations.
A square root must still be taken for any value that passes these
tests, to verify it's really a square and not one of the small fraction
-of non-squares that get through (ie. a pseudo-square to all the tested
+of non-squares that get through (i.e. a pseudo-square to all the tested
bases).
- Clearly more residue tests could be done, `mpz_perfect_square_p' only
+ Clearly more residue tests could be done, 'mpz_perfect_square_p' only
uses a compact and efficient set. Big inputs would probably benefit
from more residue testing, small inputs might be better off with less.
The assumed distribution of squares versus non-squares in the input
\1f
File: gmp.info, Node: Perfect Power Algorithm, Prev: Perfect Square Algorithm, Up: Root Extraction Algorithms
-16.5.4 Perfect Power
+15.5.4 Perfect Power
--------------------
Detecting perfect powers is required by some factorization algorithms.
-Currently `mpz_perfect_power_p' is implemented using repeated Nth root
+Currently 'mpz_perfect_power_p' is implemented using repeated Nth root
extractions, though naturally only prime roots need to be considered.
(*Note Nth Root Algorithm::.)
\1f
File: gmp.info, Node: Radix Conversion Algorithms, Next: Other Algorithms, Prev: Root Extraction Algorithms, Up: Algorithms
-16.6 Radix Conversion
+15.6 Radix Conversion
=====================
Radix conversions are less important than other algorithms. A program
\1f
File: gmp.info, Node: Binary to Radix, Next: Radix to Binary, Prev: Radix Conversion Algorithms, Up: Radix Conversion Algorithms
-16.6.1 Binary to Radix
+15.6.1 Binary to Radix
----------------------
-Conversions from binary to a power-of-2 radix use a simple and fast
-O(N) bit extraction algorithm.
+Conversions from binary to a power-of-2 radix use a simple and fast O(N)
+bit extraction algorithm.
Conversions from binary to other radices use one of two algorithms.
-Sizes below `GET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method.
+Sizes below 'GET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method.
Repeated divisions by b^n are made, where b is the radix and n is the
biggest power that fits in a limb. But instead of simply using the
remainder r from such divisions, an extra divide step is done to give a
case code is provided for decimal, allowing multiplications by 10 to
optimize to shifts and adds.
- Above `GET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is
+ Above 'GET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is
used. For an input t, powers b^(n*2^i) of the radix are calculated,
until a power between t and sqrt(t) is reached. t is then divided by
that largest power, giving a quotient which is the digits above that
power, and a remainder which is those below. These two parts are in
-turn divided by the second highest power, and so on recursively. When
-a piece has been divided down to less than `GET_STR_DC_THRESHOLD'
-limbs, the basecase algorithm described above is used.
+turn divided by the second highest power, and so on recursively. When a
+piece has been divided down to less than 'GET_STR_DC_THRESHOLD' limbs,
+the basecase algorithm described above is used.
- The advantage of this algorithm is that big divisions can make use
-of the sub-quadratic divide and conquer division (*note Divide and
-Conquer Division::), and big divisions tend to have less overheads than
-lots of separate single limb divisions anyway. But in any case the
-cost of calculating the powers b^(n*2^i) must first be overcome.
+ The advantage of this algorithm is that big divisions can make use of
+the sub-quadratic divide and conquer division (*note Divide and Conquer
+Division::), and big divisions tend to have less overheads than lots of
+separate single limb divisions anyway. But in any case the cost of
+calculating the powers b^(n*2^i) must first be overcome.
- `GET_STR_PRECOMPUTE_THRESHOLD' and `GET_STR_DC_THRESHOLD' represent
+ 'GET_STR_PRECOMPUTE_THRESHOLD' and 'GET_STR_DC_THRESHOLD' represent
the same basic thing, the point where it becomes worth doing a big
-division to cut the input in half. `GET_STR_PRECOMPUTE_THRESHOLD'
+division to cut the input in half. 'GET_STR_PRECOMPUTE_THRESHOLD'
includes the cost of calculating the radix power required, whereas
-`GET_STR_DC_THRESHOLD' assumes that's already available, which is the
+'GET_STR_DC_THRESHOLD' assumes that's already available, which is the
case when recursing.
Since the base case produces digits from least to most significant
but they want to be stored from most to least, it's necessary to
calculate in advance how many digits there will be, or at least be sure
not to underestimate that. For GMP the number of input bits is
-multiplied by `chars_per_bit_exactly' from `mp_bases', rounding up.
-The result is either correct or one too big.
+multiplied by 'chars_per_bit_exactly' from 'mp_bases', rounding up. The
+result is either correct or one too big.
Examining some of the high bits of the input could increase the
chance of getting the exact number of digits, but an exact result every
time would not be practical, since in general the difference between
numbers 100... and 99... is only in the last few bits and the work to
-identify 99... might well be almost as much as a full conversion.
-
- `mpf_get_str' doesn't currently use the algorithm described here, it
-multiplies or divides by a power of b to move the radix point to the
-just above the highest non-zero digit (or at worst one above that
-location), then multiplies by b^n to bring out digits. This is O(N^2)
-and is certainly not optimal.
+identify 99... might well be almost as much as a full conversion.
The r/b^n scheme described above for using multiplications to bring
out digits might be useful for more than a single limb. Some brief
Another possible improvement for the sub-quadratic part would be to
arrange for radix powers that balanced the sizes of quotient and
-remainder produced, ie. the highest power would be an b^(n*k)
+remainder produced, i.e. the highest power would be an b^(n*k)
approximately equal to sqrt(t), not restricted to a 2^i factor. That
ought to smooth out a graph of times against sizes, but may or may not
be a net speedup.
\1f
File: gmp.info, Node: Radix to Binary, Prev: Binary to Radix, Up: Radix Conversion Algorithms
-16.6.2 Radix to Binary
+15.6.2 Radix to Binary
----------------------
*This section needs to be rewritten, it currently describes the
O(N) bitwise concatenation algorithm.
Conversions from other radices use one of two algorithms. Sizes
-below `SET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method. Groups
+below 'SET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method. Groups
of n digits are converted to limbs, where n is the biggest power of the
base b which will fit in a limb, then those groups are accumulated into
-the result by multiplying by b^n and adding. This saves
-multi-precision operations, as per Knuth section 4.4 part E (*note
-References::). Some special case code is provided for decimal, giving
-the compiler a chance to optimize multiplications by 10.
+the result by multiplying by b^n and adding. This saves multi-precision
+operations, as per Knuth section 4.4 part E (*note References::). Some
+special case code is provided for decimal, giving the compiler a chance
+to optimize multiplications by 10.
- Above `SET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is
+ Above 'SET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is
used. First groups of n digits are converted into limbs. Then adjacent
limbs are combined into limb pairs with x*b^n+y, where x and y are the
limbs. Adjacent limb pairs are combined into quads similarly with
-x*b^(2n)+y. This continues until a single block remains, that being
-the result.
+x*b^(2n)+y. This continues until a single block remains, that being the
+result.
The advantage of this method is that the multiplications for each x
are big blocks, allowing Karatsuba and higher algorithms to be used.
But the cost of calculating the powers b^(n*2^i) must be overcome.
-`SET_STR_PRECOMPUTE_THRESHOLD' usually ends up quite big, around 5000
+'SET_STR_PRECOMPUTE_THRESHOLD' usually ends up quite big, around 5000
digits, and on some processors much bigger still.
- `SET_STR_PRECOMPUTE_THRESHOLD' is based on the input digits (and
+ 'SET_STR_PRECOMPUTE_THRESHOLD' is based on the input digits (and
tuned for decimal), though it might be better based on a limb count, so
as to be independent of the base. But that sort of count isn't used by
the base case and so would need some sort of initial calculation or
estimate.
- The main reason `SET_STR_PRECOMPUTE_THRESHOLD' is so much bigger
-than the corresponding `GET_STR_PRECOMPUTE_THRESHOLD' is that
-`mpn_mul_1' is much faster than `mpn_divrem_1' (often by a factor of 5,
-or more).
+ The main reason 'SET_STR_PRECOMPUTE_THRESHOLD' is so much bigger than
+the corresponding 'GET_STR_PRECOMPUTE_THRESHOLD' is that 'mpn_mul_1' is
+much faster than 'mpn_divrem_1' (often by a factor of 5, or more).
\1f
File: gmp.info, Node: Other Algorithms, Next: Assembly Coding, Prev: Radix Conversion Algorithms, Up: Algorithms
-16.7 Other Algorithms
+15.7 Other Algorithms
=====================
* Menu:
\1f
File: gmp.info, Node: Prime Testing Algorithm, Next: Factorial Algorithm, Prev: Other Algorithms, Up: Other Algorithms
-16.7.1 Prime Testing
+15.7.1 Prime Testing
--------------------
-The primality testing in `mpz_probab_prime_p' (*note Number Theoretic
+The primality testing in 'mpz_probab_prime_p' (*note Number Theoretic
Functions::) first does some trial division by small factors and then
uses the Miller-Rabin probabilistic primality testing algorithm, as
described in Knuth section 4.5.4 algorithm P (*note References::).
prime, if not then n is definitely composite.
Any prime n will pass the test, but some composites do too. Such
-composites are known as strong pseudoprimes to base x. No n is a
-strong pseudoprime to more than 1/4 of all bases (see Knuth exercise
-22), hence with x chosen at random there's no more than a 1/4 chance a
-"probable prime" will in fact be composite.
+composites are known as strong pseudoprimes to base x. No n is a strong
+pseudoprime to more than 1/4 of all bases (see Knuth exercise 22), hence
+with x chosen at random there's no more than a 1/4 chance a "probable
+prime" will in fact be composite.
In fact strong pseudoprimes are quite rare, making the test much more
powerful than this analysis would suggest, but 1/4 is all that's proven
\1f
File: gmp.info, Node: Factorial Algorithm, Next: Binomial Coefficients Algorithm, Prev: Prime Testing Algorithm, Up: Other Algorithms
-16.7.2 Factorial
+15.7.2 Factorial
----------------
-Factorials are calculated by a combination of removal of twos,
-powering, and binary splitting. The procedure can be best illustrated
+Factorials are calculated by a combination of two algorithms. An idea
+is shared among them: to compute the odd part of the factorial; a final
+step takes account of the power of 2 term, by shifting.
+
+ For small n, the odd factor of n! is computed with the simple
+observation that it is equal to the product of all positive odd numbers
+smaller than n times the odd factor of [n/2]!, where [x] is the integer
+part of x, and so on recursively. The procedure can be best illustrated
with an example,
- 23! = 1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.18.19.20.21.22.23
+ 23! = (23.21.19.17.15.13.11.9.7.5.3)(11.9.7.5.3)(5.3)2^{19}
+
+ Current code collects all the factors in a single list, with a loop
+and no recursion, and compute the product, with no special care for
+repeated chunks.
-has factors of two removed,
+ When n is larger, computation pass trough prime sieving. An helper
+function is used, as suggested by Peter Luschny:
- 23! = 2^19.1.1.3.1.5.3.7.1.9.5.11.3.13.7.15.1.17.9.19.5.21.11.23
+ n
+ -----
+ n! | | L(p,n)
+ msf(n) = -------------- = | | p
+ [n/2]!^2.2^k p=3
-and the resulting terms collected up according to their multiplicity,
+ Where p ranges on odd prime numbers. The exponent k is chosen to
+obtain an odd integer number: k is the number of 1 bits in the binary
+representation of [n/2]. The function L(p,n) can be defined as zero
+when p is composite, and, for any prime p, it is computed with:
- 23! = 2^19.(3.5)^3.(7.9.11)^2.(13.15.17.19.21.23)
+ ---
+ \ n
+ L(p,n) = / [---] mod 2 <= log (n) .
+ --- p^i p
+ i>0
- Each sequence such as 13.15.17.19.21.23 is evaluated by splitting
-into every second term, as for instance (13.17.21).(15.19.23), and the
-same recursively on each half. This is implemented iteratively using
-some bit twiddling.
+ With this helper function, we are able to compute the odd part of n!
+using the recursion implied by n!=[n/2]!^2*msf(n)*2^k. The recursion
+stops using the small-n algorithm on some [n/2^i].
+
+ Both the above algorithms use binary splitting to compute the product
+of many small factors. At first as many products as possible are
+accumulated in a single register, generating a list of factors that fit
+in a machine word. This list is then split into halves, and the product
+is computed recursively.
Such splitting is more efficient than repeated Nx1 multiplies since
it forms big multiplies, allowing Karatsuba and higher algorithms to be
-used. And even below the Karatsuba threshold a big block of work can
-be more efficient for the basecase algorithm.
-
- Splitting into subsequences of every second term keeps the resulting
-products more nearly equal in size than would the simpler approach of
-say taking the first half and second half of the sequence. Nearly
-equal products are more efficient for the current multiply
-implementation.
+used. And even below the Karatsuba threshold a big block of work can be
+more efficient for the basecase algorithm.
\1f
File: gmp.info, Node: Binomial Coefficients Algorithm, Next: Fibonacci Numbers Algorithm, Prev: Factorial Algorithm, Up: Other Algorithms
-16.7.3 Binomial Coefficients
+15.7.3 Binomial Coefficients
----------------------------
Binomial coefficients C(n,k) are calculated by first arranging k <= n/2
It's easy to show that each denominator i will divide the product so
far, so the exact division algorithm is used (*note Exact Division::).
- The numerators n-k+i and denominators i are first accumulated into
-as many fit a limb, to save multi-precision operations, though for
-`mpz_bin_ui' this applies only to the divisors, since n is an `mpz_t'
+ The numerators n-k+i and denominators i are first accumulated into as
+many fit a limb, to save multi-precision operations, though for
+'mpz_bin_ui' this applies only to the divisors, since n is an 'mpz_t'
and n-k+i in general won't fit in a limb at all.
\1f
File: gmp.info, Node: Fibonacci Numbers Algorithm, Next: Lucas Numbers Algorithm, Prev: Binomial Coefficients Algorithm, Up: Other Algorithms
-16.7.4 Fibonacci Numbers
+15.7.4 Fibonacci Numbers
------------------------
-The Fibonacci functions `mpz_fib_ui' and `mpz_fib2_ui' are designed for
+The Fibonacci functions 'mpz_fib_ui' and 'mpz_fib2_ui' are designed for
calculating isolated F[n] or F[n],F[n-1] values efficiently.
- For small n, a table of single limb values in `__gmp_fib_table' is
-used. On a 32-bit limb this goes up to F[47], or on a 64-bit limb up
-to F[93]. For convenience the table starts at F[-1].
+ For small n, a table of single limb values in '__gmp_fib_table' is
+used. On a 32-bit limb this goes up to F[47], or on a 64-bit limb up to
+F[93]. For convenience the table starts at F[-1].
Beyond the table, values are generated with a binary powering
algorithm, calculating a pair F[n] and F[n-1] working from high to low
code for just those doesn't seem worthwhile. If they really mattered
it'd be better to extend the data table.
- Using a table avoids lots of calculations on small numbers, and
-makes small n go fast. A bigger table would make more small n go fast,
-it's just a question of balancing size against desired speed. For GMP
-the code is kept compact, with the emphasis primarily on a good
-powering algorithm.
+ Using a table avoids lots of calculations on small numbers, and makes
+small n go fast. A bigger table would make more small n go fast, it's
+just a question of balancing size against desired speed. For GMP the
+code is kept compact, with the emphasis primarily on a good powering
+algorithm.
- `mpz_fib2_ui' returns both F[n] and F[n-1], but `mpz_fib_ui' is only
+ 'mpz_fib2_ui' returns both F[n] and F[n-1], but 'mpz_fib_ui' is only
interested in F[n]. In this case the last step of the algorithm can
become one multiply instead of two squares. One of the following two
formulas is used, according as n is odd or even.
F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k
F[2k+1] here is the same as above, just rearranged to be a multiply.
-For interest, the 2*(-1)^k term both here and above can be applied
-just to the low limb of the calculation, without a carry or borrow into
+For interest, the 2*(-1)^k term both here and above can be applied just
+to the low limb of the calculation, without a carry or borrow into
further limbs, which saves some code size. See comments with
-`mpz_fib_ui' and the internal `mpn_fib2_ui' for how this is done.
+'mpz_fib_ui' and the internal 'mpn_fib2_ui' for how this is done.
\1f
File: gmp.info, Node: Lucas Numbers Algorithm, Next: Random Number Algorithms, Prev: Fibonacci Numbers Algorithm, Up: Other Algorithms
-16.7.5 Lucas Numbers
+15.7.5 Lucas Numbers
--------------------
-`mpz_lucnum2_ui' derives a pair of Lucas numbers from a pair of
+'mpz_lucnum2_ui' derives a pair of Lucas numbers from a pair of
Fibonacci numbers with the following simple formulas.
L[k] = F[k] + 2*F[k-1]
L[k-1] = 2*F[k] - F[k-1]
- `mpz_lucnum_ui' is only interested in L[n], and some work can be
+ 'mpz_lucnum_ui' is only interested in L[n], and some work can be
saved. Trailing zero bits on n can be handled with a single square
each.
L[2k] = L[k]^2 - 2*(-1)^k
And the lowest 1 bit can be handled with one multiply of a pair of
-Fibonacci numbers, similar to what `mpz_fib_ui' does.
+Fibonacci numbers, similar to what 'mpz_fib_ui' does.
L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k
\1f
File: gmp.info, Node: Random Number Algorithms, Prev: Lucas Numbers Algorithm, Up: Other Algorithms
-16.7.6 Random Numbers
+15.7.6 Random Numbers
---------------------
-For the `urandomb' functions, random numbers are generated simply by
+For the 'urandomb' functions, random numbers are generated simply by
concatenating bits produced by the generator. As long as the generator
has good randomness properties this will produce well-distributed N bit
numbers.
- For the `urandomm' functions, random numbers in a range 0<=R<N are
+ For the 'urandomm' functions, random numbers in a range 0<=R<N are
generated by taking values R of ceil(log2(N)) bits each until one
-satisfies R<N. This will normally require only one or two attempts,
-but the attempts are limited in case the generator is somehow
-degenerate and produces only 1 bits or similar.
+satisfies R<N. This will normally require only one or two attempts, but
+the attempts are limited in case the generator is somehow degenerate and
+produces only 1 bits or similar.
The Mersenne Twister generator is by Matsumoto and Nishimura (*note
References::). It has a non-repeating period of 2^19937-1, which is a
-Mersenne prime, hence the name of the generator. The state is 624
-words of 32-bits each, which is iterated with one XOR and shift for each
+Mersenne prime, hence the name of the generator. The state is 624 words
+of 32-bits each, which is iterated with one XOR and shift for each
32-bit word generated, making the algorithm very fast. Randomness
properties are also very good and this is the default algorithm used by
GMP.
Linear congruential generators are described in many text books, for
instance Knuth volume 2 (*note References::). With a modulus M and
-parameters A and C, a integer state S is iterated by the formula S <-
-A*S+C mod M. At each step the new state is a linear function of the
+parameters A and C, an integer state S is iterated by the formula S <-
+A*S+C mod M. At each step the new state is a linear function of the
previous, mod M, hence the name of the generator.
In GMP only moduli of the form 2^N are supported, and the current
implementation is not as well optimized as it could be. Overheads are
-significant when N is small, and when N is large clearly the multiply
-at each step will become slow. This is not a big concern, since the
+significant when N is small, and when N is large clearly the multiply at
+each step will become slow. This is not a big concern, since the
Mersenne Twister generator is better in every respect and is therefore
recommended for all normal applications.
For both generators the current state can be deduced by observing
enough output and applying some linear algebra (over GF(2) in the case
-of the Mersenne Twister). This generally means raw output is
-unsuitable for cryptographic applications without further hashing or
-the like.
+of the Mersenne Twister). This generally means raw output is unsuitable
+for cryptographic applications without further hashing or the like.
\1f
File: gmp.info, Node: Assembly Coding, Prev: Other Algorithms, Up: Algorithms
-16.8 Assembly Coding
+15.8 Assembly Coding
====================
-The assembly subroutines in GMP are the most significant source of
-speed at small to moderate sizes. At larger sizes algorithm selection
-becomes more important, but of course speedups in low level routines
-will still speed up everything proportionally.
+The assembly subroutines in GMP are the most significant source of speed
+at small to moderate sizes. At larger sizes algorithm selection becomes
+more important, but of course speedups in low level routines will still
+speed up everything proportionally.
Carry handling and widening multiplies that are important for GMP
-can't be easily expressed in C. GCC `asm' blocks help a lot and are
-provided in `longlong.h', but hand coding low level routines invariably
+can't be easily expressed in C. GCC 'asm' blocks help a lot and are
+provided in 'longlong.h', but hand coding low level routines invariably
offers a speedup over generic C by a factor of anything from 2 to 10.
* Menu:
\1f
File: gmp.info, Node: Assembly Code Organisation, Next: Assembly Basics, Prev: Assembly Coding, Up: Assembly Coding
-16.8.1 Code Organisation
+15.8.1 Code Organisation
------------------------
-The various `mpn' subdirectories contain machine-dependent code, written
-in C or assembly. The `mpn/generic' subdirectory contains default code,
+The various 'mpn' subdirectories contain machine-dependent code, written
+in C or assembly. The 'mpn/generic' subdirectory contains default code,
used when there's no machine-specific version of a particular file.
- Each `mpn' subdirectory is for an ISA family. Generally 32-bit and
+ Each 'mpn' subdirectory is for an ISA family. Generally 32-bit and
64-bit variants in a family cannot share code and have separate
directories. Within a family further subdirectories may exist for CPU
variants.
- In each directory a `nails' subdirectory may exist, holding code with
-nails support for that CPU variant. A `NAILS_SUPPORT' directive in each
+ In each directory a 'nails' subdirectory may exist, holding code with
+nails support for that CPU variant. A 'NAILS_SUPPORT' directive in each
file indicates the nails values the code handles. Nails code only
exists where it's faster, or promises to be faster, than plain code.
-There's no effort put into nails if they're not going to enhance a
-given CPU.
+There's no effort put into nails if they're not going to enhance a given
+CPU.
\1f
File: gmp.info, Node: Assembly Basics, Next: Assembly Carry Propagation, Prev: Assembly Code Organisation, Up: Assembly Coding
-16.8.2 Assembly Basics
+15.8.2 Assembly Basics
----------------------
-`mpn_addmul_1' and `mpn_submul_1' are the most important routines for
+'mpn_addmul_1' and 'mpn_submul_1' are the most important routines for
overall GMP performance. All multiplications and divisions come down to
-repeated calls to these. `mpn_add_n', `mpn_sub_n', `mpn_lshift' and
-`mpn_rshift' are next most important.
+repeated calls to these. 'mpn_add_n', 'mpn_sub_n', 'mpn_lshift' and
+'mpn_rshift' are next most important.
On some CPUs assembly versions of the internal functions
-`mpn_mul_basecase' and `mpn_sqr_basecase' give significant speedups,
+'mpn_mul_basecase' and 'mpn_sqr_basecase' give significant speedups,
mainly through avoiding function call overheads. They can also
potentially make better use of a wide superscalar processor, as can
-bigger primitives like `mpn_addmul_2' or `mpn_addmul_4'.
+bigger primitives like 'mpn_addmul_2' or 'mpn_addmul_4'.
The restrictions on overlaps between sources and destinations (*note
Low-level Functions::) are designed to facilitate a variety of
-implementations. For example, knowing `mpn_add_n' won't have partly
+implementations. For example, knowing 'mpn_add_n' won't have partly
overlapping sources and destination means reading can be done far ahead
of writing on superscalar processors, and loops can be vectorized on a
vector processor, depending on the carry handling.
\1f
File: gmp.info, Node: Assembly Carry Propagation, Next: Assembly Cache Handling, Prev: Assembly Basics, Up: Assembly Coding
-16.8.3 Carry Propagation
+15.8.3 Carry Propagation
------------------------
The problem that presents most challenges in GMP is propagating carries
-from one limb to the next. In functions like `mpn_addmul_1' and
-`mpn_add_n', carries are the only dependencies between limb operations.
+from one limb to the next. In functions like 'mpn_addmul_1' and
+'mpn_add_n', carries are the only dependencies between limb operations.
- On processors with carry flags, a straightforward CISC style `adc' is
-generally best. AMD K6 `mpn_addmul_1' however is an example of an
+ On processors with carry flags, a straightforward CISC style 'adc' is
+generally best. AMD K6 'mpn_addmul_1' however is an example of an
unusual set of circumstances where a branch works out better.
- On RISC processors generally an add and compare for overflow is
-used. This sort of thing can be seen in `mpn/generic/aors_n.c'. Some
-carry propagation schemes require 4 instructions, meaning at least 4
-cycles per limb, but other schemes may use just 1 or 2. On wide
-superscalar processors performance may be completely determined by the
-number of dependent instructions between carry-in and carry-out for
-each limb.
+ On RISC processors generally an add and compare for overflow is used.
+This sort of thing can be seen in 'mpn/generic/aors_n.c'. Some carry
+propagation schemes require 4 instructions, meaning at least 4 cycles
+per limb, but other schemes may use just 1 or 2. On wide superscalar
+processors performance may be completely determined by the number of
+dependent instructions between carry-in and carry-out for each limb.
On vector processors good use can be made of the fact that a carry
bit only very rarely propagates more than one limb. When adding a
single bit to a limb, there's only a carry out if that limb was
-`0xFF...FF' which on random data will be only 1 in 2^mp_bits_per_limb.
-`mpn/cray/add_n.c' is an example of this, it adds all limbs in
-parallel, adds one set of carry bits in parallel and then only rarely
-needs to fall through to a loop propagating further carries.
-
- On the x86s, GCC (as of version 2.95.2) doesn't generate
-particularly good code for the RISC style idioms that are necessary to
-handle carry bits in C. Often conditional jumps are generated where
-`adc' or `sbb' forms would be better. And so unfortunately almost any
-loop involving carry bits needs to be coded in assembly for best
-results.
+'0xFF...FF' which on random data will be only 1 in 2^mp_bits_per_limb.
+'mpn/cray/add_n.c' is an example of this, it adds all limbs in parallel,
+adds one set of carry bits in parallel and then only rarely needs to
+fall through to a loop propagating further carries.
+
+ On the x86s, GCC (as of version 2.95.2) doesn't generate particularly
+good code for the RISC style idioms that are necessary to handle carry
+bits in C. Often conditional jumps are generated where 'adc' or 'sbb'
+forms would be better. And so unfortunately almost any loop involving
+carry bits needs to be coded in assembly for best results.
\1f
File: gmp.info, Node: Assembly Cache Handling, Next: Assembly Functional Units, Prev: Assembly Carry Propagation, Up: Assembly Coding
-16.8.4 Cache Handling
+15.8.4 Cache Handling
---------------------
GMP aims to perform well both on operands that fit entirely in L1 cache
and those which don't.
- Basic routines like `mpn_add_n' or `mpn_lshift' are often used on
+ Basic routines like 'mpn_add_n' or 'mpn_lshift' are often used on
large operands, so L2 and main memory performance is important for them.
-`mpn_mul_1' and `mpn_addmul_1' are mostly used for multiply and square
+'mpn_mul_1' and 'mpn_addmul_1' are mostly used for multiply and square
basecases, so L1 performance matters most for them, unless assembly
-versions of `mpn_mul_basecase' and `mpn_sqr_basecase' exist, in which
+versions of 'mpn_mul_basecase' and 'mpn_sqr_basecase' exist, in which
case the remaining uses are mostly for larger operands.
For L2 or main memory operands, memory access times will almost
certainly be more than the calculation time. The aim therefore is to
maximize memory throughput, by starting a load of the next cache line
-while processing the contents of the previous one. Clearly this is
-only possible if the chip has a lock-up free cache or some sort of
-prefetch instruction. Most current chips have both these features.
+while processing the contents of the previous one. Clearly this is only
+possible if the chip has a lock-up free cache or some sort of prefetch
+instruction. Most current chips have both these features.
Prefetching sources combines well with loop unrolling, since a
prefetch can be initiated once per unrolled loop (or more than once if
On CPUs without write-allocate caches, prefetching destinations will
ensure individual stores don't go further down the cache hierarchy,
limiting bandwidth. Of course for calculations which are slow anyway,
-like `mpn_divrem_1', write-throughs might be fine.
+like 'mpn_divrem_1', write-throughs might be fine.
The distance ahead to prefetch will be determined by memory latency
versus throughput. The aim of course is to have data arriving
continuously, at peak throughput. Some CPUs have limits on the number
of fetches or prefetches in progress.
- If a special prefetch instruction doesn't exist then a plain load
-can be used, but in that case care must be taken not to attempt to read
-past the end of an operand, since that might produce a segmentation
+ If a special prefetch instruction doesn't exist then a plain load can
+be used, but in that case care must be taken not to attempt to read past
+the end of an operand, since that might produce a segmentation
violation.
Some CPUs or systems have hardware that detects sequential memory
\1f
File: gmp.info, Node: Assembly Functional Units, Next: Assembly Floating Point, Prev: Assembly Cache Handling, Up: Assembly Coding
-16.8.5 Functional Units
+15.8.5 Functional Units
-----------------------
When choosing an approach for an assembly loop, consideration is given
Loop control will generally require a counter and pointer updates,
costing as much as 5 instructions, plus any delays a branch introduces.
CPU addressing modes might reduce pointer updates, perhaps by allowing
-just one updating pointer and others expressed as offsets from it, or
-on CISC chips with all addressing done with the loop counter as a
-scaled index.
+just one updating pointer and others expressed as offsets from it, or on
+CISC chips with all addressing done with the loop counter as a scaled
+index.
The final loop control cost can be amortised by processing several
limbs in each iteration (*note Assembly Loop Unrolling::). This at
least ensures loop control isn't a big fraction the work done.
- Memory throughput is always a limit. If perhaps only one load or
-one store can be done per cycle then 3 cycles/limb will the top speed
-for "binary" operations like `mpn_add_n', and any code achieving that
-is optimal.
+ Memory throughput is always a limit. If perhaps only one load or one
+store can be done per cycle then 3 cycles/limb will the top speed for
+"binary" operations like 'mpn_add_n', and any code achieving that is
+optimal.
Integer resources can be freed up by having the loop counter in a
float register, or by pressing the float units into use for some
\1f
File: gmp.info, Node: Assembly Floating Point, Next: Assembly SIMD Instructions, Prev: Assembly Functional Units, Up: Assembly Coding
-16.8.6 Floating Point
+15.8.6 Floating Point
---------------------
Floating point arithmetic is used in GMP for multiplications on CPUs
-with poor integer multipliers. It's mostly useful for `mpn_mul_1',
-`mpn_addmul_1' and `mpn_submul_1' on 64-bit machines, and
-`mpn_mul_basecase' on both 32-bit and 64-bit machines.
+with poor integer multipliers. It's mostly useful for 'mpn_mul_1',
+'mpn_addmul_1' and 'mpn_submul_1' on 64-bit machines, and
+'mpn_mul_basecase' on both 32-bit and 64-bit machines.
With IEEE 53-bit double precision floats, integer multiplications
producing up to 53 bits will give exact results. Breaking a 64x64
-multiplication into eight 16x32->48 bit pieces is convenient. With
-some care though six 21x32->53 bit products can be used, if one of the
-lower two 21-bit pieces also uses the sign bit.
+multiplication into eight 16x32->48 bit pieces is convenient. With some
+care though six 21x32->53 bit products can be used, if one of the lower
+two 21-bit pieces also uses the sign bit.
- For the `mpn_mul_1' family of functions on a 64-bit machine, the
-invariant single limb is split at the start, into 3 or 4 pieces.
-Inside the loop, the bignum operand is split into 32-bit pieces. Fast
+ For the 'mpn_mul_1' family of functions on a 64-bit machine, the
+invariant single limb is split at the start, into 3 or 4 pieces. Inside
+the loop, the bignum operand is split into 32-bit pieces. Fast
conversion of these unsigned 32-bit pieces to floating point is highly
machine-dependent. In some cases, reading the data into the integer
-unit, zero-extending to 64-bits, then transferring to the floating
-point unit back via memory is the only option.
+unit, zero-extending to 64-bits, then transferring to the floating point
+unit back via memory is the only option.
- Converting partial products back to 64-bit limbs is usually best
-done as a signed conversion. Since all values are smaller than 2^53,
-signed and unsigned are the same, but most processors lack unsigned
+ Converting partial products back to 64-bit limbs is usually best done
+as a signed conversion. Since all values are smaller than 2^53, signed
+and unsigned are the same, but most processors lack unsigned
conversions.
- Here is a diagram showing 16x32 bit products for an `mpn_mul_1' or
-`mpn_addmul_1' with a 64-bit limb. The single limb operand V is split
+ Here is a diagram showing 16x32 bit products for an 'mpn_mul_1' or
+'mpn_addmul_1' with a 64-bit limb. The single limb operand V is split
into four 16-bit parts. The multi-limb operand U is split in the loop
into two 32-bit parts.
| u32 x v48 | r80
+-----------+
- p32 and r32 can be summed using floating-point addition, and
-likewise p48 and r48. p00 and p16 can be summed with r64 and r80 from
-the previous iteration.
+ p32 and r32 can be summed using floating-point addition, and likewise
+p48 and r48. p00 and p16 can be summed with r64 and r80 from the
+previous iteration.
For each loop then, four 49-bit quantities are transferred to the
integer unit, aligned as follows,
\1f
File: gmp.info, Node: Assembly SIMD Instructions, Next: Assembly Software Pipelining, Prev: Assembly Floating Point, Up: Assembly Coding
-16.8.7 SIMD Instructions
+15.8.7 SIMD Instructions
------------------------
The single-instruction multiple-data support in current microprocessors
SIMD multiplications of say four 16x16 bit multiplies only do as much
work as one 32x32 from GMP's point of view, and need some shifts and
-adds besides. But of course if say the SIMD form is fully pipelined
-and uses less instruction decoding then it may still be worthwhile.
+adds besides. But of course if say the SIMD form is fully pipelined and
+uses less instruction decoding then it may still be worthwhile.
- On the x86 chips, MMX has so far found a use in `mpn_rshift' and
-`mpn_lshift', and is used in a special case for 16-bit multipliers in
-the P55 `mpn_mul_1'. SSE2 is used for Pentium 4 `mpn_mul_1',
-`mpn_addmul_1', and `mpn_submul_1'.
+ On the x86 chips, MMX has so far found a use in 'mpn_rshift' and
+'mpn_lshift', and is used in a special case for 16-bit multipliers in
+the P55 'mpn_mul_1'. SSE2 is used for Pentium 4 'mpn_mul_1',
+'mpn_addmul_1', and 'mpn_submul_1'.
\1f
File: gmp.info, Node: Assembly Software Pipelining, Next: Assembly Loop Unrolling, Prev: Assembly SIMD Instructions, Up: Assembly Coding
-16.8.8 Software Pipelining
+15.8.8 Software Pipelining
--------------------------
Software pipelining consists of scheduling instructions around the
branch point in a loop. For example a loop might issue a load not for
-use in the present iteration but the next, thereby allowing extra
-cycles for the data to arrive from memory.
+use in the present iteration but the next, thereby allowing extra cycles
+for the data to arrive from memory.
Naturally this is wanted only when doing things like loads or
multiplies that take several cycles to complete, and only where a CPU
each stage and each loop iteration moves them along one stage. This is
like juggling.
- If the latency of some instruction is greater than the loop time
-then it will be necessary to unroll, so one register has a result ready
-to use while another (or multiple others) are still in progress.
-(*note Assembly Loop Unrolling::).
+ If the latency of some instruction is greater than the loop time then
+it will be necessary to unroll, so one register has a result ready to
+use while another (or multiple others) are still in progress. (*note
+Assembly Loop Unrolling::).
\1f
File: gmp.info, Node: Assembly Loop Unrolling, Next: Assembly Writing Guide, Prev: Assembly Software Pipelining, Up: Assembly Coding
-16.8.9 Loop Unrolling
+15.8.9 Loop Unrolling
---------------------
Loop unrolling consists of replicating code so that several limbs are
processed in each loop. At a minimum this reduces loop overheads by a
corresponding factor, but it can also allow better register usage, for
example alternately using one register combination and then another.
-Judicious use of `m4' macros can help avoid lots of duplication in the
+Judicious use of 'm4' macros can help avoid lots of duplication in the
source code.
Any amount of unrolling can be handled with a loop counter that's
decremented by N each time, stopping when the remaining count is less
than the further N the loop will process. Or by subtracting N at the
-start, the termination condition becomes when the counter C is less
-than 0 (and the count of remaining limbs is C+N).
+start, the termination condition becomes when the counter C is less than
+0 (and the count of remaining limbs is C+N).
Alternately for a power of 2 unroll the loop count and remainder can
-be established with a shift and mask. This is convenient if also
-making a computed jump into the middle of a large loop.
+be established with a shift and mask. This is convenient if also making
+a computed jump into the middle of a large loop.
The limbs not a multiple of the unrolling can be handled in various
ways, for example
* A simple loop at the end (or the start) to process the excess.
- Care will be wanted that it isn't too much slower than the
- unrolled part.
+ Care will be wanted that it isn't too much slower than the unrolled
+ part.
* A set of binary tests, for example after an 8-limb unrolling, test
for 4 more limbs to process, then a further 2 more or not, and
finally 1 more or not. This will probably take more code space
than a simple loop.
- * A `switch' statement, providing separate code for each possible
+ * A 'switch' statement, providing separate code for each possible
excess, for example an 8-limb unrolling would have separate code
for 0 remaining, 1 remaining, etc, up to 7 remaining. This might
take a lot of code, but may be the best way to optimize all cases
\1f
File: gmp.info, Node: Assembly Writing Guide, Prev: Assembly Loop Unrolling, Up: Assembly Coding
-16.8.10 Writing Guide
+15.8.10 Writing Guide
---------------------
This is a guide to writing software pipelined loops for processing limb
greatly simplify later steps.
Then note for each instruction the functional unit and/or issue port
-requirements. If an instruction can use either of two units, like U0
-or U1 then make a category "U0/U1". Count the total using each unit
-(or combined unit), and count all instructions.
+requirements. If an instruction can use either of two units, like U0 or
+U1 then make a category "U0/U1". Count the total using each unit (or
+combined unit), and count all instructions.
Figure out from those counts the best possible loop time. The goal
will be to find a perfect schedule where instruction latencies are
factor, or perhaps a particular functional unit. It might be possible
to tweak the instructions to help the limiting factor.
- Suppose the loop time is N, then make N issue buckets, with the
-final loop branch at the end of the last. Now fill the buckets with
-dummy instructions using the functional units desired. Run this to
-make sure the intended speed is reached.
+ Suppose the loop time is N, then make N issue buckets, with the final
+loop branch at the end of the last. Now fill the buckets with dummy
+instructions using the functional units desired. Run this to make sure
+the intended speed is reached.
Now replace the dummy instructions with the real instructions from
-the slow but correct loop you started with. The first will typically
-be a load instruction. Then the instruction using that value is placed
-in a bucket an appropriate distance down. Run the loop again, to check
-it still runs at target speed.
+the slow but correct loop you started with. The first will typically be
+a load instruction. Then the instruction using that value is placed in
+a bucket an appropriate distance down. Run the loop again, to check it
+still runs at target speed.
Keep placing instructions, frequently measuring the loop. After a
-few you will need to wrap around from the last bucket back to the top
-of the loop. If you used the new-register for new-value strategy above
+few you will need to wrap around from the last bucket back to the top of
+the loop. If you used the new-register for new-value strategy above
then there will be no register conflicts. If not then take care not to
clobber something already in use. Changing registers at this time is
very error prone.
and at the end replicate and delete those whose results are unwanted
(including any further loads).
- The loop will have a minimum number of limbs loaded and processed,
-so the feed-in code must test if the request size is smaller and skip
+ The loop will have a minimum number of limbs loaded and processed, so
+the feed-in code must test if the request size is smaller and skip
either to a suitable part of the wind-down or to special code for small
sizes.
\1f
File: gmp.info, Node: Internals, Next: Contributors, Prev: Algorithms, Up: Top
-17 Internals
+16 Internals
************
*This chapter is provided only for informational purposes and the
\1f
File: gmp.info, Node: Integer Internals, Next: Rational Internals, Prev: Internals, Up: Internals
-17.1 Integer Internals
+16.1 Integer Internals
======================
-`mpz_t' variables represent integers using sign and magnitude, in space
+'mpz_t' variables represent integers using sign and magnitude, in space
dynamically allocated and reallocated. The fields are as follows.
-`_mp_size'
+'_mp_size'
The number of limbs, or the negative of that when representing a
- negative integer. Zero is represented by `_mp_size' set to zero,
- in which case the `_mp_d' data is unused.
+ negative integer. Zero is represented by '_mp_size' set to zero,
+ in which case the '_mp_d' data is undefined.
-`_mp_d'
+'_mp_d'
A pointer to an array of limbs which is the magnitude. These are
- stored "little endian" as per the `mpn' functions, so `_mp_d[0]'
- is the least significant limb and `_mp_d[ABS(_mp_size)-1]' is the
- most significant. Whenever `_mp_size' is non-zero, the most
- significant limb is non-zero.
-
- Currently there's always at least one limb allocated, so for
- instance `mpz_set_ui' never needs to reallocate, and `mpz_get_ui'
- can fetch `_mp_d[0]' unconditionally (though its value is then
- only wanted if `_mp_size' is non-zero).
-
-`_mp_alloc'
- `_mp_alloc' is the number of limbs currently allocated at `_mp_d',
- and naturally `_mp_alloc >= ABS(_mp_size)'. When an `mpz' routine
- is about to (or might be about to) increase `_mp_size', it checks
- `_mp_alloc' to see whether there's enough space, and reallocates
- if not. `MPZ_REALLOC' is generally used for this.
-
- The various bitwise logical functions like `mpz_and' behave as if
+ stored "little endian" as per the 'mpn' functions, so '_mp_d[0]' is
+ the least significant limb and '_mp_d[ABS(_mp_size)-1]' is the most
+ significant. Whenever '_mp_size' is non-zero, the most significant
+ limb is non-zero.
+
+ Currently there's always at least one readable limb, so for
+ instance 'mpz_get_ui' can fetch '_mp_d[0]' unconditionally (though
+ its value is undefined if '_mp_size' is zero).
+
+'_mp_alloc'
+ '_mp_alloc' is the number of limbs currently allocated at '_mp_d',
+ and normally '_mp_alloc >= ABS(_mp_size)'. When an 'mpz' routine
+ is about to (or might be about to) increase '_mp_size', it checks
+ '_mp_alloc' to see whether there's enough space, and reallocates if
+ not. 'MPZ_REALLOC' is generally used for this.
+
+ 'mpz_t' variables initialised with the 'mpz_roinit_n' function or
+ the 'MPZ_ROINIT_N' macro have '_mp_alloc = 0' but can have a
+ non-zero '_mp_size'. They can only be used as read-only constants.
+ See *note Integer Special Functions:: for details.
+
+ The various bitwise logical functions like 'mpz_and' behave as if
negative values were twos complement. But sign and magnitude is always
used internally, and necessary adjustments are made during the
calculations. Sometimes this isn't pretty, but sign and magnitude are
best for other routines.
- Some internal temporary variables are setup with `MPZ_TMP_INIT' and
-these have `_mp_d' space obtained from `TMP_ALLOC' rather than the
-memory allocation functions. Care is taken to ensure that these are
-big enough that no reallocation is necessary (since it would have
+ Some internal temporary variables are setup with 'MPZ_TMP_INIT' and
+these have '_mp_d' space obtained from 'TMP_ALLOC' rather than the
+memory allocation functions. Care is taken to ensure that these are big
+enough that no reallocation is necessary (since it would have
unpredictable consequences).
- `_mp_size' and `_mp_alloc' are `int', although `mp_size_t' is
-usually a `long'. This is done to make the fields just 32 bits on some
-64 bits systems, thereby saving a few bytes of data space but still
-providing plenty of range.
+ '_mp_size' and '_mp_alloc' are 'int', although 'mp_size_t' is usually
+a 'long'. This is done to make the fields just 32 bits on some 64 bits
+systems, thereby saving a few bytes of data space but still providing
+plenty of range.
\1f
File: gmp.info, Node: Rational Internals, Next: Float Internals, Prev: Integer Internals, Up: Internals
-17.2 Rational Internals
+16.2 Rational Internals
=======================
-`mpq_t' variables represent rationals using an `mpz_t' numerator and
+'mpq_t' variables represent rationals using an 'mpz_t' numerator and
denominator (*note Integer Internals::).
- The canonical form adopted is denominator positive (and non-zero),
-no common factors between numerator and denominator, and zero uniquely
+ The canonical form adopted is denominator positive (and non-zero), no
+common factors between numerator and denominator, and zero uniquely
represented as 0/1.
It's believed that casting out common factors at each stage of a
calculation is best in general. A GCD is an O(N^2) operation so it's
-better to do a few small ones immediately than to delay and have to do
-a big one later. Knowing the numerator and denominator have no common
-factors can be used for example in `mpq_mul' to make only two cross
-GCDs necessary, not four.
+better to do a few small ones immediately than to delay and have to do a
+big one later. Knowing the numerator and denominator have no common
+factors can be used for example in 'mpq_mul' to make only two cross GCDs
+necessary, not four.
This general approach to common factors is badly sub-optimal in the
presence of simple factorizations or little prospect for cancellation,
-but GMP has no way to know when this will occur. As per *Note
-Efficiency::, that's left to applications. The `mpq_t' framework might
-still suit, with `mpq_numref' and `mpq_denref' for direct access to the
-numerator and denominator, or of course `mpz_t' variables can be used
+but GMP has no way to know when this will occur. As per *note
+Efficiency::, that's left to applications. The 'mpq_t' framework might
+still suit, with 'mpq_numref' and 'mpq_denref' for direct access to the
+numerator and denominator, or of course 'mpz_t' variables can be used
directly.
\1f
File: gmp.info, Node: Float Internals, Next: Raw Output Internals, Prev: Rational Internals, Up: Internals
-17.3 Float Internals
+16.3 Float Internals
====================
Efficient calculation is the primary aim of GMP floats and the use of
whole limbs and simple rounding facilitates this.
- `mpf_t' floats have a variable precision mantissa and a single
+ 'mpf_t' floats have a variable precision mantissa and a single
machine word signed exponent. The mantissa is represented using sign
and magnitude.
<-------- _mp_size --------->
+
The fields are as follows.
-`_mp_size'
+'_mp_size'
The number of limbs currently in use, or the negative of that when
- representing a negative value. Zero is represented by `_mp_size'
- and `_mp_exp' both set to zero, and in that case the `_mp_d' data
- is unused. (In the future `_mp_exp' might be undefined when
+ representing a negative value. Zero is represented by '_mp_size'
+ and '_mp_exp' both set to zero, and in that case the '_mp_d' data
+ is unused. (In the future '_mp_exp' might be undefined when
representing zero.)
-`_mp_prec'
+'_mp_prec'
The precision of the mantissa, in limbs. In any calculation the
- aim is to produce `_mp_prec' limbs of result (the most significant
+ aim is to produce '_mp_prec' limbs of result (the most significant
being non-zero).
-`_mp_d'
+'_mp_d'
A pointer to the array of limbs which is the absolute value of the
- mantissa. These are stored "little endian" as per the `mpn'
- functions, so `_mp_d[0]' is the least significant limb and
- `_mp_d[ABS(_mp_size)-1]' the most significant.
+ mantissa. These are stored "little endian" as per the 'mpn'
+ functions, so '_mp_d[0]' is the least significant limb and
+ '_mp_d[ABS(_mp_size)-1]' the most significant.
The most significant limb is always non-zero, but there are no
other restrictions on its value, in particular the highest 1 bit
can be anywhere within the limb.
- `_mp_prec+1' limbs are allocated to `_mp_d', the extra limb being
+ '_mp_prec+1' limbs are allocated to '_mp_d', the extra limb being
for convenience (see below). There are no reallocations during a
- calculation, only in a change of precision with `mpf_set_prec'.
+ calculation, only in a change of precision with 'mpf_set_prec'.
-`_mp_exp'
+'_mp_exp'
The exponent, in limbs, determining the location of the implied
radix point. Zero means the radix point is just above the most
significant limb. Positive values mean a radix point offset
Naturally the exponent can be any value, it doesn't have to fall
within the limbs as the diagram shows, it can be a long way above
or a long way below. Limbs other than those included in the
- `{_mp_d,_mp_size}' data are treated as zero.
+ '{_mp_d,_mp_size}' data are treated as zero.
- The `_mp_size' and `_mp_prec' fields are `int', although the
-`mp_size_t' type is usually a `long'. The `_mp_exp' field is usually
-`long'. This is done to make some fields just 32 bits on some 64 bits
+ The '_mp_size' and '_mp_prec' fields are 'int', although the
+'mp_size_t' type is usually a 'long'. The '_mp_exp' field is usually
+'long'. This is done to make some fields just 32 bits on some 64 bits
systems, thereby saving a few bytes of data space but still providing
plenty of precision and a very large range.
The following various points should be noted.
Low Zeros
- The least significant limbs `_mp_d[0]' etc can be zero, though
- such low zeros can always be ignored. Routines likely to produce
- low zeros check and avoid them to save time in subsequent
- calculations, but for most routines they're quite unlikely and
- aren't checked.
+ The least significant limbs '_mp_d[0]' etc can be zero, though such
+ low zeros can always be ignored. Routines likely to produce low
+ zeros check and avoid them to save time in subsequent calculations,
+ but for most routines they're quite unlikely and aren't checked.
Mantissa Size Range
- The `_mp_size' count of limbs in use can be less than `_mp_prec' if
+ The '_mp_size' count of limbs in use can be less than '_mp_prec' if
the value can be represented in less. This means low precision
- values or small integers stored in a high precision `mpf_t' can
+ values or small integers stored in a high precision 'mpf_t' can
still be operated on efficiently.
- `_mp_size' can also be greater than `_mp_prec'. Firstly a value is
- allowed to use all of the `_mp_prec+1' limbs available at `_mp_d',
- and secondly when `mpf_set_prec_raw' lowers `_mp_prec' it leaves
- `_mp_size' unchanged and so the size can be arbitrarily bigger than
- `_mp_prec'.
+ '_mp_size' can also be greater than '_mp_prec'. Firstly a value is
+ allowed to use all of the '_mp_prec+1' limbs available at '_mp_d',
+ and secondly when 'mpf_set_prec_raw' lowers '_mp_prec' it leaves
+ '_mp_size' unchanged and so the size can be arbitrarily bigger than
+ '_mp_prec'.
Rounding
- All rounding is done on limb boundaries. Calculating `_mp_prec'
+ All rounding is done on limb boundaries. Calculating '_mp_prec'
limbs with the high non-zero will ensure the application requested
minimum precision is obtained.
- The use of simple "trunc" rounding towards zero is efficient,
- since there's no need to examine extra limbs and increment or
- decrement.
+ The use of simple "trunc" rounding towards zero is efficient, since
+ there's no need to examine extra limbs and increment or decrement.
Bit Shifts
Since the exponent is in limbs, there are no bit shifts in basic
- operations like `mpf_add' and `mpf_mul'. When differing exponents
+ operations like 'mpf_add' and 'mpf_mul'. When differing exponents
are encountered all that's needed is to adjust pointers to line up
the relevant limbs.
- Of course `mpf_mul_2exp' and `mpf_div_2exp' will require bit
+ Of course 'mpf_mul_2exp' and 'mpf_div_2exp' will require bit
shifts, but the choice is between an exponent in limbs which
requires shifts there, or one in bits which requires them almost
everywhere else.
-Use of `_mp_prec+1' Limbs
- The extra limb on `_mp_d' (`_mp_prec+1' rather than just
- `_mp_prec') helps when an `mpf' routine might get a carry from its
- operation. `mpf_add' for instance will do an `mpn_add' of
- `_mp_prec' limbs. If there's no carry then that's the result, but
+Use of '_mp_prec+1' Limbs
+ The extra limb on '_mp_d' ('_mp_prec+1' rather than just
+ '_mp_prec') helps when an 'mpf' routine might get a carry from its
+ operation. 'mpf_add' for instance will do an 'mpn_add' of
+ '_mp_prec' limbs. If there's no carry then that's the result, but
if there is a carry then it's stored in the extra limb of space and
- `_mp_size' becomes `_mp_prec+1'.
+ '_mp_size' becomes '_mp_prec+1'.
- Whenever `_mp_prec+1' limbs are held in a variable, the low limb
- is not needed for the intended precision, only the `_mp_prec' high
+ Whenever '_mp_prec+1' limbs are held in a variable, the low limb is
+ not needed for the intended precision, only the '_mp_prec' high
limbs. But zeroing it out or moving the rest down is unnecessary.
Subsequent routines reading the value will simply take the high
- limbs they need, and this will be `_mp_prec' if their target has
+ limbs they need, and this will be '_mp_prec' if their target has
that same precision. This is no more than a pointer adjustment,
and must be checked anyway since the destination precision can be
different from the sources.
- Copy functions like `mpf_set' will retain a full `_mp_prec+1' limbs
- if available. This ensures that a variable which has `_mp_size'
- equal to `_mp_prec+1' will get its full exact value copied.
- Strictly speaking this is unnecessary since only `_mp_prec' limbs
+ Copy functions like 'mpf_set' will retain a full '_mp_prec+1' limbs
+ if available. This ensures that a variable which has '_mp_size'
+ equal to '_mp_prec+1' will get its full exact value copied.
+ Strictly speaking this is unnecessary since only '_mp_prec' limbs
are needed for the application's requested precision, but it's
- considered that an `mpf_set' from one variable into another of the
+ considered that an 'mpf_set' from one variable into another of the
same precision ought to produce an exact copy.
Application Precisions
- `__GMPF_BITS_TO_PREC' converts an application requested precision
- to an `_mp_prec'. The value in bits is rounded up to a whole limb
+ '__GMPF_BITS_TO_PREC' converts an application requested precision
+ to an '_mp_prec'. The value in bits is rounded up to a whole limb
then an extra limb is added since the most significant limb of
- `_mp_d' is only non-zero and therefore might contain only one bit.
+ '_mp_d' is only non-zero and therefore might contain only one bit.
- `__GMPF_PREC_TO_BITS' does the reverse conversion, and removes the
- extra limb from `_mp_prec' before converting to bits. The net
- effect of reading back with `mpf_get_prec' is simply the precision
- rounded up to a multiple of `mp_bits_per_limb'.
+ '__GMPF_PREC_TO_BITS' does the reverse conversion, and removes the
+ extra limb from '_mp_prec' before converting to bits. The net
+ effect of reading back with 'mpf_get_prec' is simply the precision
+ rounded up to a multiple of 'mp_bits_per_limb'.
Note that the extra limb added here for the high only being
- non-zero is in addition to the extra limb allocated to `_mp_d'.
- For example with a 32-bit limb, an application request for 250
- bits will be rounded up to 8 limbs, then an extra added for the
- high being only non-zero, giving an `_mp_prec' of 9. `_mp_d' then
- gets 10 limbs allocated. Reading back with `mpf_get_prec' will
- take `_mp_prec' subtract 1 limb and multiply by 32, giving 256
- bits.
+ non-zero is in addition to the extra limb allocated to '_mp_d'.
+ For example with a 32-bit limb, an application request for 250 bits
+ will be rounded up to 8 limbs, then an extra added for the high
+ being only non-zero, giving an '_mp_prec' of 9. '_mp_d' then gets
+ 10 limbs allocated. Reading back with 'mpf_get_prec' will take
+ '_mp_prec' subtract 1 limb and multiply by 32, giving 256 bits.
Strictly speaking, the fact the high limb has at least one bit
means that a float with, say, 3 limbs of 32-bits each will be
- holding at least 65 bits, but for the purposes of `mpf_t' it's
+ holding at least 65 bits, but for the purposes of 'mpf_t' it's
considered simply to be 64 bits, a nice multiple of the limb size.
\1f
File: gmp.info, Node: Raw Output Internals, Next: C++ Interface Internals, Prev: Float Internals, Up: Internals
-17.4 Raw Output Internals
+16.4 Raw Output Internals
=========================
-`mpz_out_raw' uses the following format.
+'mpz_out_raw' uses the following format.
+------+------------------------+
| size | data bytes |
+------+------------------------+
The size is 4 bytes written most significant byte first, being the
-number of subsequent data bytes, or the twos complement negative of
-that when a negative integer is represented. The data bytes are the
-absolute value of the integer, written most significant byte first.
+number of subsequent data bytes, or the twos complement negative of that
+when a negative integer is represented. The data bytes are the absolute
+value of the integer, written most significant byte first.
The most significant data byte is always non-zero, so the output is
the same on all systems, irrespective of limb size.
In GMP 1, leading zero bytes were written to pad the data bytes to a
-multiple of the limb size. `mpz_inp_raw' will still accept this, for
+multiple of the limb size. 'mpz_inp_raw' will still accept this, for
compatibility.
The use of "big endian" for both the size and data fields is
deliberate, it makes the data easy to read in a hex dump of a file.
Unfortunately it also means that the limb data must be reversed when
-reading or writing, so neither a big endian nor little endian system
-can just read and write `_mp_d'.
+reading or writing, so neither a big endian nor little endian system can
+just read and write '_mp_d'.
\1f
File: gmp.info, Node: C++ Interface Internals, Prev: Raw Output Internals, Up: Internals
-17.5 C++ Interface Internals
+16.5 C++ Interface Internals
============================
A system of expression templates is used to ensure something like
-`a=b+c' turns into a simple call to `mpz_add' etc. For `mpf_class' the
+'a=b+c' turns into a simple call to 'mpz_add' etc. For 'mpf_class' the
scheme also ensures the precision of the final destination is used for
-any temporaries within a statement like `f=w*x+y*z'. These are
+any temporaries within a statement like 'f=w*x+y*z'. These are
important features which a naive implementation cannot provide.
A simplified description of the scheme follows. The true scheme is
struct __gmp_binary_plus
{
- static void eval(mpf_t f, mpf_t g, mpf_t h) { mpf_add(f, g, h); }
+ static void eval(mpf_t f, const mpf_t g, const mpf_t h)
+ {
+ mpf_add(f, g, h);
+ }
};
And an "additive expression" object,
<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >(f, g);
}
- The seemingly redundant `__gmp_expr<__gmp_binary_expr<...>>' is used
+ The seemingly redundant '__gmp_expr<__gmp_binary_expr<...>>' is used
to encapsulate any possible kind of expression into a single template
-type. In fact even `mpf_class' etc are `typedef' specializations of
-`__gmp_expr'.
+type. In fact even 'mpf_class' etc are 'typedef' specializations of
+'__gmp_expr'.
- Next we define assignment of `__gmp_expr' to `mpf_class'.
+ Next we define assignment of '__gmp_expr' to 'mpf_class'.
template <class T>
mpf_class & mpf_class::operator=(const __gmp_expr<T> &expr)
Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t());
}
- where `expr.val1' and `expr.val2' are references to the expression's
-operands (here `expr' is the `__gmp_binary_expr' stored within the
-`__gmp_expr').
+ where 'expr.val1' and 'expr.val2' are references to the expression's
+operands (here 'expr' is the '__gmp_binary_expr' stored within the
+'__gmp_expr').
This way, the expression is actually evaluated only at the time of
-assignment, when the required precision (that of `f') is known.
-Furthermore the target `mpf_t' is now available, thus we can call
-`mpf_add' directly with `f' as the output argument.
+assignment, when the required precision (that of 'f') is known.
+Furthermore the target 'mpf_t' is now available, thus we can call
+'mpf_add' directly with 'f' as the output argument.
Compound expressions are handled by defining operators taking
subexpressions as their arguments, like this:
(expr1, expr2);
}
- And the corresponding specializations of `__gmp_expr::eval':
+ And the corresponding specializations of '__gmp_expr::eval':
template <class T, class U, class Op>
void __gmp_expr
}
The expression is thus recursively evaluated to any level of
-complexity and all subexpressions are evaluated to the precision of `f'.
+complexity and all subexpressions are evaluated to the precision of 'f'.
\1f
File: gmp.info, Node: Contributors, Next: References, Prev: Internals, Up: Top
Appendix A Contributors
***********************
-Torbjo"rn Granlund wrote the original GMP library and is still the main
+Torbjörn Granlund wrote the original GMP library and is still the main
developer. Code not explicitly attributed to others, was contributed by
-Torbjo"rn. Several other individuals and organizations have contributed
-GMP. Here is a list in chronological order on first contribution:
+Torbjörn. Several other individuals and organizations have contributed
+GMP. Here is a list in chronological order on first contribution:
- Gunnar Sjo"din and Hans Riesel helped with mathematical problems in
+ Gunnar Sjödin and Hans Riesel helped with mathematical problems in
early versions of the library.
Richard Stallman helped with the interface design and revised the
the library and made creative suggestions.
John Amanatides of York University in Canada contributed the function
-`mpz_probab_prime_p'.
+'mpz_probab_prime_p'.
Paul Zimmermann wrote the REDC-based mpz_powm code, the
-Scho"nhage-Strassen FFT multiply code, and the Karatsuba square root
+Schönhage-Strassen FFT multiply code, and the Karatsuba square root
code. He also improved the Toom3 code for GMP 4.2. Paul sparked the
-development of GMP 2, with his comparisons between bignum packages.
-The ECMNET project Paul is organizing was a driving force behind many
-of the optimizations in GMP 3. Paul also wrote the new GMP 4.3 nth
-root code (with Torbjo"rn).
+development of GMP 2, with his comparisons between bignum packages. The
+ECMNET project Paul is organizing was a driving force behind many of the
+optimizations in GMP 3. Paul also wrote the new GMP 4.3 nth root code
+(with Torbjörn).
Ken Weber (Kent State University, Universidade Federal do Rio Grande
-do Sul) contributed now defunct versions of `mpz_gcd', `mpz_divexact',
-`mpn_gcd', and `mpn_bdivmod', partially supported by CNPq (Brazil)
-grant 301314194-2.
+do Sul) contributed now defunct versions of 'mpz_gcd', 'mpz_divexact',
+'mpn_gcd', and 'mpn_bdivmod', partially supported by CNPq (Brazil) grant
+301314194-2.
Per Bothner of Cygnus Support helped to set up GMP to use Cygnus'
configure. He has also made valuable suggestions and tested numerous
intermediary releases.
- Joachim Hollman was involved in the design of the `mpf' interface,
-and in the `mpz' design revisions for version 2.
+ Joachim Hollman was involved in the design of the 'mpf' interface,
+and in the 'mpz' design revisions for version 2.
- Bennet Yee contributed the initial versions of `mpz_jacobi' and
-`mpz_legendre'.
+ Bennet Yee contributed the initial versions of 'mpz_jacobi' and
+'mpz_legendre'.
- Andreas Schwab contributed the files `mpn/m68k/lshift.S' and
-`mpn/m68k/rshift.S' (now in `.asm' form).
+ Andreas Schwab contributed the files 'mpn/m68k/lshift.S' and
+'mpn/m68k/rshift.S' (now in '.asm' form).
Robert Harley of Inria, France and David Seal of ARM, England,
suggested clever improvements for population count. Robert also wrote
GMP 3, and contributed the ARM assembly code.
Torsten Ekedahl of the Mathematical department of Stockholm
-University provided significant inspiration during several phases of
-the GMP development. His mathematical expertise helped improve several
+University provided significant inspiration during several phases of the
+GMP development. His mathematical expertise helped improve several
algorithms.
Linus Nordberg wrote the new configure system based on autoconf and
implemented the new random functions.
- Kevin Ryde worked on a large number of things: optimized x86 code,
-m4 asm macros, parameter tuning, speed measuring, the configure system,
+ Kevin Ryde worked on a large number of things: optimized x86 code, m4
+asm macros, parameter tuning, speed measuring, the configure system,
function inlining, divisibility tests, bit scanning, Jacobi symbols,
Fibonacci and Lucas number functions, printf and scanf functions, perl
-interface, demo expression parser, the algorithms chapter in the
-manual, `gmpasm-mode.el', and various miscellaneous improvements
-elsewhere.
+interface, demo expression parser, the algorithms chapter in the manual,
+'gmpasm-mode.el', and various miscellaneous improvements elsewhere.
Kent Boortz made the Mac OS 9 port.
Steve Root helped write the optimized alpha 21264 assembly code.
- Gerardo Ballabio wrote the `gmpxx.h' C++ class interface and the C++
-`istream' input routines.
+ Gerardo Ballabio wrote the 'gmpxx.h' C++ class interface and the C++
+'istream' input routines.
- Jason Moxham rewrote `mpz_fac_ui'.
+ Jason Moxham rewrote 'mpz_fac_ui'.
Pedro Gimeno implemented the Mersenne Twister and made other random
number improvements.
- Niels Mo"ller wrote the sub-quadratic GCD and extended GCD code, the
-quadratic Hensel division code, and (with Torbjo"rn) the new divide and
-conquer division code for GMP 4.3. Niels also helped implement the new
-Toom multiply code for GMP 4.3 and implemented helper functions to
-simplify Toom evaluations for GMP 5.0. He wrote the original version
-of mpn_mulmod_bnm1.
+ Niels Möller wrote the sub-quadratic GCD, extended GCD and jacobi
+code, the quadratic Hensel division code, and (with Torbjörn) the new
+divide and conquer division code for GMP 4.3. Niels also helped
+implement the new Toom multiply code for GMP 4.3 and implemented helper
+functions to simplify Toom evaluations for GMP 5.0. He wrote the
+original version of mpn_mulmod_bnm1, and he is the main author of the
+mini-gmp package used for gmp bootstrapping.
Alberto Zanoni and Marco Bodrato suggested the unbalanced multiply
strategy, and found the optimal strategies for evaluation and
interpolation in Toom multiplication.
- Marco Bodrato helped implement the new Toom multiply code for GMP
-4.3 and implemented most of the new Toom multiply and squaring code for
-5.0. He is the main author of the current mpn_mulmod_bnm1 and
-mpn_mullo_n. Marco also wrote the functions mpn_invert and
-mpn_invertappr.
+ Marco Bodrato helped implement the new Toom multiply code for GMP 4.3
+and implemented most of the new Toom multiply and squaring code for 5.0.
+He is the main author of the current mpn_mulmod_bnm1, mpn_mullo_n, and
+mpn_sqrlo. Marco also wrote the functions mpn_invert and
+mpn_invertappr, and improved the speed of integer root extraction. He
+is the author of mini-mpq, an additional layer to mini-gmp; of most of
+the combinatorial functions and the BPSW primality testing
+implementation, for both the main library and the mini-gmp package.
- David Harvey suggested the internal function `mpn_bdiv_dbm1',
+ David Harvey suggested the internal function 'mpn_bdiv_dbm1',
implementing division relevant to Toom multiplication. He also worked
on fast assembly sequences, in particular on a fast AMD64
-`mpn_mul_basecase'.
+'mpn_mul_basecase'. He wrote the internal middle product functions
+'mpn_mulmid_basecase', 'mpn_toom42_mulmid', 'mpn_mulmid_n' and related
+helper routines.
+
+ Martin Boij wrote 'mpn_perfect_power_p'.
+
+ Marc Glisse improved 'gmpxx.h': use fewer temporaries (faster),
+specializations of 'numeric_limits' and 'common_type', C++11 features
+(move constructors, explicit bool conversion, UDL), make the conversion
+from 'mpq_class' to 'mpz_class' explicit, optimize operations where one
+argument is a small compile-time constant, replace some heap allocations
+by stack allocations. He also fixed the eofbit handling of C++ streams,
+and removed one division from 'mpq/aors.c'.
+
+ David S Miller wrote assembly code for SPARC T3 and T4.
- Martin Boij wrote `mpn_perfect_power_p'.
+ Mark Sofroniou cleaned up the types of mul_fft.c, letting it work for
+huge operands.
+
+ Ulrich Weigand ported GMP to the powerpc64le ABI.
(This list is chronological, not ordered after significance. If you
have contributed to GMP but are not listed above, please tell
supported in part by the ESPRIT-BRA (Basic Research Activities) 6846
project POSSO (POlynomial System SOlving).
- The development of GMP 2, 3, and 4 was supported in part by the IDA
+ The development of GMP 2, 3, and 4.0 was supported in part by the IDA
Center for Computing Sciences.
- Thanks go to Hans Thorsen for donating an SGI system for the GMP
-test system environment.
+ The development of GMP 4.3, 5.0, and 5.1 was supported in part by the
+Swedish Foundation for Strategic Research.
+
+ Thanks go to Hans Thorsen for donating an SGI system for the GMP test
+system environment.
\1f
File: gmp.info, Node: References, Next: GNU Free Documentation License, Prev: Contributors, Up: Top
* Richard Crandall and Carl Pomerance, "Prime Numbers: A
Computational Perspective", 2nd edition, Springer-Verlag, 2005.
- `http://math.dartmouth.edu/~carlp/'
+ <https://www.math.dartmouth.edu/~carlp/>
* Henri Cohen, "A Course in Computational Algebraic Number Theory",
Graduate Texts in Mathematics number 138, Springer-Verlag, 1993.
- `http://www.math.u-bordeaux.fr/~cohen/'
+ <https://www.math.u-bordeaux.fr/~cohen/>
* Donald E. Knuth, "The Art of Computer Programming", volume 2,
"Seminumerical Algorithms", 3rd edition, Addison-Wesley, 1998.
- `http://www-cs-faculty.stanford.edu/~knuth/taocp.html'
+ <https://www-cs-faculty.stanford.edu/~knuth/taocp.html>
* John D. Lipson, "Elements of Algebra and Algebraic Computing", The
Benjamin Cummings Publishing Company Inc, 1981.
* Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone,
"Handbook of Applied Cryptography",
- `http://www.cacr.math.uwaterloo.ca/hac/'
+ <http://www.cacr.math.uwaterloo.ca/hac/>
- * Richard M. Stallman and the GCC Developer Community, "Using the
- GNU Compiler Collection", Free Software Foundation, 2008,
- available online `http://gcc.gnu.org/onlinedocs/', and in the GCC
- package `ftp://ftp.gnu.org/gnu/gcc/'
+ * Richard M. Stallman and the GCC Developer Community, "Using the GNU
+ Compiler Collection", Free Software Foundation, 2008, available
+ online <https://gcc.gnu.org/onlinedocs/>, and in the GCC package
+ <https://ftp.gnu.org/gnu/gcc/>
B.2 Papers
==========
* Yves Bertot, Nicolas Magaud and Paul Zimmermann, "A Proof of GMP
Square Root", Journal of Automated Reasoning, volume 29, 2002, pp.
- 225-252. Also available online as INRIA Research Report 4475,
- June 2001, `http://www.inria.fr/rrrt/rr-4475.html'
+ 225-252. Also available online as INRIA Research Report 4475, June
+ 2002, <https://hal.inria.fr/docs/00/07/21/13/PDF/RR-4475.pdf>
* Christoph Burnikel and Joachim Ziegler, "Fast Recursive Division",
Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022,
- `http://data.mpi-sb.mpg.de/internet/reports.nsf/NumberView/1998-1-022'
+ <https://www.mpi-inf.mpg.de/~ziegler/TechRep.ps.gz>
- * Torbjo"rn Granlund and Peter L. Montgomery, "Division by Invariant
+ * Torbjörn Granlund and Peter L. Montgomery, "Division by Invariant
Integers using Multiplication", in Proceedings of the SIGPLAN
PLDI'94 Conference, June 1994. Also available
- `ftp://ftp.cwi.nl/pub/pmontgom/divcnst.psa4.gz' (and .psl.gz).
+ <https://gmplib.org/~tege/divcnst-pldi94.pdf>.
- * Niels Mo"ller and Torbjo"rn Granlund, "Improved division by
- invariant integers", to appear.
+ * Niels Möller and Torbjörn Granlund, "Improved division by invariant
+ integers", IEEE Transactions on Computers, 11 June 2010.
+ <https://gmplib.org/~tege/division-paper.pdf>
- * Torbjo"rn Granlund and Niels Mo"ller, "Division of integers large
- and small", to appear.
+ * Torbjörn Granlund and Niels Möller, "Division of integers large and
+ small", to appear.
* Tudor Jebelean, "An algorithm for exact division", Journal of
Symbolic Computation, volume 15, 1993, pp. 169-180. Research
report version available
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz'
+ <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz>
* Tudor Jebelean, "Exact Division with Karatsuba Complexity -
Extended Abstract", RISC-Linz technical report 96-31,
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz'
+ <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz>
* Tudor Jebelean, "Practical Integer Division with Karatsuba
Complexity", ISSAC 97, pp. 339-341. Technical report available
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz'
+ <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz>
* Tudor Jebelean, "A Generalization of the Binary GCD Algorithm",
ISSAC 93, pp. 111-116. Technical report version available
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz'
+ <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz>
- * Tudor Jebelean, "A Double-Digit Lehmer-Euclid Algorithm for
- Finding the GCD of Long Integers", Journal of Symbolic
- Computation, volume 19, 1995, pp. 145-157. Technical report
- version also available
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz'
+ * Tudor Jebelean, "A Double-Digit Lehmer-Euclid Algorithm for Finding
+ the GCD of Long Integers", Journal of Symbolic Computation, volume
+ 19, 1995, pp. 145-157. Technical report version also available
+ <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz>
* Werner Krandick and Tudor Jebelean, "Bidirectional Exact Integer
Division", Journal of Symbolic Computation, volume 21, 1996, pp.
441-455. Early technical report version also available
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz'
+ <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz>
* Makoto Matsumoto and Takuji Nishimura, "Mersenne Twister: A
623-dimensionally equidistributed uniform pseudorandom number
generator", ACM Transactions on Modelling and Computer Simulation,
volume 8, January 1998, pp. 3-30. Available online
- `http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/ARTICLES/mt.ps.gz'
- (or .pdf)
+ <http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/ARTICLES/mt.pdf>
* R. Moenck and A. Borodin, "Fast Modular Transforms via Division",
Proceedings of the 13th Annual IEEE Symposium on Switching and
Modular Transforms", Journal of Computer and System Sciences,
volume 8, number 3, June 1974, pp. 366-386.
- * Niels Mo"ller, "On Scho"nhage's algorithm and subquadratic integer
- GCD computation", in Mathematics of Computation, volume 77,
- January 2008, pp. 589-607.
+ * Niels Möller, "On Schönhage's algorithm and subquadratic integer
+ GCD computation", in Mathematics of Computation, volume 77, January
+ 2008, pp. 589-607,
+ <https://www.ams.org/journals/mcom/2008-77-261/S0025-5718-07-02017-0/home.html>
* Peter L. Montgomery, "Modular Multiplication Without Trial
Division", in Mathematics of Computation, volume 44, number 170,
April 1985.
- * Arnold Scho"nhage and Volker Strassen, "Schnelle Multiplikation
+ * Arnold Schönhage and Volker Strassen, "Schnelle Multiplikation
grosser Zahlen", Computing 7, 1971, pp. 281-292.
* Kenneth Weber, "The accelerated integer GCD algorithm", ACM
1995, pp. 111-122.
* Paul Zimmermann, "Karatsuba Square Root", INRIA Research Report
- 3805, November 1999, `http://www.inria.fr/rrrt/rr-3805.html'
+ 3805, November 1999,
+ <https://hal.inria.fr/inria-00072854/PDF/RR-3805.pdf>
* Paul Zimmermann, "A Proof of GMP Fast Division and Square Root
Implementations",
- `http://www.loria.fr/~zimmerma/papers/proof-div-sqrt.ps.gz'
+ <https://homepages.loria.fr/PZimmermann/papers/proof-div-sqrt.ps.gz>
* Dan Zuras, "On Squaring and Multiplying Large Integers", ARITH-11:
IEEE Symposium on Computer Arithmetic, 1993, pp. 260 to 271.
IEEE Transactions on Computers, volume 43, number 8, August 1994,
pp. 899-908.
+ * Niels Möller, "Efficient computation of the Jacobi symbol",
+ <https://arxiv.org/abs/1907.07795>
+
\1f
File: gmp.info, Node: GNU Free Documentation License, Next: Concept Index, Prev: References, Up: Top
Version 1.3, 3 November 2008
- Copyright (C) 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc.
- `http://fsf.org/'
+ Copyright © 2000-2002, 2007, 2008 Free Software Foundation, Inc.
+ <http://fsf.org/>
Everyone is permitted to copy and distribute verbatim copies
of this license document, but changing it is not allowed.
free program should come with manuals providing the same freedoms
that the software does. But this License is not limited to
software manuals; it can be used for any textual work, regardless
- of subject matter or whether it is published as a printed book.
- We recommend this License principally for works whose purpose is
+ of subject matter or whether it is published as a printed book. We
+ recommend this License principally for works whose purpose is
instruction or reference.
1. APPLICABILITY AND DEFINITIONS
This License applies to any manual or other work, in any medium,
- that contains a notice placed by the copyright holder saying it
- can be distributed under the terms of this License. Such a notice
+ that contains a notice placed by the copyright holder saying it can
+ be distributed under the terms of this License. Such a notice
grants a world-wide, royalty-free license, unlimited in duration,
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"Document", below, refers to any such manual or work. Any member
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- accept the license if you copy, modify or distribute the work in a
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+ of the public is a licensee, and is addressed as "you". You accept
+ the license if you copy, modify or distribute the work in a way
+ requiring permission under copyright law.
A "Modified Version" of the Document means any work containing the
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The "Invariant Sections" are certain Secondary Sections whose
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+ titles are designated, as being those of Invariant Sections, in the
+ notice that says that the Document is released under this License.
+ If a section does not fit the above definition of Secondary then it
+ is not allowed to be designated as Invariant. The Document may
+ contain zero Invariant Sections. If the Document does not identify
+ any Invariant Sections then there are none.
The "Cover Texts" are certain short passages of text that are
listed, as Front-Cover Texts or Back-Cover Texts, in the notice
A "Transparent" copy of the Document means a machine-readable copy,
represented in a format whose specification is available to the
general public, that is suitable for revising the document
- straightforwardly with generic text editors or (for images
- composed of pixels) generic paint programs or (for drawings) some
- widely available drawing editor, and that is suitable for input to
- text formatters or for automatic translation to a variety of
- formats suitable for input to text formatters. A copy made in an
- otherwise Transparent file format whose markup, or absence of
- markup, has been arranged to thwart or discourage subsequent
- modification by readers is not Transparent. An image format is
- not Transparent if used for any substantial amount of text. A
- copy that is not "Transparent" is called "Opaque".
+ straightforwardly with generic text editors or (for images composed
+ of pixels) generic paint programs or (for drawings) some widely
+ available drawing editor, and that is suitable for input to text
+ formatters or for automatic translation to a variety of formats
+ suitable for input to text formatters. A copy made in an otherwise
+ Transparent file format whose markup, or absence of markup, has
+ been arranged to thwart or discourage subsequent modification by
+ readers is not Transparent. An image format is not Transparent if
+ used for any substantial amount of text. A copy that is not
+ "Transparent" is called "Opaque".
Examples of suitable formats for Transparent copies include plain
ASCII without markup, Texinfo input format, LaTeX input format,
- SGML or XML using a publicly available DTD, and
- standard-conforming simple HTML, PostScript or PDF designed for
- human modification. Examples of transparent image formats include
- PNG, XCF and JPG. Opaque formats include proprietary formats that
- can be read and edited only by proprietary word processors, SGML or
- XML for which the DTD and/or processing tools are not generally
- available, and the machine-generated HTML, PostScript or PDF
- produced by some word processors for output purposes only.
+ SGML or XML using a publicly available DTD, and standard-conforming
+ simple HTML, PostScript or PDF designed for human modification.
+ Examples of transparent image formats include PNG, XCF and JPG.
+ Opaque formats include proprietary formats that can be read and
+ edited only by proprietary word processors, SGML or XML for which
+ the DTD and/or processing tools are not generally available, and
+ the machine-generated HTML, PostScript or PDF produced by some word
+ processors for output purposes only.
The "Title Page" means, for a printed book, the title page itself,
plus such following pages as are needed to hold, legibly, the
may not use technical measures to obstruct or control the reading
or further copying of the copies you make or distribute. However,
you may accept compensation in exchange for copies. If you
- distribute a large enough number of copies you must also follow
- the conditions in section 3.
+ distribute a large enough number of copies you must also follow the
+ conditions in section 3.
You may also lend copies, under the same conditions stated above,
and you may publicly display copies.
these Cover Texts: Front-Cover Texts on the front cover, and
Back-Cover Texts on the back cover. Both covers must also clearly
and legibly identify you as the publisher of these copies. The
- front cover must present the full title with all words of the
- title equally prominent and visible. You may add other material
- on the covers in addition. Copying with changes limited to the
- covers, as long as they preserve the title of the Document and
- satisfy these conditions, can be treated as verbatim copying in
- other respects.
+ front cover must present the full title with all words of the title
+ equally prominent and visible. You may add other material on the
+ covers in addition. Copying with changes limited to the covers, as
+ long as they preserve the title of the Document and satisfy these
+ conditions, can be treated as verbatim copying in other respects.
If the required texts for either cover are too voluminous to fit
legibly, you should put the first ones listed (as many as fit
adjacent pages.
If you publish or distribute Opaque copies of the Document
- numbering more than 100, you must either include a
- machine-readable Transparent copy along with each Opaque copy, or
- state in or with each Opaque copy a computer-network location from
- which the general network-using public has access to download
- using public-standard network protocols a complete Transparent
- copy of the Document, free of added material. If you use the
- latter option, you must take reasonably prudent steps, when you
- begin distribution of Opaque copies in quantity, to ensure that
- this Transparent copy will remain thus accessible at the stated
- location until at least one year after the last time you
- distribute an Opaque copy (directly or through your agents or
- retailers) of that edition to the public.
+ numbering more than 100, you must either include a machine-readable
+ Transparent copy along with each Opaque copy, or state in or with
+ each Opaque copy a computer-network location from which the general
+ network-using public has access to download using public-standard
+ network protocols a complete Transparent copy of the Document, free
+ of added material. If you use the latter option, you must take
+ reasonably prudent steps, when you begin distribution of Opaque
+ copies in quantity, to ensure that this Transparent copy will
+ remain thus accessible at the stated location until at least one
+ year after the last time you distribute an Opaque copy (directly or
+ through your agents or retailers) of that edition to the public.
It is requested, but not required, that you contact the authors of
- the Document well before redistributing any large number of
- copies, to give them a chance to provide you with an updated
- version of the Document.
+ the Document well before redistributing any large number of copies,
+ to give them a chance to provide you with an updated version of the
+ Document.
4. MODIFICATIONS
You may copy and distribute a Modified Version of the Document
under the conditions of sections 2 and 3 above, provided that you
- release the Modified Version under precisely this License, with
- the Modified Version filling the role of the Document, thus
- licensing distribution and modification of the Modified Version to
- whoever possesses a copy of it. In addition, you must do these
- things in the Modified Version:
+ release the Modified Version under precisely this License, with the
+ Modified Version filling the role of the Document, thus licensing
+ distribution and modification of the Modified Version to whoever
+ possesses a copy of it. In addition, you must do these things in
+ the Modified Version:
A. Use in the Title Page (and on the covers, if any) a title
- distinct from that of the Document, and from those of
- previous versions (which should, if there were any, be listed
- in the History section of the Document). You may use the
- same title as a previous version if the original publisher of
- that version gives permission.
+ distinct from that of the Document, and from those of previous
+ versions (which should, if there were any, be listed in the
+ History section of the Document). You may use the same title
+ as a previous version if the original publisher of that
+ version gives permission.
B. List on the Title Page, as authors, one or more persons or
entities responsible for authorship of the modifications in
I. Preserve the section Entitled "History", Preserve its Title,
and add to it an item stating at least the title, year, new
- authors, and publisher of the Modified Version as given on
- the Title Page. If there is no section Entitled "History" in
- the Document, create one stating the title, year, authors,
- and publisher of the Document as given on its Title Page,
- then add an item describing the Modified Version as stated in
- the previous sentence.
+ authors, and publisher of the Modified Version as given on the
+ Title Page. If there is no section Entitled "History" in the
+ Document, create one stating the title, year, authors, and
+ publisher of the Document as given on its Title Page, then add
+ an item describing the Modified Version as stated in the
+ previous sentence.
J. Preserve the network location, if any, given in the Document
for public access to a Transparent copy of the Document, and
likewise the network locations given in the Document for
- previous versions it was based on. These may be placed in
- the "History" section. You may omit a network location for a
- work that was published at least four years before the
- Document itself, or if the original publisher of the version
- it refers to gives permission.
+ previous versions it was based on. These may be placed in the
+ "History" section. You may omit a network location for a work
+ that was published at least four years before the Document
+ itself, or if the original publisher of the version it refers
+ to gives permission.
K. For any section Entitled "Acknowledgements" or "Dedications",
- Preserve the Title of the section, and preserve in the
- section all the substance and tone of each of the contributor
+ Preserve the Title of the section, and preserve in the section
+ all the substance and tone of each of the contributor
acknowledgements and/or dedications given therein.
- L. Preserve all the Invariant Sections of the Document,
- unaltered in their text and in their titles. Section numbers
- or the equivalent are not considered part of the section
- titles.
+ L. Preserve all the Invariant Sections of the Document, unaltered
+ in their text and in their titles. Section numbers or the
+ equivalent are not considered part of the section titles.
M. Delete any section Entitled "Endorsements". Such a section
may not be included in the Modified Version.
If the Modified Version includes new front-matter sections or
appendices that qualify as Secondary Sections and contain no
- material copied from the Document, you may at your option
- designate some or all of these sections as invariant. To do this,
- add their titles to the list of Invariant Sections in the Modified
- Version's license notice. These titles must be distinct from any
- other section titles.
+ material copied from the Document, you may at your option designate
+ some or all of these sections as invariant. To do this, add their
+ titles to the list of Invariant Sections in the Modified Version's
+ license notice. These titles must be distinct from any other
+ section titles.
You may add a section Entitled "Endorsements", provided it contains
nothing but endorsements of your Modified Version by various
definition of a standard.
You may add a passage of up to five words as a Front-Cover Text,
- and a passage of up to 25 words as a Back-Cover Text, to the end
- of the list of Cover Texts in the Modified Version. Only one
- passage of Front-Cover Text and one of Back-Cover Text may be
- added by (or through arrangements made by) any one entity. If the
- Document already includes a cover text for the same cover,
- previously added by you or by arrangement made by the same entity
- you are acting on behalf of, you may not add another; but you may
- replace the old one, on explicit permission from the previous
- publisher that added the old one.
+ and a passage of up to 25 words as a Back-Cover Text, to the end of
+ the list of Cover Texts in the Modified Version. Only one passage
+ of Front-Cover Text and one of Back-Cover Text may be added by (or
+ through arrangements made by) any one entity. If the Document
+ already includes a cover text for the same cover, previously added
+ by you or by arrangement made by the same entity you are acting on
+ behalf of, you may not add another; but you may replace the old
+ one, on explicit permission from the previous publisher that added
+ the old one.
The author(s) and publisher(s) of the Document do not by this
License give permission to use their names for publicity for or to
You may combine the Document with other documents released under
this License, under the terms defined in section 4 above for
- modified versions, provided that you include in the combination
- all of the Invariant Sections of all of the original documents,
+ modified versions, provided that you include in the combination all
+ of the Invariant Sections of all of the original documents,
unmodified, and list them all as Invariant Sections of your
combined work in its license notice, and that you preserve all
their Warranty Disclaimers.
documents released under this License, and replace the individual
copies of this License in the various documents with a single copy
that is included in the collection, provided that you follow the
- rules of this License for verbatim copying of each of the
- documents in all other respects.
+ rules of this License for verbatim copying of each of the documents
+ in all other respects.
You may extract a single document from such a collection, and
distribute it individually under this License, provided you insert
- a copy of this License into the extracted document, and follow
- this License in all other respects regarding verbatim copying of
- that document.
+ a copy of this License into the extracted document, and follow this
+ License in all other respects regarding verbatim copying of that
+ document.
7. AGGREGATION WITH INDEPENDENT WORKS
A compilation of the Document or its derivatives with other
- separate and independent documents or works, in or on a volume of
- a storage or distribution medium, is called an "aggregate" if the
+ separate and independent documents or works, in or on a volume of a
+ storage or distribution medium, is called an "aggregate" if the
copyright resulting from the compilation is not used to limit the
legal rights of the compilation's users beyond what the individual
works permit. When the Document is included in an aggregate, this
However, if you cease all violation of this License, then your
license from a particular copyright holder is reinstated (a)
- provisionally, unless and until the copyright holder explicitly
- and finally terminates your license, and (b) permanently, if the
+ provisionally, unless and until the copyright holder explicitly and
+ finally terminates your license, and (b) permanently, if the
copyright holder fails to notify you of the violation by some
reasonable means prior to 60 days after the cessation.
after your receipt of the notice.
Termination of your rights under this section does not terminate
- the licenses of parties who have received copies or rights from
- you under this License. If your rights have been terminated and
- not permanently reinstated, receipt of a copy of some or all of
- the same material does not give you any rights to use it.
+ the licenses of parties who have received copies or rights from you
+ under this License. If your rights have been terminated and not
+ permanently reinstated, receipt of a copy of some or all of the
+ same material does not give you any rights to use it.
- 10. FUTURE REVISIONS OF THIS LICENSE
+ 10. FUTURE REVISIONS OF THIS LICENSE
The Free Software Foundation may publish new, revised versions of
the GNU Free Documentation License from time to time. Such new
versions will be similar in spirit to the present version, but may
differ in detail to address new problems or concerns. See
- `http://www.gnu.org/copyleft/'.
+ <https://www.gnu.org/copyleft/>.
Each version of the License is given a distinguishing version
number. If the Document specifies that a particular numbered
version of this License "or any later version" applies to it, you
have the option of following the terms and conditions either of
that specified version or of any later version that has been
- published (not as a draft) by the Free Software Foundation. If
- the Document does not specify a version number of this License,
- you may choose any version ever published (not as a draft) by the
- Free Software Foundation. If the Document specifies that a proxy
- can decide which future versions of this License can be used, that
+ published (not as a draft) by the Free Software Foundation. If the
+ Document does not specify a version number of this License, you may
+ choose any version ever published (not as a draft) by the Free
+ Software Foundation. If the Document specifies that a proxy can
+ decide which future versions of this License can be used, that
proxy's public statement of acceptance of a version permanently
authorizes you to choose that version for the Document.
- 11. RELICENSING
+ 11. RELICENSING
"Massive Multiauthor Collaboration Site" (or "MMC Site") means any
World Wide Web server that publishes copyrightable works and also
site under CC-BY-SA on the same site at any time before August 1,
2009, provided the MMC is eligible for relicensing.
-
ADDENDUM: How to use this License for your documents
====================================================
Free Documentation License''.
If you have Invariant Sections, Front-Cover Texts and Back-Cover
-Texts, replace the "with...Texts." line with this:
+Texts, replace the "with...Texts." line with this:
with the Invariant Sections being LIST THEIR TITLES, with
the Front-Cover Texts being LIST, and with the Back-Cover Texts
situation.
If your document contains nontrivial examples of program code, we
-recommend releasing these examples in parallel under your choice of
-free software license, such as the GNU General Public License, to
-permit their use in free software.
+recommend releasing these examples in parallel under your choice of free
+software license, such as the GNU General Public License, to permit
+their use in free software.
\1f
File: gmp.info, Node: Concept Index, Next: Function Index, Prev: GNU Free Documentation License, Up: Top
* #include: Headers and Libraries.
(line 6)
-* --build: Build Options. (line 52)
-* --disable-fft: Build Options. (line 317)
-* --disable-shared: Build Options. (line 45)
-* --disable-static: Build Options. (line 45)
-* --enable-alloca: Build Options. (line 278)
-* --enable-assert: Build Options. (line 327)
-* --enable-cxx: Build Options. (line 230)
-* --enable-fat: Build Options. (line 164)
-* --enable-mpbsd: Build Options. (line 322)
+* --build: Build Options. (line 51)
+* --disable-fft: Build Options. (line 307)
+* --disable-shared: Build Options. (line 44)
+* --disable-static: Build Options. (line 44)
+* --enable-alloca: Build Options. (line 273)
+* --enable-assert: Build Options. (line 313)
+* --enable-cxx: Build Options. (line 225)
+* --enable-fat: Build Options. (line 160)
+* --enable-profiling: Build Options. (line 317)
* --enable-profiling <1>: Profiling. (line 6)
-* --enable-profiling: Build Options. (line 331)
* --exec-prefix: Build Options. (line 32)
-* --host: Build Options. (line 66)
+* --host: Build Options. (line 65)
* --prefix: Build Options. (line 32)
* -finstrument-functions: Profiling. (line 66)
* 2exp functions: Efficiency. (line 43)
* 68000: Notes for Particular Systems.
- (line 80)
+ (line 94)
* 80x86: Notes for Particular Systems.
- (line 126)
-* ABI <1>: Build Options. (line 171)
-* ABI: ABI and ISA. (line 6)
-* About this manual: Introduction to GMP. (line 58)
+ (line 150)
+* ABI: Build Options. (line 167)
+* ABI <1>: ABI and ISA. (line 6)
+* About this manual: Introduction to GMP. (line 57)
* AC_CHECK_LIB: Autoconf. (line 11)
-* AIX <1>: ABI and ISA. (line 184)
-* AIX <2>: Notes for Particular Systems.
+* AIX: ABI and ISA. (line 174)
+* AIX <1>: Notes for Particular Systems.
(line 7)
-* AIX: ABI and ISA. (line 169)
* Algorithms: Algorithms. (line 6)
-* alloca: Build Options. (line 278)
+* alloca: Build Options. (line 273)
* Allocation of memory: Custom Allocation. (line 6)
* AMD64: ABI and ISA. (line 44)
-* Anonymous FTP of latest version: Introduction to GMP. (line 38)
+* Anonymous FTP of latest version: Introduction to GMP. (line 37)
* Application Binary Interface: ABI and ISA. (line 6)
-* Arithmetic functions <1>: Float Arithmetic. (line 6)
-* Arithmetic functions <2>: Integer Arithmetic. (line 6)
-* Arithmetic functions: Rational Arithmetic. (line 6)
+* Arithmetic functions: Integer Arithmetic. (line 6)
+* Arithmetic functions <1>: Rational Arithmetic. (line 6)
+* Arithmetic functions <2>: Float Arithmetic. (line 6)
* ARM: Notes for Particular Systems.
(line 20)
* Assembly cache handling: Assembly Cache Handling.
(line 6)
* Assembly writing guide: Assembly Writing Guide.
(line 6)
-* Assertion checking <1>: Debugging. (line 79)
-* Assertion checking: Build Options. (line 327)
-* Assignment functions <1>: Assigning Floats. (line 6)
-* Assignment functions <2>: Initializing Rationals.
+* Assertion checking: Build Options. (line 313)
+* Assertion checking <1>: Debugging. (line 74)
+* Assignment functions: Assigning Integers. (line 6)
+* Assignment functions <1>: Simultaneous Integer Init & Assign.
(line 6)
-* Assignment functions <3>: Simultaneous Integer Init & Assign.
+* Assignment functions <2>: Initializing Rationals.
(line 6)
+* Assignment functions <3>: Assigning Floats. (line 6)
* Assignment functions <4>: Simultaneous Float Init & Assign.
(line 6)
-* Assignment functions: Assigning Integers. (line 6)
* Autoconf: Autoconf. (line 6)
* Basics: GMP Basics. (line 6)
-* Berkeley MP compatible functions <1>: Build Options. (line 322)
-* Berkeley MP compatible functions: BSD Compatible Functions.
- (line 6)
* Binomial coefficient algorithm: Binomial Coefficients Algorithm.
(line 6)
* Binomial coefficient functions: Number Theoretic Functions.
- (line 100)
+ (line 128)
* Binutils strip: Known Build Problems.
(line 28)
* Bit manipulation functions: Integer Logic and Bit Fiddling.
(line 6)
* Bit scanning functions: Integer Logic and Bit Fiddling.
- (line 38)
-* Bit shift left: Integer Arithmetic. (line 35)
-* Bit shift right: Integer Division. (line 53)
+ (line 39)
+* Bit shift left: Integer Arithmetic. (line 38)
+* Bit shift right: Integer Division. (line 74)
* Bits per limb: Useful Macros and Constants.
(line 7)
-* BSD MP compatible functions <1>: Build Options. (line 322)
-* BSD MP compatible functions: BSD Compatible Functions.
- (line 6)
* Bug reporting: Reporting Bugs. (line 6)
* Build directory: Build Options. (line 19)
* Build notes for binary packaging: Notes for Package Builds.
* Build options: Build Options. (line 6)
* Build problems known: Known Build Problems.
(line 6)
-* Build system: Build Options. (line 52)
+* Build system: Build Options. (line 51)
* Building GMP: Installing GMP. (line 6)
* Bus error: Debugging. (line 7)
-* C compiler: Build Options. (line 182)
-* C++ compiler: Build Options. (line 254)
+* C compiler: Build Options. (line 178)
+* C++ compiler: Build Options. (line 249)
* C++ interface: C++ Class Interface. (line 6)
* C++ interface internals: C++ Interface Internals.
(line 6)
* C++ istream input: C++ Formatted Input. (line 6)
* C++ ostream output: C++ Formatted Output.
(line 6)
-* C++ support: Build Options. (line 230)
-* CC: Build Options. (line 182)
-* CC_FOR_BUILD: Build Options. (line 217)
-* CFLAGS: Build Options. (line 182)
-* Checker: Debugging. (line 115)
-* checkergcc: Debugging. (line 122)
+* C++ support: Build Options. (line 225)
+* CC: Build Options. (line 178)
+* CC_FOR_BUILD: Build Options. (line 212)
+* CFLAGS: Build Options. (line 178)
+* Checker: Debugging. (line 110)
+* checkergcc: Debugging. (line 117)
* Code organisation: Assembly Code Organisation.
(line 6)
* Compaq C++: Notes for Particular Systems.
(line 25)
-* Comparison functions <1>: Integer Comparisons. (line 6)
-* Comparison functions <2>: Comparing Rationals. (line 6)
-* Comparison functions: Float Comparison. (line 6)
+* Comparison functions: Integer Comparisons. (line 6)
+* Comparison functions <1>: Comparing Rationals. (line 6)
+* Comparison functions <2>: Float Comparison. (line 6)
* Compatibility with older versions: Compatibility with older versions.
(line 6)
* Conditions for copying GNU MP: Copying. (line 6)
* Configuring GMP: Installing GMP. (line 6)
-* Congruence algorithm: Exact Remainder. (line 29)
-* Congruence functions: Integer Division. (line 124)
+* Congruence algorithm: Exact Remainder. (line 30)
+* Congruence functions: Integer Division. (line 150)
* Constants: Useful Macros and Constants.
(line 6)
* Contributors: Contributors. (line 6)
(line 6)
* Conventions for variables: Variable Conventions.
(line 6)
-* Conversion functions <1>: Converting Integers. (line 6)
-* Conversion functions <2>: Converting Floats. (line 6)
-* Conversion functions: Rational Conversions.
+* Conversion functions: Converting Integers. (line 6)
+* Conversion functions <1>: Rational Conversions.
(line 6)
+* Conversion functions <2>: Converting Floats. (line 6)
* Copying conditions: Copying. (line 6)
-* CPPFLAGS: Build Options. (line 208)
-* CPU types <1>: Introduction to GMP. (line 24)
-* CPU types: Build Options. (line 108)
-* Cross compiling: Build Options. (line 66)
+* CPPFLAGS: Build Options. (line 204)
+* CPU types: Introduction to GMP. (line 24)
+* CPU types <1>: Build Options. (line 107)
+* Cross compiling: Build Options. (line 65)
+* Cryptography functions, low-level: Low-level Functions. (line 507)
* Custom allocation: Custom Allocation. (line 6)
-* CXX: Build Options. (line 254)
-* CXXFLAGS: Build Options. (line 254)
+* CXX: Build Options. (line 249)
+* CXXFLAGS: Build Options. (line 249)
* Cygwin: Notes for Particular Systems.
- (line 43)
+ (line 57)
* Darwin: Known Build Problems.
(line 51)
* Debugging: Debugging. (line 6)
(line 6)
* Digits in an integer: Miscellaneous Integer Functions.
(line 23)
-* Divisibility algorithm: Exact Remainder. (line 29)
-* Divisibility functions: Integer Division. (line 124)
+* Divisibility algorithm: Exact Remainder. (line 30)
+* Divisibility functions: Integer Division. (line 136)
+* Divisibility functions <1>: Integer Division. (line 150)
* Divisibility testing: Efficiency. (line 91)
* Division algorithms: Division Algorithms. (line 6)
-* Division functions <1>: Rational Arithmetic. (line 22)
-* Division functions <2>: Integer Division. (line 6)
-* Division functions: Float Arithmetic. (line 33)
-* DJGPP <1>: Notes for Particular Systems.
- (line 43)
-* DJGPP: Known Build Problems.
+* Division functions: Integer Division. (line 6)
+* Division functions <1>: Rational Arithmetic. (line 24)
+* Division functions <2>: Float Arithmetic. (line 33)
+* DJGPP: Notes for Particular Systems.
+ (line 57)
+* DJGPP <1>: Known Build Problems.
(line 18)
* DLLs: Notes for Particular Systems.
- (line 56)
-* DocBook: Build Options. (line 354)
-* Documentation formats: Build Options. (line 347)
+ (line 70)
+* DocBook: Build Options. (line 340)
+* Documentation formats: Build Options. (line 333)
* Documentation license: GNU Free Documentation License.
(line 6)
-* DVI: Build Options. (line 350)
+* DVI: Build Options. (line 336)
* Efficiency: Efficiency. (line 6)
* Emacs: Emacs. (line 6)
-* Exact division functions: Integer Division. (line 102)
+* Exact division functions: Integer Division. (line 125)
* Exact remainder: Exact Remainder. (line 6)
* Example programs: Demonstration Programs.
(line 6)
* Exec prefix: Build Options. (line 32)
+* Execution profiling: Build Options. (line 317)
* Execution profiling <1>: Profiling. (line 6)
-* Execution profiling: Build Options. (line 331)
-* Exponentiation functions <1>: Integer Exponentiation.
+* Exponentiation functions: Integer Exponentiation.
(line 6)
-* Exponentiation functions: Float Arithmetic. (line 41)
+* Exponentiation functions <1>: Float Arithmetic. (line 41)
* Export: Integer Import and Export.
(line 45)
* Expression parsing demo: Demonstration Programs.
- (line 18)
+ (line 15)
+* Expression parsing demo <1>: Demonstration Programs.
+ (line 17)
+* Expression parsing demo <2>: Demonstration Programs.
+ (line 19)
* Extended GCD: Number Theoretic Functions.
- (line 45)
+ (line 47)
* Factor removal functions: Number Theoretic Functions.
- (line 90)
+ (line 108)
* Factorial algorithm: Factorial Algorithm. (line 6)
* Factorial functions: Number Theoretic Functions.
- (line 95)
+ (line 116)
* Factorization demo: Demonstration Programs.
- (line 25)
+ (line 22)
* Fast Fourier Transform: FFT Multiplication. (line 6)
-* Fat binary: Build Options. (line 164)
+* Fat binary: Build Options. (line 160)
+* FFT multiplication: Build Options. (line 307)
* FFT multiplication <1>: FFT Multiplication. (line 6)
-* FFT multiplication: Build Options. (line 317)
* Fibonacci number algorithm: Fibonacci Numbers Algorithm.
(line 6)
* Fibonacci sequence functions: Number Theoretic Functions.
- (line 108)
+ (line 136)
* Float arithmetic functions: Float Arithmetic. (line 6)
+* Float assignment functions: Assigning Floats. (line 6)
* Float assignment functions <1>: Simultaneous Float Init & Assign.
(line 6)
-* Float assignment functions: Assigning Floats. (line 6)
* Float comparison functions: Float Comparison. (line 6)
* Float conversion functions: Converting Floats. (line 6)
* Float functions: Floating-point Functions.
(line 6)
+* Float initialization functions: Initializing Floats. (line 6)
* Float initialization functions <1>: Simultaneous Float Init & Assign.
(line 6)
-* Float initialization functions: Initializing Floats. (line 6)
* Float input and output functions: I/O of Floats. (line 6)
* Float internals: Float Internals. (line 6)
* Float miscellaneous functions: Miscellaneous Float Functions.
(line 27)
* Float rounding functions: Miscellaneous Float Functions.
(line 9)
-* Float sign tests: Float Comparison. (line 33)
+* Float sign tests: Float Comparison. (line 34)
* Floating point mode: Notes for Particular Systems.
(line 34)
* Floating-point functions: Floating-point Functions.
* Formatted output: Formatted Output. (line 6)
* Free Documentation License: GNU Free Documentation License.
(line 6)
-* frexp <1>: Converting Floats. (line 23)
-* frexp: Converting Integers. (line 42)
-* FTP of latest version: Introduction to GMP. (line 38)
+* FreeBSD: Notes for Particular Systems.
+ (line 43)
+* FreeBSD <1>: Notes for Particular Systems.
+ (line 52)
+* frexp: Converting Integers. (line 43)
+* frexp <1>: Converting Floats. (line 24)
+* FTP of latest version: Introduction to GMP. (line 37)
* Function classes: Function Classes. (line 6)
* FunctionCheck: Profiling. (line 77)
-* GCC Checker: Debugging. (line 115)
+* GCC Checker: Debugging. (line 110)
* GCD algorithms: Greatest Common Divisor Algorithms.
(line 6)
* GCD extended: Number Theoretic Functions.
- (line 45)
+ (line 47)
* GCD functions: Number Theoretic Functions.
(line 30)
-* GDB: Debugging. (line 58)
-* Generic C: Build Options. (line 153)
+* GDB: Debugging. (line 53)
+* Generic C: Build Options. (line 151)
* GMP Perl module: Demonstration Programs.
- (line 35)
+ (line 28)
* GMP version number: Useful Macros and Constants.
(line 12)
* gmp.h: Headers and Libraries.
(line 6)
* gmpxx.h: C++ Interface General.
(line 8)
-* GNU Debugger: Debugging. (line 58)
+* GNU Debugger: Debugging. (line 53)
* GNU Free Documentation License: GNU Free Documentation License.
(line 6)
* GNU strip: Known Build Problems.
(line 34)
* Headers: Headers and Libraries.
(line 6)
-* Heap problems: Debugging. (line 24)
-* Home page: Introduction to GMP. (line 34)
-* Host system: Build Options. (line 66)
-* HP-UX: ABI and ISA. (line 107)
-* HPPA: ABI and ISA. (line 68)
-* I/O functions <1>: I/O of Integers. (line 6)
-* I/O functions <2>: I/O of Rationals. (line 6)
-* I/O functions: I/O of Floats. (line 6)
+* Heap problems: Debugging. (line 23)
+* Home page: Introduction to GMP. (line 33)
+* Host system: Build Options. (line 65)
+* HP-UX: ABI and ISA. (line 76)
+* HP-UX <1>: ABI and ISA. (line 114)
+* HPPA: ABI and ISA. (line 76)
+* I/O functions: I/O of Integers. (line 6)
+* I/O functions <1>: I/O of Rationals. (line 6)
+* I/O functions <2>: I/O of Floats. (line 6)
* i386: Notes for Particular Systems.
- (line 126)
-* IA-64: ABI and ISA. (line 107)
+ (line 150)
+* IA-64: ABI and ISA. (line 114)
* Import: Integer Import and Export.
(line 11)
* In-place operations: Efficiency. (line 57)
* Include files: Headers and Libraries.
(line 6)
* info-lookup-symbol: Emacs. (line 6)
-* Initialization functions <1>: Initializing Integers.
+* Initialization functions: Initializing Integers.
(line 6)
-* Initialization functions <2>: Initializing Rationals.
+* Initialization functions <1>: Simultaneous Integer Init & Assign.
(line 6)
-* Initialization functions <3>: Random State Initialization.
+* Initialization functions <2>: Initializing Rationals.
(line 6)
+* Initialization functions <3>: Initializing Floats. (line 6)
* Initialization functions <4>: Simultaneous Float Init & Assign.
(line 6)
-* Initialization functions <5>: Simultaneous Integer Init & Assign.
+* Initialization functions <5>: Random State Initialization.
(line 6)
-* Initialization functions: Initializing Floats. (line 6)
* Initializing and clearing: Efficiency. (line 21)
-* Input functions <1>: I/O of Integers. (line 6)
-* Input functions <2>: I/O of Rationals. (line 6)
-* Input functions <3>: I/O of Floats. (line 6)
-* Input functions: Formatted Input Functions.
+* Input functions: I/O of Integers. (line 6)
+* Input functions <1>: I/O of Rationals. (line 6)
+* Input functions <2>: I/O of Floats. (line 6)
+* Input functions <3>: Formatted Input Functions.
(line 6)
* Install prefix: Build Options. (line 32)
* Installing GMP: Installing GMP. (line 6)
* Integer: Nomenclature and Types.
(line 6)
* Integer arithmetic functions: Integer Arithmetic. (line 6)
+* Integer assignment functions: Assigning Integers. (line 6)
* Integer assignment functions <1>: Simultaneous Integer Init & Assign.
(line 6)
-* Integer assignment functions: Assigning Integers. (line 6)
* Integer bit manipulation functions: Integer Logic and Bit Fiddling.
(line 6)
* Integer comparison functions: Integer Comparisons. (line 6)
* Integer functions: Integer Functions. (line 6)
* Integer import: Integer Import and Export.
(line 11)
-* Integer initialization functions <1>: Simultaneous Integer Init & Assign.
- (line 6)
* Integer initialization functions: Initializing Integers.
(line 6)
+* Integer initialization functions <1>: Simultaneous Integer Init & Assign.
+ (line 6)
* Integer input and output functions: I/O of Integers. (line 6)
* Integer internals: Integer Internals. (line 6)
* Integer logical functions: Integer Logic and Bit Fiddling.
* Integer special functions: Integer Special Functions.
(line 6)
* Interix: Notes for Particular Systems.
- (line 51)
+ (line 65)
* Internals: Internals. (line 6)
* Introduction: Introduction to GMP. (line 6)
* Inverse modulo functions: Number Theoretic Functions.
- (line 60)
+ (line 74)
+* IRIX: ABI and ISA. (line 139)
* IRIX <1>: Known Build Problems.
(line 38)
-* IRIX: ABI and ISA. (line 132)
* ISA: ABI and ISA. (line 6)
* istream input: C++ Formatted Input. (line 6)
* Jacobi symbol algorithm: Jacobi Symbol. (line 6)
* Jacobi symbol functions: Number Theoretic Functions.
- (line 66)
+ (line 83)
* Karatsuba multiplication: Karatsuba Multiplication.
(line 6)
* Karatsuba square root algorithm: Square Root Algorithm.
(line 6)
* Kronecker symbol functions: Number Theoretic Functions.
- (line 78)
+ (line 95)
* Language bindings: Language Bindings. (line 6)
-* Latest version of GMP: Introduction to GMP. (line 38)
+* Latest version of GMP: Introduction to GMP. (line 37)
* LCM functions: Number Theoretic Functions.
- (line 55)
+ (line 68)
* Least common multiple functions: Number Theoretic Functions.
- (line 55)
+ (line 68)
* Legendre symbol functions: Number Theoretic Functions.
- (line 69)
+ (line 86)
* libgmp: Headers and Libraries.
(line 22)
* libgmpxx: Headers and Libraries.
* Linear congruential algorithm: Random Number Algorithms.
(line 25)
* Linear congruential random numbers: Random State Initialization.
+ (line 18)
+* Linear congruential random numbers <1>: Random State Initialization.
(line 32)
* Linking: Headers and Libraries.
(line 22)
* Logical functions: Integer Logic and Bit Fiddling.
(line 6)
* Low-level functions: Low-level Functions. (line 6)
+* Low-level functions for cryptography: Low-level Functions. (line 507)
* Lucas number algorithm: Lucas Numbers Algorithm.
(line 6)
* Lucas number functions: Number Theoretic Functions.
- (line 119)
+ (line 147)
* MacOS X: Known Build Problems.
(line 51)
-* Mailing lists: Introduction to GMP. (line 45)
-* Malloc debugger: Debugging. (line 30)
-* Malloc problems: Debugging. (line 24)
+* Mailing lists: Introduction to GMP. (line 44)
+* Malloc debugger: Debugging. (line 29)
+* Malloc problems: Debugging. (line 23)
* Memory allocation: Custom Allocation. (line 6)
* Memory management: Memory Management. (line 6)
* Mersenne twister algorithm: Random Number Algorithms.
* Mersenne twister random numbers: Random State Initialization.
(line 13)
* MINGW: Notes for Particular Systems.
- (line 43)
-* MIPS: ABI and ISA. (line 132)
+ (line 57)
+* MIPS: ABI and ISA. (line 139)
* Miscellaneous float functions: Miscellaneous Float Functions.
(line 6)
* Miscellaneous integer functions: Miscellaneous Integer Functions.
(line 6)
* MMX: Notes for Particular Systems.
- (line 132)
+ (line 156)
* Modular inverse functions: Number Theoretic Functions.
- (line 60)
+ (line 74)
* Most significant bit: Miscellaneous Integer Functions.
(line 34)
-* mp.h: BSD Compatible Functions.
- (line 21)
-* MPN_PATH: Build Options. (line 335)
+* MPN_PATH: Build Options. (line 321)
* MS Windows: Notes for Particular Systems.
- (line 56)
+ (line 57)
+* MS Windows <1>: Notes for Particular Systems.
+ (line 70)
* MS-DOS: Notes for Particular Systems.
- (line 43)
+ (line 57)
* Multi-threading: Reentrancy. (line 6)
* Multiplication algorithms: Multiplication Algorithms.
(line 6)
-* Nails: Low-level Functions. (line 478)
-* Native compilation: Build Options. (line 52)
+* Nails: Low-level Functions. (line 686)
+* Native compilation: Build Options. (line 51)
+* NetBSD: Notes for Particular Systems.
+ (line 100)
* NeXT: Known Build Problems.
(line 57)
* Next prime function: Number Theoretic Functions.
(line 6)
* Non-Unix systems: Build Options. (line 11)
* Nth root algorithm: Nth Root Algorithm. (line 6)
-* Number sequences: Efficiency. (line 147)
+* Number sequences: Efficiency. (line 145)
* Number theoretic functions: Number Theoretic Functions.
(line 6)
* Numerator and denominator: Applying Integer Functions.
(line 6)
* obstack output: Formatted Output Functions.
- (line 81)
+ (line 79)
* OpenBSD: Notes for Particular Systems.
- (line 86)
+ (line 109)
* Optimizing performance: Performance optimization.
(line 6)
* ostream output: C++ Formatted Output.
(line 6)
* Other languages: Language Bindings. (line 6)
-* Output functions <1>: I/O of Floats. (line 6)
-* Output functions <2>: I/O of Rationals. (line 6)
+* Output functions: I/O of Integers. (line 6)
+* Output functions <1>: I/O of Rationals. (line 6)
+* Output functions <2>: I/O of Floats. (line 6)
* Output functions <3>: Formatted Output Functions.
(line 6)
-* Output functions: I/O of Integers. (line 6)
* Packaged builds: Notes for Package Builds.
(line 6)
* Parameter conventions: Parameter Conventions.
(line 6)
* Parsing expressions demo: Demonstration Programs.
- (line 21)
+ (line 15)
+* Parsing expressions demo <1>: Demonstration Programs.
+ (line 17)
+* Parsing expressions demo <2>: Demonstration Programs.
+ (line 19)
* Particular systems: Notes for Particular Systems.
(line 6)
* Past GMP versions: Compatibility with older versions.
(line 6)
-* PDF: Build Options. (line 350)
+* PDF: Build Options. (line 336)
* Perfect power algorithm: Perfect Power Algorithm.
(line 6)
-* Perfect power functions: Integer Roots. (line 27)
+* Perfect power functions: Integer Roots. (line 28)
* Perfect square algorithm: Perfect Square Algorithm.
(line 6)
-* Perfect square functions: Integer Roots. (line 36)
+* Perfect square functions: Integer Roots. (line 37)
* perl: Demonstration Programs.
- (line 35)
+ (line 28)
* Perl module: Demonstration Programs.
- (line 35)
-* Postscript: Build Options. (line 350)
+ (line 28)
+* Postscript: Build Options. (line 336)
+* Power/PowerPC: Notes for Particular Systems.
+ (line 115)
* Power/PowerPC <1>: Known Build Problems.
(line 63)
-* Power/PowerPC: Notes for Particular Systems.
- (line 92)
* Powering algorithms: Powering Algorithms. (line 6)
-* Powering functions <1>: Float Arithmetic. (line 41)
* Powering functions: Integer Exponentiation.
(line 6)
-* PowerPC: ABI and ISA. (line 167)
+* Powering functions <1>: Float Arithmetic. (line 41)
+* PowerPC: ABI and ISA. (line 173)
* Precision of floats: Floating-point Functions.
(line 6)
* Precision of hardware floating point: Notes for Particular Systems.
(line 6)
* Prime testing functions: Number Theoretic Functions.
(line 7)
+* Primorial functions: Number Theoretic Functions.
+ (line 121)
* printf formatted output: Formatted Output. (line 6)
* Probable prime testing functions: Number Theoretic Functions.
(line 7)
(line 6)
* Random number algorithms: Random Number Algorithms.
(line 6)
-* Random number functions <1>: Integer Random Numbers.
+* Random number functions: Integer Random Numbers.
(line 6)
-* Random number functions <2>: Miscellaneous Float Functions.
+* Random number functions <1>: Miscellaneous Float Functions.
(line 27)
-* Random number functions: Random Number Functions.
+* Random number functions <2>: Random Number Functions.
(line 6)
* Random number seeding: Random State Seeding.
(line 6)
(line 6)
* Random state: Nomenclature and Types.
(line 46)
-* Rational arithmetic: Efficiency. (line 113)
+* Rational arithmetic: Efficiency. (line 111)
* Rational arithmetic functions: Rational Arithmetic. (line 6)
* Rational assignment functions: Initializing Rationals.
(line 6)
(line 6)
* Rational numerator and denominator: Applying Integer Functions.
(line 6)
-* Rational sign tests: Comparing Rationals. (line 27)
+* Rational sign tests: Comparing Rationals. (line 28)
* Raw output internals: Raw Output Internals.
(line 6)
* Reallocations: Efficiency. (line 30)
* Reentrancy: Reentrancy. (line 6)
-* References: References. (line 6)
+* References: References. (line 5)
* Remove factor functions: Number Theoretic Functions.
- (line 90)
+ (line 108)
* Reporting bugs: Reporting Bugs. (line 6)
* Root extraction algorithm: Nth Root Algorithm. (line 6)
* Root extraction algorithms: Root Extraction Algorithms.
(line 6)
-* Root extraction functions <1>: Float Arithmetic. (line 37)
* Root extraction functions: Integer Roots. (line 6)
-* Root testing functions: Integer Roots. (line 36)
+* Root extraction functions <1>: Float Arithmetic. (line 37)
+* Root testing functions: Integer Roots. (line 28)
+* Root testing functions <1>: Integer Roots. (line 37)
* Rounding functions: Miscellaneous Float Functions.
(line 9)
* Sample programs: Demonstration Programs.
(line 6)
* Scan bit functions: Integer Logic and Bit Fiddling.
- (line 38)
+ (line 39)
* scanf formatted input: Formatted Input. (line 6)
* SCO: Known Build Problems.
(line 38)
* Sequent Symmetry: Known Build Problems.
(line 68)
* Services for Unix: Notes for Particular Systems.
- (line 51)
+ (line 65)
* Shared library versioning: Notes for Package Builds.
(line 9)
-* Sign tests <1>: Float Comparison. (line 33)
-* Sign tests <2>: Integer Comparisons. (line 28)
-* Sign tests: Comparing Rationals. (line 27)
+* Sign tests: Integer Comparisons. (line 28)
+* Sign tests <1>: Comparing Rationals. (line 28)
+* Sign tests <2>: Float Comparison. (line 34)
* Size in digits: Miscellaneous Integer Functions.
(line 23)
* Small operands: Efficiency. (line 7)
-* Solaris <1>: ABI and ISA. (line 201)
-* Solaris: Known Build Problems.
- (line 78)
+* Solaris: ABI and ISA. (line 204)
+* Solaris <1>: Known Build Problems.
+ (line 72)
+* Solaris <2>: Known Build Problems.
+ (line 77)
* Sparc: Notes for Particular Systems.
- (line 108)
-* Sparc V9: ABI and ISA. (line 201)
+ (line 127)
+* Sparc <1>: Notes for Particular Systems.
+ (line 132)
+* Sparc V9: ABI and ISA. (line 204)
* Special integer functions: Integer Special Functions.
(line 6)
* Square root algorithm: Square Root Algorithm.
(line 6)
* SSE2: Notes for Particular Systems.
- (line 132)
-* Stack backtrace: Debugging. (line 50)
+ (line 156)
+* Stack backtrace: Debugging. (line 45)
+* Stack overflow: Build Options. (line 273)
* Stack overflow <1>: Debugging. (line 7)
-* Stack overflow: Build Options. (line 278)
* Static linking: Efficiency. (line 14)
* stdarg.h: Headers and Libraries.
(line 17)
(line 11)
* Stripped libraries: Known Build Problems.
(line 28)
-* Sun: ABI and ISA. (line 201)
+* Sun: ABI and ISA. (line 204)
* SunOS: Notes for Particular Systems.
- (line 120)
+ (line 144)
* Systems: Notes for Particular Systems.
(line 6)
-* Temporary memory: Build Options. (line 278)
-* Texinfo: Build Options. (line 347)
-* Text input/output: Efficiency. (line 153)
+* Temporary memory: Build Options. (line 273)
+* Texinfo: Build Options. (line 333)
+* Text input/output: Efficiency. (line 151)
* Thread safety: Reentrancy. (line 6)
-* Toom multiplication <1>: Other Multiplication.
+* Toom multiplication: Toom 3-Way Multiplication.
(line 6)
-* Toom multiplication <2>: Toom 4-Way Multiplication.
+* Toom multiplication <1>: Toom 4-Way Multiplication.
(line 6)
-* Toom multiplication: Toom 3-Way Multiplication.
+* Toom multiplication <2>: Higher degree Toom'n'half.
+ (line 6)
+* Toom multiplication <3>: Other Multiplication.
(line 6)
* Types: Nomenclature and Types.
(line 6)
(line 6)
* User-defined precision: Floating-point Functions.
(line 6)
-* Valgrind: Debugging. (line 130)
+* Valgrind: Debugging. (line 125)
* Variable conventions: Variable Conventions.
(line 6)
* Version number: Useful Macros and Constants.
(line 12)
-* Web page: Introduction to GMP. (line 34)
+* Web page: Introduction to GMP. (line 33)
* Windows: Notes for Particular Systems.
- (line 56)
+ (line 57)
+* Windows <1>: Notes for Particular Systems.
+ (line 70)
* x86: Notes for Particular Systems.
- (line 126)
+ (line 150)
* x87: Notes for Particular Systems.
(line 34)
-* XML: Build Options. (line 354)
+* XML: Build Options. (line 340)
\1f
File: gmp.info, Node: Function Index, Prev: Concept Index, Up: Top
\0\b[index\0\b]
* Menu:
+* _mpz_realloc: Integer Special Functions.
+ (line 13)
* __GMP_CC: Useful Macros and Constants.
- (line 23)
+ (line 22)
* __GMP_CFLAGS: Useful Macros and Constants.
- (line 24)
+ (line 23)
* __GNU_MP_VERSION: Useful Macros and Constants.
- (line 10)
+ (line 9)
* __GNU_MP_VERSION_MINOR: Useful Macros and Constants.
- (line 11)
+ (line 10)
* __GNU_MP_VERSION_PATCHLEVEL: Useful Macros and Constants.
- (line 12)
-* _mpz_realloc: Integer Special Functions.
- (line 51)
+ (line 11)
+* abs: C++ Interface Integers.
+ (line 46)
* abs <1>: C++ Interface Rationals.
- (line 43)
-* abs <2>: C++ Interface Integers.
- (line 42)
-* abs: C++ Interface Floats.
- (line 70)
+ (line 47)
+* abs <2>: C++ Interface Floats.
+ (line 82)
* ceil: C++ Interface Floats.
- (line 71)
-* cmp <1>: C++ Interface Floats.
- (line 72)
+ (line 83)
+* cmp: C++ Interface Integers.
+ (line 47)
+* cmp <1>: C++ Interface Integers.
+ (line 48)
* cmp <2>: C++ Interface Rationals.
- (line 44)
-* cmp <3>: C++ Interface Integers.
- (line 44)
-* cmp: C++ Interface Rationals.
- (line 45)
+ (line 48)
+* cmp <3>: C++ Interface Rationals.
+ (line 49)
+* cmp <4>: C++ Interface Floats.
+ (line 84)
+* cmp <5>: C++ Interface Floats.
+ (line 85)
+* factorial: C++ Interface Integers.
+ (line 71)
+* fibonacci: C++ Interface Integers.
+ (line 75)
* floor: C++ Interface Floats.
- (line 80)
-* gcd: BSD Compatible Functions.
- (line 82)
+ (line 95)
+* gcd: C++ Interface Integers.
+ (line 68)
* gmp_asprintf: Formatted Output Functions.
- (line 65)
+ (line 63)
* gmp_errno: Random State Initialization.
- (line 55)
+ (line 56)
* GMP_ERROR_INVALID_ARGUMENT: Random State Initialization.
- (line 55)
+ (line 56)
* GMP_ERROR_UNSUPPORTED_ARGUMENT: Random State Initialization.
- (line 55)
+ (line 56)
* gmp_fprintf: Formatted Output Functions.
- (line 29)
+ (line 28)
* gmp_fscanf: Formatted Input Functions.
- (line 25)
-* GMP_LIMB_BITS: Low-level Functions. (line 508)
-* GMP_NAIL_BITS: Low-level Functions. (line 506)
-* GMP_NAIL_MASK: Low-level Functions. (line 516)
-* GMP_NUMB_BITS: Low-level Functions. (line 507)
-* GMP_NUMB_MASK: Low-level Functions. (line 517)
-* GMP_NUMB_MAX: Low-level Functions. (line 525)
+ (line 24)
+* GMP_LIMB_BITS: Low-level Functions. (line 714)
+* GMP_NAIL_BITS: Low-level Functions. (line 712)
+* GMP_NAIL_MASK: Low-level Functions. (line 722)
+* GMP_NUMB_BITS: Low-level Functions. (line 713)
+* GMP_NUMB_MASK: Low-level Functions. (line 723)
+* GMP_NUMB_MAX: Low-level Functions. (line 731)
* gmp_obstack_printf: Formatted Output Functions.
- (line 79)
+ (line 75)
* gmp_obstack_vprintf: Formatted Output Functions.
- (line 81)
+ (line 77)
* gmp_printf: Formatted Output Functions.
- (line 24)
-* GMP_RAND_ALG_DEFAULT: Random State Initialization.
- (line 49)
-* GMP_RAND_ALG_LC: Random State Initialization.
- (line 49)
+ (line 23)
* gmp_randclass: C++ Interface Random Numbers.
- (line 7)
+ (line 6)
* gmp_randclass::get_f: C++ Interface Random Numbers.
+ (line 44)
+* gmp_randclass::get_f <1>: C++ Interface Random Numbers.
(line 45)
* gmp_randclass::get_z_bits: C++ Interface Random Numbers.
- (line 39)
+ (line 37)
+* gmp_randclass::get_z_bits <1>: C++ Interface Random Numbers.
+ (line 38)
* gmp_randclass::get_z_range: C++ Interface Random Numbers.
- (line 42)
+ (line 41)
* gmp_randclass::gmp_randclass: C++ Interface Random Numbers.
- (line 13)
+ (line 11)
+* gmp_randclass::gmp_randclass <1>: C++ Interface Random Numbers.
+ (line 26)
* gmp_randclass::seed: C++ Interface Random Numbers.
+ (line 32)
+* gmp_randclass::seed <1>: C++ Interface Random Numbers.
(line 33)
* gmp_randclear: Random State Initialization.
(line 62)
* gmp_randinit: Random State Initialization.
- (line 47)
+ (line 45)
* gmp_randinit_default: Random State Initialization.
- (line 7)
+ (line 6)
* gmp_randinit_lc_2exp: Random State Initialization.
- (line 18)
+ (line 16)
* gmp_randinit_lc_2exp_size: Random State Initialization.
- (line 32)
+ (line 30)
* gmp_randinit_mt: Random State Initialization.
- (line 13)
+ (line 12)
* gmp_randinit_set: Random State Initialization.
- (line 43)
+ (line 41)
* gmp_randseed: Random State Seeding.
- (line 7)
+ (line 6)
* gmp_randseed_ui: Random State Seeding.
- (line 9)
+ (line 8)
* gmp_randstate_t: Nomenclature and Types.
(line 46)
+* GMP_RAND_ALG_DEFAULT: Random State Initialization.
+ (line 50)
+* GMP_RAND_ALG_LC: Random State Initialization.
+ (line 50)
* gmp_scanf: Formatted Input Functions.
- (line 21)
+ (line 20)
* gmp_snprintf: Formatted Output Functions.
- (line 46)
+ (line 44)
* gmp_sprintf: Formatted Output Functions.
- (line 34)
+ (line 33)
* gmp_sscanf: Formatted Input Functions.
- (line 29)
+ (line 28)
* gmp_urandomb_ui: Random State Miscellaneous.
- (line 8)
+ (line 6)
* gmp_urandomm_ui: Random State Miscellaneous.
- (line 14)
+ (line 12)
* gmp_vasprintf: Formatted Output Functions.
- (line 66)
+ (line 64)
* gmp_version: Useful Macros and Constants.
(line 18)
* gmp_vfprintf: Formatted Output Functions.
- (line 30)
+ (line 29)
* gmp_vfscanf: Formatted Input Functions.
- (line 26)
-* gmp_vprintf: Formatted Output Functions.
(line 25)
+* gmp_vprintf: Formatted Output Functions.
+ (line 24)
* gmp_vscanf: Formatted Input Functions.
- (line 22)
+ (line 21)
* gmp_vsnprintf: Formatted Output Functions.
- (line 48)
+ (line 46)
* gmp_vsprintf: Formatted Output Functions.
- (line 35)
+ (line 34)
* gmp_vsscanf: Formatted Input Functions.
- (line 31)
-* hypot: C++ Interface Floats.
- (line 81)
-* itom: BSD Compatible Functions.
(line 29)
-* madd: BSD Compatible Functions.
- (line 43)
-* mcmp: BSD Compatible Functions.
- (line 85)
-* mdiv: BSD Compatible Functions.
- (line 53)
-* mfree: BSD Compatible Functions.
- (line 105)
-* min: BSD Compatible Functions.
- (line 89)
-* MINT: BSD Compatible Functions.
- (line 21)
-* mout: BSD Compatible Functions.
- (line 94)
-* move: BSD Compatible Functions.
- (line 39)
-* mp_bitcnt_t: Nomenclature and Types.
- (line 42)
-* mp_bits_per_limb: Useful Macros and Constants.
- (line 7)
-* mp_exp_t: Nomenclature and Types.
- (line 27)
-* mp_get_memory_functions: Custom Allocation. (line 93)
-* mp_limb_t: Nomenclature and Types.
- (line 31)
-* mp_set_memory_functions: Custom Allocation. (line 21)
-* mp_size_t: Nomenclature and Types.
- (line 37)
-* mpf_abs: Float Arithmetic. (line 47)
-* mpf_add: Float Arithmetic. (line 7)
-* mpf_add_ui: Float Arithmetic. (line 9)
+* hypot: C++ Interface Floats.
+ (line 96)
+* lcm: C++ Interface Integers.
+ (line 69)
+* mpf_abs: Float Arithmetic. (line 46)
+* mpf_add: Float Arithmetic. (line 6)
+* mpf_add_ui: Float Arithmetic. (line 7)
* mpf_ceil: Miscellaneous Float Functions.
- (line 7)
+ (line 6)
* mpf_class: C++ Interface General.
- (line 20)
+ (line 19)
* mpf_class::fits_sint_p: C++ Interface Floats.
- (line 74)
+ (line 87)
* mpf_class::fits_slong_p: C++ Interface Floats.
- (line 75)
+ (line 88)
* mpf_class::fits_sshort_p: C++ Interface Floats.
- (line 76)
+ (line 89)
* mpf_class::fits_uint_p: C++ Interface Floats.
- (line 77)
+ (line 91)
* mpf_class::fits_ulong_p: C++ Interface Floats.
- (line 78)
+ (line 92)
* mpf_class::fits_ushort_p: C++ Interface Floats.
- (line 79)
+ (line 93)
* mpf_class::get_d: C++ Interface Floats.
- (line 82)
+ (line 98)
* mpf_class::get_mpf_t: C++ Interface General.
- (line 66)
+ (line 65)
* mpf_class::get_prec: C++ Interface Floats.
- (line 100)
+ (line 120)
* mpf_class::get_si: C++ Interface Floats.
- (line 83)
+ (line 99)
* mpf_class::get_str: C++ Interface Floats.
- (line 85)
+ (line 100)
* mpf_class::get_ui: C++ Interface Floats.
- (line 86)
+ (line 102)
* mpf_class::mpf_class: C++ Interface Floats.
- (line 38)
+ (line 11)
+* mpf_class::mpf_class <1>: C++ Interface Floats.
+ (line 12)
+* mpf_class::mpf_class <2>: C++ Interface Floats.
+ (line 32)
+* mpf_class::mpf_class <3>: C++ Interface Floats.
+ (line 33)
+* mpf_class::mpf_class <4>: C++ Interface Floats.
+ (line 41)
+* mpf_class::mpf_class <5>: C++ Interface Floats.
+ (line 42)
+* mpf_class::mpf_class <6>: C++ Interface Floats.
+ (line 44)
+* mpf_class::mpf_class <7>: C++ Interface Floats.
+ (line 45)
* mpf_class::operator=: C++ Interface Floats.
- (line 47)
+ (line 59)
* mpf_class::set_prec: C++ Interface Floats.
- (line 101)
+ (line 121)
* mpf_class::set_prec_raw: C++ Interface Floats.
- (line 102)
+ (line 122)
* mpf_class::set_str: C++ Interface Floats.
- (line 88)
-* mpf_clear: Initializing Floats. (line 37)
-* mpf_clears: Initializing Floats. (line 41)
-* mpf_cmp: Float Comparison. (line 7)
+ (line 104)
+* mpf_class::set_str <1>: C++ Interface Floats.
+ (line 105)
+* mpf_class::swap: C++ Interface Floats.
+ (line 109)
+* mpf_clear: Initializing Floats. (line 36)
+* mpf_clears: Initializing Floats. (line 40)
+* mpf_cmp: Float Comparison. (line 6)
* mpf_cmp_d: Float Comparison. (line 8)
* mpf_cmp_si: Float Comparison. (line 10)
* mpf_cmp_ui: Float Comparison. (line 9)
-* mpf_div: Float Arithmetic. (line 29)
+* mpf_cmp_z: Float Comparison. (line 7)
+* mpf_div: Float Arithmetic. (line 28)
* mpf_div_2exp: Float Arithmetic. (line 53)
-* mpf_div_ui: Float Arithmetic. (line 33)
+* mpf_div_ui: Float Arithmetic. (line 31)
* mpf_eq: Float Comparison. (line 17)
* mpf_fits_sint_p: Miscellaneous Float Functions.
- (line 20)
+ (line 19)
* mpf_fits_slong_p: Miscellaneous Float Functions.
- (line 18)
+ (line 17)
* mpf_fits_sshort_p: Miscellaneous Float Functions.
- (line 22)
+ (line 21)
* mpf_fits_uint_p: Miscellaneous Float Functions.
- (line 19)
+ (line 18)
* mpf_fits_ulong_p: Miscellaneous Float Functions.
- (line 17)
+ (line 16)
* mpf_fits_ushort_p: Miscellaneous Float Functions.
- (line 21)
+ (line 20)
* mpf_floor: Miscellaneous Float Functions.
- (line 8)
-* mpf_get_d: Converting Floats. (line 7)
-* mpf_get_d_2exp: Converting Floats. (line 16)
-* mpf_get_default_prec: Initializing Floats. (line 12)
-* mpf_get_prec: Initializing Floats. (line 62)
+ (line 7)
+* mpf_get_d: Converting Floats. (line 6)
+* mpf_get_default_prec: Initializing Floats. (line 11)
+* mpf_get_d_2exp: Converting Floats. (line 15)
+* mpf_get_prec: Initializing Floats. (line 61)
* mpf_get_si: Converting Floats. (line 27)
-* mpf_get_str: Converting Floats. (line 37)
+* mpf_get_str: Converting Floats. (line 36)
* mpf_get_ui: Converting Floats. (line 28)
-* mpf_init: Initializing Floats. (line 19)
-* mpf_init2: Initializing Floats. (line 26)
+* mpf_init: Initializing Floats. (line 18)
+* mpf_init2: Initializing Floats. (line 25)
+* mpf_inits: Initializing Floats. (line 30)
* mpf_init_set: Simultaneous Float Init & Assign.
- (line 16)
+ (line 15)
* mpf_init_set_d: Simultaneous Float Init & Assign.
- (line 19)
-* mpf_init_set_si: Simultaneous Float Init & Assign.
(line 18)
+* mpf_init_set_si: Simultaneous Float Init & Assign.
+ (line 17)
* mpf_init_set_str: Simultaneous Float Init & Assign.
- (line 25)
+ (line 24)
* mpf_init_set_ui: Simultaneous Float Init & Assign.
- (line 17)
-* mpf_inits: Initializing Floats. (line 31)
-* mpf_inp_str: I/O of Floats. (line 37)
+ (line 16)
+* mpf_inp_str: I/O of Floats. (line 38)
* mpf_integer_p: Miscellaneous Float Functions.
- (line 14)
-* mpf_mul: Float Arithmetic. (line 19)
-* mpf_mul_2exp: Float Arithmetic. (line 50)
-* mpf_mul_ui: Float Arithmetic. (line 21)
-* mpf_neg: Float Arithmetic. (line 44)
+ (line 13)
+* mpf_mul: Float Arithmetic. (line 18)
+* mpf_mul_2exp: Float Arithmetic. (line 49)
+* mpf_mul_ui: Float Arithmetic. (line 19)
+* mpf_neg: Float Arithmetic. (line 43)
* mpf_out_str: I/O of Floats. (line 17)
-* mpf_pow_ui: Float Arithmetic. (line 41)
+* mpf_pow_ui: Float Arithmetic. (line 39)
* mpf_random2: Miscellaneous Float Functions.
- (line 36)
-* mpf_reldiff: Float Comparison. (line 29)
-* mpf_set: Assigning Floats. (line 10)
-* mpf_set_d: Assigning Floats. (line 13)
-* mpf_set_default_prec: Initializing Floats. (line 7)
-* mpf_set_prec: Initializing Floats. (line 65)
-* mpf_set_prec_raw: Initializing Floats. (line 72)
-* mpf_set_q: Assigning Floats. (line 15)
-* mpf_set_si: Assigning Floats. (line 12)
-* mpf_set_str: Assigning Floats. (line 18)
-* mpf_set_ui: Assigning Floats. (line 11)
-* mpf_set_z: Assigning Floats. (line 14)
+ (line 35)
+* mpf_reldiff: Float Comparison. (line 28)
+* mpf_set: Assigning Floats. (line 9)
+* mpf_set_d: Assigning Floats. (line 12)
+* mpf_set_default_prec: Initializing Floats. (line 6)
+* mpf_set_prec: Initializing Floats. (line 64)
+* mpf_set_prec_raw: Initializing Floats. (line 71)
+* mpf_set_q: Assigning Floats. (line 14)
+* mpf_set_si: Assigning Floats. (line 11)
+* mpf_set_str: Assigning Floats. (line 17)
+* mpf_set_ui: Assigning Floats. (line 10)
+* mpf_set_z: Assigning Floats. (line 13)
* mpf_sgn: Float Comparison. (line 33)
-* mpf_sqrt: Float Arithmetic. (line 36)
-* mpf_sqrt_ui: Float Arithmetic. (line 37)
-* mpf_sub: Float Arithmetic. (line 12)
-* mpf_sub_ui: Float Arithmetic. (line 16)
-* mpf_swap: Assigning Floats. (line 52)
+* mpf_sqrt: Float Arithmetic. (line 35)
+* mpf_sqrt_ui: Float Arithmetic. (line 36)
+* mpf_sub: Float Arithmetic. (line 11)
+* mpf_sub_ui: Float Arithmetic. (line 14)
+* mpf_swap: Assigning Floats. (line 50)
* mpf_t: Nomenclature and Types.
(line 21)
* mpf_trunc: Miscellaneous Float Functions.
- (line 9)
-* mpf_ui_div: Float Arithmetic. (line 31)
-* mpf_ui_sub: Float Arithmetic. (line 14)
+ (line 8)
+* mpf_ui_div: Float Arithmetic. (line 29)
+* mpf_ui_sub: Float Arithmetic. (line 12)
* mpf_urandomb: Miscellaneous Float Functions.
- (line 27)
-* mpn_add: Low-level Functions. (line 69)
-* mpn_add_1: Low-level Functions. (line 64)
-* mpn_add_n: Low-level Functions. (line 54)
+ (line 25)
+* mpn_add: Low-level Functions. (line 67)
* mpn_addmul_1: Low-level Functions. (line 148)
-* mpn_and_n: Low-level Functions. (line 420)
-* mpn_andn_n: Low-level Functions. (line 435)
-* mpn_cmp: Low-level Functions. (line 284)
-* mpn_com: Low-level Functions. (line 460)
-* mpn_copyd: Low-level Functions. (line 469)
-* mpn_copyi: Low-level Functions. (line 465)
-* mpn_divexact_by3: Low-level Functions. (line 229)
-* mpn_divexact_by3c: Low-level Functions. (line 231)
-* mpn_divmod: Low-level Functions. (line 224)
-* mpn_divmod_1: Low-level Functions. (line 208)
-* mpn_divrem: Low-level Functions. (line 182)
-* mpn_divrem_1: Low-level Functions. (line 206)
-* mpn_gcd: Low-level Functions. (line 289)
-* mpn_gcd_1: Low-level Functions. (line 299)
-* mpn_gcdext: Low-level Functions. (line 305)
-* mpn_get_str: Low-level Functions. (line 346)
-* mpn_hamdist: Low-level Functions. (line 410)
-* mpn_ior_n: Low-level Functions. (line 425)
-* mpn_iorn_n: Low-level Functions. (line 440)
-* mpn_lshift: Low-level Functions. (line 260)
-* mpn_mod_1: Low-level Functions. (line 255)
+* mpn_add_1: Low-level Functions. (line 62)
+* mpn_add_n: Low-level Functions. (line 52)
+* mpn_andn_n: Low-level Functions. (line 462)
+* mpn_and_n: Low-level Functions. (line 447)
+* mpn_cmp: Low-level Functions. (line 293)
+* mpn_cnd_add_n: Low-level Functions. (line 540)
+* mpn_cnd_sub_n: Low-level Functions. (line 542)
+* mpn_cnd_swap: Low-level Functions. (line 567)
+* mpn_com: Low-level Functions. (line 487)
+* mpn_copyd: Low-level Functions. (line 496)
+* mpn_copyi: Low-level Functions. (line 492)
+* mpn_divexact_1: Low-level Functions. (line 231)
+* mpn_divexact_by3: Low-level Functions. (line 238)
+* mpn_divexact_by3c: Low-level Functions. (line 240)
+* mpn_divmod: Low-level Functions. (line 226)
+* mpn_divmod_1: Low-level Functions. (line 210)
+* mpn_divrem: Low-level Functions. (line 183)
+* mpn_divrem_1: Low-level Functions. (line 208)
+* mpn_gcd: Low-level Functions. (line 301)
+* mpn_gcdext: Low-level Functions. (line 316)
+* mpn_gcd_1: Low-level Functions. (line 311)
+* mpn_get_str: Low-level Functions. (line 371)
+* mpn_hamdist: Low-level Functions. (line 436)
+* mpn_iorn_n: Low-level Functions. (line 467)
+* mpn_ior_n: Low-level Functions. (line 452)
+* mpn_lshift: Low-level Functions. (line 269)
+* mpn_mod_1: Low-level Functions. (line 264)
* mpn_mul: Low-level Functions. (line 114)
* mpn_mul_1: Low-level Functions. (line 133)
* mpn_mul_n: Low-level Functions. (line 103)
-* mpn_nand_n: Low-level Functions. (line 445)
-* mpn_neg: Low-level Functions. (line 98)
-* mpn_nior_n: Low-level Functions. (line 450)
-* mpn_perfect_square_p: Low-level Functions. (line 416)
-* mpn_popcount: Low-level Functions. (line 406)
-* mpn_random: Low-level Functions. (line 395)
-* mpn_random2: Low-level Functions. (line 396)
-* mpn_rshift: Low-level Functions. (line 272)
-* mpn_scan0: Low-level Functions. (line 380)
-* mpn_scan1: Low-level Functions. (line 388)
-* mpn_set_str: Low-level Functions. (line 361)
+* mpn_nand_n: Low-level Functions. (line 472)
+* mpn_neg: Low-level Functions. (line 96)
+* mpn_nior_n: Low-level Functions. (line 477)
+* mpn_perfect_square_p: Low-level Functions. (line 442)
+* mpn_popcount: Low-level Functions. (line 432)
+* mpn_random: Low-level Functions. (line 422)
+* mpn_random2: Low-level Functions. (line 423)
+* mpn_rshift: Low-level Functions. (line 281)
+* mpn_scan0: Low-level Functions. (line 406)
+* mpn_scan1: Low-level Functions. (line 414)
+* mpn_sec_add_1: Low-level Functions. (line 553)
+* mpn_sec_div_qr: Low-level Functions. (line 630)
+* mpn_sec_div_qr_itch: Low-level Functions. (line 633)
+* mpn_sec_div_r: Low-level Functions. (line 649)
+* mpn_sec_div_r_itch: Low-level Functions. (line 651)
+* mpn_sec_invert: Low-level Functions. (line 665)
+* mpn_sec_invert_itch: Low-level Functions. (line 667)
+* mpn_sec_mul: Low-level Functions. (line 574)
+* mpn_sec_mul_itch: Low-level Functions. (line 577)
+* mpn_sec_powm: Low-level Functions. (line 604)
+* mpn_sec_powm_itch: Low-level Functions. (line 607)
+* mpn_sec_sqr: Low-level Functions. (line 590)
+* mpn_sec_sqr_itch: Low-level Functions. (line 592)
+* mpn_sec_sub_1: Low-level Functions. (line 555)
+* mpn_sec_tabselect: Low-level Functions. (line 622)
+* mpn_set_str: Low-level Functions. (line 386)
+* mpn_sizeinbase: Low-level Functions. (line 364)
* mpn_sqr: Low-level Functions. (line 125)
-* mpn_sqrtrem: Low-level Functions. (line 328)
-* mpn_sub: Low-level Functions. (line 90)
-* mpn_sub_1: Low-level Functions. (line 85)
-* mpn_sub_n: Low-level Functions. (line 76)
-* mpn_submul_1: Low-level Functions. (line 159)
-* mpn_tdiv_qr: Low-level Functions. (line 171)
-* mpn_xnor_n: Low-level Functions. (line 455)
-* mpn_xor_n: Low-level Functions. (line 430)
-* mpn_zero: Low-level Functions. (line 472)
-* mpq_abs: Rational Arithmetic. (line 31)
-* mpq_add: Rational Arithmetic. (line 7)
+* mpn_sqrtrem: Low-level Functions. (line 346)
+* mpn_sub: Low-level Functions. (line 88)
+* mpn_submul_1: Low-level Functions. (line 160)
+* mpn_sub_1: Low-level Functions. (line 83)
+* mpn_sub_n: Low-level Functions. (line 74)
+* mpn_tdiv_qr: Low-level Functions. (line 172)
+* mpn_xnor_n: Low-level Functions. (line 482)
+* mpn_xor_n: Low-level Functions. (line 457)
+* mpn_zero: Low-level Functions. (line 500)
+* mpn_zero_p: Low-level Functions. (line 298)
+* mpq_abs: Rational Arithmetic. (line 33)
+* mpq_add: Rational Arithmetic. (line 6)
* mpq_canonicalize: Rational Number Functions.
- (line 22)
+ (line 21)
* mpq_class: C++ Interface General.
- (line 19)
+ (line 18)
* mpq_class::canonicalize: C++ Interface Rationals.
- (line 37)
+ (line 41)
* mpq_class::get_d: C++ Interface Rationals.
- (line 46)
+ (line 51)
* mpq_class::get_den: C++ Interface Rationals.
- (line 58)
+ (line 67)
* mpq_class::get_den_mpz_t: C++ Interface Rationals.
- (line 68)
+ (line 77)
* mpq_class::get_mpq_t: C++ Interface General.
- (line 65)
+ (line 64)
* mpq_class::get_num: C++ Interface Rationals.
- (line 57)
+ (line 66)
* mpq_class::get_num_mpz_t: C++ Interface Rationals.
- (line 67)
+ (line 76)
* mpq_class::get_str: C++ Interface Rationals.
- (line 47)
+ (line 52)
* mpq_class::mpq_class: C++ Interface Rationals.
- (line 22)
+ (line 9)
+* mpq_class::mpq_class <1>: C++ Interface Rationals.
+ (line 10)
+* mpq_class::mpq_class <2>: C++ Interface Rationals.
+ (line 21)
+* mpq_class::mpq_class <3>: C++ Interface Rationals.
+ (line 26)
+* mpq_class::mpq_class <4>: C++ Interface Rationals.
+ (line 28)
* mpq_class::set_str: C++ Interface Rationals.
- (line 49)
+ (line 54)
+* mpq_class::set_str <1>: C++ Interface Rationals.
+ (line 55)
+* mpq_class::swap: C++ Interface Rationals.
+ (line 58)
* mpq_clear: Initializing Rationals.
- (line 16)
+ (line 15)
* mpq_clears: Initializing Rationals.
- (line 20)
-* mpq_cmp: Comparing Rationals. (line 7)
-* mpq_cmp_si: Comparing Rationals. (line 17)
-* mpq_cmp_ui: Comparing Rationals. (line 15)
+ (line 19)
+* mpq_cmp: Comparing Rationals. (line 6)
+* mpq_cmp_si: Comparing Rationals. (line 16)
+* mpq_cmp_ui: Comparing Rationals. (line 14)
+* mpq_cmp_z: Comparing Rationals. (line 7)
* mpq_denref: Applying Integer Functions.
- (line 18)
+ (line 16)
* mpq_div: Rational Arithmetic. (line 22)
-* mpq_div_2exp: Rational Arithmetic. (line 25)
+* mpq_div_2exp: Rational Arithmetic. (line 26)
* mpq_equal: Comparing Rationals. (line 33)
* mpq_get_d: Rational Conversions.
- (line 7)
+ (line 6)
* mpq_get_den: Applying Integer Functions.
- (line 24)
+ (line 22)
* mpq_get_num: Applying Integer Functions.
- (line 23)
+ (line 21)
* mpq_get_str: Rational Conversions.
- (line 22)
+ (line 21)
* mpq_init: Initializing Rationals.
- (line 7)
+ (line 6)
* mpq_inits: Initializing Rationals.
- (line 12)
-* mpq_inp_str: I/O of Rationals. (line 23)
-* mpq_inv: Rational Arithmetic. (line 34)
-* mpq_mul: Rational Arithmetic. (line 15)
+ (line 11)
+* mpq_inp_str: I/O of Rationals. (line 32)
+* mpq_inv: Rational Arithmetic. (line 36)
+* mpq_mul: Rational Arithmetic. (line 14)
* mpq_mul_2exp: Rational Arithmetic. (line 18)
-* mpq_neg: Rational Arithmetic. (line 28)
+* mpq_neg: Rational Arithmetic. (line 30)
* mpq_numref: Applying Integer Functions.
- (line 17)
-* mpq_out_str: I/O of Rationals. (line 15)
+ (line 15)
+* mpq_out_str: I/O of Rationals. (line 17)
* mpq_set: Initializing Rationals.
- (line 24)
+ (line 23)
* mpq_set_d: Rational Conversions.
- (line 17)
+ (line 16)
* mpq_set_den: Applying Integer Functions.
- (line 26)
+ (line 24)
* mpq_set_f: Rational Conversions.
- (line 18)
+ (line 17)
* mpq_set_num: Applying Integer Functions.
- (line 25)
+ (line 23)
* mpq_set_si: Initializing Rationals.
- (line 31)
+ (line 29)
* mpq_set_str: Initializing Rationals.
- (line 36)
+ (line 35)
* mpq_set_ui: Initializing Rationals.
- (line 29)
+ (line 27)
* mpq_set_z: Initializing Rationals.
- (line 25)
+ (line 24)
* mpq_sgn: Comparing Rationals. (line 27)
-* mpq_sub: Rational Arithmetic. (line 11)
+* mpq_sub: Rational Arithmetic. (line 10)
* mpq_swap: Initializing Rationals.
- (line 56)
+ (line 54)
* mpq_t: Nomenclature and Types.
(line 16)
-* mpz_abs: Integer Arithmetic. (line 42)
-* mpz_add: Integer Arithmetic. (line 7)
-* mpz_add_ui: Integer Arithmetic. (line 9)
-* mpz_addmul: Integer Arithmetic. (line 25)
-* mpz_addmul_ui: Integer Arithmetic. (line 27)
+* mpz_2fac_ui: Number Theoretic Functions.
+ (line 113)
+* mpz_abs: Integer Arithmetic. (line 44)
+* mpz_add: Integer Arithmetic. (line 6)
+* mpz_addmul: Integer Arithmetic. (line 24)
+* mpz_addmul_ui: Integer Arithmetic. (line 26)
+* mpz_add_ui: Integer Arithmetic. (line 7)
* mpz_and: Integer Logic and Bit Fiddling.
- (line 11)
+ (line 10)
* mpz_array_init: Integer Special Functions.
- (line 11)
+ (line 9)
* mpz_bin_ui: Number Theoretic Functions.
- (line 98)
+ (line 124)
* mpz_bin_uiui: Number Theoretic Functions.
- (line 100)
-* mpz_cdiv_q: Integer Division. (line 13)
-* mpz_cdiv_q_2exp: Integer Division. (line 24)
-* mpz_cdiv_q_ui: Integer Division. (line 17)
-* mpz_cdiv_qr: Integer Division. (line 15)
+ (line 126)
+* mpz_cdiv_q: Integer Division. (line 12)
+* mpz_cdiv_qr: Integer Division. (line 14)
* mpz_cdiv_qr_ui: Integer Division. (line 21)
-* mpz_cdiv_r: Integer Division. (line 14)
-* mpz_cdiv_r_2exp: Integer Division. (line 25)
+* mpz_cdiv_q_2exp: Integer Division. (line 26)
+* mpz_cdiv_q_ui: Integer Division. (line 17)
+* mpz_cdiv_r: Integer Division. (line 13)
+* mpz_cdiv_r_2exp: Integer Division. (line 29)
* mpz_cdiv_r_ui: Integer Division. (line 19)
* mpz_cdiv_ui: Integer Division. (line 23)
* mpz_class: C++ Interface General.
- (line 18)
+ (line 17)
+* mpz_class::factorial: C++ Interface Integers.
+ (line 70)
+* mpz_class::fibonacci: C++ Interface Integers.
+ (line 74)
* mpz_class::fits_sint_p: C++ Interface Integers.
- (line 45)
+ (line 50)
* mpz_class::fits_slong_p: C++ Interface Integers.
- (line 46)
+ (line 51)
* mpz_class::fits_sshort_p: C++ Interface Integers.
- (line 47)
+ (line 52)
* mpz_class::fits_uint_p: C++ Interface Integers.
- (line 48)
+ (line 54)
* mpz_class::fits_ulong_p: C++ Interface Integers.
- (line 49)
+ (line 55)
* mpz_class::fits_ushort_p: C++ Interface Integers.
- (line 50)
+ (line 56)
* mpz_class::get_d: C++ Interface Integers.
- (line 51)
+ (line 58)
* mpz_class::get_mpz_t: C++ Interface General.
- (line 64)
+ (line 63)
* mpz_class::get_si: C++ Interface Integers.
- (line 52)
+ (line 59)
* mpz_class::get_str: C++ Interface Integers.
- (line 53)
+ (line 60)
* mpz_class::get_ui: C++ Interface Integers.
- (line 54)
+ (line 61)
* mpz_class::mpz_class: C++ Interface Integers.
- (line 7)
+ (line 6)
+* mpz_class::mpz_class <1>: C++ Interface Integers.
+ (line 14)
+* mpz_class::mpz_class <2>: C++ Interface Integers.
+ (line 19)
+* mpz_class::mpz_class <3>: C++ Interface Integers.
+ (line 21)
+* mpz_class::primorial: C++ Interface Integers.
+ (line 72)
* mpz_class::set_str: C++ Interface Integers.
- (line 56)
+ (line 63)
+* mpz_class::set_str <1>: C++ Interface Integers.
+ (line 64)
+* mpz_class::swap: C++ Interface Integers.
+ (line 77)
* mpz_clear: Initializing Integers.
- (line 44)
-* mpz_clears: Initializing Integers.
(line 48)
+* mpz_clears: Initializing Integers.
+ (line 52)
* mpz_clrbit: Integer Logic and Bit Fiddling.
(line 54)
-* mpz_cmp: Integer Comparisons. (line 7)
-* mpz_cmp_d: Integer Comparisons. (line 8)
-* mpz_cmp_si: Integer Comparisons. (line 9)
-* mpz_cmp_ui: Integer Comparisons. (line 10)
-* mpz_cmpabs: Integer Comparisons. (line 18)
-* mpz_cmpabs_d: Integer Comparisons. (line 19)
-* mpz_cmpabs_ui: Integer Comparisons. (line 20)
+* mpz_cmp: Integer Comparisons. (line 6)
+* mpz_cmpabs: Integer Comparisons. (line 17)
+* mpz_cmpabs_d: Integer Comparisons. (line 18)
+* mpz_cmpabs_ui: Integer Comparisons. (line 19)
+* mpz_cmp_d: Integer Comparisons. (line 7)
+* mpz_cmp_si: Integer Comparisons. (line 8)
+* mpz_cmp_ui: Integer Comparisons. (line 9)
* mpz_com: Integer Logic and Bit Fiddling.
- (line 20)
+ (line 19)
* mpz_combit: Integer Logic and Bit Fiddling.
(line 57)
-* mpz_congruent_2exp_p: Integer Division. (line 124)
-* mpz_congruent_p: Integer Division. (line 121)
-* mpz_congruent_ui_p: Integer Division. (line 123)
-* mpz_divexact: Integer Division. (line 101)
-* mpz_divexact_ui: Integer Division. (line 102)
-* mpz_divisible_2exp_p: Integer Division. (line 112)
-* mpz_divisible_p: Integer Division. (line 110)
-* mpz_divisible_ui_p: Integer Division. (line 111)
+* mpz_congruent_2exp_p: Integer Division. (line 148)
+* mpz_congruent_p: Integer Division. (line 144)
+* mpz_congruent_ui_p: Integer Division. (line 146)
+* mpz_divexact: Integer Division. (line 122)
+* mpz_divexact_ui: Integer Division. (line 123)
+* mpz_divisible_2exp_p: Integer Division. (line 135)
+* mpz_divisible_p: Integer Division. (line 132)
+* mpz_divisible_ui_p: Integer Division. (line 133)
* mpz_even_p: Miscellaneous Integer Functions.
- (line 18)
+ (line 17)
* mpz_export: Integer Import and Export.
- (line 45)
+ (line 43)
* mpz_fac_ui: Number Theoretic Functions.
- (line 95)
-* mpz_fdiv_q: Integer Division. (line 27)
-* mpz_fdiv_q_2exp: Integer Division. (line 38)
-* mpz_fdiv_q_ui: Integer Division. (line 31)
-* mpz_fdiv_qr: Integer Division. (line 29)
-* mpz_fdiv_qr_ui: Integer Division. (line 35)
-* mpz_fdiv_r: Integer Division. (line 28)
-* mpz_fdiv_r_2exp: Integer Division. (line 39)
-* mpz_fdiv_r_ui: Integer Division. (line 33)
-* mpz_fdiv_ui: Integer Division. (line 37)
+ (line 112)
+* mpz_fdiv_q: Integer Division. (line 33)
+* mpz_fdiv_qr: Integer Division. (line 35)
+* mpz_fdiv_qr_ui: Integer Division. (line 42)
+* mpz_fdiv_q_2exp: Integer Division. (line 47)
+* mpz_fdiv_q_ui: Integer Division. (line 38)
+* mpz_fdiv_r: Integer Division. (line 34)
+* mpz_fdiv_r_2exp: Integer Division. (line 50)
+* mpz_fdiv_r_ui: Integer Division. (line 40)
+* mpz_fdiv_ui: Integer Division. (line 44)
* mpz_fib2_ui: Number Theoretic Functions.
- (line 108)
+ (line 134)
* mpz_fib_ui: Number Theoretic Functions.
- (line 106)
+ (line 133)
* mpz_fits_sint_p: Miscellaneous Integer Functions.
- (line 10)
+ (line 9)
* mpz_fits_slong_p: Miscellaneous Integer Functions.
- (line 8)
+ (line 7)
* mpz_fits_sshort_p: Miscellaneous Integer Functions.
- (line 12)
+ (line 11)
* mpz_fits_uint_p: Miscellaneous Integer Functions.
- (line 9)
+ (line 8)
* mpz_fits_ulong_p: Miscellaneous Integer Functions.
- (line 7)
+ (line 6)
* mpz_fits_ushort_p: Miscellaneous Integer Functions.
- (line 11)
+ (line 10)
* mpz_gcd: Number Theoretic Functions.
- (line 30)
-* mpz_gcd_ui: Number Theoretic Functions.
- (line 35)
+ (line 29)
* mpz_gcdext: Number Theoretic Functions.
(line 45)
-* mpz_get_d: Converting Integers. (line 27)
-* mpz_get_d_2exp: Converting Integers. (line 35)
-* mpz_get_si: Converting Integers. (line 18)
-* mpz_get_str: Converting Integers. (line 46)
-* mpz_get_ui: Converting Integers. (line 11)
+* mpz_gcd_ui: Number Theoretic Functions.
+ (line 35)
* mpz_getlimbn: Integer Special Functions.
- (line 60)
+ (line 22)
+* mpz_get_d: Converting Integers. (line 26)
+* mpz_get_d_2exp: Converting Integers. (line 34)
+* mpz_get_si: Converting Integers. (line 17)
+* mpz_get_str: Converting Integers. (line 46)
+* mpz_get_ui: Converting Integers. (line 10)
* mpz_hamdist: Integer Logic and Bit Fiddling.
- (line 29)
+ (line 28)
* mpz_import: Integer Import and Export.
- (line 11)
+ (line 9)
* mpz_init: Initializing Integers.
- (line 26)
+ (line 25)
* mpz_init2: Initializing Integers.
- (line 33)
+ (line 32)
+* mpz_inits: Initializing Integers.
+ (line 28)
* mpz_init_set: Simultaneous Integer Init & Assign.
- (line 27)
+ (line 26)
* mpz_init_set_d: Simultaneous Integer Init & Assign.
- (line 30)
-* mpz_init_set_si: Simultaneous Integer Init & Assign.
(line 29)
+* mpz_init_set_si: Simultaneous Integer Init & Assign.
+ (line 28)
* mpz_init_set_str: Simultaneous Integer Init & Assign.
- (line 34)
+ (line 33)
* mpz_init_set_ui: Simultaneous Integer Init & Assign.
- (line 28)
-* mpz_inits: Initializing Integers.
- (line 29)
-* mpz_inp_raw: I/O of Integers. (line 59)
-* mpz_inp_str: I/O of Integers. (line 28)
+ (line 27)
+* mpz_inp_raw: I/O of Integers. (line 61)
+* mpz_inp_str: I/O of Integers. (line 30)
* mpz_invert: Number Theoretic Functions.
- (line 60)
+ (line 72)
* mpz_ior: Integer Logic and Bit Fiddling.
- (line 14)
+ (line 13)
* mpz_jacobi: Number Theoretic Functions.
- (line 66)
+ (line 82)
* mpz_kronecker: Number Theoretic Functions.
- (line 74)
+ (line 90)
* mpz_kronecker_si: Number Theoretic Functions.
- (line 75)
+ (line 91)
* mpz_kronecker_ui: Number Theoretic Functions.
- (line 76)
+ (line 92)
* mpz_lcm: Number Theoretic Functions.
- (line 54)
+ (line 65)
* mpz_lcm_ui: Number Theoretic Functions.
- (line 55)
+ (line 66)
* mpz_legendre: Number Theoretic Functions.
- (line 69)
+ (line 85)
+* mpz_limbs_finish: Integer Special Functions.
+ (line 47)
+* mpz_limbs_modify: Integer Special Functions.
+ (line 40)
+* mpz_limbs_read: Integer Special Functions.
+ (line 34)
+* mpz_limbs_write: Integer Special Functions.
+ (line 39)
* mpz_lucnum2_ui: Number Theoretic Functions.
- (line 119)
+ (line 145)
* mpz_lucnum_ui: Number Theoretic Functions.
- (line 117)
-* mpz_mod: Integer Division. (line 91)
-* mpz_mod_ui: Integer Division. (line 93)
-* mpz_mul: Integer Arithmetic. (line 19)
-* mpz_mul_2exp: Integer Arithmetic. (line 35)
-* mpz_mul_si: Integer Arithmetic. (line 20)
-* mpz_mul_ui: Integer Arithmetic. (line 22)
-* mpz_neg: Integer Arithmetic. (line 39)
+ (line 144)
+* mpz_mfac_uiui: Number Theoretic Functions.
+ (line 114)
+* mpz_mod: Integer Division. (line 112)
+* mpz_mod_ui: Integer Division. (line 113)
+* mpz_mul: Integer Arithmetic. (line 18)
+* mpz_mul_2exp: Integer Arithmetic. (line 36)
+* mpz_mul_si: Integer Arithmetic. (line 19)
+* mpz_mul_ui: Integer Arithmetic. (line 20)
+* mpz_neg: Integer Arithmetic. (line 41)
* mpz_nextprime: Number Theoretic Functions.
- (line 23)
+ (line 22)
* mpz_odd_p: Miscellaneous Integer Functions.
- (line 17)
-* mpz_out_raw: I/O of Integers. (line 43)
-* mpz_out_str: I/O of Integers. (line 16)
+ (line 16)
+* mpz_out_raw: I/O of Integers. (line 45)
+* mpz_out_str: I/O of Integers. (line 17)
* mpz_perfect_power_p: Integer Roots. (line 27)
* mpz_perfect_square_p: Integer Roots. (line 36)
* mpz_popcount: Integer Logic and Bit Fiddling.
- (line 23)
-* mpz_pow_ui: Integer Exponentiation.
- (line 31)
+ (line 22)
* mpz_powm: Integer Exponentiation.
- (line 8)
+ (line 6)
* mpz_powm_sec: Integer Exponentiation.
- (line 18)
+ (line 16)
* mpz_powm_ui: Integer Exponentiation.
- (line 10)
+ (line 8)
+* mpz_pow_ui: Integer Exponentiation.
+ (line 29)
+* mpz_primorial_ui: Number Theoretic Functions.
+ (line 120)
* mpz_probab_prime_p: Number Theoretic Functions.
- (line 7)
+ (line 6)
* mpz_random: Integer Random Numbers.
- (line 42)
+ (line 41)
* mpz_random2: Integer Random Numbers.
- (line 51)
+ (line 50)
* mpz_realloc2: Initializing Integers.
- (line 52)
+ (line 56)
* mpz_remove: Number Theoretic Functions.
- (line 90)
-* mpz_root: Integer Roots. (line 7)
-* mpz_rootrem: Integer Roots. (line 13)
+ (line 106)
+* mpz_roinit_n: Integer Special Functions.
+ (line 67)
+* MPZ_ROINIT_N: Integer Special Functions.
+ (line 83)
+* mpz_root: Integer Roots. (line 6)
+* mpz_rootrem: Integer Roots. (line 12)
* mpz_rrandomb: Integer Random Numbers.
- (line 31)
+ (line 29)
* mpz_scan0: Integer Logic and Bit Fiddling.
- (line 37)
+ (line 35)
* mpz_scan1: Integer Logic and Bit Fiddling.
- (line 38)
-* mpz_set: Assigning Integers. (line 10)
-* mpz_set_d: Assigning Integers. (line 13)
-* mpz_set_f: Assigning Integers. (line 15)
-* mpz_set_q: Assigning Integers. (line 14)
-* mpz_set_si: Assigning Integers. (line 12)
-* mpz_set_str: Assigning Integers. (line 21)
-* mpz_set_ui: Assigning Integers. (line 11)
+ (line 37)
+* mpz_set: Assigning Integers. (line 9)
* mpz_setbit: Integer Logic and Bit Fiddling.
(line 51)
-* mpz_sgn: Integer Comparisons. (line 28)
-* mpz_si_kronecker: Number Theoretic Functions.
- (line 77)
+* mpz_set_d: Assigning Integers. (line 12)
+* mpz_set_f: Assigning Integers. (line 14)
+* mpz_set_q: Assigning Integers. (line 13)
+* mpz_set_si: Assigning Integers. (line 11)
+* mpz_set_str: Assigning Integers. (line 20)
+* mpz_set_ui: Assigning Integers. (line 10)
+* mpz_sgn: Integer Comparisons. (line 27)
* mpz_size: Integer Special Functions.
- (line 68)
+ (line 30)
* mpz_sizeinbase: Miscellaneous Integer Functions.
- (line 23)
+ (line 22)
+* mpz_si_kronecker: Number Theoretic Functions.
+ (line 93)
* mpz_sqrt: Integer Roots. (line 17)
* mpz_sqrtrem: Integer Roots. (line 20)
-* mpz_sub: Integer Arithmetic. (line 12)
-* mpz_sub_ui: Integer Arithmetic. (line 14)
+* mpz_sub: Integer Arithmetic. (line 11)
* mpz_submul: Integer Arithmetic. (line 30)
* mpz_submul_ui: Integer Arithmetic. (line 32)
-* mpz_swap: Assigning Integers. (line 37)
+* mpz_sub_ui: Integer Arithmetic. (line 12)
+* mpz_swap: Assigning Integers. (line 36)
* mpz_t: Nomenclature and Types.
(line 6)
-* mpz_tdiv_q: Integer Division. (line 41)
-* mpz_tdiv_q_2exp: Integer Division. (line 52)
-* mpz_tdiv_q_ui: Integer Division. (line 45)
-* mpz_tdiv_qr: Integer Division. (line 43)
-* mpz_tdiv_qr_ui: Integer Division. (line 49)
-* mpz_tdiv_r: Integer Division. (line 42)
-* mpz_tdiv_r_2exp: Integer Division. (line 53)
-* mpz_tdiv_r_ui: Integer Division. (line 47)
-* mpz_tdiv_ui: Integer Division. (line 51)
+* mpz_tdiv_q: Integer Division. (line 54)
+* mpz_tdiv_qr: Integer Division. (line 56)
+* mpz_tdiv_qr_ui: Integer Division. (line 63)
+* mpz_tdiv_q_2exp: Integer Division. (line 68)
+* mpz_tdiv_q_ui: Integer Division. (line 59)
+* mpz_tdiv_r: Integer Division. (line 55)
+* mpz_tdiv_r_2exp: Integer Division. (line 71)
+* mpz_tdiv_r_ui: Integer Division. (line 61)
+* mpz_tdiv_ui: Integer Division. (line 65)
* mpz_tstbit: Integer Logic and Bit Fiddling.
(line 60)
* mpz_ui_kronecker: Number Theoretic Functions.
- (line 78)
+ (line 94)
* mpz_ui_pow_ui: Integer Exponentiation.
- (line 33)
-* mpz_ui_sub: Integer Arithmetic. (line 16)
+ (line 31)
+* mpz_ui_sub: Integer Arithmetic. (line 14)
* mpz_urandomb: Integer Random Numbers.
- (line 14)
+ (line 12)
* mpz_urandomm: Integer Random Numbers.
- (line 23)
+ (line 21)
* mpz_xor: Integer Logic and Bit Fiddling.
- (line 17)
-* msqrt: BSD Compatible Functions.
- (line 63)
-* msub: BSD Compatible Functions.
- (line 46)
-* mtox: BSD Compatible Functions.
- (line 98)
-* mult: BSD Compatible Functions.
- (line 49)
+ (line 16)
+* mp_bitcnt_t: Nomenclature and Types.
+ (line 42)
+* mp_bits_per_limb: Useful Macros and Constants.
+ (line 7)
+* mp_exp_t: Nomenclature and Types.
+ (line 27)
+* mp_get_memory_functions: Custom Allocation. (line 86)
+* mp_limb_t: Nomenclature and Types.
+ (line 31)
+* mp_set_memory_functions: Custom Allocation. (line 14)
+* mp_size_t: Nomenclature and Types.
+ (line 37)
+* operator"": C++ Interface Integers.
+ (line 29)
+* operator"" <1>: C++ Interface Rationals.
+ (line 36)
+* operator"" <2>: C++ Interface Floats.
+ (line 55)
* operator%: C++ Interface Integers.
- (line 30)
+ (line 34)
* operator/: C++ Interface Integers.
- (line 29)
+ (line 33)
* operator<<: C++ Formatted Output.
- (line 20)
-* operator>> <1>: C++ Formatted Input. (line 11)
-* operator>>: C++ Interface Rationals.
- (line 77)
-* pow: BSD Compatible Functions.
- (line 71)
-* rpow: BSD Compatible Functions.
- (line 79)
-* sdiv: BSD Compatible Functions.
- (line 55)
+ (line 10)
+* operator<< <1>: C++ Formatted Output.
+ (line 19)
+* operator<< <2>: C++ Formatted Output.
+ (line 32)
+* operator>>: C++ Formatted Input. (line 10)
+* operator>> <1>: C++ Formatted Input. (line 13)
+* operator>> <2>: C++ Formatted Input. (line 24)
+* operator>> <3>: C++ Interface Rationals.
+ (line 86)
+* primorial: C++ Interface Integers.
+ (line 73)
+* sgn: C++ Interface Integers.
+ (line 65)
* sgn <1>: C++ Interface Rationals.
- (line 50)
-* sgn <2>: C++ Interface Integers.
- (line 57)
-* sgn: C++ Interface Floats.
- (line 89)
-* sqrt <1>: C++ Interface Integers.
- (line 58)
-* sqrt: C++ Interface Floats.
- (line 90)
+ (line 56)
+* sgn <2>: C++ Interface Floats.
+ (line 106)
+* sqrt: C++ Interface Integers.
+ (line 66)
+* sqrt <1>: C++ Interface Floats.
+ (line 107)
+* swap: C++ Interface Integers.
+ (line 78)
+* swap <1>: C++ Interface Rationals.
+ (line 59)
+* swap <2>: C++ Interface Floats.
+ (line 110)
* trunc: C++ Interface Floats.
- (line 91)
-* xtom: BSD Compatible Functions.
- (line 34)
-
+ (line 111)
projects / xonotic / xonotic.git / blobdiff