1 This is gmp.info, produced by makeinfo version 6.7 from gmp.texi.
3 This manual describes how to install and use the GNU multiple precision
4 arithmetic library, version 6.2.1.
6 Copyright 1991, 1993-2016, 2018-2020 Free Software Foundation, Inc.
8 Permission is granted to copy, distribute and/or modify this document
9 under the terms of the GNU Free Documentation License, Version 1.3 or
10 any later version published by the Free Software Foundation; with no
11 Invariant Sections, with the Front-Cover Texts being "A GNU Manual", and
12 with the Back-Cover Texts being "You have freedom to copy and modify
13 this GNU Manual, like GNU software". A copy of the license is included
14 in *note GNU Free Documentation License::.
15 INFO-DIR-SECTION GNU libraries
17 * gmp: (gmp). GNU Multiple Precision Arithmetic Library.
21 File: gmp.info, Node: Exact Remainder, Next: Small Quotient Division, Prev: Exact Division, Up: Division Algorithms
23 15.2.6 Exact Remainder
24 ----------------------
26 If the exact division algorithm is done with a full subtraction at each
27 stage and the dividend isn't a multiple of the divisor, then low zero
28 limbs are produced but with a remainder in the high limbs. For dividend
29 a, divisor d, quotient q, and b = 2^mp_bits_per_limb, this remainder r
34 n represents the number of zero limbs produced by the subtractions,
35 that being the number of limbs produced for q. r will be in the range
36 0<=r<d and can be viewed as a remainder, but one shifted up by a factor
39 Carrying out full subtractions at each stage means the same number of
40 cross products must be done as a normal division, but there's still some
41 single limb divisions saved. When d is a single limb some
42 simplifications arise, providing good speedups on a number of
45 The functions 'mpn_divexact_by3', 'mpn_modexact_1_odd' and the
46 internal 'mpn_redc_X' functions differ subtly in how they return r,
47 leading to some negations in the above formula, but all are essentially
50 Clearly r is zero when a is a multiple of d, and this leads to
51 divisibility or congruence tests which are potentially more efficient
52 than a normal division.
54 The factor of b^n on r can be ignored in a GCD when d is odd, hence
55 the use of 'mpn_modexact_1_odd' by 'mpn_gcd_1' and 'mpz_kronecker_ui'
56 etc (*note Greatest Common Divisor Algorithms::).
58 Montgomery's REDC method for modular multiplications uses operands of
59 the form of x*b^-n and y*b^-n and on calculating (x*b^-n)*(y*b^-n) uses
60 the factor of b^n in the exact remainder to reach a product in the same
61 form (x*y)*b^-n (*note Modular Powering Algorithm::).
63 Notice that r generally gives no useful information about the
64 ordinary remainder a mod d since b^n mod d could be anything. If
65 however b^n == 1 mod d, then r is the negative of the ordinary
66 remainder. This occurs whenever d is a factor of b^n-1, as for example
67 with 3 in 'mpn_divexact_by3'. For a 32 or 64 bit limb other such
68 factors include 5, 17 and 257, but no particular use has been found for
72 File: gmp.info, Node: Small Quotient Division, Prev: Exact Remainder, Up: Division Algorithms
74 15.2.7 Small Quotient Division
75 ------------------------------
77 An NxM division where the number of quotient limbs Q=N-M is small can be
80 An ordinary basecase division normalizes the divisor by shifting it
81 to make the high bit set, shifting the dividend accordingly, and
82 shifting the remainder back down at the end of the calculation. This is
83 wasteful if only a few quotient limbs are to be formed. Instead a
84 division of just the top 2*Q limbs of the dividend by the top Q limbs of
85 the divisor can be used to form a trial quotient. This requires only
86 those limbs normalized, not the whole of the divisor and dividend.
88 A multiply and subtract then applies the trial quotient to the M-Q
89 unused limbs of the divisor and N-Q dividend limbs (which includes Q
90 limbs remaining from the trial quotient division). The starting trial
91 quotient can be 1 or 2 too big, but all cases of 2 too big and most
92 cases of 1 too big are detected by first comparing the most significant
93 limbs that will arise from the subtraction. An addback is done if the
94 quotient still turns out to be 1 too big.
96 This whole procedure is essentially the same as one step of the
97 basecase algorithm done in a Q limb base, though with the trial quotient
98 test done only with the high limbs, not an entire Q limb "digit"
99 product. The correctness of this weaker test can be established by
100 following the argument of Knuth section 4.3.1 exercise 20 but with the
101 v2*q>b*r+u2 condition appropriately relaxed.
104 File: gmp.info, Node: Greatest Common Divisor Algorithms, Next: Powering Algorithms, Prev: Division Algorithms, Up: Algorithms
106 15.3 Greatest Common Divisor
107 ============================
112 * Lehmer's Algorithm::
118 File: gmp.info, Node: Binary GCD, Next: Lehmer's Algorithm, Prev: Greatest Common Divisor Algorithms, Up: Greatest Common Divisor Algorithms
123 At small sizes GMP uses an O(N^2) binary style GCD. This is described
124 in many textbooks, for example Knuth section 4.5.2 algorithm B. It
125 simply consists of successively reducing odd operands a and b using
127 a,b = abs(a-b),min(a,b)
128 strip factors of 2 from a
130 The Euclidean GCD algorithm, as per Knuth algorithms E and A,
131 repeatedly computes the quotient q = floor(a/b) and replaces a,b by v, u
132 - q v. The binary algorithm has so far been found to be faster than the
133 Euclidean algorithm everywhere. One reason the binary method does well
134 is that the implied quotient at each step is usually small, so often
135 only one or two subtractions are needed to get the same effect as a
136 division. Quotients 1, 2 and 3 for example occur 67.7% of the time, see
137 Knuth section 4.5.3 Theorem E.
139 When the implied quotient is large, meaning b is much smaller than a,
140 then a division is worthwhile. This is the basis for the initial a mod
141 b reductions in 'mpn_gcd' and 'mpn_gcd_1' (the latter for both Nx1 and
142 1x1 cases). But after that initial reduction, big quotients occur too
143 rarely to make it worth checking for them.
146 The final 1x1 GCD in 'mpn_gcd_1' is done in the generic C code as
147 described above. For two N-bit operands, the algorithm takes about 0.68
148 iterations per bit. For optimum performance some attention needs to be
149 paid to the way the factors of 2 are stripped from a.
151 Firstly it may be noted that in twos complement the number of low
152 zero bits on a-b is the same as b-a, so counting or testing can begin on
153 a-b without waiting for abs(a-b) to be determined.
155 A loop stripping low zero bits tends not to branch predict well,
156 since the condition is data dependent. But on average there's only a
157 few low zeros, so an option is to strip one or two bits arithmetically
158 then loop for more (as done for AMD K6). Or use a lookup table to get a
159 count for several bits then loop for more (as done for AMD K7). An
160 alternative approach is to keep just one of a or b odd and iterate
162 a,b = abs(a-b), min(a,b)
166 This requires about 1.25 iterations per bit, but stripping of a
167 single bit at each step avoids any branching. Repeating the bit strip
168 reduces to about 0.9 iterations per bit, which may be a worthwhile
171 Generally with the above approaches a speed of perhaps 6 cycles per
172 bit can be achieved, which is still not terribly fast with for instance
173 a 64-bit GCD taking nearly 400 cycles. It's this sort of time which
174 means it's not usually advantageous to combine a set of divisibility
177 Currently, the binary algorithm is used for GCD only when N < 3.
180 File: gmp.info, Node: Lehmer's Algorithm, Next: Subquadratic GCD, Prev: Binary GCD, Up: Greatest Common Divisor Algorithms
182 15.3.2 Lehmer's algorithm
183 -------------------------
185 Lehmer's improvement of the Euclidean algorithms is based on the
186 observation that the initial part of the quotient sequence depends only
187 on the most significant parts of the inputs. The variant of Lehmer's
188 algorithm used in GMP splits off the most significant two limbs, as
189 suggested, e.g., in "A Double-Digit Lehmer-Euclid Algorithm" by Jebelean
190 (*note References::). The quotients of two double-limb inputs are
191 collected as a 2 by 2 matrix with single-limb elements. This is done by
192 the function 'mpn_hgcd2'. The resulting matrix is applied to the inputs
193 using 'mpn_mul_1' and 'mpn_submul_1'. Each iteration usually reduces
194 the inputs by almost one limb. In the rare case of a large quotient, no
195 progress can be made by examining just the most significant two limbs,
196 and the quotient is computed using plain division.
198 The resulting algorithm is asymptotically O(N^2), just as the
199 Euclidean algorithm and the binary algorithm. The quadratic part of the
200 work are the calls to 'mpn_mul_1' and 'mpn_submul_1'. For small sizes,
201 the linear work is also significant. There are roughly N calls to the
202 'mpn_hgcd2' function. This function uses a couple of important
205 * It uses the same relaxed notion of correctness as 'mpn_hgcd' (see
206 next section). This means that when called with the most
207 significant two limbs of two large numbers, the returned matrix
208 does not always correspond exactly to the initial quotient sequence
209 for the two large numbers; the final quotient may sometimes be one
212 * It takes advantage of the fact the quotients are usually small.
213 The division operator is not used, since the corresponding
214 assembler instruction is very slow on most architectures. (This
215 code could probably be improved further, it uses many branches that
216 are unfriendly to prediction).
218 * It switches from double-limb calculations to single-limb
219 calculations half-way through, when the input numbers have been
220 reduced in size from two limbs to one and a half.
223 File: gmp.info, Node: Subquadratic GCD, Next: Extended GCD, Prev: Lehmer's Algorithm, Up: Greatest Common Divisor Algorithms
225 15.3.3 Subquadratic GCD
226 -----------------------
228 For inputs larger than 'GCD_DC_THRESHOLD', GCD is computed via the HGCD
229 (Half GCD) function, as a generalization to Lehmer's algorithm.
231 Let the inputs a,b be of size N limbs each. Put S = floor(N/2) + 1.
232 Then HGCD(a,b) returns a transformation matrix T with non-negative
233 elements, and reduced numbers (c;d) = T^{-1} (a;b). The reduced numbers
234 c,d must be larger than S limbs, while their difference abs(c-d) must
235 fit in S limbs. The matrix elements will also be of size roughly N/2.
237 The HGCD base case uses Lehmer's algorithm, but with the above stop
238 condition that returns reduced numbers and the corresponding
239 transformation matrix half-way through. For inputs larger than
240 'HGCD_THRESHOLD', HGCD is computed recursively, using the divide and
241 conquer algorithm in "On Schönhage's algorithm and subquadratic integer
242 GCD computation" by Möller (*note References::). The recursive
243 algorithm consists of these main steps.
245 * Call HGCD recursively, on the most significant N/2 limbs. Apply
246 the resulting matrix T_1 to the full numbers, reducing them to a
247 size just above 3N/2.
249 * Perform a small number of division or subtraction steps to reduce
250 the numbers to size below 3N/2. This is essential mainly for the
251 unlikely case of large quotients.
253 * Call HGCD recursively, on the most significant N/2 limbs of the
254 reduced numbers. Apply the resulting matrix T_2 to the full
255 numbers, reducing them to a size just above N/2.
257 * Compute T = T_1 T_2.
259 * Perform a small number of division and subtraction steps to satisfy
260 the requirements, and return.
262 GCD is then implemented as a loop around HGCD, similarly to Lehmer's
263 algorithm. Where Lehmer repeatedly chops off the top two limbs, calls
264 'mpn_hgcd2', and applies the resulting matrix to the full numbers, the
265 sub-quadratic GCD chops off the most significant third of the limbs (the
266 proportion is a tuning parameter, and 1/3 seems to be more efficient
267 than, e.g, 1/2), calls 'mpn_hgcd', and applies the resulting matrix.
268 Once the input numbers are reduced to size below 'GCD_DC_THRESHOLD',
269 Lehmer's algorithm is used for the rest of the work.
271 The asymptotic running time of both HGCD and GCD is O(M(N)*log(N)),
272 where M(N) is the time for multiplying two N-limb numbers.
275 File: gmp.info, Node: Extended GCD, Next: Jacobi Symbol, Prev: Subquadratic GCD, Up: Greatest Common Divisor Algorithms
280 The extended GCD function, or GCDEXT, calculates gcd(a,b) and also
281 cofactors x and y satisfying a*x+b*y=gcd(a,b). All the algorithms used
282 for plain GCD are extended to handle this case. The binary algorithm is
283 used only for single-limb GCDEXT. Lehmer's algorithm is used for sizes
284 up to 'GCDEXT_DC_THRESHOLD'. Above this threshold, GCDEXT is
285 implemented as a loop around HGCD, but with more book-keeping to keep
286 track of the cofactors. This gives the same asymptotic running time as
287 for GCD and HGCD, O(M(N)*log(N))
289 One difference to plain GCD is that while the inputs a and b are
290 reduced as the algorithm proceeds, the cofactors x and y grow in size.
291 This makes the tuning of the chopping-point more difficult. The current
292 code chops off the most significant half of the inputs for the call to
293 HGCD in the first iteration, and the most significant two thirds for the
294 remaining calls. This strategy could surely be improved. Also the stop
295 condition for the loop, where Lehmer's algorithm is invoked once the
296 inputs are reduced below 'GCDEXT_DC_THRESHOLD', could maybe be improved
297 by taking into account the current size of the cofactors.
300 File: gmp.info, Node: Jacobi Symbol, Prev: Extended GCD, Up: Greatest Common Divisor Algorithms
307 Initially if either operand fits in a single limb, a reduction is
308 done with either 'mpn_mod_1' or 'mpn_modexact_1_odd', followed by the
309 binary algorithm on a single limb. The binary algorithm is well suited
310 to a single limb, and the whole calculation in this case is quite
313 For inputs larger than 'GCD_DC_THRESHOLD', 'mpz_jacobi',
314 'mpz_legendre' and 'mpz_kronecker' are computed via the HGCD (Half GCD)
315 function, as a generalization to Lehmer's algorithm.
317 Most GCD algorithms reduce a and b by repeatatily computing the
318 quotient q = floor(a/b) and iteratively replacing
322 Different algorithms use different methods for calculating q, but the
323 core algorithm is the same if we use *note Lehmer's Algorithm:: or *note
324 HGCD: Subquadratic GCD.
326 At each step it is possible to compute if the reduction inverts the
327 Jacobi symbol based on the two least significant bits of A and B. For
328 more details see "Efficient computation of the Jacobi symbol" by Möller
329 (*note References::).
331 A small set of bits is thus used to track state
332 * current sign of result (1 bit)
334 * two least significant bits of A and B (4 bits)
336 * a pointer to which input is currently the denominator (1 bit)
338 In all the routines sign changes for the result are accumulated using
339 fast bit twiddling which avoids conditional jumps.
341 The final result is calculated after verifying the inputs are coprime
342 (GCD = 1) by raising (-1)^e
344 Much of the HGCD code is shared directly with the HGCD
345 implementations, such as the 2x2 matrix calculation, *Note Lehmer's
346 Algorithm:: basecase and 'GCD_DC_THRESHOLD'.
348 The asymptotic running time is O(M(N)*log(N)), where M(N) is the time
349 for multiplying two N-limb numbers.
352 File: gmp.info, Node: Powering Algorithms, Next: Root Extraction Algorithms, Prev: Greatest Common Divisor Algorithms, Up: Algorithms
354 15.4 Powering Algorithms
355 ========================
359 * Normal Powering Algorithm::
360 * Modular Powering Algorithm::
363 File: gmp.info, Node: Normal Powering Algorithm, Next: Modular Powering Algorithm, Prev: Powering Algorithms, Up: Powering Algorithms
365 15.4.1 Normal Powering
366 ----------------------
368 Normal 'mpz' or 'mpf' powering uses a simple binary algorithm,
369 successively squaring and then multiplying by the base when a 1 bit is
370 seen in the exponent, as per Knuth section 4.6.3. The "left to right"
371 variant described there is used rather than algorithm A, since it's just
372 as easy and can be done with somewhat less temporary memory.
375 File: gmp.info, Node: Modular Powering Algorithm, Prev: Normal Powering Algorithm, Up: Powering Algorithms
377 15.4.2 Modular Powering
378 -----------------------
380 Modular powering is implemented using a 2^k-ary sliding window
381 algorithm, as per "Handbook of Applied Cryptography" algorithm 14.85
382 (*note References::). k is chosen according to the size of the
383 exponent. Larger exponents use larger values of k, the choice being
384 made to minimize the average number of multiplications that must
385 supplement the squaring.
387 The modular multiplies and squarings use either a simple division or
388 the REDC method by Montgomery (*note References::). REDC is a little
389 faster, essentially saving N single limb divisions in a fashion similar
390 to an exact remainder (*note Exact Remainder::).
393 File: gmp.info, Node: Root Extraction Algorithms, Next: Radix Conversion Algorithms, Prev: Powering Algorithms, Up: Algorithms
395 15.5 Root Extraction Algorithms
396 ===============================
400 * Square Root Algorithm::
401 * Nth Root Algorithm::
402 * Perfect Square Algorithm::
403 * Perfect Power Algorithm::
406 File: gmp.info, Node: Square Root Algorithm, Next: Nth Root Algorithm, Prev: Root Extraction Algorithms, Up: Root Extraction Algorithms
411 Square roots are taken using the "Karatsuba Square Root" algorithm by
412 Paul Zimmermann (*note References::).
414 An input n is split into four parts of k bits each, so with b=2^k we
415 have n = a3*b^3 + a2*b^2 + a1*b + a0. Part a3 must be "normalized" so
416 that either the high or second highest bit is set. In GMP, k is kept on
417 a limb boundary and the input is left shifted (by an even number of
420 The square root of the high two parts is taken, by recursive
421 application of the algorithm (bottoming out in a one-limb Newton's
424 s1,r1 = sqrtrem (a3*b + a2)
426 This is an approximation to the desired root and is extended by a
427 division to give s,r,
429 q,u = divrem (r1*b + a1, 2*s1)
433 The normalization requirement on a3 means at this point s is either
434 correct or 1 too big. r is negative in the latter case, so
440 The algorithm is expressed in a divide and conquer form, but as noted
441 in the paper it can also be viewed as a discrete variant of Newton's
442 method, or as a variation on the schoolboy method (no longer taught) for
443 square roots two digits at a time.
445 If the remainder r is not required then usually only a few high limbs
446 of r and u need to be calculated to determine whether an adjustment to s
447 is required. This optimization is not currently implemented.
449 In the Karatsuba multiplication range this algorithm is
450 O(1.5*M(N/2)), where M(n) is the time to multiply two numbers of n
451 limbs. In the FFT multiplication range this grows to a bound of
452 O(6*M(N/2)). In practice a factor of about 1.5 to 1.8 is found in the
453 Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range.
455 The algorithm does all its calculations in integers and the resulting
456 'mpn_sqrtrem' is used for both 'mpz_sqrt' and 'mpf_sqrt'. The extended
457 precision given by 'mpf_sqrt_ui' is obtained by padding with zero limbs.
460 File: gmp.info, Node: Nth Root Algorithm, Next: Perfect Square Algorithm, Prev: Square Root Algorithm, Up: Root Extraction Algorithms
465 Integer Nth roots are taken using Newton's method with the following
466 iteration, where A is the input and n is the root to be taken.
469 a[i+1] = - * ( --------- + (n-1)*a[i] )
472 The initial approximation a[1] is generated bitwise by successively
473 powering a trial root with or without new 1 bits, aiming to be just
474 above the true root. The iteration converges quadratically when started
475 from a good approximation. When n is large more initial bits are needed
476 to get good convergence. The current implementation is not particularly
480 File: gmp.info, Node: Perfect Square Algorithm, Next: Perfect Power Algorithm, Prev: Nth Root Algorithm, Up: Root Extraction Algorithms
482 15.5.3 Perfect Square
483 ---------------------
485 A significant fraction of non-squares can be quickly identified by
486 checking whether the input is a quadratic residue modulo small integers.
488 'mpz_perfect_square_p' first tests the input mod 256, which means
489 just examining the low byte. Only 44 different values occur for squares
490 mod 256, so 82.8% of inputs can be immediately identified as
493 On a 32-bit system similar tests are done mod 9, 5, 7, 13 and 17, for
494 a total 99.25% of inputs identified as non-squares. On a 64-bit system
495 97 is tested too, for a total 99.62%.
497 These moduli are chosen because they're factors of 2^24-1 (or 2^48-1
498 for 64-bits), and such a remainder can be quickly taken just using
499 additions (see 'mpn_mod_34lsub1').
501 When nails are in use moduli are instead selected by the 'gen-psqr.c'
502 program and applied with an 'mpn_mod_1'. The same 2^24-1 or 2^48-1
503 could be done with nails using some extra bit shifts, but this is not
504 currently implemented.
506 In any case each modulus is applied to the 'mpn_mod_34lsub1' or
507 'mpn_mod_1' remainder and a table lookup identifies non-squares. By
508 using a "modexact" style calculation, and suitably permuted tables, just
509 one multiply each is required, see the code for details. Moduli are
510 also combined to save operations, so long as the lookup tables don't
511 become too big. 'gen-psqr.c' does all the pre-calculations.
513 A square root must still be taken for any value that passes these
514 tests, to verify it's really a square and not one of the small fraction
515 of non-squares that get through (i.e. a pseudo-square to all the tested
518 Clearly more residue tests could be done, 'mpz_perfect_square_p' only
519 uses a compact and efficient set. Big inputs would probably benefit
520 from more residue testing, small inputs might be better off with less.
521 The assumed distribution of squares versus non-squares in the input
522 would affect such considerations.
525 File: gmp.info, Node: Perfect Power Algorithm, Prev: Perfect Square Algorithm, Up: Root Extraction Algorithms
530 Detecting perfect powers is required by some factorization algorithms.
531 Currently 'mpz_perfect_power_p' is implemented using repeated Nth root
532 extractions, though naturally only prime roots need to be considered.
533 (*Note Nth Root Algorithm::.)
535 If a prime divisor p with multiplicity e can be found, then only
536 roots which are divisors of e need to be considered, much reducing the
537 work necessary. To this end divisibility by a set of small primes is
541 File: gmp.info, Node: Radix Conversion Algorithms, Next: Other Algorithms, Prev: Root Extraction Algorithms, Up: Algorithms
543 15.6 Radix Conversion
544 =====================
546 Radix conversions are less important than other algorithms. A program
547 dominated by conversions should probably use a different data
556 File: gmp.info, Node: Binary to Radix, Next: Radix to Binary, Prev: Radix Conversion Algorithms, Up: Radix Conversion Algorithms
558 15.6.1 Binary to Radix
559 ----------------------
561 Conversions from binary to a power-of-2 radix use a simple and fast O(N)
562 bit extraction algorithm.
564 Conversions from binary to other radices use one of two algorithms.
565 Sizes below 'GET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method.
566 Repeated divisions by b^n are made, where b is the radix and n is the
567 biggest power that fits in a limb. But instead of simply using the
568 remainder r from such divisions, an extra divide step is done to give a
569 fractional limb representing r/b^n. The digits of r can then be
570 extracted using multiplications by b rather than divisions. Special
571 case code is provided for decimal, allowing multiplications by 10 to
572 optimize to shifts and adds.
574 Above 'GET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is
575 used. For an input t, powers b^(n*2^i) of the radix are calculated,
576 until a power between t and sqrt(t) is reached. t is then divided by
577 that largest power, giving a quotient which is the digits above that
578 power, and a remainder which is those below. These two parts are in
579 turn divided by the second highest power, and so on recursively. When a
580 piece has been divided down to less than 'GET_STR_DC_THRESHOLD' limbs,
581 the basecase algorithm described above is used.
583 The advantage of this algorithm is that big divisions can make use of
584 the sub-quadratic divide and conquer division (*note Divide and Conquer
585 Division::), and big divisions tend to have less overheads than lots of
586 separate single limb divisions anyway. But in any case the cost of
587 calculating the powers b^(n*2^i) must first be overcome.
589 'GET_STR_PRECOMPUTE_THRESHOLD' and 'GET_STR_DC_THRESHOLD' represent
590 the same basic thing, the point where it becomes worth doing a big
591 division to cut the input in half. 'GET_STR_PRECOMPUTE_THRESHOLD'
592 includes the cost of calculating the radix power required, whereas
593 'GET_STR_DC_THRESHOLD' assumes that's already available, which is the
596 Since the base case produces digits from least to most significant
597 but they want to be stored from most to least, it's necessary to
598 calculate in advance how many digits there will be, or at least be sure
599 not to underestimate that. For GMP the number of input bits is
600 multiplied by 'chars_per_bit_exactly' from 'mp_bases', rounding up. The
601 result is either correct or one too big.
603 Examining some of the high bits of the input could increase the
604 chance of getting the exact number of digits, but an exact result every
605 time would not be practical, since in general the difference between
606 numbers 100... and 99... is only in the last few bits and the work to
607 identify 99... might well be almost as much as a full conversion.
609 The r/b^n scheme described above for using multiplications to bring
610 out digits might be useful for more than a single limb. Some brief
611 experiments with it on the base case when recursing didn't give a
612 noticeable improvement, but perhaps that was only due to the
613 implementation. Something similar would work for the sub-quadratic
614 divisions too, though there would be the cost of calculating a bigger
617 Another possible improvement for the sub-quadratic part would be to
618 arrange for radix powers that balanced the sizes of quotient and
619 remainder produced, i.e. the highest power would be an b^(n*k)
620 approximately equal to sqrt(t), not restricted to a 2^i factor. That
621 ought to smooth out a graph of times against sizes, but may or may not
625 File: gmp.info, Node: Radix to Binary, Prev: Binary to Radix, Up: Radix Conversion Algorithms
627 15.6.2 Radix to Binary
628 ----------------------
630 *This section needs to be rewritten, it currently describes the
631 algorithms used before GMP 4.3.*
633 Conversions from a power-of-2 radix into binary use a simple and fast
634 O(N) bitwise concatenation algorithm.
636 Conversions from other radices use one of two algorithms. Sizes
637 below 'SET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method. Groups
638 of n digits are converted to limbs, where n is the biggest power of the
639 base b which will fit in a limb, then those groups are accumulated into
640 the result by multiplying by b^n and adding. This saves multi-precision
641 operations, as per Knuth section 4.4 part E (*note References::). Some
642 special case code is provided for decimal, giving the compiler a chance
643 to optimize multiplications by 10.
645 Above 'SET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is
646 used. First groups of n digits are converted into limbs. Then adjacent
647 limbs are combined into limb pairs with x*b^n+y, where x and y are the
648 limbs. Adjacent limb pairs are combined into quads similarly with
649 x*b^(2n)+y. This continues until a single block remains, that being the
652 The advantage of this method is that the multiplications for each x
653 are big blocks, allowing Karatsuba and higher algorithms to be used.
654 But the cost of calculating the powers b^(n*2^i) must be overcome.
655 'SET_STR_PRECOMPUTE_THRESHOLD' usually ends up quite big, around 5000
656 digits, and on some processors much bigger still.
658 'SET_STR_PRECOMPUTE_THRESHOLD' is based on the input digits (and
659 tuned for decimal), though it might be better based on a limb count, so
660 as to be independent of the base. But that sort of count isn't used by
661 the base case and so would need some sort of initial calculation or
664 The main reason 'SET_STR_PRECOMPUTE_THRESHOLD' is so much bigger than
665 the corresponding 'GET_STR_PRECOMPUTE_THRESHOLD' is that 'mpn_mul_1' is
666 much faster than 'mpn_divrem_1' (often by a factor of 5, or more).
669 File: gmp.info, Node: Other Algorithms, Next: Assembly Coding, Prev: Radix Conversion Algorithms, Up: Algorithms
671 15.7 Other Algorithms
672 =====================
676 * Prime Testing Algorithm::
677 * Factorial Algorithm::
678 * Binomial Coefficients Algorithm::
679 * Fibonacci Numbers Algorithm::
680 * Lucas Numbers Algorithm::
681 * Random Number Algorithms::
684 File: gmp.info, Node: Prime Testing Algorithm, Next: Factorial Algorithm, Prev: Other Algorithms, Up: Other Algorithms
689 The primality testing in 'mpz_probab_prime_p' (*note Number Theoretic
690 Functions::) first does some trial division by small factors and then
691 uses the Miller-Rabin probabilistic primality testing algorithm, as
692 described in Knuth section 4.5.4 algorithm P (*note References::).
694 For an odd input n, and with n = q*2^k+1 where q is odd, this
695 algorithm selects a random base x and tests whether x^q mod n is 1 or
696 -1, or an x^(q*2^j) mod n is 1, for 1<=j<=k. If so then n is probably
697 prime, if not then n is definitely composite.
699 Any prime n will pass the test, but some composites do too. Such
700 composites are known as strong pseudoprimes to base x. No n is a strong
701 pseudoprime to more than 1/4 of all bases (see Knuth exercise 22), hence
702 with x chosen at random there's no more than a 1/4 chance a "probable
703 prime" will in fact be composite.
705 In fact strong pseudoprimes are quite rare, making the test much more
706 powerful than this analysis would suggest, but 1/4 is all that's proven
710 File: gmp.info, Node: Factorial Algorithm, Next: Binomial Coefficients Algorithm, Prev: Prime Testing Algorithm, Up: Other Algorithms
715 Factorials are calculated by a combination of two algorithms. An idea
716 is shared among them: to compute the odd part of the factorial; a final
717 step takes account of the power of 2 term, by shifting.
719 For small n, the odd factor of n! is computed with the simple
720 observation that it is equal to the product of all positive odd numbers
721 smaller than n times the odd factor of [n/2]!, where [x] is the integer
722 part of x, and so on recursively. The procedure can be best illustrated
725 23! = (23.21.19.17.15.13.11.9.7.5.3)(11.9.7.5.3)(5.3)2^{19}
727 Current code collects all the factors in a single list, with a loop
728 and no recursion, and compute the product, with no special care for
731 When n is larger, computation pass trough prime sieving. An helper
732 function is used, as suggested by Peter Luschny:
737 msf(n) = -------------- = | | p
740 Where p ranges on odd prime numbers. The exponent k is chosen to
741 obtain an odd integer number: k is the number of 1 bits in the binary
742 representation of [n/2]. The function L(p,n) can be defined as zero
743 when p is composite, and, for any prime p, it is computed with:
747 L(p,n) = / [---] mod 2 <= log (n) .
751 With this helper function, we are able to compute the odd part of n!
752 using the recursion implied by n!=[n/2]!^2*msf(n)*2^k. The recursion
753 stops using the small-n algorithm on some [n/2^i].
755 Both the above algorithms use binary splitting to compute the product
756 of many small factors. At first as many products as possible are
757 accumulated in a single register, generating a list of factors that fit
758 in a machine word. This list is then split into halves, and the product
759 is computed recursively.
761 Such splitting is more efficient than repeated Nx1 multiplies since
762 it forms big multiplies, allowing Karatsuba and higher algorithms to be
763 used. And even below the Karatsuba threshold a big block of work can be
764 more efficient for the basecase algorithm.
767 File: gmp.info, Node: Binomial Coefficients Algorithm, Next: Fibonacci Numbers Algorithm, Prev: Factorial Algorithm, Up: Other Algorithms
769 15.7.3 Binomial Coefficients
770 ----------------------------
772 Binomial coefficients C(n,k) are calculated by first arranging k <= n/2
773 using C(n,k) = C(n,n-k) if necessary, and then evaluating the following
774 product simply from i=2 to i=k.
777 C(n,k) = (n-k+1) * prod -------
780 It's easy to show that each denominator i will divide the product so
781 far, so the exact division algorithm is used (*note Exact Division::).
783 The numerators n-k+i and denominators i are first accumulated into as
784 many fit a limb, to save multi-precision operations, though for
785 'mpz_bin_ui' this applies only to the divisors, since n is an 'mpz_t'
786 and n-k+i in general won't fit in a limb at all.
789 File: gmp.info, Node: Fibonacci Numbers Algorithm, Next: Lucas Numbers Algorithm, Prev: Binomial Coefficients Algorithm, Up: Other Algorithms
791 15.7.4 Fibonacci Numbers
792 ------------------------
794 The Fibonacci functions 'mpz_fib_ui' and 'mpz_fib2_ui' are designed for
795 calculating isolated F[n] or F[n],F[n-1] values efficiently.
797 For small n, a table of single limb values in '__gmp_fib_table' is
798 used. On a 32-bit limb this goes up to F[47], or on a 64-bit limb up to
799 F[93]. For convenience the table starts at F[-1].
801 Beyond the table, values are generated with a binary powering
802 algorithm, calculating a pair F[n] and F[n-1] working from high to low
803 across the bits of n. The formulas used are
805 F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k
806 F[2k-1] = F[k]^2 + F[k-1]^2
808 F[2k] = F[2k+1] - F[2k-1]
810 At each step, k is the high b bits of n. If the next bit of n is 0
811 then F[2k],F[2k-1] is used, or if it's a 1 then F[2k+1],F[2k] is used,
812 and the process repeated until all bits of n are incorporated. Notice
813 these formulas require just two squares per bit of n.
815 It'd be possible to handle the first few n above the single limb
816 table with simple additions, using the defining Fibonacci recurrence
817 F[k+1]=F[k]+F[k-1], but this is not done since it usually turns out to
818 be faster for only about 10 or 20 values of n, and including a block of
819 code for just those doesn't seem worthwhile. If they really mattered
820 it'd be better to extend the data table.
822 Using a table avoids lots of calculations on small numbers, and makes
823 small n go fast. A bigger table would make more small n go fast, it's
824 just a question of balancing size against desired speed. For GMP the
825 code is kept compact, with the emphasis primarily on a good powering
828 'mpz_fib2_ui' returns both F[n] and F[n-1], but 'mpz_fib_ui' is only
829 interested in F[n]. In this case the last step of the algorithm can
830 become one multiply instead of two squares. One of the following two
831 formulas is used, according as n is odd or even.
833 F[2k] = F[k]*(F[k]+2F[k-1])
835 F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k
837 F[2k+1] here is the same as above, just rearranged to be a multiply.
838 For interest, the 2*(-1)^k term both here and above can be applied just
839 to the low limb of the calculation, without a carry or borrow into
840 further limbs, which saves some code size. See comments with
841 'mpz_fib_ui' and the internal 'mpn_fib2_ui' for how this is done.
844 File: gmp.info, Node: Lucas Numbers Algorithm, Next: Random Number Algorithms, Prev: Fibonacci Numbers Algorithm, Up: Other Algorithms
849 'mpz_lucnum2_ui' derives a pair of Lucas numbers from a pair of
850 Fibonacci numbers with the following simple formulas.
852 L[k] = F[k] + 2*F[k-1]
853 L[k-1] = 2*F[k] - F[k-1]
855 'mpz_lucnum_ui' is only interested in L[n], and some work can be
856 saved. Trailing zero bits on n can be handled with a single square
859 L[2k] = L[k]^2 - 2*(-1)^k
861 And the lowest 1 bit can be handled with one multiply of a pair of
862 Fibonacci numbers, similar to what 'mpz_fib_ui' does.
864 L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k
867 File: gmp.info, Node: Random Number Algorithms, Prev: Lucas Numbers Algorithm, Up: Other Algorithms
869 15.7.6 Random Numbers
870 ---------------------
872 For the 'urandomb' functions, random numbers are generated simply by
873 concatenating bits produced by the generator. As long as the generator
874 has good randomness properties this will produce well-distributed N bit
877 For the 'urandomm' functions, random numbers in a range 0<=R<N are
878 generated by taking values R of ceil(log2(N)) bits each until one
879 satisfies R<N. This will normally require only one or two attempts, but
880 the attempts are limited in case the generator is somehow degenerate and
881 produces only 1 bits or similar.
883 The Mersenne Twister generator is by Matsumoto and Nishimura (*note
884 References::). It has a non-repeating period of 2^19937-1, which is a
885 Mersenne prime, hence the name of the generator. The state is 624 words
886 of 32-bits each, which is iterated with one XOR and shift for each
887 32-bit word generated, making the algorithm very fast. Randomness
888 properties are also very good and this is the default algorithm used by
891 Linear congruential generators are described in many text books, for
892 instance Knuth volume 2 (*note References::). With a modulus M and
893 parameters A and C, an integer state S is iterated by the formula S <-
894 A*S+C mod M. At each step the new state is a linear function of the
895 previous, mod M, hence the name of the generator.
897 In GMP only moduli of the form 2^N are supported, and the current
898 implementation is not as well optimized as it could be. Overheads are
899 significant when N is small, and when N is large clearly the multiply at
900 each step will become slow. This is not a big concern, since the
901 Mersenne Twister generator is better in every respect and is therefore
902 recommended for all normal applications.
904 For both generators the current state can be deduced by observing
905 enough output and applying some linear algebra (over GF(2) in the case
906 of the Mersenne Twister). This generally means raw output is unsuitable
907 for cryptographic applications without further hashing or the like.
910 File: gmp.info, Node: Assembly Coding, Prev: Other Algorithms, Up: Algorithms
915 The assembly subroutines in GMP are the most significant source of speed
916 at small to moderate sizes. At larger sizes algorithm selection becomes
917 more important, but of course speedups in low level routines will still
918 speed up everything proportionally.
920 Carry handling and widening multiplies that are important for GMP
921 can't be easily expressed in C. GCC 'asm' blocks help a lot and are
922 provided in 'longlong.h', but hand coding low level routines invariably
923 offers a speedup over generic C by a factor of anything from 2 to 10.
927 * Assembly Code Organisation::
929 * Assembly Carry Propagation::
930 * Assembly Cache Handling::
931 * Assembly Functional Units::
932 * Assembly Floating Point::
933 * Assembly SIMD Instructions::
934 * Assembly Software Pipelining::
935 * Assembly Loop Unrolling::
936 * Assembly Writing Guide::
939 File: gmp.info, Node: Assembly Code Organisation, Next: Assembly Basics, Prev: Assembly Coding, Up: Assembly Coding
941 15.8.1 Code Organisation
942 ------------------------
944 The various 'mpn' subdirectories contain machine-dependent code, written
945 in C or assembly. The 'mpn/generic' subdirectory contains default code,
946 used when there's no machine-specific version of a particular file.
948 Each 'mpn' subdirectory is for an ISA family. Generally 32-bit and
949 64-bit variants in a family cannot share code and have separate
950 directories. Within a family further subdirectories may exist for CPU
953 In each directory a 'nails' subdirectory may exist, holding code with
954 nails support for that CPU variant. A 'NAILS_SUPPORT' directive in each
955 file indicates the nails values the code handles. Nails code only
956 exists where it's faster, or promises to be faster, than plain code.
957 There's no effort put into nails if they're not going to enhance a given
961 File: gmp.info, Node: Assembly Basics, Next: Assembly Carry Propagation, Prev: Assembly Code Organisation, Up: Assembly Coding
963 15.8.2 Assembly Basics
964 ----------------------
966 'mpn_addmul_1' and 'mpn_submul_1' are the most important routines for
967 overall GMP performance. All multiplications and divisions come down to
968 repeated calls to these. 'mpn_add_n', 'mpn_sub_n', 'mpn_lshift' and
969 'mpn_rshift' are next most important.
971 On some CPUs assembly versions of the internal functions
972 'mpn_mul_basecase' and 'mpn_sqr_basecase' give significant speedups,
973 mainly through avoiding function call overheads. They can also
974 potentially make better use of a wide superscalar processor, as can
975 bigger primitives like 'mpn_addmul_2' or 'mpn_addmul_4'.
977 The restrictions on overlaps between sources and destinations (*note
978 Low-level Functions::) are designed to facilitate a variety of
979 implementations. For example, knowing 'mpn_add_n' won't have partly
980 overlapping sources and destination means reading can be done far ahead
981 of writing on superscalar processors, and loops can be vectorized on a
982 vector processor, depending on the carry handling.
985 File: gmp.info, Node: Assembly Carry Propagation, Next: Assembly Cache Handling, Prev: Assembly Basics, Up: Assembly Coding
987 15.8.3 Carry Propagation
988 ------------------------
990 The problem that presents most challenges in GMP is propagating carries
991 from one limb to the next. In functions like 'mpn_addmul_1' and
992 'mpn_add_n', carries are the only dependencies between limb operations.
994 On processors with carry flags, a straightforward CISC style 'adc' is
995 generally best. AMD K6 'mpn_addmul_1' however is an example of an
996 unusual set of circumstances where a branch works out better.
998 On RISC processors generally an add and compare for overflow is used.
999 This sort of thing can be seen in 'mpn/generic/aors_n.c'. Some carry
1000 propagation schemes require 4 instructions, meaning at least 4 cycles
1001 per limb, but other schemes may use just 1 or 2. On wide superscalar
1002 processors performance may be completely determined by the number of
1003 dependent instructions between carry-in and carry-out for each limb.
1005 On vector processors good use can be made of the fact that a carry
1006 bit only very rarely propagates more than one limb. When adding a
1007 single bit to a limb, there's only a carry out if that limb was
1008 '0xFF...FF' which on random data will be only 1 in 2^mp_bits_per_limb.
1009 'mpn/cray/add_n.c' is an example of this, it adds all limbs in parallel,
1010 adds one set of carry bits in parallel and then only rarely needs to
1011 fall through to a loop propagating further carries.
1013 On the x86s, GCC (as of version 2.95.2) doesn't generate particularly
1014 good code for the RISC style idioms that are necessary to handle carry
1015 bits in C. Often conditional jumps are generated where 'adc' or 'sbb'
1016 forms would be better. And so unfortunately almost any loop involving
1017 carry bits needs to be coded in assembly for best results.
1020 File: gmp.info, Node: Assembly Cache Handling, Next: Assembly Functional Units, Prev: Assembly Carry Propagation, Up: Assembly Coding
1022 15.8.4 Cache Handling
1023 ---------------------
1025 GMP aims to perform well both on operands that fit entirely in L1 cache
1026 and those which don't.
1028 Basic routines like 'mpn_add_n' or 'mpn_lshift' are often used on
1029 large operands, so L2 and main memory performance is important for them.
1030 'mpn_mul_1' and 'mpn_addmul_1' are mostly used for multiply and square
1031 basecases, so L1 performance matters most for them, unless assembly
1032 versions of 'mpn_mul_basecase' and 'mpn_sqr_basecase' exist, in which
1033 case the remaining uses are mostly for larger operands.
1035 For L2 or main memory operands, memory access times will almost
1036 certainly be more than the calculation time. The aim therefore is to
1037 maximize memory throughput, by starting a load of the next cache line
1038 while processing the contents of the previous one. Clearly this is only
1039 possible if the chip has a lock-up free cache or some sort of prefetch
1040 instruction. Most current chips have both these features.
1042 Prefetching sources combines well with loop unrolling, since a
1043 prefetch can be initiated once per unrolled loop (or more than once if
1044 the loop covers more than one cache line).
1046 On CPUs without write-allocate caches, prefetching destinations will
1047 ensure individual stores don't go further down the cache hierarchy,
1048 limiting bandwidth. Of course for calculations which are slow anyway,
1049 like 'mpn_divrem_1', write-throughs might be fine.
1051 The distance ahead to prefetch will be determined by memory latency
1052 versus throughput. The aim of course is to have data arriving
1053 continuously, at peak throughput. Some CPUs have limits on the number
1054 of fetches or prefetches in progress.
1056 If a special prefetch instruction doesn't exist then a plain load can
1057 be used, but in that case care must be taken not to attempt to read past
1058 the end of an operand, since that might produce a segmentation
1061 Some CPUs or systems have hardware that detects sequential memory
1062 accesses and initiates suitable cache movements automatically, making
1066 File: gmp.info, Node: Assembly Functional Units, Next: Assembly Floating Point, Prev: Assembly Cache Handling, Up: Assembly Coding
1068 15.8.5 Functional Units
1069 -----------------------
1071 When choosing an approach for an assembly loop, consideration is given
1072 to what operations can execute simultaneously and what throughput can
1073 thereby be achieved. In some cases an algorithm can be tweaked to
1074 accommodate available resources.
1076 Loop control will generally require a counter and pointer updates,
1077 costing as much as 5 instructions, plus any delays a branch introduces.
1078 CPU addressing modes might reduce pointer updates, perhaps by allowing
1079 just one updating pointer and others expressed as offsets from it, or on
1080 CISC chips with all addressing done with the loop counter as a scaled
1083 The final loop control cost can be amortised by processing several
1084 limbs in each iteration (*note Assembly Loop Unrolling::). This at
1085 least ensures loop control isn't a big fraction the work done.
1087 Memory throughput is always a limit. If perhaps only one load or one
1088 store can be done per cycle then 3 cycles/limb will the top speed for
1089 "binary" operations like 'mpn_add_n', and any code achieving that is
1092 Integer resources can be freed up by having the loop counter in a
1093 float register, or by pressing the float units into use for some
1094 multiplying, perhaps doing every second limb on the float side (*note
1095 Assembly Floating Point::).
1097 Float resources can be freed up by doing carry propagation on the
1098 integer side, or even by doing integer to float conversions in integers
1099 using bit twiddling.
1102 File: gmp.info, Node: Assembly Floating Point, Next: Assembly SIMD Instructions, Prev: Assembly Functional Units, Up: Assembly Coding
1104 15.8.6 Floating Point
1105 ---------------------
1107 Floating point arithmetic is used in GMP for multiplications on CPUs
1108 with poor integer multipliers. It's mostly useful for 'mpn_mul_1',
1109 'mpn_addmul_1' and 'mpn_submul_1' on 64-bit machines, and
1110 'mpn_mul_basecase' on both 32-bit and 64-bit machines.
1112 With IEEE 53-bit double precision floats, integer multiplications
1113 producing up to 53 bits will give exact results. Breaking a 64x64
1114 multiplication into eight 16x32->48 bit pieces is convenient. With some
1115 care though six 21x32->53 bit products can be used, if one of the lower
1116 two 21-bit pieces also uses the sign bit.
1118 For the 'mpn_mul_1' family of functions on a 64-bit machine, the
1119 invariant single limb is split at the start, into 3 or 4 pieces. Inside
1120 the loop, the bignum operand is split into 32-bit pieces. Fast
1121 conversion of these unsigned 32-bit pieces to floating point is highly
1122 machine-dependent. In some cases, reading the data into the integer
1123 unit, zero-extending to 64-bits, then transferring to the floating point
1124 unit back via memory is the only option.
1126 Converting partial products back to 64-bit limbs is usually best done
1127 as a signed conversion. Since all values are smaller than 2^53, signed
1128 and unsigned are the same, but most processors lack unsigned
1133 Here is a diagram showing 16x32 bit products for an 'mpn_mul_1' or
1134 'mpn_addmul_1' with a 64-bit limb. The single limb operand V is split
1135 into four 16-bit parts. The multi-limb operand U is split in the loop
1136 into two 32-bit parts.
1139 |v48|v32|v16|v00| V operand
1143 x | u32 | u00 | U operand (one limb)
1146 ---------------------------------
1149 | u00 x v00 | p00 48-bit products
1173 p32 and r32 can be summed using floating-point addition, and likewise
1174 p48 and r48. p00 and p16 can be summed with r64 and r80 from the
1177 For each loop then, four 49-bit quantities are transferred to the
1178 integer unit, aligned as follows,
1180 |-----64bits----|-----64bits----|
1194 The challenge then is to sum these efficiently and add in a carry
1195 limb, generating a low 64-bit result limb and a high 33-bit carry limb
1196 (i48 extends 33 bits into the high half).
1199 File: gmp.info, Node: Assembly SIMD Instructions, Next: Assembly Software Pipelining, Prev: Assembly Floating Point, Up: Assembly Coding
1201 15.8.7 SIMD Instructions
1202 ------------------------
1204 The single-instruction multiple-data support in current microprocessors
1205 is aimed at signal processing algorithms where each data point can be
1206 treated more or less independently. There's generally not much support
1207 for propagating the sort of carries that arise in GMP.
1209 SIMD multiplications of say four 16x16 bit multiplies only do as much
1210 work as one 32x32 from GMP's point of view, and need some shifts and
1211 adds besides. But of course if say the SIMD form is fully pipelined and
1212 uses less instruction decoding then it may still be worthwhile.
1214 On the x86 chips, MMX has so far found a use in 'mpn_rshift' and
1215 'mpn_lshift', and is used in a special case for 16-bit multipliers in
1216 the P55 'mpn_mul_1'. SSE2 is used for Pentium 4 'mpn_mul_1',
1217 'mpn_addmul_1', and 'mpn_submul_1'.
1220 File: gmp.info, Node: Assembly Software Pipelining, Next: Assembly Loop Unrolling, Prev: Assembly SIMD Instructions, Up: Assembly Coding
1222 15.8.8 Software Pipelining
1223 --------------------------
1225 Software pipelining consists of scheduling instructions around the
1226 branch point in a loop. For example a loop might issue a load not for
1227 use in the present iteration but the next, thereby allowing extra cycles
1228 for the data to arrive from memory.
1230 Naturally this is wanted only when doing things like loads or
1231 multiplies that take several cycles to complete, and only where a CPU
1232 has multiple functional units so that other work can be done in the
1235 A pipeline with several stages will have a data value in progress at
1236 each stage and each loop iteration moves them along one stage. This is
1239 If the latency of some instruction is greater than the loop time then
1240 it will be necessary to unroll, so one register has a result ready to
1241 use while another (or multiple others) are still in progress. (*note
1242 Assembly Loop Unrolling::).
1245 File: gmp.info, Node: Assembly Loop Unrolling, Next: Assembly Writing Guide, Prev: Assembly Software Pipelining, Up: Assembly Coding
1247 15.8.9 Loop Unrolling
1248 ---------------------
1250 Loop unrolling consists of replicating code so that several limbs are
1251 processed in each loop. At a minimum this reduces loop overheads by a
1252 corresponding factor, but it can also allow better register usage, for
1253 example alternately using one register combination and then another.
1254 Judicious use of 'm4' macros can help avoid lots of duplication in the
1257 Any amount of unrolling can be handled with a loop counter that's
1258 decremented by N each time, stopping when the remaining count is less
1259 than the further N the loop will process. Or by subtracting N at the
1260 start, the termination condition becomes when the counter C is less than
1261 0 (and the count of remaining limbs is C+N).
1263 Alternately for a power of 2 unroll the loop count and remainder can
1264 be established with a shift and mask. This is convenient if also making
1265 a computed jump into the middle of a large loop.
1267 The limbs not a multiple of the unrolling can be handled in various
1270 * A simple loop at the end (or the start) to process the excess.
1271 Care will be wanted that it isn't too much slower than the unrolled
1274 * A set of binary tests, for example after an 8-limb unrolling, test
1275 for 4 more limbs to process, then a further 2 more or not, and
1276 finally 1 more or not. This will probably take more code space
1279 * A 'switch' statement, providing separate code for each possible
1280 excess, for example an 8-limb unrolling would have separate code
1281 for 0 remaining, 1 remaining, etc, up to 7 remaining. This might
1282 take a lot of code, but may be the best way to optimize all cases
1283 in combination with a deep pipelined loop.
1285 * A computed jump into the middle of the loop, thus making the first
1286 iteration handle the excess. This should make times smoothly
1287 increase with size, which is attractive, but setups for the jump
1288 and adjustments for pointers can be tricky and could become quite
1289 difficult in combination with deep pipelining.
1292 File: gmp.info, Node: Assembly Writing Guide, Prev: Assembly Loop Unrolling, Up: Assembly Coding
1294 15.8.10 Writing Guide
1295 ---------------------
1297 This is a guide to writing software pipelined loops for processing limb
1298 vectors in assembly.
1300 First determine the algorithm and which instructions are needed.
1301 Code it without unrolling or scheduling, to make sure it works. On a
1302 3-operand CPU try to write each new value to a new register, this will
1303 greatly simplify later steps.
1305 Then note for each instruction the functional unit and/or issue port
1306 requirements. If an instruction can use either of two units, like U0 or
1307 U1 then make a category "U0/U1". Count the total using each unit (or
1308 combined unit), and count all instructions.
1310 Figure out from those counts the best possible loop time. The goal
1311 will be to find a perfect schedule where instruction latencies are
1312 completely hidden. The total instruction count might be the limiting
1313 factor, or perhaps a particular functional unit. It might be possible
1314 to tweak the instructions to help the limiting factor.
1316 Suppose the loop time is N, then make N issue buckets, with the final
1317 loop branch at the end of the last. Now fill the buckets with dummy
1318 instructions using the functional units desired. Run this to make sure
1319 the intended speed is reached.
1321 Now replace the dummy instructions with the real instructions from
1322 the slow but correct loop you started with. The first will typically be
1323 a load instruction. Then the instruction using that value is placed in
1324 a bucket an appropriate distance down. Run the loop again, to check it
1325 still runs at target speed.
1327 Keep placing instructions, frequently measuring the loop. After a
1328 few you will need to wrap around from the last bucket back to the top of
1329 the loop. If you used the new-register for new-value strategy above
1330 then there will be no register conflicts. If not then take care not to
1331 clobber something already in use. Changing registers at this time is
1334 The loop will overlap two or more of the original loop iterations,
1335 and the computation of one vector element result will be started in one
1336 iteration of the new loop, and completed one or several iterations
1339 The final step is to create feed-in and wind-down code for the loop.
1340 A good way to do this is to make a copy (or copies) of the loop at the
1341 start and delete those instructions which don't have valid antecedents,
1342 and at the end replicate and delete those whose results are unwanted
1343 (including any further loads).
1345 The loop will have a minimum number of limbs loaded and processed, so
1346 the feed-in code must test if the request size is smaller and skip
1347 either to a suitable part of the wind-down or to special code for small
1351 File: gmp.info, Node: Internals, Next: Contributors, Prev: Algorithms, Up: Top
1356 *This chapter is provided only for informational purposes and the
1357 various internals described here may change in future GMP releases.
1358 Applications expecting to be compatible with future releases should use
1359 only the documented interfaces described in previous chapters.*
1363 * Integer Internals::
1364 * Rational Internals::
1366 * Raw Output Internals::
1367 * C++ Interface Internals::
1370 File: gmp.info, Node: Integer Internals, Next: Rational Internals, Prev: Internals, Up: Internals
1372 16.1 Integer Internals
1373 ======================
1375 'mpz_t' variables represent integers using sign and magnitude, in space
1376 dynamically allocated and reallocated. The fields are as follows.
1379 The number of limbs, or the negative of that when representing a
1380 negative integer. Zero is represented by '_mp_size' set to zero,
1381 in which case the '_mp_d' data is undefined.
1384 A pointer to an array of limbs which is the magnitude. These are
1385 stored "little endian" as per the 'mpn' functions, so '_mp_d[0]' is
1386 the least significant limb and '_mp_d[ABS(_mp_size)-1]' is the most
1387 significant. Whenever '_mp_size' is non-zero, the most significant
1390 Currently there's always at least one readable limb, so for
1391 instance 'mpz_get_ui' can fetch '_mp_d[0]' unconditionally (though
1392 its value is undefined if '_mp_size' is zero).
1395 '_mp_alloc' is the number of limbs currently allocated at '_mp_d',
1396 and normally '_mp_alloc >= ABS(_mp_size)'. When an 'mpz' routine
1397 is about to (or might be about to) increase '_mp_size', it checks
1398 '_mp_alloc' to see whether there's enough space, and reallocates if
1399 not. 'MPZ_REALLOC' is generally used for this.
1401 'mpz_t' variables initialised with the 'mpz_roinit_n' function or
1402 the 'MPZ_ROINIT_N' macro have '_mp_alloc = 0' but can have a
1403 non-zero '_mp_size'. They can only be used as read-only constants.
1404 See *note Integer Special Functions:: for details.
1406 The various bitwise logical functions like 'mpz_and' behave as if
1407 negative values were twos complement. But sign and magnitude is always
1408 used internally, and necessary adjustments are made during the
1409 calculations. Sometimes this isn't pretty, but sign and magnitude are
1410 best for other routines.
1412 Some internal temporary variables are setup with 'MPZ_TMP_INIT' and
1413 these have '_mp_d' space obtained from 'TMP_ALLOC' rather than the
1414 memory allocation functions. Care is taken to ensure that these are big
1415 enough that no reallocation is necessary (since it would have
1416 unpredictable consequences).
1418 '_mp_size' and '_mp_alloc' are 'int', although 'mp_size_t' is usually
1419 a 'long'. This is done to make the fields just 32 bits on some 64 bits
1420 systems, thereby saving a few bytes of data space but still providing
1424 File: gmp.info, Node: Rational Internals, Next: Float Internals, Prev: Integer Internals, Up: Internals
1426 16.2 Rational Internals
1427 =======================
1429 'mpq_t' variables represent rationals using an 'mpz_t' numerator and
1430 denominator (*note Integer Internals::).
1432 The canonical form adopted is denominator positive (and non-zero), no
1433 common factors between numerator and denominator, and zero uniquely
1436 It's believed that casting out common factors at each stage of a
1437 calculation is best in general. A GCD is an O(N^2) operation so it's
1438 better to do a few small ones immediately than to delay and have to do a
1439 big one later. Knowing the numerator and denominator have no common
1440 factors can be used for example in 'mpq_mul' to make only two cross GCDs
1441 necessary, not four.
1443 This general approach to common factors is badly sub-optimal in the
1444 presence of simple factorizations or little prospect for cancellation,
1445 but GMP has no way to know when this will occur. As per *note
1446 Efficiency::, that's left to applications. The 'mpq_t' framework might
1447 still suit, with 'mpq_numref' and 'mpq_denref' for direct access to the
1448 numerator and denominator, or of course 'mpz_t' variables can be used
1452 File: gmp.info, Node: Float Internals, Next: Raw Output Internals, Prev: Rational Internals, Up: Internals
1454 16.3 Float Internals
1455 ====================
1457 Efficient calculation is the primary aim of GMP floats and the use of
1458 whole limbs and simple rounding facilitates this.
1460 'mpf_t' floats have a variable precision mantissa and a single
1461 machine word signed exponent. The mantissa is represented using sign
1465 significant significant
1469 |---- _mp_exp ---> |
1470 _____ _____ _____ _____ _____
1471 |_____|_____|_____|_____|_____|
1472 . <------------ radix point
1474 <-------- _mp_size --------->
1477 The fields are as follows.
1480 The number of limbs currently in use, or the negative of that when
1481 representing a negative value. Zero is represented by '_mp_size'
1482 and '_mp_exp' both set to zero, and in that case the '_mp_d' data
1483 is unused. (In the future '_mp_exp' might be undefined when
1487 The precision of the mantissa, in limbs. In any calculation the
1488 aim is to produce '_mp_prec' limbs of result (the most significant
1492 A pointer to the array of limbs which is the absolute value of the
1493 mantissa. These are stored "little endian" as per the 'mpn'
1494 functions, so '_mp_d[0]' is the least significant limb and
1495 '_mp_d[ABS(_mp_size)-1]' the most significant.
1497 The most significant limb is always non-zero, but there are no
1498 other restrictions on its value, in particular the highest 1 bit
1499 can be anywhere within the limb.
1501 '_mp_prec+1' limbs are allocated to '_mp_d', the extra limb being
1502 for convenience (see below). There are no reallocations during a
1503 calculation, only in a change of precision with 'mpf_set_prec'.
1506 The exponent, in limbs, determining the location of the implied
1507 radix point. Zero means the radix point is just above the most
1508 significant limb. Positive values mean a radix point offset
1509 towards the lower limbs and hence a value >= 1, as for example in
1510 the diagram above. Negative exponents mean a radix point further
1511 above the highest limb.
1513 Naturally the exponent can be any value, it doesn't have to fall
1514 within the limbs as the diagram shows, it can be a long way above
1515 or a long way below. Limbs other than those included in the
1516 '{_mp_d,_mp_size}' data are treated as zero.
1518 The '_mp_size' and '_mp_prec' fields are 'int', although the
1519 'mp_size_t' type is usually a 'long'. The '_mp_exp' field is usually
1520 'long'. This is done to make some fields just 32 bits on some 64 bits
1521 systems, thereby saving a few bytes of data space but still providing
1522 plenty of precision and a very large range.
1525 The following various points should be noted.
1528 The least significant limbs '_mp_d[0]' etc can be zero, though such
1529 low zeros can always be ignored. Routines likely to produce low
1530 zeros check and avoid them to save time in subsequent calculations,
1531 but for most routines they're quite unlikely and aren't checked.
1534 The '_mp_size' count of limbs in use can be less than '_mp_prec' if
1535 the value can be represented in less. This means low precision
1536 values or small integers stored in a high precision 'mpf_t' can
1537 still be operated on efficiently.
1539 '_mp_size' can also be greater than '_mp_prec'. Firstly a value is
1540 allowed to use all of the '_mp_prec+1' limbs available at '_mp_d',
1541 and secondly when 'mpf_set_prec_raw' lowers '_mp_prec' it leaves
1542 '_mp_size' unchanged and so the size can be arbitrarily bigger than
1546 All rounding is done on limb boundaries. Calculating '_mp_prec'
1547 limbs with the high non-zero will ensure the application requested
1548 minimum precision is obtained.
1550 The use of simple "trunc" rounding towards zero is efficient, since
1551 there's no need to examine extra limbs and increment or decrement.
1554 Since the exponent is in limbs, there are no bit shifts in basic
1555 operations like 'mpf_add' and 'mpf_mul'. When differing exponents
1556 are encountered all that's needed is to adjust pointers to line up
1559 Of course 'mpf_mul_2exp' and 'mpf_div_2exp' will require bit
1560 shifts, but the choice is between an exponent in limbs which
1561 requires shifts there, or one in bits which requires them almost
1564 Use of '_mp_prec+1' Limbs
1565 The extra limb on '_mp_d' ('_mp_prec+1' rather than just
1566 '_mp_prec') helps when an 'mpf' routine might get a carry from its
1567 operation. 'mpf_add' for instance will do an 'mpn_add' of
1568 '_mp_prec' limbs. If there's no carry then that's the result, but
1569 if there is a carry then it's stored in the extra limb of space and
1570 '_mp_size' becomes '_mp_prec+1'.
1572 Whenever '_mp_prec+1' limbs are held in a variable, the low limb is
1573 not needed for the intended precision, only the '_mp_prec' high
1574 limbs. But zeroing it out or moving the rest down is unnecessary.
1575 Subsequent routines reading the value will simply take the high
1576 limbs they need, and this will be '_mp_prec' if their target has
1577 that same precision. This is no more than a pointer adjustment,
1578 and must be checked anyway since the destination precision can be
1579 different from the sources.
1581 Copy functions like 'mpf_set' will retain a full '_mp_prec+1' limbs
1582 if available. This ensures that a variable which has '_mp_size'
1583 equal to '_mp_prec+1' will get its full exact value copied.
1584 Strictly speaking this is unnecessary since only '_mp_prec' limbs
1585 are needed for the application's requested precision, but it's
1586 considered that an 'mpf_set' from one variable into another of the
1587 same precision ought to produce an exact copy.
1589 Application Precisions
1590 '__GMPF_BITS_TO_PREC' converts an application requested precision
1591 to an '_mp_prec'. The value in bits is rounded up to a whole limb
1592 then an extra limb is added since the most significant limb of
1593 '_mp_d' is only non-zero and therefore might contain only one bit.
1595 '__GMPF_PREC_TO_BITS' does the reverse conversion, and removes the
1596 extra limb from '_mp_prec' before converting to bits. The net
1597 effect of reading back with 'mpf_get_prec' is simply the precision
1598 rounded up to a multiple of 'mp_bits_per_limb'.
1600 Note that the extra limb added here for the high only being
1601 non-zero is in addition to the extra limb allocated to '_mp_d'.
1602 For example with a 32-bit limb, an application request for 250 bits
1603 will be rounded up to 8 limbs, then an extra added for the high
1604 being only non-zero, giving an '_mp_prec' of 9. '_mp_d' then gets
1605 10 limbs allocated. Reading back with 'mpf_get_prec' will take
1606 '_mp_prec' subtract 1 limb and multiply by 32, giving 256 bits.
1608 Strictly speaking, the fact the high limb has at least one bit
1609 means that a float with, say, 3 limbs of 32-bits each will be
1610 holding at least 65 bits, but for the purposes of 'mpf_t' it's
1611 considered simply to be 64 bits, a nice multiple of the limb size.
1614 File: gmp.info, Node: Raw Output Internals, Next: C++ Interface Internals, Prev: Float Internals, Up: Internals
1616 16.4 Raw Output Internals
1617 =========================
1619 'mpz_out_raw' uses the following format.
1621 +------+------------------------+
1622 | size | data bytes |
1623 +------+------------------------+
1625 The size is 4 bytes written most significant byte first, being the
1626 number of subsequent data bytes, or the twos complement negative of that
1627 when a negative integer is represented. The data bytes are the absolute
1628 value of the integer, written most significant byte first.
1630 The most significant data byte is always non-zero, so the output is
1631 the same on all systems, irrespective of limb size.
1633 In GMP 1, leading zero bytes were written to pad the data bytes to a
1634 multiple of the limb size. 'mpz_inp_raw' will still accept this, for
1637 The use of "big endian" for both the size and data fields is
1638 deliberate, it makes the data easy to read in a hex dump of a file.
1639 Unfortunately it also means that the limb data must be reversed when
1640 reading or writing, so neither a big endian nor little endian system can
1641 just read and write '_mp_d'.
1644 File: gmp.info, Node: C++ Interface Internals, Prev: Raw Output Internals, Up: Internals
1646 16.5 C++ Interface Internals
1647 ============================
1649 A system of expression templates is used to ensure something like
1650 'a=b+c' turns into a simple call to 'mpz_add' etc. For 'mpf_class' the
1651 scheme also ensures the precision of the final destination is used for
1652 any temporaries within a statement like 'f=w*x+y*z'. These are
1653 important features which a naive implementation cannot provide.
1655 A simplified description of the scheme follows. The true scheme is
1656 complicated by the fact that expressions have different return types.
1657 For detailed information, refer to the source code.
1659 To perform an operation, say, addition, we first define a "function
1660 object" evaluating it,
1662 struct __gmp_binary_plus
1664 static void eval(mpf_t f, const mpf_t g, const mpf_t h)
1670 And an "additive expression" object,
1672 __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >
1673 operator+(const mpf_class &f, const mpf_class &g)
1676 <__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >(f, g);
1679 The seemingly redundant '__gmp_expr<__gmp_binary_expr<...>>' is used
1680 to encapsulate any possible kind of expression into a single template
1681 type. In fact even 'mpf_class' etc are 'typedef' specializations of
1684 Next we define assignment of '__gmp_expr' to 'mpf_class'.
1687 mpf_class & mpf_class::operator=(const __gmp_expr<T> &expr)
1689 expr.eval(this->get_mpf_t(), this->precision());
1694 void __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, Op> >::eval
1695 (mpf_t f, mp_bitcnt_t precision)
1697 Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t());
1700 where 'expr.val1' and 'expr.val2' are references to the expression's
1701 operands (here 'expr' is the '__gmp_binary_expr' stored within the
1704 This way, the expression is actually evaluated only at the time of
1705 assignment, when the required precision (that of 'f') is known.
1706 Furthermore the target 'mpf_t' is now available, thus we can call
1707 'mpf_add' directly with 'f' as the output argument.
1709 Compound expressions are handled by defining operators taking
1710 subexpressions as their arguments, like this:
1712 template <class T, class U>
1714 <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
1715 operator+(const __gmp_expr<T> &expr1, const __gmp_expr<U> &expr2)
1718 <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
1722 And the corresponding specializations of '__gmp_expr::eval':
1724 template <class T, class U, class Op>
1726 <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, Op> >::eval
1727 (mpf_t f, mp_bitcnt_t precision)
1729 // declare two temporaries
1730 mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision);
1731 Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t());
1734 The expression is thus recursively evaluated to any level of
1735 complexity and all subexpressions are evaluated to the precision of 'f'.
1738 File: gmp.info, Node: Contributors, Next: References, Prev: Internals, Up: Top
1740 Appendix A Contributors
1741 ***********************
1743 Torbjörn Granlund wrote the original GMP library and is still the main
1744 developer. Code not explicitly attributed to others, was contributed by
1745 Torbjörn. Several other individuals and organizations have contributed
1746 GMP. Here is a list in chronological order on first contribution:
1748 Gunnar Sjödin and Hans Riesel helped with mathematical problems in
1749 early versions of the library.
1751 Richard Stallman helped with the interface design and revised the
1752 first version of this manual.
1754 Brian Beuning and Doug Lea helped with testing of early versions of
1755 the library and made creative suggestions.
1757 John Amanatides of York University in Canada contributed the function
1758 'mpz_probab_prime_p'.
1760 Paul Zimmermann wrote the REDC-based mpz_powm code, the
1761 Schönhage-Strassen FFT multiply code, and the Karatsuba square root
1762 code. He also improved the Toom3 code for GMP 4.2. Paul sparked the
1763 development of GMP 2, with his comparisons between bignum packages. The
1764 ECMNET project Paul is organizing was a driving force behind many of the
1765 optimizations in GMP 3. Paul also wrote the new GMP 4.3 nth root code
1768 Ken Weber (Kent State University, Universidade Federal do Rio Grande
1769 do Sul) contributed now defunct versions of 'mpz_gcd', 'mpz_divexact',
1770 'mpn_gcd', and 'mpn_bdivmod', partially supported by CNPq (Brazil) grant
1773 Per Bothner of Cygnus Support helped to set up GMP to use Cygnus'
1774 configure. He has also made valuable suggestions and tested numerous
1775 intermediary releases.
1777 Joachim Hollman was involved in the design of the 'mpf' interface,
1778 and in the 'mpz' design revisions for version 2.
1780 Bennet Yee contributed the initial versions of 'mpz_jacobi' and
1783 Andreas Schwab contributed the files 'mpn/m68k/lshift.S' and
1784 'mpn/m68k/rshift.S' (now in '.asm' form).
1786 Robert Harley of Inria, France and David Seal of ARM, England,
1787 suggested clever improvements for population count. Robert also wrote
1788 highly optimized Karatsuba and 3-way Toom multiplication functions for
1789 GMP 3, and contributed the ARM assembly code.
1791 Torsten Ekedahl of the Mathematical department of Stockholm
1792 University provided significant inspiration during several phases of the
1793 GMP development. His mathematical expertise helped improve several
1796 Linus Nordberg wrote the new configure system based on autoconf and
1797 implemented the new random functions.
1799 Kevin Ryde worked on a large number of things: optimized x86 code, m4
1800 asm macros, parameter tuning, speed measuring, the configure system,
1801 function inlining, divisibility tests, bit scanning, Jacobi symbols,
1802 Fibonacci and Lucas number functions, printf and scanf functions, perl
1803 interface, demo expression parser, the algorithms chapter in the manual,
1804 'gmpasm-mode.el', and various miscellaneous improvements elsewhere.
1806 Kent Boortz made the Mac OS 9 port.
1808 Steve Root helped write the optimized alpha 21264 assembly code.
1810 Gerardo Ballabio wrote the 'gmpxx.h' C++ class interface and the C++
1811 'istream' input routines.
1813 Jason Moxham rewrote 'mpz_fac_ui'.
1815 Pedro Gimeno implemented the Mersenne Twister and made other random
1816 number improvements.
1818 Niels Möller wrote the sub-quadratic GCD, extended GCD and jacobi
1819 code, the quadratic Hensel division code, and (with Torbjörn) the new
1820 divide and conquer division code for GMP 4.3. Niels also helped
1821 implement the new Toom multiply code for GMP 4.3 and implemented helper
1822 functions to simplify Toom evaluations for GMP 5.0. He wrote the
1823 original version of mpn_mulmod_bnm1, and he is the main author of the
1824 mini-gmp package used for gmp bootstrapping.
1826 Alberto Zanoni and Marco Bodrato suggested the unbalanced multiply
1827 strategy, and found the optimal strategies for evaluation and
1828 interpolation in Toom multiplication.
1830 Marco Bodrato helped implement the new Toom multiply code for GMP 4.3
1831 and implemented most of the new Toom multiply and squaring code for 5.0.
1832 He is the main author of the current mpn_mulmod_bnm1, mpn_mullo_n, and
1833 mpn_sqrlo. Marco also wrote the functions mpn_invert and
1834 mpn_invertappr, and improved the speed of integer root extraction. He
1835 is the author of mini-mpq, an additional layer to mini-gmp; of most of
1836 the combinatorial functions and the BPSW primality testing
1837 implementation, for both the main library and the mini-gmp package.
1839 David Harvey suggested the internal function 'mpn_bdiv_dbm1',
1840 implementing division relevant to Toom multiplication. He also worked
1841 on fast assembly sequences, in particular on a fast AMD64
1842 'mpn_mul_basecase'. He wrote the internal middle product functions
1843 'mpn_mulmid_basecase', 'mpn_toom42_mulmid', 'mpn_mulmid_n' and related
1846 Martin Boij wrote 'mpn_perfect_power_p'.
1848 Marc Glisse improved 'gmpxx.h': use fewer temporaries (faster),
1849 specializations of 'numeric_limits' and 'common_type', C++11 features
1850 (move constructors, explicit bool conversion, UDL), make the conversion
1851 from 'mpq_class' to 'mpz_class' explicit, optimize operations where one
1852 argument is a small compile-time constant, replace some heap allocations
1853 by stack allocations. He also fixed the eofbit handling of C++ streams,
1854 and removed one division from 'mpq/aors.c'.
1856 David S Miller wrote assembly code for SPARC T3 and T4.
1858 Mark Sofroniou cleaned up the types of mul_fft.c, letting it work for
1861 Ulrich Weigand ported GMP to the powerpc64le ABI.
1863 (This list is chronological, not ordered after significance. If you
1864 have contributed to GMP but are not listed above, please tell
1865 <gmp-devel@gmplib.org> about the omission!)
1867 The development of floating point functions of GNU MP 2, were
1868 supported in part by the ESPRIT-BRA (Basic Research Activities) 6846
1869 project POSSO (POlynomial System SOlving).
1871 The development of GMP 2, 3, and 4.0 was supported in part by the IDA
1872 Center for Computing Sciences.
1874 The development of GMP 4.3, 5.0, and 5.1 was supported in part by the
1875 Swedish Foundation for Strategic Research.
1877 Thanks go to Hans Thorsen for donating an SGI system for the GMP test
1881 File: gmp.info, Node: References, Next: GNU Free Documentation License, Prev: Contributors, Up: Top
1883 Appendix B References
1884 *********************
1889 * Jonathan M. Borwein and Peter B. Borwein, "Pi and the AGM: A Study
1890 in Analytic Number Theory and Computational Complexity", Wiley,
1893 * Richard Crandall and Carl Pomerance, "Prime Numbers: A
1894 Computational Perspective", 2nd edition, Springer-Verlag, 2005.
1895 <https://www.math.dartmouth.edu/~carlp/>
1897 * Henri Cohen, "A Course in Computational Algebraic Number Theory",
1898 Graduate Texts in Mathematics number 138, Springer-Verlag, 1993.
1899 <https://www.math.u-bordeaux.fr/~cohen/>
1901 * Donald E. Knuth, "The Art of Computer Programming", volume 2,
1902 "Seminumerical Algorithms", 3rd edition, Addison-Wesley, 1998.
1903 <https://www-cs-faculty.stanford.edu/~knuth/taocp.html>
1905 * John D. Lipson, "Elements of Algebra and Algebraic Computing", The
1906 Benjamin Cummings Publishing Company Inc, 1981.
1908 * Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone,
1909 "Handbook of Applied Cryptography",
1910 <http://www.cacr.math.uwaterloo.ca/hac/>
1912 * Richard M. Stallman and the GCC Developer Community, "Using the GNU
1913 Compiler Collection", Free Software Foundation, 2008, available
1914 online <https://gcc.gnu.org/onlinedocs/>, and in the GCC package
1915 <https://ftp.gnu.org/gnu/gcc/>
1920 * Yves Bertot, Nicolas Magaud and Paul Zimmermann, "A Proof of GMP
1921 Square Root", Journal of Automated Reasoning, volume 29, 2002, pp.
1922 225-252. Also available online as INRIA Research Report 4475, June
1923 2002, <https://hal.inria.fr/docs/00/07/21/13/PDF/RR-4475.pdf>
1925 * Christoph Burnikel and Joachim Ziegler, "Fast Recursive Division",
1926 Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022,
1927 <https://www.mpi-inf.mpg.de/~ziegler/TechRep.ps.gz>
1929 * Torbjörn Granlund and Peter L. Montgomery, "Division by Invariant
1930 Integers using Multiplication", in Proceedings of the SIGPLAN
1931 PLDI'94 Conference, June 1994. Also available
1932 <https://gmplib.org/~tege/divcnst-pldi94.pdf>.
1934 * Niels Möller and Torbjörn Granlund, "Improved division by invariant
1935 integers", IEEE Transactions on Computers, 11 June 2010.
1936 <https://gmplib.org/~tege/division-paper.pdf>
1938 * Torbjörn Granlund and Niels Möller, "Division of integers large and
1941 * Tudor Jebelean, "An algorithm for exact division", Journal of
1942 Symbolic Computation, volume 15, 1993, pp. 169-180. Research
1943 report version available
1944 <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz>
1946 * Tudor Jebelean, "Exact Division with Karatsuba Complexity -
1947 Extended Abstract", RISC-Linz technical report 96-31,
1948 <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz>
1950 * Tudor Jebelean, "Practical Integer Division with Karatsuba
1951 Complexity", ISSAC 97, pp. 339-341. Technical report available
1952 <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz>
1954 * Tudor Jebelean, "A Generalization of the Binary GCD Algorithm",
1955 ISSAC 93, pp. 111-116. Technical report version available
1956 <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz>
1958 * Tudor Jebelean, "A Double-Digit Lehmer-Euclid Algorithm for Finding
1959 the GCD of Long Integers", Journal of Symbolic Computation, volume
1960 19, 1995, pp. 145-157. Technical report version also available
1961 <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz>
1963 * Werner Krandick and Tudor Jebelean, "Bidirectional Exact Integer
1964 Division", Journal of Symbolic Computation, volume 21, 1996, pp.
1965 441-455. Early technical report version also available
1966 <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz>
1968 * Makoto Matsumoto and Takuji Nishimura, "Mersenne Twister: A
1969 623-dimensionally equidistributed uniform pseudorandom number
1970 generator", ACM Transactions on Modelling and Computer Simulation,
1971 volume 8, January 1998, pp. 3-30. Available online
1972 <http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/ARTICLES/mt.pdf>
1974 * R. Moenck and A. Borodin, "Fast Modular Transforms via Division",
1975 Proceedings of the 13th Annual IEEE Symposium on Switching and
1976 Automata Theory, October 1972, pp. 90-96. Reprinted as "Fast
1977 Modular Transforms", Journal of Computer and System Sciences,
1978 volume 8, number 3, June 1974, pp. 366-386.
1980 * Niels Möller, "On Schönhage's algorithm and subquadratic integer
1981 GCD computation", in Mathematics of Computation, volume 77, January
1983 <https://www.ams.org/journals/mcom/2008-77-261/S0025-5718-07-02017-0/home.html>
1985 * Peter L. Montgomery, "Modular Multiplication Without Trial
1986 Division", in Mathematics of Computation, volume 44, number 170,
1989 * Arnold Schönhage and Volker Strassen, "Schnelle Multiplikation
1990 grosser Zahlen", Computing 7, 1971, pp. 281-292.
1992 * Kenneth Weber, "The accelerated integer GCD algorithm", ACM
1993 Transactions on Mathematical Software, volume 21, number 1, March
1996 * Paul Zimmermann, "Karatsuba Square Root", INRIA Research Report
1997 3805, November 1999,
1998 <https://hal.inria.fr/inria-00072854/PDF/RR-3805.pdf>
2000 * Paul Zimmermann, "A Proof of GMP Fast Division and Square Root
2002 <https://homepages.loria.fr/PZimmermann/papers/proof-div-sqrt.ps.gz>
2004 * Dan Zuras, "On Squaring and Multiplying Large Integers", ARITH-11:
2005 IEEE Symposium on Computer Arithmetic, 1993, pp. 260 to 271.
2006 Reprinted as "More on Multiplying and Squaring Large Integers",
2007 IEEE Transactions on Computers, volume 43, number 8, August 1994,
2010 * Niels Möller, "Efficient computation of the Jacobi symbol",
2011 <https://arxiv.org/abs/1907.07795>
2014 File: gmp.info, Node: GNU Free Documentation License, Next: Concept Index, Prev: References, Up: Top
2016 Appendix C GNU Free Documentation License
2017 *****************************************
2019 Version 1.3, 3 November 2008
2021 Copyright © 2000-2002, 2007, 2008 Free Software Foundation, Inc.
2024 Everyone is permitted to copy and distribute verbatim copies
2025 of this license document, but changing it is not allowed.
2029 The purpose of this License is to make a manual, textbook, or other
2030 functional and useful document "free" in the sense of freedom: to
2031 assure everyone the effective freedom to copy and redistribute it,
2032 with or without modifying it, either commercially or
2033 noncommercially. Secondarily, this License preserves for the
2034 author and publisher a way to get credit for their work, while not
2035 being considered responsible for modifications made by others.
2037 This License is a kind of "copyleft", which means that derivative
2038 works of the document must themselves be free in the same sense.
2039 It complements the GNU General Public License, which is a copyleft
2040 license designed for free software.
2042 We have designed this License in order to use it for manuals for
2043 free software, because free software needs free documentation: a
2044 free program should come with manuals providing the same freedoms
2045 that the software does. But this License is not limited to
2046 software manuals; it can be used for any textual work, regardless
2047 of subject matter or whether it is published as a printed book. We
2048 recommend this License principally for works whose purpose is
2049 instruction or reference.
2051 1. APPLICABILITY AND DEFINITIONS
2053 This License applies to any manual or other work, in any medium,
2054 that contains a notice placed by the copyright holder saying it can
2055 be distributed under the terms of this License. Such a notice
2056 grants a world-wide, royalty-free license, unlimited in duration,
2057 to use that work under the conditions stated herein. The
2058 "Document", below, refers to any such manual or work. Any member
2059 of the public is a licensee, and is addressed as "you". You accept
2060 the license if you copy, modify or distribute the work in a way
2061 requiring permission under copyright law.
2063 A "Modified Version" of the Document means any work containing the
2064 Document or a portion of it, either copied verbatim, or with
2065 modifications and/or translated into another language.
2067 A "Secondary Section" is a named appendix or a front-matter section
2068 of the Document that deals exclusively with the relationship of the
2069 publishers or authors of the Document to the Document's overall
2070 subject (or to related matters) and contains nothing that could
2071 fall directly within that overall subject. (Thus, if the Document
2072 is in part a textbook of mathematics, a Secondary Section may not
2073 explain any mathematics.) The relationship could be a matter of
2074 historical connection with the subject or with related matters, or
2075 of legal, commercial, philosophical, ethical or political position
2078 The "Invariant Sections" are certain Secondary Sections whose
2079 titles are designated, as being those of Invariant Sections, in the
2080 notice that says that the Document is released under this License.
2081 If a section does not fit the above definition of Secondary then it
2082 is not allowed to be designated as Invariant. The Document may
2083 contain zero Invariant Sections. If the Document does not identify
2084 any Invariant Sections then there are none.
2086 The "Cover Texts" are certain short passages of text that are
2087 listed, as Front-Cover Texts or Back-Cover Texts, in the notice
2088 that says that the Document is released under this License. A
2089 Front-Cover Text may be at most 5 words, and a Back-Cover Text may
2090 be at most 25 words.
2092 A "Transparent" copy of the Document means a machine-readable copy,
2093 represented in a format whose specification is available to the
2094 general public, that is suitable for revising the document
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2100 Transparent file format whose markup, or absence of markup, has
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2106 Examples of suitable formats for Transparent copies include plain
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2110 Examples of transparent image formats include PNG, XCF and JPG.
2111 Opaque formats include proprietary formats that can be read and
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2113 the DTD and/or processing tools are not generally available, and
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2115 processors for output purposes only.
2117 The "Title Page" means, for a printed book, the title page itself,
2118 plus such following pages as are needed to hold, legibly, the
2119 material this License requires to appear in the title page. For
2120 works in formats which do not have any title page as such, "Title
2121 Page" means the text near the most prominent appearance of the
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2124 The "publisher" means any person or entity that distributes copies
2125 of the Document to the public.
2127 A section "Entitled XYZ" means a named subunit of the Document
2128 whose title either is precisely XYZ or contains XYZ in parentheses
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2130 stands for a specific section name mentioned below, such as
2131 "Acknowledgements", "Dedications", "Endorsements", or "History".)
2132 To "Preserve the Title" of such a section when you modify the
2133 Document means that it remains a section "Entitled XYZ" according
2136 The Document may include Warranty Disclaimers next to the notice
2137 which states that this License applies to the Document. These
2138 Warranty Disclaimers are considered to be included by reference in
2139 this License, but only as regards disclaiming warranties: any other
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2145 You may copy and distribute the Document in any medium, either
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2152 you may accept compensation in exchange for copies. If you
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2156 You may also lend copies, under the same conditions stated above,
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2159 3. COPYING IN QUANTITY
2161 If you publish printed copies (or copies in media that commonly
2162 have printed covers) of the Document, numbering more than 100, and
2163 the Document's license notice requires Cover Texts, you must
2164 enclose the copies in covers that carry, clearly and legibly, all
2165 these Cover Texts: Front-Cover Texts on the front cover, and
2166 Back-Cover Texts on the back cover. Both covers must also clearly
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2168 front cover must present the full title with all words of the title
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2170 covers in addition. Copying with changes limited to the covers, as
2171 long as they preserve the title of the Document and satisfy these
2172 conditions, can be treated as verbatim copying in other respects.
2174 If the required texts for either cover are too voluminous to fit
2175 legibly, you should put the first ones listed (as many as fit
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2179 If you publish or distribute Opaque copies of the Document
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2181 Transparent copy along with each Opaque copy, or state in or with
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2192 It is requested, but not required, that you contact the authors of
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2194 to give them a chance to provide you with an updated version of the
2199 You may copy and distribute a Modified Version of the Document
2200 under the conditions of sections 2 and 3 above, provided that you
2201 release the Modified Version under precisely this License, with the
2202 Modified Version filling the role of the Document, thus licensing
2203 distribution and modification of the Modified Version to whoever
2204 possesses a copy of it. In addition, you must do these things in
2205 the Modified Version:
2207 A. Use in the Title Page (and on the covers, if any) a title
2208 distinct from that of the Document, and from those of previous
2209 versions (which should, if there were any, be listed in the
2210 History section of the Document). You may use the same title
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2214 B. List on the Title Page, as authors, one or more persons or
2215 entities responsible for authorship of the modifications in
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2221 C. State on the Title page the name of the publisher of the
2222 Modified Version, as the publisher.
2224 D. Preserve all the copyright notices of the Document.
2226 E. Add an appropriate copyright notice for your modifications
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2229 F. Include, immediately after the copyright notices, a license
2230 notice giving the public permission to use the Modified
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2234 G. Preserve in that license notice the full lists of Invariant
2235 Sections and required Cover Texts given in the Document's
2238 H. Include an unaltered copy of this License.
2240 I. Preserve the section Entitled "History", Preserve its Title,
2241 and add to it an item stating at least the title, year, new
2242 authors, and publisher of the Modified Version as given on the
2243 Title Page. If there is no section Entitled "History" in the
2244 Document, create one stating the title, year, authors, and
2245 publisher of the Document as given on its Title Page, then add
2246 an item describing the Modified Version as stated in the
2249 J. Preserve the network location, if any, given in the Document
2250 for public access to a Transparent copy of the Document, and
2251 likewise the network locations given in the Document for
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2253 "History" section. You may omit a network location for a work
2254 that was published at least four years before the Document
2255 itself, or if the original publisher of the version it refers
2256 to gives permission.
2258 K. For any section Entitled "Acknowledgements" or "Dedications",
2259 Preserve the Title of the section, and preserve in the section
2260 all the substance and tone of each of the contributor
2261 acknowledgements and/or dedications given therein.
2263 L. Preserve all the Invariant Sections of the Document, unaltered
2264 in their text and in their titles. Section numbers or the
2265 equivalent are not considered part of the section titles.
2267 M. Delete any section Entitled "Endorsements". Such a section
2268 may not be included in the Modified Version.
2270 N. Do not retitle any existing section to be Entitled
2271 "Endorsements" or to conflict in title with any Invariant
2274 O. Preserve any Warranty Disclaimers.
2276 If the Modified Version includes new front-matter sections or
2277 appendices that qualify as Secondary Sections and contain no
2278 material copied from the Document, you may at your option designate
2279 some or all of these sections as invariant. To do this, add their
2280 titles to the list of Invariant Sections in the Modified Version's
2281 license notice. These titles must be distinct from any other
2284 You may add a section Entitled "Endorsements", provided it contains
2285 nothing but endorsements of your Modified Version by various
2286 parties--for example, statements of peer review or that the text
2287 has been approved by an organization as the authoritative
2288 definition of a standard.
2290 You may add a passage of up to five words as a Front-Cover Text,
2291 and a passage of up to 25 words as a Back-Cover Text, to the end of
2292 the list of Cover Texts in the Modified Version. Only one passage
2293 of Front-Cover Text and one of Back-Cover Text may be added by (or
2294 through arrangements made by) any one entity. If the Document
2295 already includes a cover text for the same cover, previously added
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2298 one, on explicit permission from the previous publisher that added
2301 The author(s) and publisher(s) of the Document do not by this
2302 License give permission to use their names for publicity for or to
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2305 5. COMBINING DOCUMENTS
2307 You may combine the Document with other documents released under
2308 this License, under the terms defined in section 4 above for
2309 modified versions, provided that you include in the combination all
2310 of the Invariant Sections of all of the original documents,
2311 unmodified, and list them all as Invariant Sections of your
2312 combined work in its license notice, and that you preserve all
2313 their Warranty Disclaimers.
2315 The combined work need only contain one copy of this License, and
2316 multiple identical Invariant Sections may be replaced with a single
2317 copy. If there are multiple Invariant Sections with the same name
2318 but different contents, make the title of each such section unique
2319 by adding at the end of it, in parentheses, the name of the
2320 original author or publisher of that section if known, or else a
2321 unique number. Make the same adjustment to the section titles in
2322 the list of Invariant Sections in the license notice of the
2325 In the combination, you must combine any sections Entitled
2326 "History" in the various original documents, forming one section
2327 Entitled "History"; likewise combine any sections Entitled
2328 "Acknowledgements", and any sections Entitled "Dedications". You
2329 must delete all sections Entitled "Endorsements."
2331 6. COLLECTIONS OF DOCUMENTS
2333 You may make a collection consisting of the Document and other
2334 documents released under this License, and replace the individual
2335 copies of this License in the various documents with a single copy
2336 that is included in the collection, provided that you follow the
2337 rules of this License for verbatim copying of each of the documents
2338 in all other respects.
2340 You may extract a single document from such a collection, and
2341 distribute it individually under this License, provided you insert
2342 a copy of this License into the extracted document, and follow this
2343 License in all other respects regarding verbatim copying of that
2346 7. AGGREGATION WITH INDEPENDENT WORKS
2348 A compilation of the Document or its derivatives with other
2349 separate and independent documents or works, in or on a volume of a
2350 storage or distribution medium, is called an "aggregate" if the
2351 copyright resulting from the compilation is not used to limit the
2352 legal rights of the compilation's users beyond what the individual
2353 works permit. When the Document is included in an aggregate, this
2354 License does not apply to the other works in the aggregate which
2355 are not themselves derivative works of the Document.
2357 If the Cover Text requirement of section 3 is applicable to these
2358 copies of the Document, then if the Document is less than one half
2359 of the entire aggregate, the Document's Cover Texts may be placed
2360 on covers that bracket the Document within the aggregate, or the
2361 electronic equivalent of covers if the Document is in electronic
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2363 the whole aggregate.
2367 Translation is considered a kind of modification, so you may
2368 distribute translations of the Document under the terms of section
2369 4. Replacing Invariant Sections with translations requires special
2370 permission from their copyright holders, but you may include
2371 translations of some or all Invariant Sections in addition to the
2372 original versions of these Invariant Sections. You may include a
2373 translation of this License, and all the license notices in the
2374 Document, and any Warranty Disclaimers, provided that you also
2375 include the original English version of this License and the
2376 original versions of those notices and disclaimers. In case of a
2377 disagreement between the translation and the original version of
2378 this License or a notice or disclaimer, the original version will
2381 If a section in the Document is Entitled "Acknowledgements",
2382 "Dedications", or "History", the requirement (section 4) to
2383 Preserve its Title (section 1) will typically require changing the
2388 You may not copy, modify, sublicense, or distribute the Document
2389 except as expressly provided under this License. Any attempt
2390 otherwise to copy, modify, sublicense, or distribute it is void,
2391 and will automatically terminate your rights under this License.
2393 However, if you cease all violation of this License, then your
2394 license from a particular copyright holder is reinstated (a)
2395 provisionally, unless and until the copyright holder explicitly and
2396 finally terminates your license, and (b) permanently, if the
2397 copyright holder fails to notify you of the violation by some
2398 reasonable means prior to 60 days after the cessation.
2400 Moreover, your license from a particular copyright holder is
2401 reinstated permanently if the copyright holder notifies you of the
2402 violation by some reasonable means, this is the first time you have
2403 received notice of violation of this License (for any work) from
2404 that copyright holder, and you cure the violation prior to 30 days
2405 after your receipt of the notice.
2407 Termination of your rights under this section does not terminate
2408 the licenses of parties who have received copies or rights from you
2409 under this License. If your rights have been terminated and not
2410 permanently reinstated, receipt of a copy of some or all of the
2411 same material does not give you any rights to use it.
2413 10. FUTURE REVISIONS OF THIS LICENSE
2415 The Free Software Foundation may publish new, revised versions of
2416 the GNU Free Documentation License from time to time. Such new
2417 versions will be similar in spirit to the present version, but may
2418 differ in detail to address new problems or concerns. See
2419 <https://www.gnu.org/copyleft/>.
2421 Each version of the License is given a distinguishing version
2422 number. If the Document specifies that a particular numbered
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2424 have the option of following the terms and conditions either of
2425 that specified version or of any later version that has been
2426 published (not as a draft) by the Free Software Foundation. If the
2427 Document does not specify a version number of this License, you may
2428 choose any version ever published (not as a draft) by the Free
2429 Software Foundation. If the Document specifies that a proxy can
2430 decide which future versions of this License can be used, that
2431 proxy's public statement of acceptance of a version permanently
2432 authorizes you to choose that version for the Document.
2436 "Massive Multiauthor Collaboration Site" (or "MMC Site") means any
2437 World Wide Web server that publishes copyrightable works and also
2438 provides prominent facilities for anybody to edit those works. A
2439 public wiki that anybody can edit is an example of such a server.
2440 A "Massive Multiauthor Collaboration" (or "MMC") contained in the
2441 site means any set of copyrightable works thus published on the MMC
2444 "CC-BY-SA" means the Creative Commons Attribution-Share Alike 3.0
2445 license published by Creative Commons Corporation, a not-for-profit
2446 corporation with a principal place of business in San Francisco,
2447 California, as well as future copyleft versions of that license
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2450 "Incorporate" means to publish or republish a Document, in whole or
2451 in part, as part of another Document.
2453 An MMC is "eligible for relicensing" if it is licensed under this
2454 License, and if all works that were first published under this
2455 License somewhere other than this MMC, and subsequently
2456 incorporated in whole or in part into the MMC, (1) had no cover
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2458 to November 1, 2008.
2460 The operator of an MMC Site may republish an MMC contained in the
2461 site under CC-BY-SA on the same site at any time before August 1,
2462 2009, provided the MMC is eligible for relicensing.
2464 ADDENDUM: How to use this License for your documents
2465 ====================================================
2467 To use this License in a document you have written, include a copy of
2468 the License in the document and put the following copyright and license
2469 notices just after the title page:
2471 Copyright (C) YEAR YOUR NAME.
2472 Permission is granted to copy, distribute and/or modify this document
2473 under the terms of the GNU Free Documentation License, Version 1.3
2474 or any later version published by the Free Software Foundation;
2475 with no Invariant Sections, no Front-Cover Texts, and no Back-Cover
2476 Texts. A copy of the license is included in the section entitled ``GNU
2477 Free Documentation License''.
2479 If you have Invariant Sections, Front-Cover Texts and Back-Cover
2480 Texts, replace the "with...Texts." line with this:
2482 with the Invariant Sections being LIST THEIR TITLES, with
2483 the Front-Cover Texts being LIST, and with the Back-Cover Texts
2486 If you have Invariant Sections without Cover Texts, or some other
2487 combination of the three, merge those two alternatives to suit the
2490 If your document contains nontrivial examples of program code, we
2491 recommend releasing these examples in parallel under your choice of free
2492 software license, such as the GNU General Public License, to permit
2493 their use in free software.
2496 File: gmp.info, Node: Concept Index, Next: Function Index, Prev: GNU Free Documentation License, Up: Top
2504 * #include: Headers and Libraries.
2506 * --build: Build Options. (line 51)
2507 * --disable-fft: Build Options. (line 307)
2508 * --disable-shared: Build Options. (line 44)
2509 * --disable-static: Build Options. (line 44)
2510 * --enable-alloca: Build Options. (line 273)
2511 * --enable-assert: Build Options. (line 313)
2512 * --enable-cxx: Build Options. (line 225)
2513 * --enable-fat: Build Options. (line 160)
2514 * --enable-profiling: Build Options. (line 317)
2515 * --enable-profiling <1>: Profiling. (line 6)
2516 * --exec-prefix: Build Options. (line 32)
2517 * --host: Build Options. (line 65)
2518 * --prefix: Build Options. (line 32)
2519 * -finstrument-functions: Profiling. (line 66)
2520 * 2exp functions: Efficiency. (line 43)
2521 * 68000: Notes for Particular Systems.
2523 * 80x86: Notes for Particular Systems.
2525 * ABI: Build Options. (line 167)
2526 * ABI <1>: ABI and ISA. (line 6)
2527 * About this manual: Introduction to GMP. (line 57)
2528 * AC_CHECK_LIB: Autoconf. (line 11)
2529 * AIX: ABI and ISA. (line 174)
2530 * AIX <1>: Notes for Particular Systems.
2532 * Algorithms: Algorithms. (line 6)
2533 * alloca: Build Options. (line 273)
2534 * Allocation of memory: Custom Allocation. (line 6)
2535 * AMD64: ABI and ISA. (line 44)
2536 * Anonymous FTP of latest version: Introduction to GMP. (line 37)
2537 * Application Binary Interface: ABI and ISA. (line 6)
2538 * Arithmetic functions: Integer Arithmetic. (line 6)
2539 * Arithmetic functions <1>: Rational Arithmetic. (line 6)
2540 * Arithmetic functions <2>: Float Arithmetic. (line 6)
2541 * ARM: Notes for Particular Systems.
2543 * Assembly cache handling: Assembly Cache Handling.
2545 * Assembly carry propagation: Assembly Carry Propagation.
2547 * Assembly code organisation: Assembly Code Organisation.
2549 * Assembly coding: Assembly Coding. (line 6)
2550 * Assembly floating Point: Assembly Floating Point.
2552 * Assembly loop unrolling: Assembly Loop Unrolling.
2554 * Assembly SIMD: Assembly SIMD Instructions.
2556 * Assembly software pipelining: Assembly Software Pipelining.
2558 * Assembly writing guide: Assembly Writing Guide.
2560 * Assertion checking: Build Options. (line 313)
2561 * Assertion checking <1>: Debugging. (line 74)
2562 * Assignment functions: Assigning Integers. (line 6)
2563 * Assignment functions <1>: Simultaneous Integer Init & Assign.
2565 * Assignment functions <2>: Initializing Rationals.
2567 * Assignment functions <3>: Assigning Floats. (line 6)
2568 * Assignment functions <4>: Simultaneous Float Init & Assign.
2570 * Autoconf: Autoconf. (line 6)
2571 * Basics: GMP Basics. (line 6)
2572 * Binomial coefficient algorithm: Binomial Coefficients Algorithm.
2574 * Binomial coefficient functions: Number Theoretic Functions.
2576 * Binutils strip: Known Build Problems.
2578 * Bit manipulation functions: Integer Logic and Bit Fiddling.
2580 * Bit scanning functions: Integer Logic and Bit Fiddling.
2582 * Bit shift left: Integer Arithmetic. (line 38)
2583 * Bit shift right: Integer Division. (line 74)
2584 * Bits per limb: Useful Macros and Constants.
2586 * Bug reporting: Reporting Bugs. (line 6)
2587 * Build directory: Build Options. (line 19)
2588 * Build notes for binary packaging: Notes for Package Builds.
2590 * Build notes for particular systems: Notes for Particular Systems.
2592 * Build options: Build Options. (line 6)
2593 * Build problems known: Known Build Problems.
2595 * Build system: Build Options. (line 51)
2596 * Building GMP: Installing GMP. (line 6)
2597 * Bus error: Debugging. (line 7)
2598 * C compiler: Build Options. (line 178)
2599 * C++ compiler: Build Options. (line 249)
2600 * C++ interface: C++ Class Interface. (line 6)
2601 * C++ interface internals: C++ Interface Internals.
2603 * C++ istream input: C++ Formatted Input. (line 6)
2604 * C++ ostream output: C++ Formatted Output.
2606 * C++ support: Build Options. (line 225)
2607 * CC: Build Options. (line 178)
2608 * CC_FOR_BUILD: Build Options. (line 212)
2609 * CFLAGS: Build Options. (line 178)
2610 * Checker: Debugging. (line 110)
2611 * checkergcc: Debugging. (line 117)
2612 * Code organisation: Assembly Code Organisation.
2614 * Compaq C++: Notes for Particular Systems.
2616 * Comparison functions: Integer Comparisons. (line 6)
2617 * Comparison functions <1>: Comparing Rationals. (line 6)
2618 * Comparison functions <2>: Float Comparison. (line 6)
2619 * Compatibility with older versions: Compatibility with older versions.
2621 * Conditions for copying GNU MP: Copying. (line 6)
2622 * Configuring GMP: Installing GMP. (line 6)
2623 * Congruence algorithm: Exact Remainder. (line 30)
2624 * Congruence functions: Integer Division. (line 150)
2625 * Constants: Useful Macros and Constants.
2627 * Contributors: Contributors. (line 6)
2628 * Conventions for parameters: Parameter Conventions.
2630 * Conventions for variables: Variable Conventions.
2632 * Conversion functions: Converting Integers. (line 6)
2633 * Conversion functions <1>: Rational Conversions.
2635 * Conversion functions <2>: Converting Floats. (line 6)
2636 * Copying conditions: Copying. (line 6)
2637 * CPPFLAGS: Build Options. (line 204)
2638 * CPU types: Introduction to GMP. (line 24)
2639 * CPU types <1>: Build Options. (line 107)
2640 * Cross compiling: Build Options. (line 65)
2641 * Cryptography functions, low-level: Low-level Functions. (line 507)
2642 * Custom allocation: Custom Allocation. (line 6)
2643 * CXX: Build Options. (line 249)
2644 * CXXFLAGS: Build Options. (line 249)
2645 * Cygwin: Notes for Particular Systems.
2647 * Darwin: Known Build Problems.
2649 * Debugging: Debugging. (line 6)
2650 * Demonstration programs: Demonstration Programs.
2652 * Digits in an integer: Miscellaneous Integer Functions.
2654 * Divisibility algorithm: Exact Remainder. (line 30)
2655 * Divisibility functions: Integer Division. (line 136)
2656 * Divisibility functions <1>: Integer Division. (line 150)
2657 * Divisibility testing: Efficiency. (line 91)
2658 * Division algorithms: Division Algorithms. (line 6)
2659 * Division functions: Integer Division. (line 6)
2660 * Division functions <1>: Rational Arithmetic. (line 24)
2661 * Division functions <2>: Float Arithmetic. (line 33)
2662 * DJGPP: Notes for Particular Systems.
2664 * DJGPP <1>: Known Build Problems.
2666 * DLLs: Notes for Particular Systems.
2668 * DocBook: Build Options. (line 340)
2669 * Documentation formats: Build Options. (line 333)
2670 * Documentation license: GNU Free Documentation License.
2672 * DVI: Build Options. (line 336)
2673 * Efficiency: Efficiency. (line 6)
2674 * Emacs: Emacs. (line 6)
2675 * Exact division functions: Integer Division. (line 125)
2676 * Exact remainder: Exact Remainder. (line 6)
2677 * Example programs: Demonstration Programs.
2679 * Exec prefix: Build Options. (line 32)
2680 * Execution profiling: Build Options. (line 317)
2681 * Execution profiling <1>: Profiling. (line 6)
2682 * Exponentiation functions: Integer Exponentiation.
2684 * Exponentiation functions <1>: Float Arithmetic. (line 41)
2685 * Export: Integer Import and Export.
2687 * Expression parsing demo: Demonstration Programs.
2689 * Expression parsing demo <1>: Demonstration Programs.
2691 * Expression parsing demo <2>: Demonstration Programs.
2693 * Extended GCD: Number Theoretic Functions.
2695 * Factor removal functions: Number Theoretic Functions.
2697 * Factorial algorithm: Factorial Algorithm. (line 6)
2698 * Factorial functions: Number Theoretic Functions.
2700 * Factorization demo: Demonstration Programs.
2702 * Fast Fourier Transform: FFT Multiplication. (line 6)
2703 * Fat binary: Build Options. (line 160)
2704 * FFT multiplication: Build Options. (line 307)
2705 * FFT multiplication <1>: FFT Multiplication. (line 6)
2706 * Fibonacci number algorithm: Fibonacci Numbers Algorithm.
2708 * Fibonacci sequence functions: Number Theoretic Functions.
2710 * Float arithmetic functions: Float Arithmetic. (line 6)
2711 * Float assignment functions: Assigning Floats. (line 6)
2712 * Float assignment functions <1>: Simultaneous Float Init & Assign.
2714 * Float comparison functions: Float Comparison. (line 6)
2715 * Float conversion functions: Converting Floats. (line 6)
2716 * Float functions: Floating-point Functions.
2718 * Float initialization functions: Initializing Floats. (line 6)
2719 * Float initialization functions <1>: Simultaneous Float Init & Assign.
2721 * Float input and output functions: I/O of Floats. (line 6)
2722 * Float internals: Float Internals. (line 6)
2723 * Float miscellaneous functions: Miscellaneous Float Functions.
2725 * Float random number functions: Miscellaneous Float Functions.
2727 * Float rounding functions: Miscellaneous Float Functions.
2729 * Float sign tests: Float Comparison. (line 34)
2730 * Floating point mode: Notes for Particular Systems.
2732 * Floating-point functions: Floating-point Functions.
2734 * Floating-point number: Nomenclature and Types.
2736 * fnccheck: Profiling. (line 77)
2737 * Formatted input: Formatted Input. (line 6)
2738 * Formatted output: Formatted Output. (line 6)
2739 * Free Documentation License: GNU Free Documentation License.
2741 * FreeBSD: Notes for Particular Systems.
2743 * FreeBSD <1>: Notes for Particular Systems.
2745 * frexp: Converting Integers. (line 43)
2746 * frexp <1>: Converting Floats. (line 24)
2747 * FTP of latest version: Introduction to GMP. (line 37)
2748 * Function classes: Function Classes. (line 6)
2749 * FunctionCheck: Profiling. (line 77)
2750 * GCC Checker: Debugging. (line 110)
2751 * GCD algorithms: Greatest Common Divisor Algorithms.
2753 * GCD extended: Number Theoretic Functions.
2755 * GCD functions: Number Theoretic Functions.
2757 * GDB: Debugging. (line 53)
2758 * Generic C: Build Options. (line 151)
2759 * GMP Perl module: Demonstration Programs.
2761 * GMP version number: Useful Macros and Constants.
2763 * gmp.h: Headers and Libraries.
2765 * gmpxx.h: C++ Interface General.
2767 * GNU Debugger: Debugging. (line 53)
2768 * GNU Free Documentation License: GNU Free Documentation License.
2770 * GNU strip: Known Build Problems.
2772 * gprof: Profiling. (line 41)
2773 * Greatest common divisor algorithms: Greatest Common Divisor Algorithms.
2775 * Greatest common divisor functions: Number Theoretic Functions.
2777 * Hardware floating point mode: Notes for Particular Systems.
2779 * Headers: Headers and Libraries.
2781 * Heap problems: Debugging. (line 23)
2782 * Home page: Introduction to GMP. (line 33)
2783 * Host system: Build Options. (line 65)
2784 * HP-UX: ABI and ISA. (line 76)
2785 * HP-UX <1>: ABI and ISA. (line 114)
2786 * HPPA: ABI and ISA. (line 76)
2787 * I/O functions: I/O of Integers. (line 6)
2788 * I/O functions <1>: I/O of Rationals. (line 6)
2789 * I/O functions <2>: I/O of Floats. (line 6)
2790 * i386: Notes for Particular Systems.
2792 * IA-64: ABI and ISA. (line 114)
2793 * Import: Integer Import and Export.
2795 * In-place operations: Efficiency. (line 57)
2796 * Include files: Headers and Libraries.
2798 * info-lookup-symbol: Emacs. (line 6)
2799 * Initialization functions: Initializing Integers.
2801 * Initialization functions <1>: Simultaneous Integer Init & Assign.
2803 * Initialization functions <2>: Initializing Rationals.
2805 * Initialization functions <3>: Initializing Floats. (line 6)
2806 * Initialization functions <4>: Simultaneous Float Init & Assign.
2808 * Initialization functions <5>: Random State Initialization.
2810 * Initializing and clearing: Efficiency. (line 21)
2811 * Input functions: I/O of Integers. (line 6)
2812 * Input functions <1>: I/O of Rationals. (line 6)
2813 * Input functions <2>: I/O of Floats. (line 6)
2814 * Input functions <3>: Formatted Input Functions.
2816 * Install prefix: Build Options. (line 32)
2817 * Installing GMP: Installing GMP. (line 6)
2818 * Instruction Set Architecture: ABI and ISA. (line 6)
2819 * instrument-functions: Profiling. (line 66)
2820 * Integer: Nomenclature and Types.
2822 * Integer arithmetic functions: Integer Arithmetic. (line 6)
2823 * Integer assignment functions: Assigning Integers. (line 6)
2824 * Integer assignment functions <1>: Simultaneous Integer Init & Assign.
2826 * Integer bit manipulation functions: Integer Logic and Bit Fiddling.
2828 * Integer comparison functions: Integer Comparisons. (line 6)
2829 * Integer conversion functions: Converting Integers. (line 6)
2830 * Integer division functions: Integer Division. (line 6)
2831 * Integer exponentiation functions: Integer Exponentiation.
2833 * Integer export: Integer Import and Export.
2835 * Integer functions: Integer Functions. (line 6)
2836 * Integer import: Integer Import and Export.
2838 * Integer initialization functions: Initializing Integers.
2840 * Integer initialization functions <1>: Simultaneous Integer Init & Assign.
2842 * Integer input and output functions: I/O of Integers. (line 6)
2843 * Integer internals: Integer Internals. (line 6)
2844 * Integer logical functions: Integer Logic and Bit Fiddling.
2846 * Integer miscellaneous functions: Miscellaneous Integer Functions.
2848 * Integer random number functions: Integer Random Numbers.
2850 * Integer root functions: Integer Roots. (line 6)
2851 * Integer sign tests: Integer Comparisons. (line 28)
2852 * Integer special functions: Integer Special Functions.
2854 * Interix: Notes for Particular Systems.
2856 * Internals: Internals. (line 6)
2857 * Introduction: Introduction to GMP. (line 6)
2858 * Inverse modulo functions: Number Theoretic Functions.
2860 * IRIX: ABI and ISA. (line 139)
2861 * IRIX <1>: Known Build Problems.
2863 * ISA: ABI and ISA. (line 6)
2864 * istream input: C++ Formatted Input. (line 6)
2865 * Jacobi symbol algorithm: Jacobi Symbol. (line 6)
2866 * Jacobi symbol functions: Number Theoretic Functions.
2868 * Karatsuba multiplication: Karatsuba Multiplication.
2870 * Karatsuba square root algorithm: Square Root Algorithm.
2872 * Kronecker symbol functions: Number Theoretic Functions.
2874 * Language bindings: Language Bindings. (line 6)
2875 * Latest version of GMP: Introduction to GMP. (line 37)
2876 * LCM functions: Number Theoretic Functions.
2878 * Least common multiple functions: Number Theoretic Functions.
2880 * Legendre symbol functions: Number Theoretic Functions.
2882 * libgmp: Headers and Libraries.
2884 * libgmpxx: Headers and Libraries.
2886 * Libraries: Headers and Libraries.
2888 * Libtool: Headers and Libraries.
2890 * Libtool versioning: Notes for Package Builds.
2892 * License conditions: Copying. (line 6)
2893 * Limb: Nomenclature and Types.
2895 * Limb size: Useful Macros and Constants.
2897 * Linear congruential algorithm: Random Number Algorithms.
2899 * Linear congruential random numbers: Random State Initialization.
2901 * Linear congruential random numbers <1>: Random State Initialization.
2903 * Linking: Headers and Libraries.
2905 * Logical functions: Integer Logic and Bit Fiddling.
2907 * Low-level functions: Low-level Functions. (line 6)
2908 * Low-level functions for cryptography: Low-level Functions. (line 507)
2909 * Lucas number algorithm: Lucas Numbers Algorithm.
2911 * Lucas number functions: Number Theoretic Functions.
2913 * MacOS X: Known Build Problems.
2915 * Mailing lists: Introduction to GMP. (line 44)
2916 * Malloc debugger: Debugging. (line 29)
2917 * Malloc problems: Debugging. (line 23)
2918 * Memory allocation: Custom Allocation. (line 6)
2919 * Memory management: Memory Management. (line 6)
2920 * Mersenne twister algorithm: Random Number Algorithms.
2922 * Mersenne twister random numbers: Random State Initialization.
2924 * MINGW: Notes for Particular Systems.
2926 * MIPS: ABI and ISA. (line 139)
2927 * Miscellaneous float functions: Miscellaneous Float Functions.
2929 * Miscellaneous integer functions: Miscellaneous Integer Functions.
2931 * MMX: Notes for Particular Systems.
2933 * Modular inverse functions: Number Theoretic Functions.
2935 * Most significant bit: Miscellaneous Integer Functions.
2937 * MPN_PATH: Build Options. (line 321)
2938 * MS Windows: Notes for Particular Systems.
2940 * MS Windows <1>: Notes for Particular Systems.
2942 * MS-DOS: Notes for Particular Systems.
2944 * Multi-threading: Reentrancy. (line 6)
2945 * Multiplication algorithms: Multiplication Algorithms.
2947 * Nails: Low-level Functions. (line 686)
2948 * Native compilation: Build Options. (line 51)
2949 * NetBSD: Notes for Particular Systems.
2951 * NeXT: Known Build Problems.
2953 * Next prime function: Number Theoretic Functions.
2955 * Nomenclature: Nomenclature and Types.
2957 * Non-Unix systems: Build Options. (line 11)
2958 * Nth root algorithm: Nth Root Algorithm. (line 6)
2959 * Number sequences: Efficiency. (line 145)
2960 * Number theoretic functions: Number Theoretic Functions.
2962 * Numerator and denominator: Applying Integer Functions.
2964 * obstack output: Formatted Output Functions.
2966 * OpenBSD: Notes for Particular Systems.
2968 * Optimizing performance: Performance optimization.
2970 * ostream output: C++ Formatted Output.
2972 * Other languages: Language Bindings. (line 6)
2973 * Output functions: I/O of Integers. (line 6)
2974 * Output functions <1>: I/O of Rationals. (line 6)
2975 * Output functions <2>: I/O of Floats. (line 6)
2976 * Output functions <3>: Formatted Output Functions.
2978 * Packaged builds: Notes for Package Builds.
2980 * Parameter conventions: Parameter Conventions.
2982 * Parsing expressions demo: Demonstration Programs.
2984 * Parsing expressions demo <1>: Demonstration Programs.
2986 * Parsing expressions demo <2>: Demonstration Programs.
2988 * Particular systems: Notes for Particular Systems.
2990 * Past GMP versions: Compatibility with older versions.
2992 * PDF: Build Options. (line 336)
2993 * Perfect power algorithm: Perfect Power Algorithm.
2995 * Perfect power functions: Integer Roots. (line 28)
2996 * Perfect square algorithm: Perfect Square Algorithm.
2998 * Perfect square functions: Integer Roots. (line 37)
2999 * perl: Demonstration Programs.
3001 * Perl module: Demonstration Programs.
3003 * Postscript: Build Options. (line 336)
3004 * Power/PowerPC: Notes for Particular Systems.
3006 * Power/PowerPC <1>: Known Build Problems.
3008 * Powering algorithms: Powering Algorithms. (line 6)
3009 * Powering functions: Integer Exponentiation.
3011 * Powering functions <1>: Float Arithmetic. (line 41)
3012 * PowerPC: ABI and ISA. (line 173)
3013 * Precision of floats: Floating-point Functions.
3015 * Precision of hardware floating point: Notes for Particular Systems.
3017 * Prefix: Build Options. (line 32)
3018 * Prime testing algorithms: Prime Testing Algorithm.
3020 * Prime testing functions: Number Theoretic Functions.
3022 * Primorial functions: Number Theoretic Functions.
3024 * printf formatted output: Formatted Output. (line 6)
3025 * Probable prime testing functions: Number Theoretic Functions.
3027 * prof: Profiling. (line 24)
3028 * Profiling: Profiling. (line 6)
3029 * Radix conversion algorithms: Radix Conversion Algorithms.
3031 * Random number algorithms: Random Number Algorithms.
3033 * Random number functions: Integer Random Numbers.
3035 * Random number functions <1>: Miscellaneous Float Functions.
3037 * Random number functions <2>: Random Number Functions.
3039 * Random number seeding: Random State Seeding.
3041 * Random number state: Random State Initialization.
3043 * Random state: Nomenclature and Types.
3045 * Rational arithmetic: Efficiency. (line 111)
3046 * Rational arithmetic functions: Rational Arithmetic. (line 6)
3047 * Rational assignment functions: Initializing Rationals.
3049 * Rational comparison functions: Comparing Rationals. (line 6)
3050 * Rational conversion functions: Rational Conversions.
3052 * Rational initialization functions: Initializing Rationals.
3054 * Rational input and output functions: I/O of Rationals. (line 6)
3055 * Rational internals: Rational Internals. (line 6)
3056 * Rational number: Nomenclature and Types.
3058 * Rational number functions: Rational Number Functions.
3060 * Rational numerator and denominator: Applying Integer Functions.
3062 * Rational sign tests: Comparing Rationals. (line 28)
3063 * Raw output internals: Raw Output Internals.
3065 * Reallocations: Efficiency. (line 30)
3066 * Reentrancy: Reentrancy. (line 6)
3067 * References: References. (line 5)
3068 * Remove factor functions: Number Theoretic Functions.
3070 * Reporting bugs: Reporting Bugs. (line 6)
3071 * Root extraction algorithm: Nth Root Algorithm. (line 6)
3072 * Root extraction algorithms: Root Extraction Algorithms.
3074 * Root extraction functions: Integer Roots. (line 6)
3075 * Root extraction functions <1>: Float Arithmetic. (line 37)
3076 * Root testing functions: Integer Roots. (line 28)
3077 * Root testing functions <1>: Integer Roots. (line 37)
3078 * Rounding functions: Miscellaneous Float Functions.
3080 * Sample programs: Demonstration Programs.
3082 * Scan bit functions: Integer Logic and Bit Fiddling.
3084 * scanf formatted input: Formatted Input. (line 6)
3085 * SCO: Known Build Problems.
3087 * Seeding random numbers: Random State Seeding.
3089 * Segmentation violation: Debugging. (line 7)
3090 * Sequent Symmetry: Known Build Problems.
3092 * Services for Unix: Notes for Particular Systems.
3094 * Shared library versioning: Notes for Package Builds.
3096 * Sign tests: Integer Comparisons. (line 28)
3097 * Sign tests <1>: Comparing Rationals. (line 28)
3098 * Sign tests <2>: Float Comparison. (line 34)
3099 * Size in digits: Miscellaneous Integer Functions.
3101 * Small operands: Efficiency. (line 7)
3102 * Solaris: ABI and ISA. (line 204)
3103 * Solaris <1>: Known Build Problems.
3105 * Solaris <2>: Known Build Problems.
3107 * Sparc: Notes for Particular Systems.
3109 * Sparc <1>: Notes for Particular Systems.
3111 * Sparc V9: ABI and ISA. (line 204)
3112 * Special integer functions: Integer Special Functions.
3114 * Square root algorithm: Square Root Algorithm.
3116 * SSE2: Notes for Particular Systems.
3118 * Stack backtrace: Debugging. (line 45)
3119 * Stack overflow: Build Options. (line 273)
3120 * Stack overflow <1>: Debugging. (line 7)
3121 * Static linking: Efficiency. (line 14)
3122 * stdarg.h: Headers and Libraries.
3124 * stdio.h: Headers and Libraries.
3126 * Stripped libraries: Known Build Problems.
3128 * Sun: ABI and ISA. (line 204)
3129 * SunOS: Notes for Particular Systems.
3131 * Systems: Notes for Particular Systems.
3133 * Temporary memory: Build Options. (line 273)
3134 * Texinfo: Build Options. (line 333)
3135 * Text input/output: Efficiency. (line 151)
3136 * Thread safety: Reentrancy. (line 6)
3137 * Toom multiplication: Toom 3-Way Multiplication.
3139 * Toom multiplication <1>: Toom 4-Way Multiplication.
3141 * Toom multiplication <2>: Higher degree Toom'n'half.
3143 * Toom multiplication <3>: Other Multiplication.
3145 * Types: Nomenclature and Types.
3147 * ui and si functions: Efficiency. (line 50)
3148 * Unbalanced multiplication: Unbalanced Multiplication.
3150 * Upward compatibility: Compatibility with older versions.
3152 * Useful macros and constants: Useful Macros and Constants.
3154 * User-defined precision: Floating-point Functions.
3156 * Valgrind: Debugging. (line 125)
3157 * Variable conventions: Variable Conventions.
3159 * Version number: Useful Macros and Constants.
3161 * Web page: Introduction to GMP. (line 33)
3162 * Windows: Notes for Particular Systems.
3164 * Windows <1>: Notes for Particular Systems.
3166 * x86: Notes for Particular Systems.
3168 * x87: Notes for Particular Systems.
3170 * XML: Build Options. (line 340)
3173 File: gmp.info, Node: Function Index, Prev: Concept Index, Up: Top
3175 Function and Type Index
3176 ***********************
3181 * _mpz_realloc: Integer Special Functions.
3183 * __GMP_CC: Useful Macros and Constants.
3185 * __GMP_CFLAGS: Useful Macros and Constants.
3187 * __GNU_MP_VERSION: Useful Macros and Constants.
3189 * __GNU_MP_VERSION_MINOR: Useful Macros and Constants.
3191 * __GNU_MP_VERSION_PATCHLEVEL: Useful Macros and Constants.
3193 * abs: C++ Interface Integers.
3195 * abs <1>: C++ Interface Rationals.
3197 * abs <2>: C++ Interface Floats.
3199 * ceil: C++ Interface Floats.
3201 * cmp: C++ Interface Integers.
3203 * cmp <1>: C++ Interface Integers.
3205 * cmp <2>: C++ Interface Rationals.
3207 * cmp <3>: C++ Interface Rationals.
3209 * cmp <4>: C++ Interface Floats.
3211 * cmp <5>: C++ Interface Floats.
3213 * factorial: C++ Interface Integers.
3215 * fibonacci: C++ Interface Integers.
3217 * floor: C++ Interface Floats.
3219 * gcd: C++ Interface Integers.
3221 * gmp_asprintf: Formatted Output Functions.
3223 * gmp_errno: Random State Initialization.
3225 * GMP_ERROR_INVALID_ARGUMENT: Random State Initialization.
3227 * GMP_ERROR_UNSUPPORTED_ARGUMENT: Random State Initialization.
3229 * gmp_fprintf: Formatted Output Functions.
3231 * gmp_fscanf: Formatted Input Functions.
3233 * GMP_LIMB_BITS: Low-level Functions. (line 714)
3234 * GMP_NAIL_BITS: Low-level Functions. (line 712)
3235 * GMP_NAIL_MASK: Low-level Functions. (line 722)
3236 * GMP_NUMB_BITS: Low-level Functions. (line 713)
3237 * GMP_NUMB_MASK: Low-level Functions. (line 723)
3238 * GMP_NUMB_MAX: Low-level Functions. (line 731)
3239 * gmp_obstack_printf: Formatted Output Functions.
3241 * gmp_obstack_vprintf: Formatted Output Functions.
3243 * gmp_printf: Formatted Output Functions.
3245 * gmp_randclass: C++ Interface Random Numbers.
3247 * gmp_randclass::get_f: C++ Interface Random Numbers.
3249 * gmp_randclass::get_f <1>: C++ Interface Random Numbers.
3251 * gmp_randclass::get_z_bits: C++ Interface Random Numbers.
3253 * gmp_randclass::get_z_bits <1>: C++ Interface Random Numbers.
3255 * gmp_randclass::get_z_range: C++ Interface Random Numbers.
3257 * gmp_randclass::gmp_randclass: C++ Interface Random Numbers.
3259 * gmp_randclass::gmp_randclass <1>: C++ Interface Random Numbers.
3261 * gmp_randclass::seed: C++ Interface Random Numbers.
3263 * gmp_randclass::seed <1>: C++ Interface Random Numbers.
3265 * gmp_randclear: Random State Initialization.
3267 * gmp_randinit: Random State Initialization.
3269 * gmp_randinit_default: Random State Initialization.
3271 * gmp_randinit_lc_2exp: Random State Initialization.
3273 * gmp_randinit_lc_2exp_size: Random State Initialization.
3275 * gmp_randinit_mt: Random State Initialization.
3277 * gmp_randinit_set: Random State Initialization.
3279 * gmp_randseed: Random State Seeding.
3281 * gmp_randseed_ui: Random State Seeding.
3283 * gmp_randstate_t: Nomenclature and Types.
3285 * GMP_RAND_ALG_DEFAULT: Random State Initialization.
3287 * GMP_RAND_ALG_LC: Random State Initialization.
3289 * gmp_scanf: Formatted Input Functions.
3291 * gmp_snprintf: Formatted Output Functions.
3293 * gmp_sprintf: Formatted Output Functions.
3295 * gmp_sscanf: Formatted Input Functions.
3297 * gmp_urandomb_ui: Random State Miscellaneous.
3299 * gmp_urandomm_ui: Random State Miscellaneous.
3301 * gmp_vasprintf: Formatted Output Functions.
3303 * gmp_version: Useful Macros and Constants.
3305 * gmp_vfprintf: Formatted Output Functions.
3307 * gmp_vfscanf: Formatted Input Functions.
3309 * gmp_vprintf: Formatted Output Functions.
3311 * gmp_vscanf: Formatted Input Functions.
3313 * gmp_vsnprintf: Formatted Output Functions.
3315 * gmp_vsprintf: Formatted Output Functions.
3317 * gmp_vsscanf: Formatted Input Functions.
3319 * hypot: C++ Interface Floats.
3321 * lcm: C++ Interface Integers.
3323 * mpf_abs: Float Arithmetic. (line 46)
3324 * mpf_add: Float Arithmetic. (line 6)
3325 * mpf_add_ui: Float Arithmetic. (line 7)
3326 * mpf_ceil: Miscellaneous Float Functions.
3328 * mpf_class: C++ Interface General.
3330 * mpf_class::fits_sint_p: C++ Interface Floats.
3332 * mpf_class::fits_slong_p: C++ Interface Floats.
3334 * mpf_class::fits_sshort_p: C++ Interface Floats.
3336 * mpf_class::fits_uint_p: C++ Interface Floats.
3338 * mpf_class::fits_ulong_p: C++ Interface Floats.
3340 * mpf_class::fits_ushort_p: C++ Interface Floats.
3342 * mpf_class::get_d: C++ Interface Floats.
3344 * mpf_class::get_mpf_t: C++ Interface General.
3346 * mpf_class::get_prec: C++ Interface Floats.
3348 * mpf_class::get_si: C++ Interface Floats.
3350 * mpf_class::get_str: C++ Interface Floats.
3352 * mpf_class::get_ui: C++ Interface Floats.
3354 * mpf_class::mpf_class: C++ Interface Floats.
3356 * mpf_class::mpf_class <1>: C++ Interface Floats.
3358 * mpf_class::mpf_class <2>: C++ Interface Floats.
3360 * mpf_class::mpf_class <3>: C++ Interface Floats.
3362 * mpf_class::mpf_class <4>: C++ Interface Floats.
3364 * mpf_class::mpf_class <5>: C++ Interface Floats.
3366 * mpf_class::mpf_class <6>: C++ Interface Floats.
3368 * mpf_class::mpf_class <7>: C++ Interface Floats.
3370 * mpf_class::operator=: C++ Interface Floats.
3372 * mpf_class::set_prec: C++ Interface Floats.
3374 * mpf_class::set_prec_raw: C++ Interface Floats.
3376 * mpf_class::set_str: C++ Interface Floats.
3378 * mpf_class::set_str <1>: C++ Interface Floats.
3380 * mpf_class::swap: C++ Interface Floats.
3382 * mpf_clear: Initializing Floats. (line 36)
3383 * mpf_clears: Initializing Floats. (line 40)
3384 * mpf_cmp: Float Comparison. (line 6)
3385 * mpf_cmp_d: Float Comparison. (line 8)
3386 * mpf_cmp_si: Float Comparison. (line 10)
3387 * mpf_cmp_ui: Float Comparison. (line 9)
3388 * mpf_cmp_z: Float Comparison. (line 7)
3389 * mpf_div: Float Arithmetic. (line 28)
3390 * mpf_div_2exp: Float Arithmetic. (line 53)
3391 * mpf_div_ui: Float Arithmetic. (line 31)
3392 * mpf_eq: Float Comparison. (line 17)
3393 * mpf_fits_sint_p: Miscellaneous Float Functions.
3395 * mpf_fits_slong_p: Miscellaneous Float Functions.
3397 * mpf_fits_sshort_p: Miscellaneous Float Functions.
3399 * mpf_fits_uint_p: Miscellaneous Float Functions.
3401 * mpf_fits_ulong_p: Miscellaneous Float Functions.
3403 * mpf_fits_ushort_p: Miscellaneous Float Functions.
3405 * mpf_floor: Miscellaneous Float Functions.
3407 * mpf_get_d: Converting Floats. (line 6)
3408 * mpf_get_default_prec: Initializing Floats. (line 11)
3409 * mpf_get_d_2exp: Converting Floats. (line 15)
3410 * mpf_get_prec: Initializing Floats. (line 61)
3411 * mpf_get_si: Converting Floats. (line 27)
3412 * mpf_get_str: Converting Floats. (line 36)
3413 * mpf_get_ui: Converting Floats. (line 28)
3414 * mpf_init: Initializing Floats. (line 18)
3415 * mpf_init2: Initializing Floats. (line 25)
3416 * mpf_inits: Initializing Floats. (line 30)
3417 * mpf_init_set: Simultaneous Float Init & Assign.
3419 * mpf_init_set_d: Simultaneous Float Init & Assign.
3421 * mpf_init_set_si: Simultaneous Float Init & Assign.
3423 * mpf_init_set_str: Simultaneous Float Init & Assign.
3425 * mpf_init_set_ui: Simultaneous Float Init & Assign.
3427 * mpf_inp_str: I/O of Floats. (line 38)
3428 * mpf_integer_p: Miscellaneous Float Functions.
3430 * mpf_mul: Float Arithmetic. (line 18)
3431 * mpf_mul_2exp: Float Arithmetic. (line 49)
3432 * mpf_mul_ui: Float Arithmetic. (line 19)
3433 * mpf_neg: Float Arithmetic. (line 43)
3434 * mpf_out_str: I/O of Floats. (line 17)
3435 * mpf_pow_ui: Float Arithmetic. (line 39)
3436 * mpf_random2: Miscellaneous Float Functions.
3438 * mpf_reldiff: Float Comparison. (line 28)
3439 * mpf_set: Assigning Floats. (line 9)
3440 * mpf_set_d: Assigning Floats. (line 12)
3441 * mpf_set_default_prec: Initializing Floats. (line 6)
3442 * mpf_set_prec: Initializing Floats. (line 64)
3443 * mpf_set_prec_raw: Initializing Floats. (line 71)
3444 * mpf_set_q: Assigning Floats. (line 14)
3445 * mpf_set_si: Assigning Floats. (line 11)
3446 * mpf_set_str: Assigning Floats. (line 17)
3447 * mpf_set_ui: Assigning Floats. (line 10)
3448 * mpf_set_z: Assigning Floats. (line 13)
3449 * mpf_sgn: Float Comparison. (line 33)
3450 * mpf_sqrt: Float Arithmetic. (line 35)
3451 * mpf_sqrt_ui: Float Arithmetic. (line 36)
3452 * mpf_sub: Float Arithmetic. (line 11)
3453 * mpf_sub_ui: Float Arithmetic. (line 14)
3454 * mpf_swap: Assigning Floats. (line 50)
3455 * mpf_t: Nomenclature and Types.
3457 * mpf_trunc: Miscellaneous Float Functions.
3459 * mpf_ui_div: Float Arithmetic. (line 29)
3460 * mpf_ui_sub: Float Arithmetic. (line 12)
3461 * mpf_urandomb: Miscellaneous Float Functions.
3463 * mpn_add: Low-level Functions. (line 67)
3464 * mpn_addmul_1: Low-level Functions. (line 148)
3465 * mpn_add_1: Low-level Functions. (line 62)
3466 * mpn_add_n: Low-level Functions. (line 52)
3467 * mpn_andn_n: Low-level Functions. (line 462)
3468 * mpn_and_n: Low-level Functions. (line 447)
3469 * mpn_cmp: Low-level Functions. (line 293)
3470 * mpn_cnd_add_n: Low-level Functions. (line 540)
3471 * mpn_cnd_sub_n: Low-level Functions. (line 542)
3472 * mpn_cnd_swap: Low-level Functions. (line 567)
3473 * mpn_com: Low-level Functions. (line 487)
3474 * mpn_copyd: Low-level Functions. (line 496)
3475 * mpn_copyi: Low-level Functions. (line 492)
3476 * mpn_divexact_1: Low-level Functions. (line 231)
3477 * mpn_divexact_by3: Low-level Functions. (line 238)
3478 * mpn_divexact_by3c: Low-level Functions. (line 240)
3479 * mpn_divmod: Low-level Functions. (line 226)
3480 * mpn_divmod_1: Low-level Functions. (line 210)
3481 * mpn_divrem: Low-level Functions. (line 183)
3482 * mpn_divrem_1: Low-level Functions. (line 208)
3483 * mpn_gcd: Low-level Functions. (line 301)
3484 * mpn_gcdext: Low-level Functions. (line 316)
3485 * mpn_gcd_1: Low-level Functions. (line 311)
3486 * mpn_get_str: Low-level Functions. (line 371)
3487 * mpn_hamdist: Low-level Functions. (line 436)
3488 * mpn_iorn_n: Low-level Functions. (line 467)
3489 * mpn_ior_n: Low-level Functions. (line 452)
3490 * mpn_lshift: Low-level Functions. (line 269)
3491 * mpn_mod_1: Low-level Functions. (line 264)
3492 * mpn_mul: Low-level Functions. (line 114)
3493 * mpn_mul_1: Low-level Functions. (line 133)
3494 * mpn_mul_n: Low-level Functions. (line 103)
3495 * mpn_nand_n: Low-level Functions. (line 472)
3496 * mpn_neg: Low-level Functions. (line 96)
3497 * mpn_nior_n: Low-level Functions. (line 477)
3498 * mpn_perfect_square_p: Low-level Functions. (line 442)
3499 * mpn_popcount: Low-level Functions. (line 432)
3500 * mpn_random: Low-level Functions. (line 422)
3501 * mpn_random2: Low-level Functions. (line 423)
3502 * mpn_rshift: Low-level Functions. (line 281)
3503 * mpn_scan0: Low-level Functions. (line 406)
3504 * mpn_scan1: Low-level Functions. (line 414)
3505 * mpn_sec_add_1: Low-level Functions. (line 553)
3506 * mpn_sec_div_qr: Low-level Functions. (line 630)
3507 * mpn_sec_div_qr_itch: Low-level Functions. (line 633)
3508 * mpn_sec_div_r: Low-level Functions. (line 649)
3509 * mpn_sec_div_r_itch: Low-level Functions. (line 651)
3510 * mpn_sec_invert: Low-level Functions. (line 665)
3511 * mpn_sec_invert_itch: Low-level Functions. (line 667)
3512 * mpn_sec_mul: Low-level Functions. (line 574)
3513 * mpn_sec_mul_itch: Low-level Functions. (line 577)
3514 * mpn_sec_powm: Low-level Functions. (line 604)
3515 * mpn_sec_powm_itch: Low-level Functions. (line 607)
3516 * mpn_sec_sqr: Low-level Functions. (line 590)
3517 * mpn_sec_sqr_itch: Low-level Functions. (line 592)
3518 * mpn_sec_sub_1: Low-level Functions. (line 555)
3519 * mpn_sec_tabselect: Low-level Functions. (line 622)
3520 * mpn_set_str: Low-level Functions. (line 386)
3521 * mpn_sizeinbase: Low-level Functions. (line 364)
3522 * mpn_sqr: Low-level Functions. (line 125)
3523 * mpn_sqrtrem: Low-level Functions. (line 346)
3524 * mpn_sub: Low-level Functions. (line 88)
3525 * mpn_submul_1: Low-level Functions. (line 160)
3526 * mpn_sub_1: Low-level Functions. (line 83)
3527 * mpn_sub_n: Low-level Functions. (line 74)
3528 * mpn_tdiv_qr: Low-level Functions. (line 172)
3529 * mpn_xnor_n: Low-level Functions. (line 482)
3530 * mpn_xor_n: Low-level Functions. (line 457)
3531 * mpn_zero: Low-level Functions. (line 500)
3532 * mpn_zero_p: Low-level Functions. (line 298)
3533 * mpq_abs: Rational Arithmetic. (line 33)
3534 * mpq_add: Rational Arithmetic. (line 6)
3535 * mpq_canonicalize: Rational Number Functions.
3537 * mpq_class: C++ Interface General.
3539 * mpq_class::canonicalize: C++ Interface Rationals.
3541 * mpq_class::get_d: C++ Interface Rationals.
3543 * mpq_class::get_den: C++ Interface Rationals.
3545 * mpq_class::get_den_mpz_t: C++ Interface Rationals.
3547 * mpq_class::get_mpq_t: C++ Interface General.
3549 * mpq_class::get_num: C++ Interface Rationals.
3551 * mpq_class::get_num_mpz_t: C++ Interface Rationals.
3553 * mpq_class::get_str: C++ Interface Rationals.
3555 * mpq_class::mpq_class: C++ Interface Rationals.
3557 * mpq_class::mpq_class <1>: C++ Interface Rationals.
3559 * mpq_class::mpq_class <2>: C++ Interface Rationals.
3561 * mpq_class::mpq_class <3>: C++ Interface Rationals.
3563 * mpq_class::mpq_class <4>: C++ Interface Rationals.
3565 * mpq_class::set_str: C++ Interface Rationals.
3567 * mpq_class::set_str <1>: C++ Interface Rationals.
3569 * mpq_class::swap: C++ Interface Rationals.
3571 * mpq_clear: Initializing Rationals.
3573 * mpq_clears: Initializing Rationals.
3575 * mpq_cmp: Comparing Rationals. (line 6)
3576 * mpq_cmp_si: Comparing Rationals. (line 16)
3577 * mpq_cmp_ui: Comparing Rationals. (line 14)
3578 * mpq_cmp_z: Comparing Rationals. (line 7)
3579 * mpq_denref: Applying Integer Functions.
3581 * mpq_div: Rational Arithmetic. (line 22)
3582 * mpq_div_2exp: Rational Arithmetic. (line 26)
3583 * mpq_equal: Comparing Rationals. (line 33)
3584 * mpq_get_d: Rational Conversions.
3586 * mpq_get_den: Applying Integer Functions.
3588 * mpq_get_num: Applying Integer Functions.
3590 * mpq_get_str: Rational Conversions.
3592 * mpq_init: Initializing Rationals.
3594 * mpq_inits: Initializing Rationals.
3596 * mpq_inp_str: I/O of Rationals. (line 32)
3597 * mpq_inv: Rational Arithmetic. (line 36)
3598 * mpq_mul: Rational Arithmetic. (line 14)
3599 * mpq_mul_2exp: Rational Arithmetic. (line 18)
3600 * mpq_neg: Rational Arithmetic. (line 30)
3601 * mpq_numref: Applying Integer Functions.
3603 * mpq_out_str: I/O of Rationals. (line 17)
3604 * mpq_set: Initializing Rationals.
3606 * mpq_set_d: Rational Conversions.
3608 * mpq_set_den: Applying Integer Functions.
3610 * mpq_set_f: Rational Conversions.
3612 * mpq_set_num: Applying Integer Functions.
3614 * mpq_set_si: Initializing Rationals.
3616 * mpq_set_str: Initializing Rationals.
3618 * mpq_set_ui: Initializing Rationals.
3620 * mpq_set_z: Initializing Rationals.
3622 * mpq_sgn: Comparing Rationals. (line 27)
3623 * mpq_sub: Rational Arithmetic. (line 10)
3624 * mpq_swap: Initializing Rationals.
3626 * mpq_t: Nomenclature and Types.
3628 * mpz_2fac_ui: Number Theoretic Functions.
3630 * mpz_abs: Integer Arithmetic. (line 44)
3631 * mpz_add: Integer Arithmetic. (line 6)
3632 * mpz_addmul: Integer Arithmetic. (line 24)
3633 * mpz_addmul_ui: Integer Arithmetic. (line 26)
3634 * mpz_add_ui: Integer Arithmetic. (line 7)
3635 * mpz_and: Integer Logic and Bit Fiddling.
3637 * mpz_array_init: Integer Special Functions.
3639 * mpz_bin_ui: Number Theoretic Functions.
3641 * mpz_bin_uiui: Number Theoretic Functions.
3643 * mpz_cdiv_q: Integer Division. (line 12)
3644 * mpz_cdiv_qr: Integer Division. (line 14)
3645 * mpz_cdiv_qr_ui: Integer Division. (line 21)
3646 * mpz_cdiv_q_2exp: Integer Division. (line 26)
3647 * mpz_cdiv_q_ui: Integer Division. (line 17)
3648 * mpz_cdiv_r: Integer Division. (line 13)
3649 * mpz_cdiv_r_2exp: Integer Division. (line 29)
3650 * mpz_cdiv_r_ui: Integer Division. (line 19)
3651 * mpz_cdiv_ui: Integer Division. (line 23)
3652 * mpz_class: C++ Interface General.
3654 * mpz_class::factorial: C++ Interface Integers.
3656 * mpz_class::fibonacci: C++ Interface Integers.
3658 * mpz_class::fits_sint_p: C++ Interface Integers.
3660 * mpz_class::fits_slong_p: C++ Interface Integers.
3662 * mpz_class::fits_sshort_p: C++ Interface Integers.
3664 * mpz_class::fits_uint_p: C++ Interface Integers.
3666 * mpz_class::fits_ulong_p: C++ Interface Integers.
3668 * mpz_class::fits_ushort_p: C++ Interface Integers.
3670 * mpz_class::get_d: C++ Interface Integers.
3672 * mpz_class::get_mpz_t: C++ Interface General.
3674 * mpz_class::get_si: C++ Interface Integers.
3676 * mpz_class::get_str: C++ Interface Integers.
3678 * mpz_class::get_ui: C++ Interface Integers.
3680 * mpz_class::mpz_class: C++ Interface Integers.
3682 * mpz_class::mpz_class <1>: C++ Interface Integers.
3684 * mpz_class::mpz_class <2>: C++ Interface Integers.
3686 * mpz_class::mpz_class <3>: C++ Interface Integers.
3688 * mpz_class::primorial: C++ Interface Integers.
3690 * mpz_class::set_str: C++ Interface Integers.
3692 * mpz_class::set_str <1>: C++ Interface Integers.
3694 * mpz_class::swap: C++ Interface Integers.
3696 * mpz_clear: Initializing Integers.
3698 * mpz_clears: Initializing Integers.
3700 * mpz_clrbit: Integer Logic and Bit Fiddling.
3702 * mpz_cmp: Integer Comparisons. (line 6)
3703 * mpz_cmpabs: Integer Comparisons. (line 17)
3704 * mpz_cmpabs_d: Integer Comparisons. (line 18)
3705 * mpz_cmpabs_ui: Integer Comparisons. (line 19)
3706 * mpz_cmp_d: Integer Comparisons. (line 7)
3707 * mpz_cmp_si: Integer Comparisons. (line 8)
3708 * mpz_cmp_ui: Integer Comparisons. (line 9)
3709 * mpz_com: Integer Logic and Bit Fiddling.
3711 * mpz_combit: Integer Logic and Bit Fiddling.
3713 * mpz_congruent_2exp_p: Integer Division. (line 148)
3714 * mpz_congruent_p: Integer Division. (line 144)
3715 * mpz_congruent_ui_p: Integer Division. (line 146)
3716 * mpz_divexact: Integer Division. (line 122)
3717 * mpz_divexact_ui: Integer Division. (line 123)
3718 * mpz_divisible_2exp_p: Integer Division. (line 135)
3719 * mpz_divisible_p: Integer Division. (line 132)
3720 * mpz_divisible_ui_p: Integer Division. (line 133)
3721 * mpz_even_p: Miscellaneous Integer Functions.
3723 * mpz_export: Integer Import and Export.
3725 * mpz_fac_ui: Number Theoretic Functions.
3727 * mpz_fdiv_q: Integer Division. (line 33)
3728 * mpz_fdiv_qr: Integer Division. (line 35)
3729 * mpz_fdiv_qr_ui: Integer Division. (line 42)
3730 * mpz_fdiv_q_2exp: Integer Division. (line 47)
3731 * mpz_fdiv_q_ui: Integer Division. (line 38)
3732 * mpz_fdiv_r: Integer Division. (line 34)
3733 * mpz_fdiv_r_2exp: Integer Division. (line 50)
3734 * mpz_fdiv_r_ui: Integer Division. (line 40)
3735 * mpz_fdiv_ui: Integer Division. (line 44)
3736 * mpz_fib2_ui: Number Theoretic Functions.
3738 * mpz_fib_ui: Number Theoretic Functions.
3740 * mpz_fits_sint_p: Miscellaneous Integer Functions.
3742 * mpz_fits_slong_p: Miscellaneous Integer Functions.
3744 * mpz_fits_sshort_p: Miscellaneous Integer Functions.
3746 * mpz_fits_uint_p: Miscellaneous Integer Functions.
3748 * mpz_fits_ulong_p: Miscellaneous Integer Functions.
3750 * mpz_fits_ushort_p: Miscellaneous Integer Functions.
3752 * mpz_gcd: Number Theoretic Functions.
3754 * mpz_gcdext: Number Theoretic Functions.
3756 * mpz_gcd_ui: Number Theoretic Functions.
3758 * mpz_getlimbn: Integer Special Functions.
3760 * mpz_get_d: Converting Integers. (line 26)
3761 * mpz_get_d_2exp: Converting Integers. (line 34)
3762 * mpz_get_si: Converting Integers. (line 17)
3763 * mpz_get_str: Converting Integers. (line 46)
3764 * mpz_get_ui: Converting Integers. (line 10)
3765 * mpz_hamdist: Integer Logic and Bit Fiddling.
3767 * mpz_import: Integer Import and Export.
3769 * mpz_init: Initializing Integers.
3771 * mpz_init2: Initializing Integers.
3773 * mpz_inits: Initializing Integers.
3775 * mpz_init_set: Simultaneous Integer Init & Assign.
3777 * mpz_init_set_d: Simultaneous Integer Init & Assign.
3779 * mpz_init_set_si: Simultaneous Integer Init & Assign.
3781 * mpz_init_set_str: Simultaneous Integer Init & Assign.
3783 * mpz_init_set_ui: Simultaneous Integer Init & Assign.
3785 * mpz_inp_raw: I/O of Integers. (line 61)
3786 * mpz_inp_str: I/O of Integers. (line 30)
3787 * mpz_invert: Number Theoretic Functions.
3789 * mpz_ior: Integer Logic and Bit Fiddling.
3791 * mpz_jacobi: Number Theoretic Functions.
3793 * mpz_kronecker: Number Theoretic Functions.
3795 * mpz_kronecker_si: Number Theoretic Functions.
3797 * mpz_kronecker_ui: Number Theoretic Functions.
3799 * mpz_lcm: Number Theoretic Functions.
3801 * mpz_lcm_ui: Number Theoretic Functions.
3803 * mpz_legendre: Number Theoretic Functions.
3805 * mpz_limbs_finish: Integer Special Functions.
3807 * mpz_limbs_modify: Integer Special Functions.
3809 * mpz_limbs_read: Integer Special Functions.
3811 * mpz_limbs_write: Integer Special Functions.
3813 * mpz_lucnum2_ui: Number Theoretic Functions.
3815 * mpz_lucnum_ui: Number Theoretic Functions.
3817 * mpz_mfac_uiui: Number Theoretic Functions.
3819 * mpz_mod: Integer Division. (line 112)
3820 * mpz_mod_ui: Integer Division. (line 113)
3821 * mpz_mul: Integer Arithmetic. (line 18)
3822 * mpz_mul_2exp: Integer Arithmetic. (line 36)
3823 * mpz_mul_si: Integer Arithmetic. (line 19)
3824 * mpz_mul_ui: Integer Arithmetic. (line 20)
3825 * mpz_neg: Integer Arithmetic. (line 41)
3826 * mpz_nextprime: Number Theoretic Functions.
3828 * mpz_odd_p: Miscellaneous Integer Functions.
3830 * mpz_out_raw: I/O of Integers. (line 45)
3831 * mpz_out_str: I/O of Integers. (line 17)
3832 * mpz_perfect_power_p: Integer Roots. (line 27)
3833 * mpz_perfect_square_p: Integer Roots. (line 36)
3834 * mpz_popcount: Integer Logic and Bit Fiddling.
3836 * mpz_powm: Integer Exponentiation.
3838 * mpz_powm_sec: Integer Exponentiation.
3840 * mpz_powm_ui: Integer Exponentiation.
3842 * mpz_pow_ui: Integer Exponentiation.
3844 * mpz_primorial_ui: Number Theoretic Functions.
3846 * mpz_probab_prime_p: Number Theoretic Functions.
3848 * mpz_random: Integer Random Numbers.
3850 * mpz_random2: Integer Random Numbers.
3852 * mpz_realloc2: Initializing Integers.
3854 * mpz_remove: Number Theoretic Functions.
3856 * mpz_roinit_n: Integer Special Functions.
3858 * MPZ_ROINIT_N: Integer Special Functions.
3860 * mpz_root: Integer Roots. (line 6)
3861 * mpz_rootrem: Integer Roots. (line 12)
3862 * mpz_rrandomb: Integer Random Numbers.
3864 * mpz_scan0: Integer Logic and Bit Fiddling.
3866 * mpz_scan1: Integer Logic and Bit Fiddling.
3868 * mpz_set: Assigning Integers. (line 9)
3869 * mpz_setbit: Integer Logic and Bit Fiddling.
3871 * mpz_set_d: Assigning Integers. (line 12)
3872 * mpz_set_f: Assigning Integers. (line 14)
3873 * mpz_set_q: Assigning Integers. (line 13)
3874 * mpz_set_si: Assigning Integers. (line 11)
3875 * mpz_set_str: Assigning Integers. (line 20)
3876 * mpz_set_ui: Assigning Integers. (line 10)
3877 * mpz_sgn: Integer Comparisons. (line 27)
3878 * mpz_size: Integer Special Functions.
3880 * mpz_sizeinbase: Miscellaneous Integer Functions.
3882 * mpz_si_kronecker: Number Theoretic Functions.
3884 * mpz_sqrt: Integer Roots. (line 17)
3885 * mpz_sqrtrem: Integer Roots. (line 20)
3886 * mpz_sub: Integer Arithmetic. (line 11)
3887 * mpz_submul: Integer Arithmetic. (line 30)
3888 * mpz_submul_ui: Integer Arithmetic. (line 32)
3889 * mpz_sub_ui: Integer Arithmetic. (line 12)
3890 * mpz_swap: Assigning Integers. (line 36)
3891 * mpz_t: Nomenclature and Types.
3893 * mpz_tdiv_q: Integer Division. (line 54)
3894 * mpz_tdiv_qr: Integer Division. (line 56)
3895 * mpz_tdiv_qr_ui: Integer Division. (line 63)
3896 * mpz_tdiv_q_2exp: Integer Division. (line 68)
3897 * mpz_tdiv_q_ui: Integer Division. (line 59)
3898 * mpz_tdiv_r: Integer Division. (line 55)
3899 * mpz_tdiv_r_2exp: Integer Division. (line 71)
3900 * mpz_tdiv_r_ui: Integer Division. (line 61)
3901 * mpz_tdiv_ui: Integer Division. (line 65)
3902 * mpz_tstbit: Integer Logic and Bit Fiddling.
3904 * mpz_ui_kronecker: Number Theoretic Functions.
3906 * mpz_ui_pow_ui: Integer Exponentiation.
3908 * mpz_ui_sub: Integer Arithmetic. (line 14)
3909 * mpz_urandomb: Integer Random Numbers.
3911 * mpz_urandomm: Integer Random Numbers.
3913 * mpz_xor: Integer Logic and Bit Fiddling.
3915 * mp_bitcnt_t: Nomenclature and Types.
3917 * mp_bits_per_limb: Useful Macros and Constants.
3919 * mp_exp_t: Nomenclature and Types.
3921 * mp_get_memory_functions: Custom Allocation. (line 86)
3922 * mp_limb_t: Nomenclature and Types.
3924 * mp_set_memory_functions: Custom Allocation. (line 14)
3925 * mp_size_t: Nomenclature and Types.
3927 * operator"": C++ Interface Integers.
3929 * operator"" <1>: C++ Interface Rationals.
3931 * operator"" <2>: C++ Interface Floats.
3933 * operator%: C++ Interface Integers.
3935 * operator/: C++ Interface Integers.
3937 * operator<<: C++ Formatted Output.
3939 * operator<< <1>: C++ Formatted Output.
3941 * operator<< <2>: C++ Formatted Output.
3943 * operator>>: C++ Formatted Input. (line 10)
3944 * operator>> <1>: C++ Formatted Input. (line 13)
3945 * operator>> <2>: C++ Formatted Input. (line 24)
3946 * operator>> <3>: C++ Interface Rationals.
3948 * primorial: C++ Interface Integers.
3950 * sgn: C++ Interface Integers.
3952 * sgn <1>: C++ Interface Rationals.
3954 * sgn <2>: C++ Interface Floats.
3956 * sqrt: C++ Interface Integers.
3958 * sqrt <1>: C++ Interface Floats.
3960 * swap: C++ Interface Integers.
3962 * swap <1>: C++ Interface Rationals.
3964 * swap <2>: C++ Interface Floats.
3966 * trunc: C++ Interface Floats.
projects / xonotic / xonotic.git / blob