1This is gmp.info, produced by makeinfo version 6.1 from gmp.texi. 2 3This manual describes how to install and use the GNU multiple precision 4arithmetic library, version 6.1.2. 5 6 Copyright 1991, 1993-2016 Free Software Foundation, Inc. 7 8 Permission is granted to copy, distribute and/or modify this document 9under the terms of the GNU Free Documentation License, Version 1.3 or 10any later version published by the Free Software Foundation; with no 11Invariant Sections, with the Front-Cover Texts being "A GNU Manual", and 12with the Back-Cover Texts being "You have freedom to copy and modify 13this GNU Manual, like GNU software". A copy of the license is included 14in *note GNU Free Documentation License::. 15INFO-DIR-SECTION GNU libraries 16START-INFO-DIR-ENTRY 17* gmp: (gmp). GNU Multiple Precision Arithmetic Library. 18END-INFO-DIR-ENTRY 19 20 21File: gmp.info, Node: Exact Remainder, Next: Small Quotient Division, Prev: Exact Division, Up: Division Algorithms 22 2315.2.6 Exact Remainder 24---------------------- 25 26If the exact division algorithm is done with a full subtraction at each 27stage and the dividend isn't a multiple of the divisor, then low zero 28limbs are produced but with a remainder in the high limbs. For dividend 29a, divisor d, quotient q, and b = 2^mp_bits_per_limb, this remainder r 30is of the form 31 32 a = q*d + r*b^n 33 34 n represents the number of zero limbs produced by the subtractions, 35that being the number of limbs produced for q. r will be in the range 360<=r<d and can be viewed as a remainder, but one shifted up by a factor 37of b^n. 38 39 Carrying out full subtractions at each stage means the same number of 40cross products must be done as a normal division, but there's still some 41single limb divisions saved. When d is a single limb some 42simplifications arise, providing good speedups on a number of 43processors. 44 45 The functions 'mpn_divexact_by3', 'mpn_modexact_1_odd' and the 46internal 'mpn_redc_X' functions differ subtly in how they return r, 47leading to some negations in the above formula, but all are essentially 48the same. 49 50 Clearly r is zero when a is a multiple of d, and this leads to 51divisibility or congruence tests which are potentially more efficient 52than a normal division. 53 54 The factor of b^n on r can be ignored in a GCD when d is odd, hence 55the use of 'mpn_modexact_1_odd' by 'mpn_gcd_1' and 'mpz_kronecker_ui' 56etc (*note Greatest Common Divisor Algorithms::). 57 58 Montgomery's REDC method for modular multiplications uses operands of 59the form of x*b^-n and y*b^-n and on calculating (x*b^-n)*(y*b^-n) uses 60the factor of b^n in the exact remainder to reach a product in the same 61form (x*y)*b^-n (*note Modular Powering Algorithm::). 62 63 Notice that r generally gives no useful information about the 64ordinary remainder a mod d since b^n mod d could be anything. If 65however b^n == 1 mod d, then r is the negative of the ordinary 66remainder. This occurs whenever d is a factor of b^n-1, as for example 67with 3 in 'mpn_divexact_by3'. For a 32 or 64 bit limb other such 68factors include 5, 17 and 257, but no particular use has been found for 69this. 70 71 72File: gmp.info, Node: Small Quotient Division, Prev: Exact Remainder, Up: Division Algorithms 73 7415.2.7 Small Quotient Division 75------------------------------ 76 77An NxM division where the number of quotient limbs Q=N-M is small can be 78optimized somewhat. 79 80 An ordinary basecase division normalizes the divisor by shifting it 81to make the high bit set, shifting the dividend accordingly, and 82shifting the remainder back down at the end of the calculation. This is 83wasteful if only a few quotient limbs are to be formed. Instead a 84division of just the top 2*Q limbs of the dividend by the top Q limbs of 85the divisor can be used to form a trial quotient. This requires only 86those limbs normalized, not the whole of the divisor and dividend. 87 88 A multiply and subtract then applies the trial quotient to the M-Q 89unused limbs of the divisor and N-Q dividend limbs (which includes Q 90limbs remaining from the trial quotient division). The starting trial 91quotient can be 1 or 2 too big, but all cases of 2 too big and most 92cases of 1 too big are detected by first comparing the most significant 93limbs that will arise from the subtraction. An addback is done if the 94quotient still turns out to be 1 too big. 95 96 This whole procedure is essentially the same as one step of the 97basecase algorithm done in a Q limb base, though with the trial quotient 98test done only with the high limbs, not an entire Q limb "digit" 99product. The correctness of this weaker test can be established by 100following the argument of Knuth section 4.3.1 exercise 20 but with the 101v2*q>b*r+u2 condition appropriately relaxed. 102 103 104File: gmp.info, Node: Greatest Common Divisor Algorithms, Next: Powering Algorithms, Prev: Division Algorithms, Up: Algorithms 105 10615.3 Greatest Common Divisor 107============================ 108 109* Menu: 110 111* Binary GCD:: 112* Lehmer's Algorithm:: 113* Subquadratic GCD:: 114* Extended GCD:: 115* Jacobi Symbol:: 116 117 118File: gmp.info, Node: Binary GCD, Next: Lehmer's Algorithm, Prev: Greatest Common Divisor Algorithms, Up: Greatest Common Divisor Algorithms 119 12015.3.1 Binary GCD 121----------------- 122 123At small sizes GMP uses an O(N^2) binary style GCD. This is described 124in many textbooks, for example Knuth section 4.5.2 algorithm B. It 125simply consists of successively reducing odd operands a and b using 126 127 a,b = abs(a-b),min(a,b) 128 strip factors of 2 from a 129 130 The Euclidean GCD algorithm, as per Knuth algorithms E and A, 131repeatedly 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 133Euclidean algorithm everywhere. One reason the binary method does well 134is that the implied quotient at each step is usually small, so often 135only one or two subtractions are needed to get the same effect as a 136division. Quotients 1, 2 and 3 for example occur 67.7% of the time, see 137Knuth section 4.5.3 Theorem E. 138 139 When the implied quotient is large, meaning b is much smaller than a, 140then a division is worthwhile. This is the basis for the initial a mod 141b reductions in 'mpn_gcd' and 'mpn_gcd_1' (the latter for both Nx1 and 1421x1 cases). But after that initial reduction, big quotients occur too 143rarely to make it worth checking for them. 144 145 146 The final 1x1 GCD in 'mpn_gcd_1' is done in the generic C code as 147described above. For two N-bit operands, the algorithm takes about 0.68 148iterations per bit. For optimum performance some attention needs to be 149paid to the way the factors of 2 are stripped from a. 150 151 Firstly it may be noted that in twos complement the number of low 152zero bits on a-b is the same as b-a, so counting or testing can begin on 153a-b without waiting for abs(a-b) to be determined. 154 155 A loop stripping low zero bits tends not to branch predict well, 156since the condition is data dependent. But on average there's only a 157few low zeros, so an option is to strip one or two bits arithmetically 158then loop for more (as done for AMD K6). Or use a lookup table to get a 159count for several bits then loop for more (as done for AMD K7). An 160alternative approach is to keep just one of a or b odd and iterate 161 162 a,b = abs(a-b), min(a,b) 163 a = a/2 if even 164 b = b/2 if even 165 166 This requires about 1.25 iterations per bit, but stripping of a 167single bit at each step avoids any branching. Repeating the bit strip 168reduces to about 0.9 iterations per bit, which may be a worthwhile 169tradeoff. 170 171 Generally with the above approaches a speed of perhaps 6 cycles per 172bit can be achieved, which is still not terribly fast with for instance 173a 64-bit GCD taking nearly 400 cycles. It's this sort of time which 174means it's not usually advantageous to combine a set of divisibility 175tests into a GCD. 176 177 Currently, the binary algorithm is used for GCD only when N < 3. 178 179 180File: gmp.info, Node: Lehmer's Algorithm, Next: Subquadratic GCD, Prev: Binary GCD, Up: Greatest Common Divisor Algorithms 181 18215.3.2 Lehmer's algorithm 183------------------------- 184 185Lehmer's improvement of the Euclidean algorithms is based on the 186observation that the initial part of the quotient sequence depends only 187on the most significant parts of the inputs. The variant of Lehmer's 188algorithm used in GMP splits off the most significant two limbs, as 189suggested, e.g., in "A Double-Digit Lehmer-Euclid Algorithm" by Jebelean 190(*note References::). The quotients of two double-limb inputs are 191collected as a 2 by 2 matrix with single-limb elements. This is done by 192the function 'mpn_hgcd2'. The resulting matrix is applied to the inputs 193using 'mpn_mul_1' and 'mpn_submul_1'. Each iteration usually reduces 194the inputs by almost one limb. In the rare case of a large quotient, no 195progress can be made by examining just the most significant two limbs, 196and the quotient is computed using plain division. 197 198 The resulting algorithm is asymptotically O(N^2), just as the 199Euclidean algorithm and the binary algorithm. The quadratic part of the 200work are the calls to 'mpn_mul_1' and 'mpn_submul_1'. For small sizes, 201the linear work is also significant. There are roughly N calls to the 202'mpn_hgcd2' function. This function uses a couple of important 203optimizations: 204 205 * 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 210 off. 211 212 * 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). 217 218 * 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. 221 222 223File: gmp.info, Node: Subquadratic GCD, Next: Extended GCD, Prev: Lehmer's Algorithm, Up: Greatest Common Divisor Algorithms 224 22515.3.3 Subquadratic GCD 226----------------------- 227 228For inputs larger than 'GCD_DC_THRESHOLD', GCD is computed via the HGCD 229(Half GCD) function, as a generalization to Lehmer's algorithm. 230 231 Let the inputs a,b be of size N limbs each. Put S = floor(N/2) + 1. 232Then HGCD(a,b) returns a transformation matrix T with non-negative 233elements, and reduced numbers (c;d) = T^{-1} (a;b). The reduced numbers 234c,d must be larger than S limbs, while their difference abs(c-d) must 235fit in S limbs. The matrix elements will also be of size roughly N/2. 236 237 The HGCD base case uses Lehmer's algorithm, but with the above stop 238condition that returns reduced numbers and the corresponding 239transformation matrix half-way through. For inputs larger than 240'HGCD_THRESHOLD', HGCD is computed recursively, using the divide and 241conquer algorithm in "On Sch�nhage's algorithm and subquadratic integer 242GCD computation" by M�ller (*note References::). The recursive 243algorithm consists of these main steps. 244 245 * 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. 248 249 * 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. 252 253 * 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. 256 257 * Compute T = T_1 T_2. 258 259 * Perform a small number of division and subtraction steps to satisfy 260 the requirements, and return. 261 262 GCD is then implemented as a loop around HGCD, similarly to Lehmer's 263algorithm. Where Lehmer repeatedly chops off the top two limbs, calls 264'mpn_hgcd2', and applies the resulting matrix to the full numbers, the 265sub-quadratic GCD chops off the most significant third of the limbs (the 266proportion is a tuning parameter, and 1/3 seems to be more efficient 267than, e.g, 1/2), calls 'mpn_hgcd', and applies the resulting matrix. 268Once the input numbers are reduced to size below 'GCD_DC_THRESHOLD', 269Lehmer's algorithm is used for the rest of the work. 270 271 The asymptotic running time of both HGCD and GCD is O(M(N)*log(N)), 272where M(N) is the time for multiplying two N-limb numbers. 273 274 275File: gmp.info, Node: Extended GCD, Next: Jacobi Symbol, Prev: Subquadratic GCD, Up: Greatest Common Divisor Algorithms 276 27715.3.4 Extended GCD 278------------------- 279 280The extended GCD function, or GCDEXT, calculates gcd(a,b) and also 281cofactors x and y satisfying a*x+b*y=gcd(a,b). All the algorithms used 282for plain GCD are extended to handle this case. The binary algorithm is 283used only for single-limb GCDEXT. Lehmer's algorithm is used for sizes 284up to 'GCDEXT_DC_THRESHOLD'. Above this threshold, GCDEXT is 285implemented as a loop around HGCD, but with more book-keeping to keep 286track of the cofactors. This gives the same asymptotic running time as 287for GCD and HGCD, O(M(N)*log(N)) 288 289 One difference to plain GCD is that while the inputs a and b are 290reduced as the algorithm proceeds, the cofactors x and y grow in size. 291This makes the tuning of the chopping-point more difficult. The current 292code chops off the most significant half of the inputs for the call to 293HGCD in the first iteration, and the most significant two thirds for the 294remaining calls. This strategy could surely be improved. Also the stop 295condition for the loop, where Lehmer's algorithm is invoked once the 296inputs are reduced below 'GCDEXT_DC_THRESHOLD', could maybe be improved 297by taking into account the current size of the cofactors. 298 299 300File: gmp.info, Node: Jacobi Symbol, Prev: Extended GCD, Up: Greatest Common Divisor Algorithms 301 30215.3.5 Jacobi Symbol 303-------------------- 304 305[This section is obsolete. The current Jacobi code actually uses a very 306efficient algorithm.] 307 308 'mpz_jacobi' and 'mpz_kronecker' are currently implemented with a 309simple binary algorithm similar to that described for the GCDs (*note 310Binary GCD::). They're not very fast when both inputs are large. 311Lehmer's multi-step improvement or a binary based multi-step algorithm 312is likely to be better. 313 314 When one operand fits a single limb, and that includes 315'mpz_kronecker_ui' and friends, an initial reduction is done with either 316'mpn_mod_1' or 'mpn_modexact_1_odd', followed by the binary algorithm on 317a single limb. The binary algorithm is well suited to a single limb, 318and the whole calculation in this case is quite efficient. 319 320 In all the routines sign changes for the result are accumulated using 321some bit twiddling, avoiding table lookups or conditional jumps. 322 323 324File: gmp.info, Node: Powering Algorithms, Next: Root Extraction Algorithms, Prev: Greatest Common Divisor Algorithms, Up: Algorithms 325 32615.4 Powering Algorithms 327======================== 328 329* Menu: 330 331* Normal Powering Algorithm:: 332* Modular Powering Algorithm:: 333 334 335File: gmp.info, Node: Normal Powering Algorithm, Next: Modular Powering Algorithm, Prev: Powering Algorithms, Up: Powering Algorithms 336 33715.4.1 Normal Powering 338---------------------- 339 340Normal 'mpz' or 'mpf' powering uses a simple binary algorithm, 341successively squaring and then multiplying by the base when a 1 bit is 342seen in the exponent, as per Knuth section 4.6.3. The "left to right" 343variant described there is used rather than algorithm A, since it's just 344as easy and can be done with somewhat less temporary memory. 345 346 347File: gmp.info, Node: Modular Powering Algorithm, Prev: Normal Powering Algorithm, Up: Powering Algorithms 348 34915.4.2 Modular Powering 350----------------------- 351 352Modular powering is implemented using a 2^k-ary sliding window 353algorithm, as per "Handbook of Applied Cryptography" algorithm 14.85 354(*note References::). k is chosen according to the size of the 355exponent. Larger exponents use larger values of k, the choice being 356made to minimize the average number of multiplications that must 357supplement the squaring. 358 359 The modular multiplies and squarings use either a simple division or 360the REDC method by Montgomery (*note References::). REDC is a little 361faster, essentially saving N single limb divisions in a fashion similar 362to an exact remainder (*note Exact Remainder::). 363 364 365File: gmp.info, Node: Root Extraction Algorithms, Next: Radix Conversion Algorithms, Prev: Powering Algorithms, Up: Algorithms 366 36715.5 Root Extraction Algorithms 368=============================== 369 370* Menu: 371 372* Square Root Algorithm:: 373* Nth Root Algorithm:: 374* Perfect Square Algorithm:: 375* Perfect Power Algorithm:: 376 377 378File: gmp.info, Node: Square Root Algorithm, Next: Nth Root Algorithm, Prev: Root Extraction Algorithms, Up: Root Extraction Algorithms 379 38015.5.1 Square Root 381------------------ 382 383Square roots are taken using the "Karatsuba Square Root" algorithm by 384Paul Zimmermann (*note References::). 385 386 An input n is split into four parts of k bits each, so with b=2^k we 387have n = a3*b^3 + a2*b^2 + a1*b + a0. Part a3 must be "normalized" so 388that either the high or second highest bit is set. In GMP, k is kept on 389a limb boundary and the input is left shifted (by an even number of 390bits) to normalize. 391 392 The square root of the high two parts is taken, by recursive 393application of the algorithm (bottoming out in a one-limb Newton's 394method), 395 396 s1,r1 = sqrtrem (a3*b + a2) 397 398 This is an approximation to the desired root and is extended by a 399division to give s,r, 400 401 q,u = divrem (r1*b + a1, 2*s1) 402 s = s1*b + q 403 r = u*b + a0 - q^2 404 405 The normalization requirement on a3 means at this point s is either 406correct or 1 too big. r is negative in the latter case, so 407 408 if r < 0 then 409 r = r + 2*s - 1 410 s = s - 1 411 412 The algorithm is expressed in a divide and conquer form, but as noted 413in the paper it can also be viewed as a discrete variant of Newton's 414method, or as a variation on the schoolboy method (no longer taught) for 415square roots two digits at a time. 416 417 If the remainder r is not required then usually only a few high limbs 418of r and u need to be calculated to determine whether an adjustment to s 419is required. This optimization is not currently implemented. 420 421 In the Karatsuba multiplication range this algorithm is 422O(1.5*M(N/2)), where M(n) is the time to multiply two numbers of n 423limbs. In the FFT multiplication range this grows to a bound of 424O(6*M(N/2)). In practice a factor of about 1.5 to 1.8 is found in the 425Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range. 426 427 The algorithm does all its calculations in integers and the resulting 428'mpn_sqrtrem' is used for both 'mpz_sqrt' and 'mpf_sqrt'. The extended 429precision given by 'mpf_sqrt_ui' is obtained by padding with zero limbs. 430 431 432File: gmp.info, Node: Nth Root Algorithm, Next: Perfect Square Algorithm, Prev: Square Root Algorithm, Up: Root Extraction Algorithms 433 43415.5.2 Nth Root 435--------------- 436 437Integer Nth roots are taken using Newton's method with the following 438iteration, where A is the input and n is the root to be taken. 439 440 1 A 441 a[i+1] = - * ( --------- + (n-1)*a[i] ) 442 n a[i]^(n-1) 443 444 The initial approximation a[1] is generated bitwise by successively 445powering a trial root with or without new 1 bits, aiming to be just 446above the true root. The iteration converges quadratically when started 447from a good approximation. When n is large more initial bits are needed 448to get good convergence. The current implementation is not particularly 449well optimized. 450 451 452File: gmp.info, Node: Perfect Square Algorithm, Next: Perfect Power Algorithm, Prev: Nth Root Algorithm, Up: Root Extraction Algorithms 453 45415.5.3 Perfect Square 455--------------------- 456 457A significant fraction of non-squares can be quickly identified by 458checking whether the input is a quadratic residue modulo small integers. 459 460 'mpz_perfect_square_p' first tests the input mod 256, which means 461just examining the low byte. Only 44 different values occur for squares 462mod 256, so 82.8% of inputs can be immediately identified as 463non-squares. 464 465 On a 32-bit system similar tests are done mod 9, 5, 7, 13 and 17, for 466a total 99.25% of inputs identified as non-squares. On a 64-bit system 46797 is tested too, for a total 99.62%. 468 469 These moduli are chosen because they're factors of 2^24-1 (or 2^48-1 470for 64-bits), and such a remainder can be quickly taken just using 471additions (see 'mpn_mod_34lsub1'). 472 473 When nails are in use moduli are instead selected by the 'gen-psqr.c' 474program and applied with an 'mpn_mod_1'. The same 2^24-1 or 2^48-1 475could be done with nails using some extra bit shifts, but this is not 476currently implemented. 477 478 In any case each modulus is applied to the 'mpn_mod_34lsub1' or 479'mpn_mod_1' remainder and a table lookup identifies non-squares. By 480using a "modexact" style calculation, and suitably permuted tables, just 481one multiply each is required, see the code for details. Moduli are 482also combined to save operations, so long as the lookup tables don't 483become too big. 'gen-psqr.c' does all the pre-calculations. 484 485 A square root must still be taken for any value that passes these 486tests, to verify it's really a square and not one of the small fraction 487of non-squares that get through (i.e. a pseudo-square to all the tested 488bases). 489 490 Clearly more residue tests could be done, 'mpz_perfect_square_p' only 491uses a compact and efficient set. Big inputs would probably benefit 492from more residue testing, small inputs might be better off with less. 493The assumed distribution of squares versus non-squares in the input 494would affect such considerations. 495 496 497File: gmp.info, Node: Perfect Power Algorithm, Prev: Perfect Square Algorithm, Up: Root Extraction Algorithms 498 49915.5.4 Perfect Power 500-------------------- 501 502Detecting perfect powers is required by some factorization algorithms. 503Currently 'mpz_perfect_power_p' is implemented using repeated Nth root 504extractions, though naturally only prime roots need to be considered. 505(*Note Nth Root Algorithm::.) 506 507 If a prime divisor p with multiplicity e can be found, then only 508roots which are divisors of e need to be considered, much reducing the 509work necessary. To this end divisibility by a set of small primes is 510checked. 511 512 513File: gmp.info, Node: Radix Conversion Algorithms, Next: Other Algorithms, Prev: Root Extraction Algorithms, Up: Algorithms 514 51515.6 Radix Conversion 516===================== 517 518Radix conversions are less important than other algorithms. A program 519dominated by conversions should probably use a different data 520representation. 521 522* Menu: 523 524* Binary to Radix:: 525* Radix to Binary:: 526 527 528File: gmp.info, Node: Binary to Radix, Next: Radix to Binary, Prev: Radix Conversion Algorithms, Up: Radix Conversion Algorithms 529 53015.6.1 Binary to Radix 531---------------------- 532 533Conversions from binary to a power-of-2 radix use a simple and fast O(N) 534bit extraction algorithm. 535 536 Conversions from binary to other radices use one of two algorithms. 537Sizes below 'GET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method. 538Repeated divisions by b^n are made, where b is the radix and n is the 539biggest power that fits in a limb. But instead of simply using the 540remainder r from such divisions, an extra divide step is done to give a 541fractional limb representing r/b^n. The digits of r can then be 542extracted using multiplications by b rather than divisions. Special 543case code is provided for decimal, allowing multiplications by 10 to 544optimize to shifts and adds. 545 546 Above 'GET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is 547used. For an input t, powers b^(n*2^i) of the radix are calculated, 548until a power between t and sqrt(t) is reached. t is then divided by 549that largest power, giving a quotient which is the digits above that 550power, and a remainder which is those below. These two parts are in 551turn divided by the second highest power, and so on recursively. When a 552piece has been divided down to less than 'GET_STR_DC_THRESHOLD' limbs, 553the basecase algorithm described above is used. 554 555 The advantage of this algorithm is that big divisions can make use of 556the sub-quadratic divide and conquer division (*note Divide and Conquer 557Division::), and big divisions tend to have less overheads than lots of 558separate single limb divisions anyway. But in any case the cost of 559calculating the powers b^(n*2^i) must first be overcome. 560 561 'GET_STR_PRECOMPUTE_THRESHOLD' and 'GET_STR_DC_THRESHOLD' represent 562the same basic thing, the point where it becomes worth doing a big 563division to cut the input in half. 'GET_STR_PRECOMPUTE_THRESHOLD' 564includes the cost of calculating the radix power required, whereas 565'GET_STR_DC_THRESHOLD' assumes that's already available, which is the 566case when recursing. 567 568 Since the base case produces digits from least to most significant 569but they want to be stored from most to least, it's necessary to 570calculate in advance how many digits there will be, or at least be sure 571not to underestimate that. For GMP the number of input bits is 572multiplied by 'chars_per_bit_exactly' from 'mp_bases', rounding up. The 573result is either correct or one too big. 574 575 Examining some of the high bits of the input could increase the 576chance of getting the exact number of digits, but an exact result every 577time would not be practical, since in general the difference between 578numbers 100... and 99... is only in the last few bits and the work to 579identify 99... might well be almost as much as a full conversion. 580 581 The r/b^n scheme described above for using multiplications to bring 582out digits might be useful for more than a single limb. Some brief 583experiments with it on the base case when recursing didn't give a 584noticeable improvement, but perhaps that was only due to the 585implementation. Something similar would work for the sub-quadratic 586divisions too, though there would be the cost of calculating a bigger 587radix power. 588 589 Another possible improvement for the sub-quadratic part would be to 590arrange for radix powers that balanced the sizes of quotient and 591remainder produced, i.e. the highest power would be an b^(n*k) 592approximately equal to sqrt(t), not restricted to a 2^i factor. That 593ought to smooth out a graph of times against sizes, but may or may not 594be a net speedup. 595 596 597File: gmp.info, Node: Radix to Binary, Prev: Binary to Radix, Up: Radix Conversion Algorithms 598 59915.6.2 Radix to Binary 600---------------------- 601 602*This section needs to be rewritten, it currently describes the 603algorithms used before GMP 4.3.* 604 605 Conversions from a power-of-2 radix into binary use a simple and fast 606O(N) bitwise concatenation algorithm. 607 608 Conversions from other radices use one of two algorithms. Sizes 609below 'SET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method. Groups 610of n digits are converted to limbs, where n is the biggest power of the 611base b which will fit in a limb, then those groups are accumulated into 612the result by multiplying by b^n and adding. This saves multi-precision 613operations, as per Knuth section 4.4 part E (*note References::). Some 614special case code is provided for decimal, giving the compiler a chance 615to optimize multiplications by 10. 616 617 Above 'SET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is 618used. First groups of n digits are converted into limbs. Then adjacent 619limbs are combined into limb pairs with x*b^n+y, where x and y are the 620limbs. Adjacent limb pairs are combined into quads similarly with 621x*b^(2n)+y. This continues until a single block remains, that being the 622result. 623 624 The advantage of this method is that the multiplications for each x 625are big blocks, allowing Karatsuba and higher algorithms to be used. 626But the cost of calculating the powers b^(n*2^i) must be overcome. 627'SET_STR_PRECOMPUTE_THRESHOLD' usually ends up quite big, around 5000 628digits, and on some processors much bigger still. 629 630 'SET_STR_PRECOMPUTE_THRESHOLD' is based on the input digits (and 631tuned for decimal), though it might be better based on a limb count, so 632as to be independent of the base. But that sort of count isn't used by 633the base case and so would need some sort of initial calculation or 634estimate. 635 636 The main reason 'SET_STR_PRECOMPUTE_THRESHOLD' is so much bigger than 637the corresponding 'GET_STR_PRECOMPUTE_THRESHOLD' is that 'mpn_mul_1' is 638much faster than 'mpn_divrem_1' (often by a factor of 5, or more). 639 640 641File: gmp.info, Node: Other Algorithms, Next: Assembly Coding, Prev: Radix Conversion Algorithms, Up: Algorithms 642 64315.7 Other Algorithms 644===================== 645 646* Menu: 647 648* Prime Testing Algorithm:: 649* Factorial Algorithm:: 650* Binomial Coefficients Algorithm:: 651* Fibonacci Numbers Algorithm:: 652* Lucas Numbers Algorithm:: 653* Random Number Algorithms:: 654 655 656File: gmp.info, Node: Prime Testing Algorithm, Next: Factorial Algorithm, Prev: Other Algorithms, Up: Other Algorithms 657 65815.7.1 Prime Testing 659-------------------- 660 661The primality testing in 'mpz_probab_prime_p' (*note Number Theoretic 662Functions::) first does some trial division by small factors and then 663uses the Miller-Rabin probabilistic primality testing algorithm, as 664described in Knuth section 4.5.4 algorithm P (*note References::). 665 666 For an odd input n, and with n = q*2^k+1 where q is odd, this 667algorithm selects a random base x and tests whether x^q mod n is 1 or 668-1, or an x^(q*2^j) mod n is 1, for 1<=j<=k. If so then n is probably 669prime, if not then n is definitely composite. 670 671 Any prime n will pass the test, but some composites do too. Such 672composites are known as strong pseudoprimes to base x. No n is a strong 673pseudoprime to more than 1/4 of all bases (see Knuth exercise 22), hence 674with x chosen at random there's no more than a 1/4 chance a "probable 675prime" will in fact be composite. 676 677 In fact strong pseudoprimes are quite rare, making the test much more 678powerful than this analysis would suggest, but 1/4 is all that's proven 679for an arbitrary n. 680 681 682File: gmp.info, Node: Factorial Algorithm, Next: Binomial Coefficients Algorithm, Prev: Prime Testing Algorithm, Up: Other Algorithms 683 68415.7.2 Factorial 685---------------- 686 687Factorials are calculated by a combination of two algorithms. An idea 688is shared among them: to compute the odd part of the factorial; a final 689step takes account of the power of 2 term, by shifting. 690 691 For small n, the odd factor of n! is computed with the simple 692observation that it is equal to the product of all positive odd numbers 693smaller than n times the odd factor of [n/2]!, where [x] is the integer 694part of x, and so on recursively. The procedure can be best illustrated 695with an example, 696 697 23! = (23.21.19.17.15.13.11.9.7.5.3)(11.9.7.5.3)(5.3)2^{19} 698 699 Current code collects all the factors in a single list, with a loop 700and no recursion, and compute the product, with no special care for 701repeated chunks. 702 703 When n is larger, computation pass trough prime sieving. An helper 704function is used, as suggested by Peter Luschny: 705 706 n 707 ----- 708 n! | | L(p,n) 709 msf(n) = -------------- = | | p 710 [n/2]!^2.2^k p=3 711 712 Where p ranges on odd prime numbers. The exponent k is chosen to 713obtain an odd integer number: k is the number of 1 bits in the binary 714representation of [n/2]. The function L(p,n) can be defined as zero 715when p is composite, and, for any prime p, it is computed with: 716 717 --- 718 \ n 719 L(p,n) = / [---] mod 2 <= log (n) . 720 --- p^i p 721 i>0 722 723 With this helper function, we are able to compute the odd part of n! 724using the recursion implied by n!=[n/2]!^2*msf(n)*2^k. The recursion 725stops using the small-n algorithm on some [n/2^i]. 726 727 Both the above algorithms use binary splitting to compute the product 728of many small factors. At first as many products as possible are 729accumulated in a single register, generating a list of factors that fit 730in a machine word. This list is then split into halves, and the product 731is computed recursively. 732 733 Such splitting is more efficient than repeated Nx1 multiplies since 734it forms big multiplies, allowing Karatsuba and higher algorithms to be 735used. And even below the Karatsuba threshold a big block of work can be 736more efficient for the basecase algorithm. 737 738 739File: gmp.info, Node: Binomial Coefficients Algorithm, Next: Fibonacci Numbers Algorithm, Prev: Factorial Algorithm, Up: Other Algorithms 740 74115.7.3 Binomial Coefficients 742---------------------------- 743 744Binomial coefficients C(n,k) are calculated by first arranging k <= n/2 745using C(n,k) = C(n,n-k) if necessary, and then evaluating the following 746product simply from i=2 to i=k. 747 748 k (n-k+i) 749 C(n,k) = (n-k+1) * prod ------- 750 i=2 i 751 752 It's easy to show that each denominator i will divide the product so 753far, so the exact division algorithm is used (*note Exact Division::). 754 755 The numerators n-k+i and denominators i are first accumulated into as 756many fit a limb, to save multi-precision operations, though for 757'mpz_bin_ui' this applies only to the divisors, since n is an 'mpz_t' 758and n-k+i in general won't fit in a limb at all. 759 760 761File: gmp.info, Node: Fibonacci Numbers Algorithm, Next: Lucas Numbers Algorithm, Prev: Binomial Coefficients Algorithm, Up: Other Algorithms 762 76315.7.4 Fibonacci Numbers 764------------------------ 765 766The Fibonacci functions 'mpz_fib_ui' and 'mpz_fib2_ui' are designed for 767calculating isolated F[n] or F[n],F[n-1] values efficiently. 768 769 For small n, a table of single limb values in '__gmp_fib_table' is 770used. On a 32-bit limb this goes up to F[47], or on a 64-bit limb up to 771F[93]. For convenience the table starts at F[-1]. 772 773 Beyond the table, values are generated with a binary powering 774algorithm, calculating a pair F[n] and F[n-1] working from high to low 775across the bits of n. The formulas used are 776 777 F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k 778 F[2k-1] = F[k]^2 + F[k-1]^2 779 780 F[2k] = F[2k+1] - F[2k-1] 781 782 At each step, k is the high b bits of n. If the next bit of n is 0 783then F[2k],F[2k-1] is used, or if it's a 1 then F[2k+1],F[2k] is used, 784and the process repeated until all bits of n are incorporated. Notice 785these formulas require just two squares per bit of n. 786 787 It'd be possible to handle the first few n above the single limb 788table with simple additions, using the defining Fibonacci recurrence 789F[k+1]=F[k]+F[k-1], but this is not done since it usually turns out to 790be faster for only about 10 or 20 values of n, and including a block of 791code for just those doesn't seem worthwhile. If they really mattered 792it'd be better to extend the data table. 793 794 Using a table avoids lots of calculations on small numbers, and makes 795small n go fast. A bigger table would make more small n go fast, it's 796just a question of balancing size against desired speed. For GMP the 797code is kept compact, with the emphasis primarily on a good powering 798algorithm. 799 800 'mpz_fib2_ui' returns both F[n] and F[n-1], but 'mpz_fib_ui' is only 801interested in F[n]. In this case the last step of the algorithm can 802become one multiply instead of two squares. One of the following two 803formulas is used, according as n is odd or even. 804 805 F[2k] = F[k]*(F[k]+2F[k-1]) 806 807 F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k 808 809 F[2k+1] here is the same as above, just rearranged to be a multiply. 810For interest, the 2*(-1)^k term both here and above can be applied just 811to the low limb of the calculation, without a carry or borrow into 812further limbs, which saves some code size. See comments with 813'mpz_fib_ui' and the internal 'mpn_fib2_ui' for how this is done. 814 815 816File: gmp.info, Node: Lucas Numbers Algorithm, Next: Random Number Algorithms, Prev: Fibonacci Numbers Algorithm, Up: Other Algorithms 817 81815.7.5 Lucas Numbers 819-------------------- 820 821'mpz_lucnum2_ui' derives a pair of Lucas numbers from a pair of 822Fibonacci numbers with the following simple formulas. 823 824 L[k] = F[k] + 2*F[k-1] 825 L[k-1] = 2*F[k] - F[k-1] 826 827 'mpz_lucnum_ui' is only interested in L[n], and some work can be 828saved. Trailing zero bits on n can be handled with a single square 829each. 830 831 L[2k] = L[k]^2 - 2*(-1)^k 832 833 And the lowest 1 bit can be handled with one multiply of a pair of 834Fibonacci numbers, similar to what 'mpz_fib_ui' does. 835 836 L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k 837 838 839File: gmp.info, Node: Random Number Algorithms, Prev: Lucas Numbers Algorithm, Up: Other Algorithms 840 84115.7.6 Random Numbers 842--------------------- 843 844For the 'urandomb' functions, random numbers are generated simply by 845concatenating bits produced by the generator. As long as the generator 846has good randomness properties this will produce well-distributed N bit 847numbers. 848 849 For the 'urandomm' functions, random numbers in a range 0<=R<N are 850generated by taking values R of ceil(log2(N)) bits each until one 851satisfies R<N. This will normally require only one or two attempts, but 852the attempts are limited in case the generator is somehow degenerate and 853produces only 1 bits or similar. 854 855 The Mersenne Twister generator is by Matsumoto and Nishimura (*note 856References::). It has a non-repeating period of 2^19937-1, which is a 857Mersenne prime, hence the name of the generator. The state is 624 words 858of 32-bits each, which is iterated with one XOR and shift for each 85932-bit word generated, making the algorithm very fast. Randomness 860properties are also very good and this is the default algorithm used by 861GMP. 862 863 Linear congruential generators are described in many text books, for 864instance Knuth volume 2 (*note References::). With a modulus M and 865parameters A and C, an integer state S is iterated by the formula S <- 866A*S+C mod M. At each step the new state is a linear function of the 867previous, mod M, hence the name of the generator. 868 869 In GMP only moduli of the form 2^N are supported, and the current 870implementation is not as well optimized as it could be. Overheads are 871significant when N is small, and when N is large clearly the multiply at 872each step will become slow. This is not a big concern, since the 873Mersenne Twister generator is better in every respect and is therefore 874recommended for all normal applications. 875 876 For both generators the current state can be deduced by observing 877enough output and applying some linear algebra (over GF(2) in the case 878of the Mersenne Twister). This generally means raw output is unsuitable 879for cryptographic applications without further hashing or the like. 880 881 882File: gmp.info, Node: Assembly Coding, Prev: Other Algorithms, Up: Algorithms 883 88415.8 Assembly Coding 885==================== 886 887The assembly subroutines in GMP are the most significant source of speed 888at small to moderate sizes. At larger sizes algorithm selection becomes 889more important, but of course speedups in low level routines will still 890speed up everything proportionally. 891 892 Carry handling and widening multiplies that are important for GMP 893can't be easily expressed in C. GCC 'asm' blocks help a lot and are 894provided in 'longlong.h', but hand coding low level routines invariably 895offers a speedup over generic C by a factor of anything from 2 to 10. 896 897* Menu: 898 899* Assembly Code Organisation:: 900* Assembly Basics:: 901* Assembly Carry Propagation:: 902* Assembly Cache Handling:: 903* Assembly Functional Units:: 904* Assembly Floating Point:: 905* Assembly SIMD Instructions:: 906* Assembly Software Pipelining:: 907* Assembly Loop Unrolling:: 908* Assembly Writing Guide:: 909 910 911File: gmp.info, Node: Assembly Code Organisation, Next: Assembly Basics, Prev: Assembly Coding, Up: Assembly Coding 912 91315.8.1 Code Organisation 914------------------------ 915 916The various 'mpn' subdirectories contain machine-dependent code, written 917in C or assembly. The 'mpn/generic' subdirectory contains default code, 918used when there's no machine-specific version of a particular file. 919 920 Each 'mpn' subdirectory is for an ISA family. Generally 32-bit and 92164-bit variants in a family cannot share code and have separate 922directories. Within a family further subdirectories may exist for CPU 923variants. 924 925 In each directory a 'nails' subdirectory may exist, holding code with 926nails support for that CPU variant. A 'NAILS_SUPPORT' directive in each 927file indicates the nails values the code handles. Nails code only 928exists where it's faster, or promises to be faster, than plain code. 929There's no effort put into nails if they're not going to enhance a given 930CPU. 931 932 933File: gmp.info, Node: Assembly Basics, Next: Assembly Carry Propagation, Prev: Assembly Code Organisation, Up: Assembly Coding 934 93515.8.2 Assembly Basics 936---------------------- 937 938'mpn_addmul_1' and 'mpn_submul_1' are the most important routines for 939overall GMP performance. All multiplications and divisions come down to 940repeated calls to these. 'mpn_add_n', 'mpn_sub_n', 'mpn_lshift' and 941'mpn_rshift' are next most important. 942 943 On some CPUs assembly versions of the internal functions 944'mpn_mul_basecase' and 'mpn_sqr_basecase' give significant speedups, 945mainly through avoiding function call overheads. They can also 946potentially make better use of a wide superscalar processor, as can 947bigger primitives like 'mpn_addmul_2' or 'mpn_addmul_4'. 948 949 The restrictions on overlaps between sources and destinations (*note 950Low-level Functions::) are designed to facilitate a variety of 951implementations. For example, knowing 'mpn_add_n' won't have partly 952overlapping sources and destination means reading can be done far ahead 953of writing on superscalar processors, and loops can be vectorized on a 954vector processor, depending on the carry handling. 955 956 957File: gmp.info, Node: Assembly Carry Propagation, Next: Assembly Cache Handling, Prev: Assembly Basics, Up: Assembly Coding 958 95915.8.3 Carry Propagation 960------------------------ 961 962The problem that presents most challenges in GMP is propagating carries 963from one limb to the next. In functions like 'mpn_addmul_1' and 964'mpn_add_n', carries are the only dependencies between limb operations. 965 966 On processors with carry flags, a straightforward CISC style 'adc' is 967generally best. AMD K6 'mpn_addmul_1' however is an example of an 968unusual set of circumstances where a branch works out better. 969 970 On RISC processors generally an add and compare for overflow is used. 971This sort of thing can be seen in 'mpn/generic/aors_n.c'. Some carry 972propagation schemes require 4 instructions, meaning at least 4 cycles 973per limb, but other schemes may use just 1 or 2. On wide superscalar 974processors performance may be completely determined by the number of 975dependent instructions between carry-in and carry-out for each limb. 976 977 On vector processors good use can be made of the fact that a carry 978bit only very rarely propagates more than one limb. When adding a 979single bit to a limb, there's only a carry out if that limb was 980'0xFF...FF' which on random data will be only 1 in 2^mp_bits_per_limb. 981'mpn/cray/add_n.c' is an example of this, it adds all limbs in parallel, 982adds one set of carry bits in parallel and then only rarely needs to 983fall through to a loop propagating further carries. 984 985 On the x86s, GCC (as of version 2.95.2) doesn't generate particularly 986good code for the RISC style idioms that are necessary to handle carry 987bits in C. Often conditional jumps are generated where 'adc' or 'sbb' 988forms would be better. And so unfortunately almost any loop involving 989carry bits needs to be coded in assembly for best results. 990 991 992File: gmp.info, Node: Assembly Cache Handling, Next: Assembly Functional Units, Prev: Assembly Carry Propagation, Up: Assembly Coding 993 99415.8.4 Cache Handling 995--------------------- 996 997GMP aims to perform well both on operands that fit entirely in L1 cache 998and those which don't. 999 1000 Basic routines like 'mpn_add_n' or 'mpn_lshift' are often used on 1001large operands, so L2 and main memory performance is important for them. 1002'mpn_mul_1' and 'mpn_addmul_1' are mostly used for multiply and square 1003basecases, so L1 performance matters most for them, unless assembly 1004versions of 'mpn_mul_basecase' and 'mpn_sqr_basecase' exist, in which 1005case the remaining uses are mostly for larger operands. 1006 1007 For L2 or main memory operands, memory access times will almost 1008certainly be more than the calculation time. The aim therefore is to 1009maximize memory throughput, by starting a load of the next cache line 1010while processing the contents of the previous one. Clearly this is only 1011possible if the chip has a lock-up free cache or some sort of prefetch 1012instruction. Most current chips have both these features. 1013 1014 Prefetching sources combines well with loop unrolling, since a 1015prefetch can be initiated once per unrolled loop (or more than once if 1016the loop covers more than one cache line). 1017 1018 On CPUs without write-allocate caches, prefetching destinations will 1019ensure individual stores don't go further down the cache hierarchy, 1020limiting bandwidth. Of course for calculations which are slow anyway, 1021like 'mpn_divrem_1', write-throughs might be fine. 1022 1023 The distance ahead to prefetch will be determined by memory latency 1024versus throughput. The aim of course is to have data arriving 1025continuously, at peak throughput. Some CPUs have limits on the number 1026of fetches or prefetches in progress. 1027 1028 If a special prefetch instruction doesn't exist then a plain load can 1029be used, but in that case care must be taken not to attempt to read past 1030the end of an operand, since that might produce a segmentation 1031violation. 1032 1033 Some CPUs or systems have hardware that detects sequential memory 1034accesses and initiates suitable cache movements automatically, making 1035life easy. 1036 1037 1038File: gmp.info, Node: Assembly Functional Units, Next: Assembly Floating Point, Prev: Assembly Cache Handling, Up: Assembly Coding 1039 104015.8.5 Functional Units 1041----------------------- 1042 1043When choosing an approach for an assembly loop, consideration is given 1044to what operations can execute simultaneously and what throughput can 1045thereby be achieved. In some cases an algorithm can be tweaked to 1046accommodate available resources. 1047 1048 Loop control will generally require a counter and pointer updates, 1049costing as much as 5 instructions, plus any delays a branch introduces. 1050CPU addressing modes might reduce pointer updates, perhaps by allowing 1051just one updating pointer and others expressed as offsets from it, or on 1052CISC chips with all addressing done with the loop counter as a scaled 1053index. 1054 1055 The final loop control cost can be amortised by processing several 1056limbs in each iteration (*note Assembly Loop Unrolling::). This at 1057least ensures loop control isn't a big fraction the work done. 1058 1059 Memory throughput is always a limit. If perhaps only one load or one 1060store can be done per cycle then 3 cycles/limb will the top speed for 1061"binary" operations like 'mpn_add_n', and any code achieving that is 1062optimal. 1063 1064 Integer resources can be freed up by having the loop counter in a 1065float register, or by pressing the float units into use for some 1066multiplying, perhaps doing every second limb on the float side (*note 1067Assembly Floating Point::). 1068 1069 Float resources can be freed up by doing carry propagation on the 1070integer side, or even by doing integer to float conversions in integers 1071using bit twiddling. 1072 1073 1074File: gmp.info, Node: Assembly Floating Point, Next: Assembly SIMD Instructions, Prev: Assembly Functional Units, Up: Assembly Coding 1075 107615.8.6 Floating Point 1077--------------------- 1078 1079Floating point arithmetic is used in GMP for multiplications on CPUs 1080with poor integer multipliers. It's mostly useful for 'mpn_mul_1', 1081'mpn_addmul_1' and 'mpn_submul_1' on 64-bit machines, and 1082'mpn_mul_basecase' on both 32-bit and 64-bit machines. 1083 1084 With IEEE 53-bit double precision floats, integer multiplications 1085producing up to 53 bits will give exact results. Breaking a 64x64 1086multiplication into eight 16x32->48 bit pieces is convenient. With some 1087care though six 21x32->53 bit products can be used, if one of the lower 1088two 21-bit pieces also uses the sign bit. 1089 1090 For the 'mpn_mul_1' family of functions on a 64-bit machine, the 1091invariant single limb is split at the start, into 3 or 4 pieces. Inside 1092the loop, the bignum operand is split into 32-bit pieces. Fast 1093conversion of these unsigned 32-bit pieces to floating point is highly 1094machine-dependent. In some cases, reading the data into the integer 1095unit, zero-extending to 64-bits, then transferring to the floating point 1096unit back via memory is the only option. 1097 1098 Converting partial products back to 64-bit limbs is usually best done 1099as a signed conversion. Since all values are smaller than 2^53, signed 1100and unsigned are the same, but most processors lack unsigned 1101conversions. 1102 1103 1104 1105 Here is a diagram showing 16x32 bit products for an 'mpn_mul_1' or 1106'mpn_addmul_1' with a 64-bit limb. The single limb operand V is split 1107into four 16-bit parts. The multi-limb operand U is split in the loop 1108into two 32-bit parts. 1109 1110 +---+---+---+---+ 1111 |v48|v32|v16|v00| V operand 1112 +---+---+---+---+ 1113 1114 +-------+---+---+ 1115 x | u32 | u00 | U operand (one limb) 1116 +---------------+ 1117 1118 --------------------------------- 1119 1120 +-----------+ 1121 | u00 x v00 | p00 48-bit products 1122 +-----------+ 1123 +-----------+ 1124 | u00 x v16 | p16 1125 +-----------+ 1126 +-----------+ 1127 | u00 x v32 | p32 1128 +-----------+ 1129 +-----------+ 1130 | u00 x v48 | p48 1131 +-----------+ 1132 +-----------+ 1133 | u32 x v00 | r32 1134 +-----------+ 1135 +-----------+ 1136 | u32 x v16 | r48 1137 +-----------+ 1138 +-----------+ 1139 | u32 x v32 | r64 1140 +-----------+ 1141 +-----------+ 1142 | u32 x v48 | r80 1143 +-----------+ 1144 1145 p32 and r32 can be summed using floating-point addition, and likewise 1146p48 and r48. p00 and p16 can be summed with r64 and r80 from the 1147previous iteration. 1148 1149 For each loop then, four 49-bit quantities are transferred to the 1150integer unit, aligned as follows, 1151 1152 |-----64bits----|-----64bits----| 1153 +------------+ 1154 | p00 + r64' | i00 1155 +------------+ 1156 +------------+ 1157 | p16 + r80' | i16 1158 +------------+ 1159 +------------+ 1160 | p32 + r32 | i32 1161 +------------+ 1162 +------------+ 1163 | p48 + r48 | i48 1164 +------------+ 1165 1166 The challenge then is to sum these efficiently and add in a carry 1167limb, generating a low 64-bit result limb and a high 33-bit carry limb 1168(i48 extends 33 bits into the high half). 1169 1170 1171File: gmp.info, Node: Assembly SIMD Instructions, Next: Assembly Software Pipelining, Prev: Assembly Floating Point, Up: Assembly Coding 1172 117315.8.7 SIMD Instructions 1174------------------------ 1175 1176The single-instruction multiple-data support in current microprocessors 1177is aimed at signal processing algorithms where each data point can be 1178treated more or less independently. There's generally not much support 1179for propagating the sort of carries that arise in GMP. 1180 1181 SIMD multiplications of say four 16x16 bit multiplies only do as much 1182work as one 32x32 from GMP's point of view, and need some shifts and 1183adds besides. But of course if say the SIMD form is fully pipelined and 1184uses less instruction decoding then it may still be worthwhile. 1185 1186 On the x86 chips, MMX has so far found a use in 'mpn_rshift' and 1187'mpn_lshift', and is used in a special case for 16-bit multipliers in 1188the P55 'mpn_mul_1'. SSE2 is used for Pentium 4 'mpn_mul_1', 1189'mpn_addmul_1', and 'mpn_submul_1'. 1190 1191 1192File: gmp.info, Node: Assembly Software Pipelining, Next: Assembly Loop Unrolling, Prev: Assembly SIMD Instructions, Up: Assembly Coding 1193 119415.8.8 Software Pipelining 1195-------------------------- 1196 1197Software pipelining consists of scheduling instructions around the 1198branch point in a loop. For example a loop might issue a load not for 1199use in the present iteration but the next, thereby allowing extra cycles 1200for the data to arrive from memory. 1201 1202 Naturally this is wanted only when doing things like loads or 1203multiplies that take several cycles to complete, and only where a CPU 1204has multiple functional units so that other work can be done in the 1205meantime. 1206 1207 A pipeline with several stages will have a data value in progress at 1208each stage and each loop iteration moves them along one stage. This is 1209like juggling. 1210 1211 If the latency of some instruction is greater than the loop time then 1212it will be necessary to unroll, so one register has a result ready to 1213use while another (or multiple others) are still in progress. (*note 1214Assembly Loop Unrolling::). 1215 1216 1217File: gmp.info, Node: Assembly Loop Unrolling, Next: Assembly Writing Guide, Prev: Assembly Software Pipelining, Up: Assembly Coding 1218 121915.8.9 Loop Unrolling 1220--------------------- 1221 1222Loop unrolling consists of replicating code so that several limbs are 1223processed in each loop. At a minimum this reduces loop overheads by a 1224corresponding factor, but it can also allow better register usage, for 1225example alternately using one register combination and then another. 1226Judicious use of 'm4' macros can help avoid lots of duplication in the 1227source code. 1228 1229 Any amount of unrolling can be handled with a loop counter that's 1230decremented by N each time, stopping when the remaining count is less 1231than the further N the loop will process. Or by subtracting N at the 1232start, the termination condition becomes when the counter C is less than 12330 (and the count of remaining limbs is C+N). 1234 1235 Alternately for a power of 2 unroll the loop count and remainder can 1236be established with a shift and mask. This is convenient if also making 1237a computed jump into the middle of a large loop. 1238 1239 The limbs not a multiple of the unrolling can be handled in various 1240ways, for example 1241 1242 * A simple loop at the end (or the start) to process the excess. 1243 Care will be wanted that it isn't too much slower than the unrolled 1244 part. 1245 1246 * A set of binary tests, for example after an 8-limb unrolling, test 1247 for 4 more limbs to process, then a further 2 more or not, and 1248 finally 1 more or not. This will probably take more code space 1249 than a simple loop. 1250 1251 * A 'switch' statement, providing separate code for each possible 1252 excess, for example an 8-limb unrolling would have separate code 1253 for 0 remaining, 1 remaining, etc, up to 7 remaining. This might 1254 take a lot of code, but may be the best way to optimize all cases 1255 in combination with a deep pipelined loop. 1256 1257 * A computed jump into the middle of the loop, thus making the first 1258 iteration handle the excess. This should make times smoothly 1259 increase with size, which is attractive, but setups for the jump 1260 and adjustments for pointers can be tricky and could become quite 1261 difficult in combination with deep pipelining. 1262 1263 1264File: gmp.info, Node: Assembly Writing Guide, Prev: Assembly Loop Unrolling, Up: Assembly Coding 1265 126615.8.10 Writing Guide 1267--------------------- 1268 1269This is a guide to writing software pipelined loops for processing limb 1270vectors in assembly. 1271 1272 First determine the algorithm and which instructions are needed. 1273Code it without unrolling or scheduling, to make sure it works. On a 12743-operand CPU try to write each new value to a new register, this will 1275greatly simplify later steps. 1276 1277 Then note for each instruction the functional unit and/or issue port 1278requirements. If an instruction can use either of two units, like U0 or 1279U1 then make a category "U0/U1". Count the total using each unit (or 1280combined unit), and count all instructions. 1281 1282 Figure out from those counts the best possible loop time. The goal 1283will be to find a perfect schedule where instruction latencies are 1284completely hidden. The total instruction count might be the limiting 1285factor, or perhaps a particular functional unit. It might be possible 1286to tweak the instructions to help the limiting factor. 1287 1288 Suppose the loop time is N, then make N issue buckets, with the final 1289loop branch at the end of the last. Now fill the buckets with dummy 1290instructions using the functional units desired. Run this to make sure 1291the intended speed is reached. 1292 1293 Now replace the dummy instructions with the real instructions from 1294the slow but correct loop you started with. The first will typically be 1295a load instruction. Then the instruction using that value is placed in 1296a bucket an appropriate distance down. Run the loop again, to check it 1297still runs at target speed. 1298 1299 Keep placing instructions, frequently measuring the loop. After a 1300few you will need to wrap around from the last bucket back to the top of 1301the loop. If you used the new-register for new-value strategy above 1302then there will be no register conflicts. If not then take care not to 1303clobber something already in use. Changing registers at this time is 1304very error prone. 1305 1306 The loop will overlap two or more of the original loop iterations, 1307and the computation of one vector element result will be started in one 1308iteration of the new loop, and completed one or several iterations 1309later. 1310 1311 The final step is to create feed-in and wind-down code for the loop. 1312A good way to do this is to make a copy (or copies) of the loop at the 1313start and delete those instructions which don't have valid antecedents, 1314and at the end replicate and delete those whose results are unwanted 1315(including any further loads). 1316 1317 The loop will have a minimum number of limbs loaded and processed, so 1318the feed-in code must test if the request size is smaller and skip 1319either to a suitable part of the wind-down or to special code for small 1320sizes. 1321 1322 1323File: gmp.info, Node: Internals, Next: Contributors, Prev: Algorithms, Up: Top 1324 132516 Internals 1326************ 1327 1328*This chapter is provided only for informational purposes and the 1329various internals described here may change in future GMP releases. 1330Applications expecting to be compatible with future releases should use 1331only the documented interfaces described in previous chapters.* 1332 1333* Menu: 1334 1335* Integer Internals:: 1336* Rational Internals:: 1337* Float Internals:: 1338* Raw Output Internals:: 1339* C++ Interface Internals:: 1340 1341 1342File: gmp.info, Node: Integer Internals, Next: Rational Internals, Prev: Internals, Up: Internals 1343 134416.1 Integer Internals 1345====================== 1346 1347'mpz_t' variables represent integers using sign and magnitude, in space 1348dynamically allocated and reallocated. The fields are as follows. 1349 1350'_mp_size' 1351 The number of limbs, or the negative of that when representing a 1352 negative integer. Zero is represented by '_mp_size' set to zero, 1353 in which case the '_mp_d' data is unused. 1354 1355'_mp_d' 1356 A pointer to an array of limbs which is the magnitude. These are 1357 stored "little endian" as per the 'mpn' functions, so '_mp_d[0]' is 1358 the least significant limb and '_mp_d[ABS(_mp_size)-1]' is the most 1359 significant. Whenever '_mp_size' is non-zero, the most significant 1360 limb is non-zero. 1361 1362 Currently there's always at least one limb allocated, so for 1363 instance 'mpz_set_ui' never needs to reallocate, and 'mpz_get_ui' 1364 can fetch '_mp_d[0]' unconditionally (though its value is then only 1365 wanted if '_mp_size' is non-zero). 1366 1367'_mp_alloc' 1368 '_mp_alloc' is the number of limbs currently allocated at '_mp_d', 1369 and naturally '_mp_alloc >= ABS(_mp_size)'. When an 'mpz' routine 1370 is about to (or might be about to) increase '_mp_size', it checks 1371 '_mp_alloc' to see whether there's enough space, and reallocates if 1372 not. 'MPZ_REALLOC' is generally used for this. 1373 1374 The various bitwise logical functions like 'mpz_and' behave as if 1375negative values were twos complement. But sign and magnitude is always 1376used internally, and necessary adjustments are made during the 1377calculations. Sometimes this isn't pretty, but sign and magnitude are 1378best for other routines. 1379 1380 Some internal temporary variables are setup with 'MPZ_TMP_INIT' and 1381these have '_mp_d' space obtained from 'TMP_ALLOC' rather than the 1382memory allocation functions. Care is taken to ensure that these are big 1383enough that no reallocation is necessary (since it would have 1384unpredictable consequences). 1385 1386 '_mp_size' and '_mp_alloc' are 'int', although 'mp_size_t' is usually 1387a 'long'. This is done to make the fields just 32 bits on some 64 bits 1388systems, thereby saving a few bytes of data space but still providing 1389plenty of range. 1390 1391 1392File: gmp.info, Node: Rational Internals, Next: Float Internals, Prev: Integer Internals, Up: Internals 1393 139416.2 Rational Internals 1395======================= 1396 1397'mpq_t' variables represent rationals using an 'mpz_t' numerator and 1398denominator (*note Integer Internals::). 1399 1400 The canonical form adopted is denominator positive (and non-zero), no 1401common factors between numerator and denominator, and zero uniquely 1402represented as 0/1. 1403 1404 It's believed that casting out common factors at each stage of a 1405calculation is best in general. A GCD is an O(N^2) operation so it's 1406better to do a few small ones immediately than to delay and have to do a 1407big one later. Knowing the numerator and denominator have no common 1408factors can be used for example in 'mpq_mul' to make only two cross GCDs 1409necessary, not four. 1410 1411 This general approach to common factors is badly sub-optimal in the 1412presence of simple factorizations or little prospect for cancellation, 1413but GMP has no way to know when this will occur. As per *note 1414Efficiency::, that's left to applications. The 'mpq_t' framework might 1415still suit, with 'mpq_numref' and 'mpq_denref' for direct access to the 1416numerator and denominator, or of course 'mpz_t' variables can be used 1417directly. 1418 1419 1420File: gmp.info, Node: Float Internals, Next: Raw Output Internals, Prev: Rational Internals, Up: Internals 1421 142216.3 Float Internals 1423==================== 1424 1425Efficient calculation is the primary aim of GMP floats and the use of 1426whole limbs and simple rounding facilitates this. 1427 1428 'mpf_t' floats have a variable precision mantissa and a single 1429machine word signed exponent. The mantissa is represented using sign 1430and magnitude. 1431 1432 most least 1433 significant significant 1434 limb limb 1435 1436 _mp_d 1437 |---- _mp_exp ---> | 1438 _____ _____ _____ _____ _____ 1439 |_____|_____|_____|_____|_____| 1440 . <------------ radix point 1441 1442 <-------- _mp_size ---------> 1443 1444 1445The fields are as follows. 1446 1447'_mp_size' 1448 The number of limbs currently in use, or the negative of that when 1449 representing a negative value. Zero is represented by '_mp_size' 1450 and '_mp_exp' both set to zero, and in that case the '_mp_d' data 1451 is unused. (In the future '_mp_exp' might be undefined when 1452 representing zero.) 1453 1454'_mp_prec' 1455 The precision of the mantissa, in limbs. In any calculation the 1456 aim is to produce '_mp_prec' limbs of result (the most significant 1457 being non-zero). 1458 1459'_mp_d' 1460 A pointer to the array of limbs which is the absolute value of the 1461 mantissa. These are stored "little endian" as per the 'mpn' 1462 functions, so '_mp_d[0]' is the least significant limb and 1463 '_mp_d[ABS(_mp_size)-1]' the most significant. 1464 1465 The most significant limb is always non-zero, but there are no 1466 other restrictions on its value, in particular the highest 1 bit 1467 can be anywhere within the limb. 1468 1469 '_mp_prec+1' limbs are allocated to '_mp_d', the extra limb being 1470 for convenience (see below). There are no reallocations during a 1471 calculation, only in a change of precision with 'mpf_set_prec'. 1472 1473'_mp_exp' 1474 The exponent, in limbs, determining the location of the implied 1475 radix point. Zero means the radix point is just above the most 1476 significant limb. Positive values mean a radix point offset 1477 towards the lower limbs and hence a value >= 1, as for example in 1478 the diagram above. Negative exponents mean a radix point further 1479 above the highest limb. 1480 1481 Naturally the exponent can be any value, it doesn't have to fall 1482 within the limbs as the diagram shows, it can be a long way above 1483 or a long way below. Limbs other than those included in the 1484 '{_mp_d,_mp_size}' data are treated as zero. 1485 1486 The '_mp_size' and '_mp_prec' fields are 'int', although the 1487'mp_size_t' type is usually a 'long'. The '_mp_exp' field is usually 1488'long'. This is done to make some fields just 32 bits on some 64 bits 1489systems, thereby saving a few bytes of data space but still providing 1490plenty of precision and a very large range. 1491 1492 1493The following various points should be noted. 1494 1495Low Zeros 1496 The least significant limbs '_mp_d[0]' etc can be zero, though such 1497 low zeros can always be ignored. Routines likely to produce low 1498 zeros check and avoid them to save time in subsequent calculations, 1499 but for most routines they're quite unlikely and aren't checked. 1500 1501Mantissa Size Range 1502 The '_mp_size' count of limbs in use can be less than '_mp_prec' if 1503 the value can be represented in less. This means low precision 1504 values or small integers stored in a high precision 'mpf_t' can 1505 still be operated on efficiently. 1506 1507 '_mp_size' can also be greater than '_mp_prec'. Firstly a value is 1508 allowed to use all of the '_mp_prec+1' limbs available at '_mp_d', 1509 and secondly when 'mpf_set_prec_raw' lowers '_mp_prec' it leaves 1510 '_mp_size' unchanged and so the size can be arbitrarily bigger than 1511 '_mp_prec'. 1512 1513Rounding 1514 All rounding is done on limb boundaries. Calculating '_mp_prec' 1515 limbs with the high non-zero will ensure the application requested 1516 minimum precision is obtained. 1517 1518 The use of simple "trunc" rounding towards zero is efficient, since 1519 there's no need to examine extra limbs and increment or decrement. 1520 1521Bit Shifts 1522 Since the exponent is in limbs, there are no bit shifts in basic 1523 operations like 'mpf_add' and 'mpf_mul'. When differing exponents 1524 are encountered all that's needed is to adjust pointers to line up 1525 the relevant limbs. 1526 1527 Of course 'mpf_mul_2exp' and 'mpf_div_2exp' will require bit 1528 shifts, but the choice is between an exponent in limbs which 1529 requires shifts there, or one in bits which requires them almost 1530 everywhere else. 1531 1532Use of '_mp_prec+1' Limbs 1533 The extra limb on '_mp_d' ('_mp_prec+1' rather than just 1534 '_mp_prec') helps when an 'mpf' routine might get a carry from its 1535 operation. 'mpf_add' for instance will do an 'mpn_add' of 1536 '_mp_prec' limbs. If there's no carry then that's the result, but 1537 if there is a carry then it's stored in the extra limb of space and 1538 '_mp_size' becomes '_mp_prec+1'. 1539 1540 Whenever '_mp_prec+1' limbs are held in a variable, the low limb is 1541 not needed for the intended precision, only the '_mp_prec' high 1542 limbs. But zeroing it out or moving the rest down is unnecessary. 1543 Subsequent routines reading the value will simply take the high 1544 limbs they need, and this will be '_mp_prec' if their target has 1545 that same precision. This is no more than a pointer adjustment, 1546 and must be checked anyway since the destination precision can be 1547 different from the sources. 1548 1549 Copy functions like 'mpf_set' will retain a full '_mp_prec+1' limbs 1550 if available. This ensures that a variable which has '_mp_size' 1551 equal to '_mp_prec+1' will get its full exact value copied. 1552 Strictly speaking this is unnecessary since only '_mp_prec' limbs 1553 are needed for the application's requested precision, but it's 1554 considered that an 'mpf_set' from one variable into another of the 1555 same precision ought to produce an exact copy. 1556 1557Application Precisions 1558 '__GMPF_BITS_TO_PREC' converts an application requested precision 1559 to an '_mp_prec'. The value in bits is rounded up to a whole limb 1560 then an extra limb is added since the most significant limb of 1561 '_mp_d' is only non-zero and therefore might contain only one bit. 1562 1563 '__GMPF_PREC_TO_BITS' does the reverse conversion, and removes the 1564 extra limb from '_mp_prec' before converting to bits. The net 1565 effect of reading back with 'mpf_get_prec' is simply the precision 1566 rounded up to a multiple of 'mp_bits_per_limb'. 1567 1568 Note that the extra limb added here for the high only being 1569 non-zero is in addition to the extra limb allocated to '_mp_d'. 1570 For example with a 32-bit limb, an application request for 250 bits 1571 will be rounded up to 8 limbs, then an extra added for the high 1572 being only non-zero, giving an '_mp_prec' of 9. '_mp_d' then gets 1573 10 limbs allocated. Reading back with 'mpf_get_prec' will take 1574 '_mp_prec' subtract 1 limb and multiply by 32, giving 256 bits. 1575 1576 Strictly speaking, the fact the high limb has at least one bit 1577 means that a float with, say, 3 limbs of 32-bits each will be 1578 holding at least 65 bits, but for the purposes of 'mpf_t' it's 1579 considered simply to be 64 bits, a nice multiple of the limb size. 1580 1581 1582File: gmp.info, Node: Raw Output Internals, Next: C++ Interface Internals, Prev: Float Internals, Up: Internals 1583 158416.4 Raw Output Internals 1585========================= 1586 1587'mpz_out_raw' uses the following format. 1588 1589 +------+------------------------+ 1590 | size | data bytes | 1591 +------+------------------------+ 1592 1593 The size is 4 bytes written most significant byte first, being the 1594number of subsequent data bytes, or the twos complement negative of that 1595when a negative integer is represented. The data bytes are the absolute 1596value of the integer, written most significant byte first. 1597 1598 The most significant data byte is always non-zero, so the output is 1599the same on all systems, irrespective of limb size. 1600 1601 In GMP 1, leading zero bytes were written to pad the data bytes to a 1602multiple of the limb size. 'mpz_inp_raw' will still accept this, for 1603compatibility. 1604 1605 The use of "big endian" for both the size and data fields is 1606deliberate, it makes the data easy to read in a hex dump of a file. 1607Unfortunately it also means that the limb data must be reversed when 1608reading or writing, so neither a big endian nor little endian system can 1609just read and write '_mp_d'. 1610 1611 1612File: gmp.info, Node: C++ Interface Internals, Prev: Raw Output Internals, Up: Internals 1613 161416.5 C++ Interface Internals 1615============================ 1616 1617A system of expression templates is used to ensure something like 1618'a=b+c' turns into a simple call to 'mpz_add' etc. For 'mpf_class' the 1619scheme also ensures the precision of the final destination is used for 1620any temporaries within a statement like 'f=w*x+y*z'. These are 1621important features which a naive implementation cannot provide. 1622 1623 A simplified description of the scheme follows. The true scheme is 1624complicated by the fact that expressions have different return types. 1625For detailed information, refer to the source code. 1626 1627 To perform an operation, say, addition, we first define a "function 1628object" evaluating it, 1629 1630 struct __gmp_binary_plus 1631 { 1632 static void eval(mpf_t f, const mpf_t g, const mpf_t h) 1633 { 1634 mpf_add(f, g, h); 1635 } 1636 }; 1637 1638And an "additive expression" object, 1639 1640 __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> > 1641 operator+(const mpf_class &f, const mpf_class &g) 1642 { 1643 return __gmp_expr 1644 <__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >(f, g); 1645 } 1646 1647 The seemingly redundant '__gmp_expr<__gmp_binary_expr<...>>' is used 1648to encapsulate any possible kind of expression into a single template 1649type. In fact even 'mpf_class' etc are 'typedef' specializations of 1650'__gmp_expr'. 1651 1652 Next we define assignment of '__gmp_expr' to 'mpf_class'. 1653 1654 template <class T> 1655 mpf_class & mpf_class::operator=(const __gmp_expr<T> &expr) 1656 { 1657 expr.eval(this->get_mpf_t(), this->precision()); 1658 return *this; 1659 } 1660 1661 template <class Op> 1662 void __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, Op> >::eval 1663 (mpf_t f, mp_bitcnt_t precision) 1664 { 1665 Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t()); 1666 } 1667 1668 where 'expr.val1' and 'expr.val2' are references to the expression's 1669operands (here 'expr' is the '__gmp_binary_expr' stored within the 1670'__gmp_expr'). 1671 1672 This way, the expression is actually evaluated only at the time of 1673assignment, when the required precision (that of 'f') is known. 1674Furthermore the target 'mpf_t' is now available, thus we can call 1675'mpf_add' directly with 'f' as the output argument. 1676 1677 Compound expressions are handled by defining operators taking 1678subexpressions as their arguments, like this: 1679 1680 template <class T, class U> 1681 __gmp_expr 1682 <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> > 1683 operator+(const __gmp_expr<T> &expr1, const __gmp_expr<U> &expr2) 1684 { 1685 return __gmp_expr 1686 <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> > 1687 (expr1, expr2); 1688 } 1689 1690 And the corresponding specializations of '__gmp_expr::eval': 1691 1692 template <class T, class U, class Op> 1693 void __gmp_expr 1694 <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, Op> >::eval 1695 (mpf_t f, mp_bitcnt_t precision) 1696 { 1697 // declare two temporaries 1698 mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision); 1699 Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t()); 1700 } 1701 1702 The expression is thus recursively evaluated to any level of 1703complexity and all subexpressions are evaluated to the precision of 'f'. 1704 1705 1706File: gmp.info, Node: Contributors, Next: References, Prev: Internals, Up: Top 1707 1708Appendix A Contributors 1709*********************** 1710 1711Torbj�rn Granlund wrote the original GMP library and is still the main 1712developer. Code not explicitly attributed to others, was contributed by 1713Torbj�rn. Several other individuals and organizations have contributed 1714GMP. Here is a list in chronological order on first contribution: 1715 1716 Gunnar Sj�din and Hans Riesel helped with mathematical problems in 1717early versions of the library. 1718 1719 Richard Stallman helped with the interface design and revised the 1720first version of this manual. 1721 1722 Brian Beuning and Doug Lea helped with testing of early versions of 1723the library and made creative suggestions. 1724 1725 John Amanatides of York University in Canada contributed the function 1726'mpz_probab_prime_p'. 1727 1728 Paul Zimmermann wrote the REDC-based mpz_powm code, the 1729Sch�nhage-Strassen FFT multiply code, and the Karatsuba square root 1730code. He also improved the Toom3 code for GMP 4.2. Paul sparked the 1731development of GMP 2, with his comparisons between bignum packages. The 1732ECMNET project Paul is organizing was a driving force behind many of the 1733optimizations in GMP 3. Paul also wrote the new GMP 4.3 nth root code 1734(with Torbj�rn). 1735 1736 Ken Weber (Kent State University, Universidade Federal do Rio Grande 1737do Sul) contributed now defunct versions of 'mpz_gcd', 'mpz_divexact', 1738'mpn_gcd', and 'mpn_bdivmod', partially supported by CNPq (Brazil) grant 1739301314194-2. 1740 1741 Per Bothner of Cygnus Support helped to set up GMP to use Cygnus' 1742configure. He has also made valuable suggestions and tested numerous 1743intermediary releases. 1744 1745 Joachim Hollman was involved in the design of the 'mpf' interface, 1746and in the 'mpz' design revisions for version 2. 1747 1748 Bennet Yee contributed the initial versions of 'mpz_jacobi' and 1749'mpz_legendre'. 1750 1751 Andreas Schwab contributed the files 'mpn/m68k/lshift.S' and 1752'mpn/m68k/rshift.S' (now in '.asm' form). 1753 1754 Robert Harley of Inria, France and David Seal of ARM, England, 1755suggested clever improvements for population count. Robert also wrote 1756highly optimized Karatsuba and 3-way Toom multiplication functions for 1757GMP 3, and contributed the ARM assembly code. 1758 1759 Torsten Ekedahl of the Mathematical department of Stockholm 1760University provided significant inspiration during several phases of the 1761GMP development. His mathematical expertise helped improve several 1762algorithms. 1763 1764 Linus Nordberg wrote the new configure system based on autoconf and 1765implemented the new random functions. 1766 1767 Kevin Ryde worked on a large number of things: optimized x86 code, m4 1768asm macros, parameter tuning, speed measuring, the configure system, 1769function inlining, divisibility tests, bit scanning, Jacobi symbols, 1770Fibonacci and Lucas number functions, printf and scanf functions, perl 1771interface, demo expression parser, the algorithms chapter in the manual, 1772'gmpasm-mode.el', and various miscellaneous improvements elsewhere. 1773 1774 Kent Boortz made the Mac OS 9 port. 1775 1776 Steve Root helped write the optimized alpha 21264 assembly code. 1777 1778 Gerardo Ballabio wrote the 'gmpxx.h' C++ class interface and the C++ 1779'istream' input routines. 1780 1781 Jason Moxham rewrote 'mpz_fac_ui'. 1782 1783 Pedro Gimeno implemented the Mersenne Twister and made other random 1784number improvements. 1785 1786 Niels M�ller wrote the sub-quadratic GCD, extended GCD and jacobi 1787code, the quadratic Hensel division code, and (with Torbj�rn) the new 1788divide and conquer division code for GMP 4.3. Niels also helped 1789implement the new Toom multiply code for GMP 4.3 and implemented helper 1790functions to simplify Toom evaluations for GMP 5.0. He wrote the 1791original version of mpn_mulmod_bnm1, and he is the main author of the 1792mini-gmp package used for gmp bootstrapping. 1793 1794 Alberto Zanoni and Marco Bodrato suggested the unbalanced multiply 1795strategy, and found the optimal strategies for evaluation and 1796interpolation in Toom multiplication. 1797 1798 Marco Bodrato helped implement the new Toom multiply code for GMP 4.3 1799and implemented most of the new Toom multiply and squaring code for 5.0. 1800He is the main author of the current mpn_mulmod_bnm1, mpn_mullo_n, and 1801mpn_sqrlo. Marco also wrote the functions mpn_invert and 1802mpn_invertappr, and improved the speed of integer root extraction. He 1803is the author of the current combinatorial functions: binomial, 1804factorial, multifactorial, primorial. 1805 1806 David Harvey suggested the internal function 'mpn_bdiv_dbm1', 1807implementing division relevant to Toom multiplication. He also worked 1808on fast assembly sequences, in particular on a fast AMD64 1809'mpn_mul_basecase'. He wrote the internal middle product functions 1810'mpn_mulmid_basecase', 'mpn_toom42_mulmid', 'mpn_mulmid_n' and related 1811helper routines. 1812 1813 Martin Boij wrote 'mpn_perfect_power_p'. 1814 1815 Marc Glisse improved 'gmpxx.h': use fewer temporaries (faster), 1816specializations of 'numeric_limits' and 'common_type', C++11 features 1817(move constructors, explicit bool conversion, UDL), make the conversion 1818from 'mpq_class' to 'mpz_class' explicit, optimize operations where one 1819argument is a small compile-time constant, replace some heap allocations 1820by stack allocations. He also fixed the eofbit handling of C++ streams, 1821and removed one division from 'mpq/aors.c'. 1822 1823 David S Miller wrote assembly code for SPARC T3 and T4. 1824 1825 Mark Sofroniou cleaned up the types of mul_fft.c, letting it work for 1826huge operands. 1827 1828 Ulrich Weigand ported GMP to the powerpc64le ABI. 1829 1830 (This list is chronological, not ordered after significance. If you 1831have contributed to GMP but are not listed above, please tell 1832<gmp-devel@gmplib.org> about the omission!) 1833 1834 The development of floating point functions of GNU MP 2, were 1835supported in part by the ESPRIT-BRA (Basic Research Activities) 6846 1836project POSSO (POlynomial System SOlving). 1837 1838 The development of GMP 2, 3, and 4.0 was supported in part by the IDA 1839Center for Computing Sciences. 1840 1841 The development of GMP 4.3, 5.0, and 5.1 was supported in part by the 1842Swedish Foundation for Strategic Research. 1843 1844 Thanks go to Hans Thorsen for donating an SGI system for the GMP test 1845system environment. 1846 1847 1848File: gmp.info, Node: References, Next: GNU Free Documentation License, Prev: Contributors, Up: Top 1849 1850Appendix B References 1851********************* 1852 1853B.1 Books 1854========= 1855 1856 * Jonathan M. Borwein and Peter B. Borwein, "Pi and the AGM: A Study 1857 in Analytic Number Theory and Computational Complexity", Wiley, 1858 1998. 1859 1860 * Richard Crandall and Carl Pomerance, "Prime Numbers: A 1861 Computational Perspective", 2nd edition, Springer-Verlag, 2005. 1862 <http://www.math.dartmouth.edu/~carlp/> 1863 1864 * Henri Cohen, "A Course in Computational Algebraic Number Theory", 1865 Graduate Texts in Mathematics number 138, Springer-Verlag, 1993. 1866 <http://www.math.u-bordeaux.fr/~cohen/> 1867 1868 * Donald E. Knuth, "The Art of Computer Programming", volume 2, 1869 "Seminumerical Algorithms", 3rd edition, Addison-Wesley, 1998. 1870 <http://www-cs-faculty.stanford.edu/~knuth/taocp.html> 1871 1872 * John D. Lipson, "Elements of Algebra and Algebraic Computing", The 1873 Benjamin Cummings Publishing Company Inc, 1981. 1874 1875 * Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, 1876 "Handbook of Applied Cryptography", 1877 <http://www.cacr.math.uwaterloo.ca/hac/> 1878 1879 * Richard M. Stallman and the GCC Developer Community, "Using the GNU 1880 Compiler Collection", Free Software Foundation, 2008, available 1881 online <https://gcc.gnu.org/onlinedocs/>, and in the GCC package 1882 <https://ftp.gnu.org/gnu/gcc/> 1883 1884B.2 Papers 1885========== 1886 1887 * Yves Bertot, Nicolas Magaud and Paul Zimmermann, "A Proof of GMP 1888 Square Root", Journal of Automated Reasoning, volume 29, 2002, pp. 1889 225-252. Also available online as INRIA Research Report 4475, June 1890 2002, <http://hal.inria.fr/docs/00/07/21/13/PDF/RR-4475.pdf> 1891 1892 * Christoph Burnikel and Joachim Ziegler, "Fast Recursive Division", 1893 Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022, 1894 <http://data.mpi-sb.mpg.de/internet/reports.nsf/NumberView/1998-1-022> 1895 1896 * Torbj�rn Granlund and Peter L. Montgomery, "Division by Invariant 1897 Integers using Multiplication", in Proceedings of the SIGPLAN 1898 PLDI'94 Conference, June 1994. Also available 1899 <https://gmplib.org/~tege/divcnst-pldi94.pdf>. 1900 1901 * Niels M�ller and Torbj�rn Granlund, "Improved division by invariant 1902 integers", IEEE Transactions on Computers, 11 June 2010. 1903 <https://gmplib.org/~tege/division-paper.pdf> 1904 1905 * Torbj�rn Granlund and Niels M�ller, "Division of integers large and 1906 small", to appear. 1907 1908 * Tudor Jebelean, "An algorithm for exact division", Journal of 1909 Symbolic Computation, volume 15, 1993, pp. 169-180. Research 1910 report version available 1911 <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz> 1912 1913 * Tudor Jebelean, "Exact Division with Karatsuba Complexity - 1914 Extended Abstract", RISC-Linz technical report 96-31, 1915 <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz> 1916 1917 * Tudor Jebelean, "Practical Integer Division with Karatsuba 1918 Complexity", ISSAC 97, pp. 339-341. Technical report available 1919 <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz> 1920 1921 * Tudor Jebelean, "A Generalization of the Binary GCD Algorithm", 1922 ISSAC 93, pp. 111-116. Technical report version available 1923 <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz> 1924 1925 * Tudor Jebelean, "A Double-Digit Lehmer-Euclid Algorithm for Finding 1926 the GCD of Long Integers", Journal of Symbolic Computation, volume 1927 19, 1995, pp. 145-157. Technical report version also available 1928 <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz> 1929 1930 * Werner Krandick and Tudor Jebelean, "Bidirectional Exact Integer 1931 Division", Journal of Symbolic Computation, volume 21, 1996, pp. 1932 441-455. Early technical report version also available 1933 <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz> 1934 1935 * Makoto Matsumoto and Takuji Nishimura, "Mersenne Twister: A 1936 623-dimensionally equidistributed uniform pseudorandom number 1937 generator", ACM Transactions on Modelling and Computer Simulation, 1938 volume 8, January 1998, pp. 3-30. Available online 1939 <http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/ARTICLES/mt.ps.gz> 1940 (or .pdf) 1941 1942 * R. Moenck and A. Borodin, "Fast Modular Transforms via Division", 1943 Proceedings of the 13th Annual IEEE Symposium on Switching and 1944 Automata Theory, October 1972, pp. 90-96. Reprinted as "Fast 1945 Modular Transforms", Journal of Computer and System Sciences, 1946 volume 8, number 3, June 1974, pp. 366-386. 1947 1948 * Niels M�ller, "On Sch�nhage's algorithm and subquadratic integer 1949 GCD computation", in Mathematics of Computation, volume 77, January 1950 2008, pp. 589-607. 1951 1952 * Peter L. Montgomery, "Modular Multiplication Without Trial 1953 Division", in Mathematics of Computation, volume 44, number 170, 1954 April 1985. 1955 1956 * Arnold Sch�nhage and Volker Strassen, "Schnelle Multiplikation 1957 grosser Zahlen", Computing 7, 1971, pp. 281-292. 1958 1959 * Kenneth Weber, "The accelerated integer GCD algorithm", ACM 1960 Transactions on Mathematical Software, volume 21, number 1, March 1961 1995, pp. 111-122. 1962 1963 * Paul Zimmermann, "Karatsuba Square Root", INRIA Research Report 1964 3805, November 1999, 1965 <http://hal.inria.fr/inria-00072854/PDF/RR-3805.pdf> 1966 1967 * Paul Zimmermann, "A Proof of GMP Fast Division and Square Root 1968 Implementations", 1969 <http://www.loria.fr/~zimmerma/papers/proof-div-sqrt.ps.gz> 1970 1971 * Dan Zuras, "On Squaring and Multiplying Large Integers", ARITH-11: 1972 IEEE Symposium on Computer Arithmetic, 1993, pp. 260 to 271. 1973 Reprinted as "More on Multiplying and Squaring Large Integers", 1974 IEEE Transactions on Computers, volume 43, number 8, August 1994, 1975 pp. 899-908. 1976 1977 1978File: gmp.info, Node: GNU Free Documentation License, Next: Concept Index, Prev: References, Up: Top 1979 1980Appendix C GNU Free Documentation License 1981***************************************** 1982 1983 Version 1.3, 3 November 2008 1984 1985 Copyright � 2000-2002, 2007, 2008 Free Software Foundation, Inc. 1986 <http://fsf.org/> 1987 1988 Everyone is permitted to copy and distribute verbatim copies 1989 of this license document, but changing it is not allowed. 1990 1991 0. PREAMBLE 1992 1993 The purpose of this License is to make a manual, textbook, or other 1994 functional and useful document "free" in the sense of freedom: to 1995 assure everyone the effective freedom to copy and redistribute it, 1996 with or without modifying it, either commercially or 1997 noncommercially. Secondarily, this License preserves for the 1998 author and publisher a way to get credit for their work, while not 1999 being considered responsible for modifications made by others. 2000 2001 This License is a kind of "copyleft", which means that derivative 2002 works of the document must themselves be free in the same sense. 2003 It complements the GNU General Public License, which is a copyleft 2004 license designed for free software. 2005 2006 We have designed this License in order to use it for manuals for 2007 free software, because free software needs free documentation: a 2008 free program should come with manuals providing the same freedoms 2009 that the software does. 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If your rights have been terminated and not 2374 permanently reinstated, receipt of a copy of some or all of the 2375 same material does not give you any rights to use it. 2376 2377 10. FUTURE REVISIONS OF THIS LICENSE 2378 2379 The Free Software Foundation may publish new, revised versions of 2380 the GNU Free Documentation License from time to time. Such new 2381 versions will be similar in spirit to the present version, but may 2382 differ in detail to address new problems or concerns. See 2383 <https://www.gnu.org/copyleft/>. 2384 2385 Each version of the License is given a distinguishing version 2386 number. If the Document specifies that a particular numbered 2387 version of this License "or any later version" applies to it, you 2388 have the option of following the terms and conditions either of 2389 that specified version or of any later version that has been 2390 published (not as a draft) by the Free Software Foundation. If the 2391 Document does not specify a version number of this License, you may 2392 choose any version ever published (not as a draft) by the Free 2393 Software Foundation. If the Document specifies that a proxy can 2394 decide which future versions of this License can be used, that 2395 proxy's public statement of acceptance of a version permanently 2396 authorizes you to choose that version for the Document. 2397 2398 11. RELICENSING 2399 2400 "Massive Multiauthor Collaboration Site" (or "MMC Site") means any 2401 World Wide Web server that publishes copyrightable works and also 2402 provides prominent facilities for anybody to edit those works. A 2403 public wiki that anybody can edit is an example of such a server. 2404 A "Massive Multiauthor Collaboration" (or "MMC") contained in the 2405 site means any set of copyrightable works thus published on the MMC 2406 site. 2407 2408 "CC-BY-SA" means the Creative Commons Attribution-Share Alike 3.0 2409 license published by Creative Commons Corporation, a not-for-profit 2410 corporation with a principal place of business in San Francisco, 2411 California, as well as future copyleft versions of that license 2412 published by that same organization. 2413 2414 "Incorporate" means to publish or republish a Document, in whole or 2415 in part, as part of another Document. 2416 2417 An MMC is "eligible for relicensing" if it is licensed under this 2418 License, and if all works that were first published under this 2419 License somewhere other than this MMC, and subsequently 2420 incorporated in whole or in part into the MMC, (1) had no cover 2421 texts or invariant sections, and (2) were thus incorporated prior 2422 to November 1, 2008. 2423 2424 The operator of an MMC Site may republish an MMC contained in the 2425 site under CC-BY-SA on the same site at any time before August 1, 2426 2009, provided the MMC is eligible for relicensing. 2427 2428ADDENDUM: How to use this License for your documents 2429==================================================== 2430 2431To use this License in a document you have written, include a copy of 2432the License in the document and put the following copyright and license 2433notices just after the title page: 2434 2435 Copyright (C) YEAR YOUR NAME. 2436 Permission is granted to copy, distribute and/or modify this document 2437 under the terms of the GNU Free Documentation License, Version 1.3 2438 or any later version published by the Free Software Foundation; 2439 with no Invariant Sections, no Front-Cover Texts, and no Back-Cover 2440 Texts. A copy of the license is included in the section entitled ``GNU 2441 Free Documentation License''. 2442 2443 If you have Invariant Sections, Front-Cover Texts and Back-Cover 2444Texts, replace the "with...Texts." line with this: 2445 2446 with the Invariant Sections being LIST THEIR TITLES, with 2447 the Front-Cover Texts being LIST, and with the Back-Cover Texts 2448 being LIST. 2449 2450 If you have Invariant Sections without Cover Texts, or some other 2451combination of the three, merge those two alternatives to suit the 2452situation. 2453 2454 If your document contains nontrivial examples of program code, we 2455recommend releasing these examples in parallel under your choice of free 2456software license, such as the GNU General Public License, to permit 2457their use in free software. 2458 2459 2460File: gmp.info, Node: Concept Index, Next: Function Index, Prev: GNU Free Documentation License, Up: Top 2461 2462Concept Index 2463************* 2464 2465[index] 2466* Menu: 2467 2468* #include: Headers and Libraries. 2469 (line 6) 2470* --build: Build Options. (line 51) 2471* --disable-fft: Build Options. (line 307) 2472* --disable-shared: Build Options. (line 44) 2473* --disable-static: Build Options. (line 44) 2474* --enable-alloca: Build Options. (line 273) 2475* --enable-assert: Build Options. (line 313) 2476* --enable-cxx: Build Options. (line 225) 2477* --enable-fat: Build Options. (line 160) 2478* --enable-profiling: Build Options. (line 317) 2479* --enable-profiling <1>: Profiling. (line 6) 2480* --exec-prefix: Build Options. (line 32) 2481* --host: Build Options. (line 65) 2482* --prefix: Build Options. (line 32) 2483* -finstrument-functions: Profiling. (line 66) 2484* 2exp functions: Efficiency. (line 43) 2485* 68000: Notes for Particular Systems. 2486 (line 94) 2487* 80x86: Notes for Particular Systems. 2488 (line 150) 2489* ABI: Build Options. (line 167) 2490* ABI <1>: ABI and ISA. (line 6) 2491* About this manual: Introduction to GMP. (line 57) 2492* AC_CHECK_LIB: Autoconf. (line 11) 2493* AIX: ABI and ISA. (line 174) 2494* AIX <1>: Notes for Particular Systems. 2495 (line 7) 2496* Algorithms: Algorithms. (line 6) 2497* alloca: Build Options. (line 273) 2498* Allocation of memory: Custom Allocation. (line 6) 2499* AMD64: ABI and ISA. (line 44) 2500* Anonymous FTP of latest version: Introduction to GMP. (line 37) 2501* Application Binary Interface: ABI and ISA. (line 6) 2502* Arithmetic functions: Integer Arithmetic. (line 6) 2503* Arithmetic functions <1>: Rational Arithmetic. (line 6) 2504* Arithmetic functions <2>: Float Arithmetic. (line 6) 2505* ARM: Notes for Particular Systems. 2506 (line 20) 2507* Assembly cache handling: Assembly Cache Handling. 2508 (line 6) 2509* Assembly carry propagation: Assembly Carry Propagation. 2510 (line 6) 2511* Assembly code organisation: Assembly Code Organisation. 2512 (line 6) 2513* Assembly coding: Assembly Coding. (line 6) 2514* Assembly floating Point: Assembly Floating Point. 2515 (line 6) 2516* Assembly loop unrolling: Assembly Loop Unrolling. 2517 (line 6) 2518* Assembly SIMD: Assembly SIMD Instructions. 2519 (line 6) 2520* Assembly software pipelining: Assembly Software Pipelining. 2521 (line 6) 2522* Assembly writing guide: Assembly Writing Guide. 2523 (line 6) 2524* Assertion checking: Build Options. (line 313) 2525* Assertion checking <1>: Debugging. (line 78) 2526* Assignment functions: Assigning Integers. (line 6) 2527* Assignment functions <1>: Simultaneous Integer Init & Assign. 2528 (line 6) 2529* Assignment functions <2>: Initializing Rationals. 2530 (line 6) 2531* Assignment functions <3>: Assigning Floats. (line 6) 2532* Assignment functions <4>: Simultaneous Float Init & Assign. 2533 (line 6) 2534* Autoconf: Autoconf. (line 6) 2535* Basics: GMP Basics. (line 6) 2536* Binomial coefficient algorithm: Binomial Coefficients Algorithm. 2537 (line 6) 2538* Binomial coefficient functions: Number Theoretic Functions. 2539 (line 124) 2540* Binutils strip: Known Build Problems. 2541 (line 28) 2542* Bit manipulation functions: Integer Logic and Bit Fiddling. 2543 (line 6) 2544* Bit scanning functions: Integer Logic and Bit Fiddling. 2545 (line 39) 2546* Bit shift left: Integer Arithmetic. (line 38) 2547* Bit shift right: Integer Division. (line 74) 2548* Bits per limb: Useful Macros and Constants. 2549 (line 7) 2550* Bug reporting: Reporting Bugs. (line 6) 2551* Build directory: Build Options. (line 19) 2552* Build notes for binary packaging: Notes for Package Builds. 2553 (line 6) 2554* Build notes for particular systems: Notes for Particular Systems. 2555 (line 6) 2556* Build options: Build Options. (line 6) 2557* Build problems known: Known Build Problems. 2558 (line 6) 2559* Build system: Build Options. (line 51) 2560* Building GMP: Installing GMP. (line 6) 2561* Bus error: Debugging. (line 7) 2562* C compiler: Build Options. (line 178) 2563* C++ compiler: Build Options. (line 249) 2564* C++ interface: C++ Class Interface. (line 6) 2565* C++ interface internals: C++ Interface Internals. 2566 (line 6) 2567* C++ istream input: C++ Formatted Input. (line 6) 2568* C++ ostream output: C++ Formatted Output. 2569 (line 6) 2570* C++ support: Build Options. (line 225) 2571* CC: Build Options. (line 178) 2572* CC_FOR_BUILD: Build Options. (line 212) 2573* CFLAGS: Build Options. (line 178) 2574* Checker: Debugging. (line 114) 2575* checkergcc: Debugging. (line 121) 2576* Code organisation: Assembly Code Organisation. 2577 (line 6) 2578* Compaq C++: Notes for Particular Systems. 2579 (line 25) 2580* Comparison functions: Integer Comparisons. (line 6) 2581* Comparison functions <1>: Comparing Rationals. (line 6) 2582* Comparison functions <2>: Float Comparison. (line 6) 2583* Compatibility with older versions: Compatibility with older versions. 2584 (line 6) 2585* Conditions for copying GNU MP: Copying. (line 6) 2586* Configuring GMP: Installing GMP. (line 6) 2587* Congruence algorithm: Exact Remainder. (line 30) 2588* Congruence functions: Integer Division. (line 150) 2589* Constants: Useful Macros and Constants. 2590 (line 6) 2591* Contributors: Contributors. (line 6) 2592* Conventions for parameters: Parameter Conventions. 2593 (line 6) 2594* Conventions for variables: Variable Conventions. 2595 (line 6) 2596* Conversion functions: Converting Integers. (line 6) 2597* Conversion functions <1>: Rational Conversions. 2598 (line 6) 2599* Conversion functions <2>: Converting Floats. (line 6) 2600* Copying conditions: Copying. (line 6) 2601* CPPFLAGS: Build Options. (line 204) 2602* CPU types: Introduction to GMP. (line 24) 2603* CPU types <1>: Build Options. (line 107) 2604* Cross compiling: Build Options. (line 65) 2605* Cryptography functions, low-level: Low-level Functions. (line 507) 2606* Custom allocation: Custom Allocation. (line 6) 2607* CXX: Build Options. (line 249) 2608* CXXFLAGS: Build Options. (line 249) 2609* Cygwin: Notes for Particular Systems. 2610 (line 57) 2611* Darwin: Known Build Problems. 2612 (line 51) 2613* Debugging: Debugging. (line 6) 2614* Demonstration programs: Demonstration Programs. 2615 (line 6) 2616* Digits in an integer: Miscellaneous Integer Functions. 2617 (line 23) 2618* Divisibility algorithm: Exact Remainder. (line 30) 2619* Divisibility functions: Integer Division. (line 136) 2620* Divisibility functions <1>: Integer Division. (line 150) 2621* Divisibility testing: Efficiency. (line 91) 2622* Division algorithms: Division Algorithms. (line 6) 2623* Division functions: Integer Division. (line 6) 2624* Division functions <1>: Rational Arithmetic. (line 24) 2625* Division functions <2>: Float Arithmetic. (line 33) 2626* DJGPP: Notes for Particular Systems. 2627 (line 57) 2628* DJGPP <1>: Known Build Problems. 2629 (line 18) 2630* DLLs: Notes for Particular Systems. 2631 (line 70) 2632* DocBook: Build Options. (line 340) 2633* Documentation formats: Build Options. (line 333) 2634* Documentation license: GNU Free Documentation License. 2635 (line 6) 2636* DVI: Build Options. (line 336) 2637* Efficiency: Efficiency. (line 6) 2638* Emacs: Emacs. (line 6) 2639* Exact division functions: Integer Division. (line 125) 2640* Exact remainder: Exact Remainder. (line 6) 2641* Example programs: Demonstration Programs. 2642 (line 6) 2643* Exec prefix: Build Options. (line 32) 2644* Execution profiling: Build Options. (line 317) 2645* Execution profiling <1>: Profiling. (line 6) 2646* Exponentiation functions: Integer Exponentiation. 2647 (line 6) 2648* Exponentiation functions <1>: Float Arithmetic. (line 41) 2649* Export: Integer Import and Export. 2650 (line 45) 2651* Expression parsing demo: Demonstration Programs. 2652 (line 15) 2653* Expression parsing demo <1>: Demonstration Programs. 2654 (line 17) 2655* Expression parsing demo <2>: Demonstration Programs. 2656 (line 19) 2657* Extended GCD: Number Theoretic Functions. 2658 (line 43) 2659* Factor removal functions: Number Theoretic Functions. 2660 (line 104) 2661* Factorial algorithm: Factorial Algorithm. (line 6) 2662* Factorial functions: Number Theoretic Functions. 2663 (line 112) 2664* Factorization demo: Demonstration Programs. 2665 (line 22) 2666* Fast Fourier Transform: FFT Multiplication. (line 6) 2667* Fat binary: Build Options. (line 160) 2668* FFT multiplication: Build Options. (line 307) 2669* FFT multiplication <1>: FFT Multiplication. (line 6) 2670* Fibonacci number algorithm: Fibonacci Numbers Algorithm. 2671 (line 6) 2672* Fibonacci sequence functions: Number Theoretic Functions. 2673 (line 132) 2674* Float arithmetic functions: Float Arithmetic. (line 6) 2675* Float assignment functions: Assigning Floats. (line 6) 2676* Float assignment functions <1>: Simultaneous Float Init & Assign. 2677 (line 6) 2678* Float comparison functions: Float Comparison. (line 6) 2679* Float conversion functions: Converting Floats. (line 6) 2680* Float functions: Floating-point Functions. 2681 (line 6) 2682* Float initialization functions: Initializing Floats. (line 6) 2683* Float initialization functions <1>: Simultaneous Float Init & Assign. 2684 (line 6) 2685* Float input and output functions: I/O of Floats. (line 6) 2686* Float internals: Float Internals. (line 6) 2687* Float miscellaneous functions: Miscellaneous Float Functions. 2688 (line 6) 2689* Float random number functions: Miscellaneous Float Functions. 2690 (line 27) 2691* Float rounding functions: Miscellaneous Float Functions. 2692 (line 9) 2693* Float sign tests: Float Comparison. (line 34) 2694* Floating point mode: Notes for Particular Systems. 2695 (line 34) 2696* Floating-point functions: Floating-point Functions. 2697 (line 6) 2698* Floating-point number: Nomenclature and Types. 2699 (line 21) 2700* fnccheck: Profiling. (line 77) 2701* Formatted input: Formatted Input. (line 6) 2702* Formatted output: Formatted Output. (line 6) 2703* Free Documentation License: GNU Free Documentation License. 2704 (line 6) 2705* FreeBSD: Notes for Particular Systems. 2706 (line 43) 2707* FreeBSD <1>: Notes for Particular Systems. 2708 (line 52) 2709* frexp: Converting Integers. (line 43) 2710* frexp <1>: Converting Floats. (line 24) 2711* FTP of latest version: Introduction to GMP. (line 37) 2712* Function classes: Function Classes. (line 6) 2713* FunctionCheck: Profiling. (line 77) 2714* GCC Checker: Debugging. (line 114) 2715* GCD algorithms: Greatest Common Divisor Algorithms. 2716 (line 6) 2717* GCD extended: Number Theoretic Functions. 2718 (line 43) 2719* GCD functions: Number Theoretic Functions. 2720 (line 26) 2721* GDB: Debugging. (line 57) 2722* Generic C: Build Options. (line 151) 2723* GMP Perl module: Demonstration Programs. 2724 (line 28) 2725* GMP version number: Useful Macros and Constants. 2726 (line 12) 2727* gmp.h: Headers and Libraries. 2728 (line 6) 2729* gmpxx.h: C++ Interface General. 2730 (line 8) 2731* GNU Debugger: Debugging. (line 57) 2732* GNU Free Documentation License: GNU Free Documentation License. 2733 (line 6) 2734* GNU strip: Known Build Problems. 2735 (line 28) 2736* gprof: Profiling. (line 41) 2737* Greatest common divisor algorithms: Greatest Common Divisor Algorithms. 2738 (line 6) 2739* Greatest common divisor functions: Number Theoretic Functions. 2740 (line 26) 2741* Hardware floating point mode: Notes for Particular Systems. 2742 (line 34) 2743* Headers: Headers and Libraries. 2744 (line 6) 2745* Heap problems: Debugging. (line 23) 2746* Home page: Introduction to GMP. (line 33) 2747* Host system: Build Options. (line 65) 2748* HP-UX: ABI and ISA. (line 76) 2749* HP-UX <1>: ABI and ISA. (line 114) 2750* HPPA: ABI and ISA. (line 76) 2751* I/O functions: I/O of Integers. (line 6) 2752* I/O functions <1>: I/O of Rationals. (line 6) 2753* I/O functions <2>: I/O of Floats. (line 6) 2754* i386: Notes for Particular Systems. 2755 (line 150) 2756* IA-64: ABI and ISA. (line 114) 2757* Import: Integer Import and Export. 2758 (line 11) 2759* In-place operations: Efficiency. (line 57) 2760* Include files: Headers and Libraries. 2761 (line 6) 2762* info-lookup-symbol: Emacs. (line 6) 2763* Initialization functions: Initializing Integers. 2764 (line 6) 2765* Initialization functions <1>: Simultaneous Integer Init & Assign. 2766 (line 6) 2767* Initialization functions <2>: Initializing Rationals. 2768 (line 6) 2769* Initialization functions <3>: Initializing Floats. (line 6) 2770* Initialization functions <4>: Simultaneous Float Init & Assign. 2771 (line 6) 2772* Initialization functions <5>: Random State Initialization. 2773 (line 6) 2774* Initializing and clearing: Efficiency. (line 21) 2775* Input functions: I/O of Integers. (line 6) 2776* Input functions <1>: I/O of Rationals. (line 6) 2777* Input functions <2>: I/O of Floats. (line 6) 2778* Input functions <3>: Formatted Input Functions. 2779 (line 6) 2780* Install prefix: Build Options. (line 32) 2781* Installing GMP: Installing GMP. (line 6) 2782* Instruction Set Architecture: ABI and ISA. (line 6) 2783* instrument-functions: Profiling. (line 66) 2784* Integer: Nomenclature and Types. 2785 (line 6) 2786* Integer arithmetic functions: Integer Arithmetic. (line 6) 2787* Integer assignment functions: Assigning Integers. (line 6) 2788* Integer assignment functions <1>: Simultaneous Integer Init & Assign. 2789 (line 6) 2790* Integer bit manipulation functions: Integer Logic and Bit Fiddling. 2791 (line 6) 2792* Integer comparison functions: Integer Comparisons. (line 6) 2793* Integer conversion functions: Converting Integers. (line 6) 2794* Integer division functions: Integer Division. (line 6) 2795* Integer exponentiation functions: Integer Exponentiation. 2796 (line 6) 2797* Integer export: Integer Import and Export. 2798 (line 45) 2799* Integer functions: Integer Functions. (line 6) 2800* Integer import: Integer Import and Export. 2801 (line 11) 2802* Integer initialization functions: Initializing Integers. 2803 (line 6) 2804* Integer initialization functions <1>: Simultaneous Integer Init & Assign. 2805 (line 6) 2806* Integer input and output functions: I/O of Integers. (line 6) 2807* Integer internals: Integer Internals. (line 6) 2808* Integer logical functions: Integer Logic and Bit Fiddling. 2809 (line 6) 2810* Integer miscellaneous functions: Miscellaneous Integer Functions. 2811 (line 6) 2812* Integer random number functions: Integer Random Numbers. 2813 (line 6) 2814* Integer root functions: Integer Roots. (line 6) 2815* Integer sign tests: Integer Comparisons. (line 28) 2816* Integer special functions: Integer Special Functions. 2817 (line 6) 2818* Interix: Notes for Particular Systems. 2819 (line 65) 2820* Internals: Internals. (line 6) 2821* Introduction: Introduction to GMP. (line 6) 2822* Inverse modulo functions: Number Theoretic Functions. 2823 (line 70) 2824* IRIX: ABI and ISA. (line 139) 2825* IRIX <1>: Known Build Problems. 2826 (line 38) 2827* ISA: ABI and ISA. (line 6) 2828* istream input: C++ Formatted Input. (line 6) 2829* Jacobi symbol algorithm: Jacobi Symbol. (line 6) 2830* Jacobi symbol functions: Number Theoretic Functions. 2831 (line 79) 2832* Karatsuba multiplication: Karatsuba Multiplication. 2833 (line 6) 2834* Karatsuba square root algorithm: Square Root Algorithm. 2835 (line 6) 2836* Kronecker symbol functions: Number Theoretic Functions. 2837 (line 91) 2838* Language bindings: Language Bindings. (line 6) 2839* Latest version of GMP: Introduction to GMP. (line 37) 2840* LCM functions: Number Theoretic Functions. 2841 (line 64) 2842* Least common multiple functions: Number Theoretic Functions. 2843 (line 64) 2844* Legendre symbol functions: Number Theoretic Functions. 2845 (line 82) 2846* libgmp: Headers and Libraries. 2847 (line 22) 2848* libgmpxx: Headers and Libraries. 2849 (line 27) 2850* Libraries: Headers and Libraries. 2851 (line 22) 2852* Libtool: Headers and Libraries. 2853 (line 33) 2854* Libtool versioning: Notes for Package Builds. 2855 (line 9) 2856* License conditions: Copying. (line 6) 2857* Limb: Nomenclature and Types. 2858 (line 31) 2859* Limb size: Useful Macros and Constants. 2860 (line 7) 2861* Linear congruential algorithm: Random Number Algorithms. 2862 (line 25) 2863* Linear congruential random numbers: Random State Initialization. 2864 (line 18) 2865* Linear congruential random numbers <1>: Random State Initialization. 2866 (line 32) 2867* Linking: Headers and Libraries. 2868 (line 22) 2869* Logical functions: Integer Logic and Bit Fiddling. 2870 (line 6) 2871* Low-level functions: Low-level Functions. (line 6) 2872* Low-level functions for cryptography: Low-level Functions. (line 507) 2873* Lucas number algorithm: Lucas Numbers Algorithm. 2874 (line 6) 2875* Lucas number functions: Number Theoretic Functions. 2876 (line 143) 2877* MacOS X: Known Build Problems. 2878 (line 51) 2879* Mailing lists: Introduction to GMP. (line 44) 2880* Malloc debugger: Debugging. (line 29) 2881* Malloc problems: Debugging. (line 23) 2882* Memory allocation: Custom Allocation. (line 6) 2883* Memory management: Memory Management. (line 6) 2884* Mersenne twister algorithm: Random Number Algorithms. 2885 (line 17) 2886* Mersenne twister random numbers: Random State Initialization. 2887 (line 13) 2888* MINGW: Notes for Particular Systems. 2889 (line 57) 2890* MIPS: ABI and ISA. (line 139) 2891* Miscellaneous float functions: Miscellaneous Float Functions. 2892 (line 6) 2893* Miscellaneous integer functions: Miscellaneous Integer Functions. 2894 (line 6) 2895* MMX: Notes for Particular Systems. 2896 (line 156) 2897* Modular inverse functions: Number Theoretic Functions. 2898 (line 70) 2899* Most significant bit: Miscellaneous Integer Functions. 2900 (line 34) 2901* MPN_PATH: Build Options. (line 321) 2902* MS Windows: Notes for Particular Systems. 2903 (line 57) 2904* MS Windows <1>: Notes for Particular Systems. 2905 (line 70) 2906* MS-DOS: Notes for Particular Systems. 2907 (line 57) 2908* Multi-threading: Reentrancy. (line 6) 2909* Multiplication algorithms: Multiplication Algorithms. 2910 (line 6) 2911* Nails: Low-level Functions. (line 685) 2912* Native compilation: Build Options. (line 51) 2913* NetBSD: Notes for Particular Systems. 2914 (line 100) 2915* NeXT: Known Build Problems. 2916 (line 57) 2917* Next prime function: Number Theoretic Functions. 2918 (line 19) 2919* Nomenclature: Nomenclature and Types. 2920 (line 6) 2921* Non-Unix systems: Build Options. (line 11) 2922* Nth root algorithm: Nth Root Algorithm. (line 6) 2923* Number sequences: Efficiency. (line 145) 2924* Number theoretic functions: Number Theoretic Functions. 2925 (line 6) 2926* Numerator and denominator: Applying Integer Functions. 2927 (line 6) 2928* obstack output: Formatted Output Functions. 2929 (line 79) 2930* OpenBSD: Notes for Particular Systems. 2931 (line 109) 2932* Optimizing performance: Performance optimization. 2933 (line 6) 2934* ostream output: C++ Formatted Output. 2935 (line 6) 2936* Other languages: Language Bindings. (line 6) 2937* Output functions: I/O of Integers. (line 6) 2938* Output functions <1>: I/O of Rationals. (line 6) 2939* Output functions <2>: I/O of Floats. (line 6) 2940* Output functions <3>: Formatted Output Functions. 2941 (line 6) 2942* Packaged builds: Notes for Package Builds. 2943 (line 6) 2944* Parameter conventions: Parameter Conventions. 2945 (line 6) 2946* Parsing expressions demo: Demonstration Programs. 2947 (line 15) 2948* Parsing expressions demo <1>: Demonstration Programs. 2949 (line 17) 2950* Parsing expressions demo <2>: Demonstration Programs. 2951 (line 19) 2952* Particular systems: Notes for Particular Systems. 2953 (line 6) 2954* Past GMP versions: Compatibility with older versions. 2955 (line 6) 2956* PDF: Build Options. (line 336) 2957* Perfect power algorithm: Perfect Power Algorithm. 2958 (line 6) 2959* Perfect power functions: Integer Roots. (line 28) 2960* Perfect square algorithm: Perfect Square Algorithm. 2961 (line 6) 2962* Perfect square functions: Integer Roots. (line 37) 2963* perl: Demonstration Programs. 2964 (line 28) 2965* Perl module: Demonstration Programs. 2966 (line 28) 2967* Postscript: Build Options. (line 336) 2968* Power/PowerPC: Notes for Particular Systems. 2969 (line 115) 2970* Power/PowerPC <1>: Known Build Problems. 2971 (line 63) 2972* Powering algorithms: Powering Algorithms. (line 6) 2973* Powering functions: Integer Exponentiation. 2974 (line 6) 2975* Powering functions <1>: Float Arithmetic. (line 41) 2976* PowerPC: ABI and ISA. (line 173) 2977* Precision of floats: Floating-point Functions. 2978 (line 6) 2979* Precision of hardware floating point: Notes for Particular Systems. 2980 (line 34) 2981* Prefix: Build Options. (line 32) 2982* Prime testing algorithms: Prime Testing Algorithm. 2983 (line 6) 2984* Prime testing functions: Number Theoretic Functions. 2985 (line 7) 2986* Primorial functions: Number Theoretic Functions. 2987 (line 117) 2988* printf formatted output: Formatted Output. (line 6) 2989* Probable prime testing functions: Number Theoretic Functions. 2990 (line 7) 2991* prof: Profiling. (line 24) 2992* Profiling: Profiling. (line 6) 2993* Radix conversion algorithms: Radix Conversion Algorithms. 2994 (line 6) 2995* Random number algorithms: Random Number Algorithms. 2996 (line 6) 2997* Random number functions: Integer Random Numbers. 2998 (line 6) 2999* Random number functions <1>: Miscellaneous Float Functions. 3000 (line 27) 3001* Random number functions <2>: Random Number Functions. 3002 (line 6) 3003* Random number seeding: Random State Seeding. 3004 (line 6) 3005* Random number state: Random State Initialization. 3006 (line 6) 3007* Random state: Nomenclature and Types. 3008 (line 46) 3009* Rational arithmetic: Efficiency. (line 111) 3010* Rational arithmetic functions: Rational Arithmetic. (line 6) 3011* Rational assignment functions: Initializing Rationals. 3012 (line 6) 3013* Rational comparison functions: Comparing Rationals. (line 6) 3014* Rational conversion functions: Rational Conversions. 3015 (line 6) 3016* Rational initialization functions: Initializing Rationals. 3017 (line 6) 3018* Rational input and output functions: I/O of Rationals. (line 6) 3019* Rational internals: Rational Internals. (line 6) 3020* Rational number: Nomenclature and Types. 3021 (line 16) 3022* Rational number functions: Rational Number Functions. 3023 (line 6) 3024* Rational numerator and denominator: Applying Integer Functions. 3025 (line 6) 3026* Rational sign tests: Comparing Rationals. (line 28) 3027* Raw output internals: Raw Output Internals. 3028 (line 6) 3029* Reallocations: Efficiency. (line 30) 3030* Reentrancy: Reentrancy. (line 6) 3031* References: References. (line 5) 3032* Remove factor functions: Number Theoretic Functions. 3033 (line 104) 3034* Reporting bugs: Reporting Bugs. (line 6) 3035* Root extraction algorithm: Nth Root Algorithm. (line 6) 3036* Root extraction algorithms: Root Extraction Algorithms. 3037 (line 6) 3038* Root extraction functions: Integer Roots. (line 6) 3039* Root extraction functions <1>: Float Arithmetic. (line 37) 3040* Root testing functions: Integer Roots. (line 28) 3041* Root testing functions <1>: Integer Roots. (line 37) 3042* Rounding functions: Miscellaneous Float Functions. 3043 (line 9) 3044* Sample programs: Demonstration Programs. 3045 (line 6) 3046* Scan bit functions: Integer Logic and Bit Fiddling. 3047 (line 39) 3048* scanf formatted input: Formatted Input. (line 6) 3049* SCO: Known Build Problems. 3050 (line 38) 3051* Seeding random numbers: Random State Seeding. 3052 (line 6) 3053* Segmentation violation: Debugging. (line 7) 3054* Sequent Symmetry: Known Build Problems. 3055 (line 68) 3056* Services for Unix: Notes for Particular Systems. 3057 (line 65) 3058* Shared library versioning: Notes for Package Builds. 3059 (line 9) 3060* Sign tests: Integer Comparisons. (line 28) 3061* Sign tests <1>: Comparing Rationals. (line 28) 3062* Sign tests <2>: Float Comparison. (line 34) 3063* Size in digits: Miscellaneous Integer Functions. 3064 (line 23) 3065* Small operands: Efficiency. (line 7) 3066* Solaris: ABI and ISA. (line 204) 3067* Solaris <1>: Known Build Problems. 3068 (line 72) 3069* Solaris <2>: Known Build Problems. 3070 (line 77) 3071* Sparc: Notes for Particular Systems. 3072 (line 127) 3073* Sparc <1>: Notes for Particular Systems. 3074 (line 132) 3075* Sparc V9: ABI and ISA. (line 204) 3076* Special integer functions: Integer Special Functions. 3077 (line 6) 3078* Square root algorithm: Square Root Algorithm. 3079 (line 6) 3080* SSE2: Notes for Particular Systems. 3081 (line 156) 3082* Stack backtrace: Debugging. (line 49) 3083* Stack overflow: Build Options. (line 273) 3084* Stack overflow <1>: Debugging. (line 7) 3085* Static linking: Efficiency. (line 14) 3086* stdarg.h: Headers and Libraries. 3087 (line 17) 3088* stdio.h: Headers and Libraries. 3089 (line 11) 3090* Stripped libraries: Known Build Problems. 3091 (line 28) 3092* Sun: ABI and ISA. (line 204) 3093* SunOS: Notes for Particular Systems. 3094 (line 144) 3095* Systems: Notes for Particular Systems. 3096 (line 6) 3097* Temporary memory: Build Options. (line 273) 3098* Texinfo: Build Options. (line 333) 3099* Text input/output: Efficiency. (line 151) 3100* Thread safety: Reentrancy. (line 6) 3101* Toom multiplication: Toom 3-Way Multiplication. 3102 (line 6) 3103* Toom multiplication <1>: Toom 4-Way Multiplication. 3104 (line 6) 3105* Toom multiplication <2>: Higher degree Toom'n'half. 3106 (line 6) 3107* Toom multiplication <3>: Other Multiplication. 3108 (line 6) 3109* Types: Nomenclature and Types. 3110 (line 6) 3111* ui and si functions: Efficiency. (line 50) 3112* Unbalanced multiplication: Unbalanced Multiplication. 3113 (line 6) 3114* Upward compatibility: Compatibility with older versions. 3115 (line 6) 3116* Useful macros and constants: Useful Macros and Constants. 3117 (line 6) 3118* User-defined precision: Floating-point Functions. 3119 (line 6) 3120* Valgrind: Debugging. (line 129) 3121* Variable conventions: Variable Conventions. 3122 (line 6) 3123* Version number: Useful Macros and Constants. 3124 (line 12) 3125* Web page: Introduction to GMP. (line 33) 3126* Windows: Notes for Particular Systems. 3127 (line 57) 3128* Windows <1>: Notes for Particular Systems. 3129 (line 70) 3130* x86: Notes for Particular Systems. 3131 (line 150) 3132* x87: Notes for Particular Systems. 3133 (line 34) 3134* XML: Build Options. (line 340) 3135 3136 3137File: gmp.info, Node: Function Index, Prev: Concept Index, Up: Top 3138 3139Function and Type Index 3140*********************** 3141 3142[index] 3143* Menu: 3144 3145* _mpz_realloc: Integer Special Functions. 3146 (line 13) 3147* __GMP_CC: Useful Macros and Constants. 3148 (line 22) 3149* __GMP_CFLAGS: Useful Macros and Constants. 3150 (line 23) 3151* __GNU_MP_VERSION: Useful Macros and Constants. 3152 (line 9) 3153* __GNU_MP_VERSION_MINOR: Useful Macros and Constants. 3154 (line 10) 3155* __GNU_MP_VERSION_PATCHLEVEL: Useful Macros and Constants. 3156 (line 11) 3157* abs: C++ Interface Integers. 3158 (line 46) 3159* abs <1>: C++ Interface Rationals. 3160 (line 47) 3161* abs <2>: C++ Interface Floats. 3162 (line 82) 3163* ceil: C++ Interface Floats. 3164 (line 83) 3165* cmp: C++ Interface Integers. 3166 (line 47) 3167* cmp <1>: C++ Interface Integers. 3168 (line 48) 3169* cmp <2>: C++ Interface Rationals. 3170 (line 48) 3171* cmp <3>: C++ Interface Rationals. 3172 (line 49) 3173* cmp <4>: C++ Interface Floats. 3174 (line 84) 3175* cmp <5>: C++ Interface Floats. 3176 (line 85) 3177* floor: C++ Interface Floats. 3178 (line 95) 3179* gcd: C++ Interface Integers. 3180 (line 68) 3181* gmp_asprintf: Formatted Output Functions. 3182 (line 63) 3183* gmp_errno: Random State Initialization. 3184 (line 56) 3185* GMP_ERROR_INVALID_ARGUMENT: Random State Initialization. 3186 (line 56) 3187* GMP_ERROR_UNSUPPORTED_ARGUMENT: Random State Initialization. 3188 (line 56) 3189* gmp_fprintf: Formatted Output Functions. 3190 (line 28) 3191* gmp_fscanf: Formatted Input Functions. 3192 (line 24) 3193* GMP_LIMB_BITS: Low-level Functions. (line 713) 3194* GMP_NAIL_BITS: Low-level Functions. (line 711) 3195* GMP_NAIL_MASK: Low-level Functions. (line 721) 3196* GMP_NUMB_BITS: Low-level Functions. (line 712) 3197* GMP_NUMB_MASK: Low-level Functions. (line 722) 3198* GMP_NUMB_MAX: Low-level Functions. (line 730) 3199* gmp_obstack_printf: Formatted Output Functions. 3200 (line 75) 3201* gmp_obstack_vprintf: Formatted Output Functions. 3202 (line 77) 3203* gmp_printf: Formatted Output Functions. 3204 (line 23) 3205* gmp_randclass: C++ Interface Random Numbers. 3206 (line 6) 3207* gmp_randclass::get_f: C++ Interface Random Numbers. 3208 (line 44) 3209* gmp_randclass::get_f <1>: C++ Interface Random Numbers. 3210 (line 45) 3211* gmp_randclass::get_z_bits: C++ Interface Random Numbers. 3212 (line 37) 3213* gmp_randclass::get_z_bits <1>: C++ Interface Random Numbers. 3214 (line 38) 3215* gmp_randclass::get_z_range: C++ Interface Random Numbers. 3216 (line 41) 3217* gmp_randclass::gmp_randclass: C++ Interface Random Numbers. 3218 (line 11) 3219* gmp_randclass::gmp_randclass <1>: C++ Interface Random Numbers. 3220 (line 26) 3221* gmp_randclass::seed: C++ Interface Random Numbers. 3222 (line 32) 3223* gmp_randclass::seed <1>: C++ Interface Random Numbers. 3224 (line 33) 3225* gmp_randclear: Random State Initialization. 3226 (line 62) 3227* gmp_randinit: Random State Initialization. 3228 (line 45) 3229* gmp_randinit_default: Random State Initialization. 3230 (line 6) 3231* gmp_randinit_lc_2exp: Random State Initialization. 3232 (line 16) 3233* gmp_randinit_lc_2exp_size: Random State Initialization. 3234 (line 30) 3235* gmp_randinit_mt: Random State Initialization. 3236 (line 12) 3237* gmp_randinit_set: Random State Initialization. 3238 (line 41) 3239* gmp_randseed: Random State Seeding. 3240 (line 6) 3241* gmp_randseed_ui: Random State Seeding. 3242 (line 8) 3243* gmp_randstate_t: Nomenclature and Types. 3244 (line 46) 3245* GMP_RAND_ALG_DEFAULT: Random State Initialization. 3246 (line 50) 3247* GMP_RAND_ALG_LC: Random State Initialization. 3248 (line 50) 3249* gmp_scanf: Formatted Input Functions. 3250 (line 20) 3251* gmp_snprintf: Formatted Output Functions. 3252 (line 44) 3253* gmp_sprintf: Formatted Output Functions. 3254 (line 33) 3255* gmp_sscanf: Formatted Input Functions. 3256 (line 28) 3257* gmp_urandomb_ui: Random State Miscellaneous. 3258 (line 6) 3259* gmp_urandomm_ui: Random State Miscellaneous. 3260 (line 12) 3261* gmp_vasprintf: Formatted Output Functions. 3262 (line 64) 3263* gmp_version: Useful Macros and Constants. 3264 (line 18) 3265* gmp_vfprintf: Formatted Output Functions. 3266 (line 29) 3267* gmp_vfscanf: Formatted Input Functions. 3268 (line 25) 3269* gmp_vprintf: Formatted Output Functions. 3270 (line 24) 3271* gmp_vscanf: Formatted Input Functions. 3272 (line 21) 3273* gmp_vsnprintf: Formatted Output Functions. 3274 (line 46) 3275* gmp_vsprintf: Formatted Output Functions. 3276 (line 34) 3277* gmp_vsscanf: Formatted Input Functions. 3278 (line 29) 3279* hypot: C++ Interface Floats. 3280 (line 96) 3281* lcm: C++ Interface Integers. 3282 (line 69) 3283* mpf_abs: Float Arithmetic. (line 46) 3284* mpf_add: Float Arithmetic. (line 6) 3285* mpf_add_ui: Float Arithmetic. (line 7) 3286* mpf_ceil: Miscellaneous Float Functions. 3287 (line 6) 3288* mpf_class: C++ Interface General. 3289 (line 19) 3290* mpf_class::fits_sint_p: C++ Interface Floats. 3291 (line 87) 3292* mpf_class::fits_slong_p: C++ Interface Floats. 3293 (line 88) 3294* mpf_class::fits_sshort_p: C++ Interface Floats. 3295 (line 89) 3296* mpf_class::fits_uint_p: C++ Interface Floats. 3297 (line 91) 3298* mpf_class::fits_ulong_p: C++ Interface Floats. 3299 (line 92) 3300* mpf_class::fits_ushort_p: C++ Interface Floats. 3301 (line 93) 3302* mpf_class::get_d: C++ Interface Floats. 3303 (line 98) 3304* mpf_class::get_mpf_t: C++ Interface General. 3305 (line 65) 3306* mpf_class::get_prec: C++ Interface Floats. 3307 (line 120) 3308* mpf_class::get_si: C++ Interface Floats. 3309 (line 99) 3310* mpf_class::get_str: C++ Interface Floats. 3311 (line 100) 3312* mpf_class::get_ui: C++ Interface Floats. 3313 (line 102) 3314* mpf_class::mpf_class: C++ Interface Floats. 3315 (line 11) 3316* mpf_class::mpf_class <1>: C++ Interface Floats. 3317 (line 12) 3318* mpf_class::mpf_class <2>: C++ Interface Floats. 3319 (line 32) 3320* mpf_class::mpf_class <3>: C++ Interface Floats. 3321 (line 33) 3322* mpf_class::mpf_class <4>: C++ Interface Floats. 3323 (line 41) 3324* mpf_class::mpf_class <5>: C++ Interface Floats. 3325 (line 42) 3326* mpf_class::mpf_class <6>: C++ Interface Floats. 3327 (line 44) 3328* mpf_class::mpf_class <7>: C++ Interface Floats. 3329 (line 45) 3330* mpf_class::operator=: C++ Interface Floats. 3331 (line 59) 3332* mpf_class::set_prec: C++ Interface Floats. 3333 (line 121) 3334* mpf_class::set_prec_raw: C++ Interface Floats. 3335 (line 122) 3336* mpf_class::set_str: C++ Interface Floats. 3337 (line 104) 3338* mpf_class::set_str <1>: C++ Interface Floats. 3339 (line 105) 3340* mpf_class::swap: C++ Interface Floats. 3341 (line 109) 3342* mpf_clear: Initializing Floats. (line 36) 3343* mpf_clears: Initializing Floats. (line 40) 3344* mpf_cmp: Float Comparison. (line 6) 3345* mpf_cmp_d: Float Comparison. (line 8) 3346* mpf_cmp_si: Float Comparison. (line 10) 3347* mpf_cmp_ui: Float Comparison. (line 9) 3348* mpf_cmp_z: Float Comparison. (line 7) 3349* mpf_div: Float Arithmetic. (line 28) 3350* mpf_div_2exp: Float Arithmetic. (line 53) 3351* mpf_div_ui: Float Arithmetic. (line 31) 3352* mpf_eq: Float Comparison. (line 17) 3353* mpf_fits_sint_p: Miscellaneous Float Functions. 3354 (line 19) 3355* mpf_fits_slong_p: Miscellaneous Float Functions. 3356 (line 17) 3357* mpf_fits_sshort_p: Miscellaneous Float Functions. 3358 (line 21) 3359* mpf_fits_uint_p: Miscellaneous Float Functions. 3360 (line 18) 3361* mpf_fits_ulong_p: Miscellaneous Float Functions. 3362 (line 16) 3363* mpf_fits_ushort_p: Miscellaneous Float Functions. 3364 (line 20) 3365* mpf_floor: Miscellaneous Float Functions. 3366 (line 7) 3367* mpf_get_d: Converting Floats. (line 6) 3368* mpf_get_default_prec: Initializing Floats. (line 11) 3369* mpf_get_d_2exp: Converting Floats. (line 15) 3370* mpf_get_prec: Initializing Floats. (line 61) 3371* mpf_get_si: Converting Floats. (line 27) 3372* mpf_get_str: Converting Floats. (line 36) 3373* mpf_get_ui: Converting Floats. (line 28) 3374* mpf_init: Initializing Floats. (line 18) 3375* mpf_init2: Initializing Floats. (line 25) 3376* mpf_inits: Initializing Floats. (line 30) 3377* mpf_init_set: Simultaneous Float Init & Assign. 3378 (line 15) 3379* mpf_init_set_d: Simultaneous Float Init & Assign. 3380 (line 18) 3381* mpf_init_set_si: Simultaneous Float Init & Assign. 3382 (line 17) 3383* mpf_init_set_str: Simultaneous Float Init & Assign. 3384 (line 24) 3385* mpf_init_set_ui: Simultaneous Float Init & Assign. 3386 (line 16) 3387* mpf_inp_str: I/O of Floats. (line 38) 3388* mpf_integer_p: Miscellaneous Float Functions. 3389 (line 13) 3390* mpf_mul: Float Arithmetic. (line 18) 3391* mpf_mul_2exp: Float Arithmetic. (line 49) 3392* mpf_mul_ui: Float Arithmetic. (line 19) 3393* mpf_neg: Float Arithmetic. (line 43) 3394* mpf_out_str: I/O of Floats. (line 17) 3395* mpf_pow_ui: Float Arithmetic. (line 39) 3396* mpf_random2: Miscellaneous Float Functions. 3397 (line 35) 3398* mpf_reldiff: Float Comparison. (line 28) 3399* mpf_set: Assigning Floats. (line 9) 3400* mpf_set_d: Assigning Floats. (line 12) 3401* mpf_set_default_prec: Initializing Floats. (line 6) 3402* mpf_set_prec: Initializing Floats. (line 64) 3403* mpf_set_prec_raw: Initializing Floats. (line 71) 3404* mpf_set_q: Assigning Floats. (line 14) 3405* mpf_set_si: Assigning Floats. (line 11) 3406* mpf_set_str: Assigning Floats. (line 17) 3407* mpf_set_ui: Assigning Floats. (line 10) 3408* mpf_set_z: Assigning Floats. (line 13) 3409* mpf_sgn: Float Comparison. (line 33) 3410* mpf_sqrt: Float Arithmetic. (line 35) 3411* mpf_sqrt_ui: Float Arithmetic. (line 36) 3412* mpf_sub: Float Arithmetic. (line 11) 3413* mpf_sub_ui: Float Arithmetic. (line 14) 3414* mpf_swap: Assigning Floats. (line 50) 3415* mpf_t: Nomenclature and Types. 3416 (line 21) 3417* mpf_trunc: Miscellaneous Float Functions. 3418 (line 8) 3419* mpf_ui_div: Float Arithmetic. (line 29) 3420* mpf_ui_sub: Float Arithmetic. (line 12) 3421* mpf_urandomb: Miscellaneous Float Functions. 3422 (line 25) 3423* mpn_add: Low-level Functions. (line 67) 3424* mpn_addmul_1: Low-level Functions. (line 148) 3425* mpn_add_1: Low-level Functions. (line 62) 3426* mpn_add_n: Low-level Functions. (line 52) 3427* mpn_andn_n: Low-level Functions. (line 462) 3428* mpn_and_n: Low-level Functions. (line 447) 3429* mpn_cmp: Low-level Functions. (line 293) 3430* mpn_cnd_add_n: Low-level Functions. (line 540) 3431* mpn_cnd_sub_n: Low-level Functions. (line 542) 3432* mpn_cnd_swap: Low-level Functions. (line 567) 3433* mpn_com: Low-level Functions. (line 487) 3434* mpn_copyd: Low-level Functions. (line 496) 3435* mpn_copyi: Low-level Functions. (line 492) 3436* mpn_divexact_1: Low-level Functions. (line 231) 3437* mpn_divexact_by3: Low-level Functions. (line 238) 3438* mpn_divexact_by3c: Low-level Functions. (line 240) 3439* mpn_divmod: Low-level Functions. (line 226) 3440* mpn_divmod_1: Low-level Functions. (line 210) 3441* mpn_divrem: Low-level Functions. (line 183) 3442* mpn_divrem_1: Low-level Functions. (line 208) 3443* mpn_gcd: Low-level Functions. (line 301) 3444* mpn_gcdext: Low-level Functions. (line 316) 3445* mpn_gcd_1: Low-level Functions. (line 311) 3446* mpn_get_str: Low-level Functions. (line 371) 3447* mpn_hamdist: Low-level Functions. (line 436) 3448* mpn_iorn_n: Low-level Functions. (line 467) 3449* mpn_ior_n: Low-level Functions. (line 452) 3450* mpn_lshift: Low-level Functions. (line 269) 3451* mpn_mod_1: Low-level Functions. (line 264) 3452* mpn_mul: Low-level Functions. (line 114) 3453* mpn_mul_1: Low-level Functions. (line 133) 3454* mpn_mul_n: Low-level Functions. (line 103) 3455* mpn_nand_n: Low-level Functions. (line 472) 3456* mpn_neg: Low-level Functions. (line 96) 3457* mpn_nior_n: Low-level Functions. (line 477) 3458* mpn_perfect_square_p: Low-level Functions. (line 442) 3459* mpn_popcount: Low-level Functions. (line 432) 3460* mpn_random: Low-level Functions. (line 422) 3461* mpn_random2: Low-level Functions. (line 423) 3462* mpn_rshift: Low-level Functions. (line 281) 3463* mpn_scan0: Low-level Functions. (line 406) 3464* mpn_scan1: Low-level Functions. (line 414) 3465* mpn_sec_add_1: Low-level Functions. (line 553) 3466* mpn_sec_div_qr: Low-level Functions. (line 629) 3467* mpn_sec_div_qr_itch: Low-level Functions. (line 632) 3468* mpn_sec_div_r: Low-level Functions. (line 648) 3469* mpn_sec_div_r_itch: Low-level Functions. (line 650) 3470* mpn_sec_invert: Low-level Functions. (line 664) 3471* mpn_sec_invert_itch: Low-level Functions. (line 666) 3472* mpn_sec_mul: Low-level Functions. (line 574) 3473* mpn_sec_mul_itch: Low-level Functions. (line 577) 3474* mpn_sec_powm: Low-level Functions. (line 604) 3475* mpn_sec_powm_itch: Low-level Functions. (line 607) 3476* mpn_sec_sqr: Low-level Functions. (line 590) 3477* mpn_sec_sqr_itch: Low-level Functions. (line 592) 3478* mpn_sec_sub_1: Low-level Functions. (line 555) 3479* mpn_sec_tabselect: Low-level Functions. (line 621) 3480* mpn_set_str: Low-level Functions. (line 386) 3481* mpn_sizeinbase: Low-level Functions. (line 364) 3482* mpn_sqr: Low-level Functions. (line 125) 3483* mpn_sqrtrem: Low-level Functions. (line 346) 3484* mpn_sub: Low-level Functions. (line 88) 3485* mpn_submul_1: Low-level Functions. (line 160) 3486* mpn_sub_1: Low-level Functions. (line 83) 3487* mpn_sub_n: Low-level Functions. (line 74) 3488* mpn_tdiv_qr: Low-level Functions. (line 172) 3489* mpn_xnor_n: Low-level Functions. (line 482) 3490* mpn_xor_n: Low-level Functions. (line 457) 3491* mpn_zero: Low-level Functions. (line 500) 3492* mpn_zero_p: Low-level Functions. (line 298) 3493* mpq_abs: Rational Arithmetic. (line 33) 3494* mpq_add: Rational Arithmetic. (line 6) 3495* mpq_canonicalize: Rational Number Functions. 3496 (line 21) 3497* mpq_class: C++ Interface General. 3498 (line 18) 3499* mpq_class::canonicalize: C++ Interface Rationals. 3500 (line 41) 3501* mpq_class::get_d: C++ Interface Rationals. 3502 (line 51) 3503* mpq_class::get_den: C++ Interface Rationals. 3504 (line 67) 3505* mpq_class::get_den_mpz_t: C++ Interface Rationals. 3506 (line 77) 3507* mpq_class::get_mpq_t: C++ Interface General. 3508 (line 64) 3509* mpq_class::get_num: C++ Interface Rationals. 3510 (line 66) 3511* mpq_class::get_num_mpz_t: C++ Interface Rationals. 3512 (line 76) 3513* mpq_class::get_str: C++ Interface Rationals. 3514 (line 52) 3515* mpq_class::mpq_class: C++ Interface Rationals. 3516 (line 9) 3517* mpq_class::mpq_class <1>: C++ Interface Rationals. 3518 (line 10) 3519* mpq_class::mpq_class <2>: C++ Interface Rationals. 3520 (line 21) 3521* mpq_class::mpq_class <3>: C++ Interface Rationals. 3522 (line 26) 3523* mpq_class::mpq_class <4>: C++ Interface Rationals. 3524 (line 28) 3525* mpq_class::set_str: C++ Interface Rationals. 3526 (line 54) 3527* mpq_class::set_str <1>: C++ Interface Rationals. 3528 (line 55) 3529* mpq_class::swap: C++ Interface Rationals. 3530 (line 58) 3531* mpq_clear: Initializing Rationals. 3532 (line 15) 3533* mpq_clears: Initializing Rationals. 3534 (line 19) 3535* mpq_cmp: Comparing Rationals. (line 6) 3536* mpq_cmp_si: Comparing Rationals. (line 16) 3537* mpq_cmp_ui: Comparing Rationals. (line 14) 3538* mpq_cmp_z: Comparing Rationals. (line 7) 3539* mpq_denref: Applying Integer Functions. 3540 (line 16) 3541* mpq_div: Rational Arithmetic. (line 22) 3542* mpq_div_2exp: Rational Arithmetic. (line 26) 3543* mpq_equal: Comparing Rationals. (line 33) 3544* mpq_get_d: Rational Conversions. 3545 (line 6) 3546* mpq_get_den: Applying Integer Functions. 3547 (line 22) 3548* mpq_get_num: Applying Integer Functions. 3549 (line 21) 3550* mpq_get_str: Rational Conversions. 3551 (line 21) 3552* mpq_init: Initializing Rationals. 3553 (line 6) 3554* mpq_inits: Initializing Rationals. 3555 (line 11) 3556* mpq_inp_str: I/O of Rationals. (line 26) 3557* mpq_inv: Rational Arithmetic. (line 36) 3558* mpq_mul: Rational Arithmetic. (line 14) 3559* mpq_mul_2exp: Rational Arithmetic. (line 18) 3560* mpq_neg: Rational Arithmetic. (line 30) 3561* mpq_numref: Applying Integer Functions. 3562 (line 15) 3563* mpq_out_str: I/O of Rationals. (line 17) 3564* mpq_set: Initializing Rationals. 3565 (line 23) 3566* mpq_set_d: Rational Conversions. 3567 (line 16) 3568* mpq_set_den: Applying Integer Functions. 3569 (line 24) 3570* mpq_set_f: Rational Conversions. 3571 (line 17) 3572* mpq_set_num: Applying Integer Functions. 3573 (line 23) 3574* mpq_set_si: Initializing Rationals. 3575 (line 29) 3576* mpq_set_str: Initializing Rationals. 3577 (line 35) 3578* mpq_set_ui: Initializing Rationals. 3579 (line 27) 3580* mpq_set_z: Initializing Rationals. 3581 (line 24) 3582* mpq_sgn: Comparing Rationals. (line 27) 3583* mpq_sub: Rational Arithmetic. (line 10) 3584* mpq_swap: Initializing Rationals. 3585 (line 54) 3586* mpq_t: Nomenclature and Types. 3587 (line 16) 3588* mpz_2fac_ui: Number Theoretic Functions. 3589 (line 109) 3590* mpz_abs: Integer Arithmetic. (line 44) 3591* mpz_add: Integer Arithmetic. (line 6) 3592* mpz_addmul: Integer Arithmetic. (line 24) 3593* mpz_addmul_ui: Integer Arithmetic. (line 26) 3594* mpz_add_ui: Integer Arithmetic. (line 7) 3595* mpz_and: Integer Logic and Bit Fiddling. 3596 (line 10) 3597* mpz_array_init: Integer Special Functions. 3598 (line 9) 3599* mpz_bin_ui: Number Theoretic Functions. 3600 (line 120) 3601* mpz_bin_uiui: Number Theoretic Functions. 3602 (line 122) 3603* mpz_cdiv_q: Integer Division. (line 12) 3604* mpz_cdiv_qr: Integer Division. (line 14) 3605* mpz_cdiv_qr_ui: Integer Division. (line 21) 3606* mpz_cdiv_q_2exp: Integer Division. (line 26) 3607* mpz_cdiv_q_ui: Integer Division. (line 17) 3608* mpz_cdiv_r: Integer Division. (line 13) 3609* mpz_cdiv_r_2exp: Integer Division. (line 29) 3610* mpz_cdiv_r_ui: Integer Division. (line 19) 3611* mpz_cdiv_ui: Integer Division. (line 23) 3612* mpz_class: C++ Interface General. 3613 (line 17) 3614* mpz_class::fits_sint_p: C++ Interface Integers. 3615 (line 50) 3616* mpz_class::fits_slong_p: C++ Interface Integers. 3617 (line 51) 3618* mpz_class::fits_sshort_p: C++ Interface Integers. 3619 (line 52) 3620* mpz_class::fits_uint_p: C++ Interface Integers. 3621 (line 54) 3622* mpz_class::fits_ulong_p: C++ Interface Integers. 3623 (line 55) 3624* mpz_class::fits_ushort_p: C++ Interface Integers. 3625 (line 56) 3626* mpz_class::get_d: C++ Interface Integers. 3627 (line 58) 3628* mpz_class::get_mpz_t: C++ Interface General. 3629 (line 63) 3630* mpz_class::get_si: C++ Interface Integers. 3631 (line 59) 3632* mpz_class::get_str: C++ Interface Integers. 3633 (line 60) 3634* mpz_class::get_ui: C++ Interface Integers. 3635 (line 61) 3636* mpz_class::mpz_class: C++ Interface Integers. 3637 (line 6) 3638* mpz_class::mpz_class <1>: C++ Interface Integers. 3639 (line 14) 3640* mpz_class::mpz_class <2>: C++ Interface Integers. 3641 (line 19) 3642* mpz_class::mpz_class <3>: C++ Interface Integers. 3643 (line 21) 3644* mpz_class::set_str: C++ Interface Integers. 3645 (line 63) 3646* mpz_class::set_str <1>: C++ Interface Integers. 3647 (line 64) 3648* mpz_class::swap: C++ Interface Integers. 3649 (line 71) 3650* mpz_clear: Initializing Integers. 3651 (line 48) 3652* mpz_clears: Initializing Integers. 3653 (line 52) 3654* mpz_clrbit: Integer Logic and Bit Fiddling. 3655 (line 54) 3656* mpz_cmp: Integer Comparisons. (line 6) 3657* mpz_cmpabs: Integer Comparisons. (line 17) 3658* mpz_cmpabs_d: Integer Comparisons. (line 18) 3659* mpz_cmpabs_ui: Integer Comparisons. (line 19) 3660* mpz_cmp_d: Integer Comparisons. (line 7) 3661* mpz_cmp_si: Integer Comparisons. (line 8) 3662* mpz_cmp_ui: Integer Comparisons. (line 9) 3663* mpz_com: Integer Logic and Bit Fiddling. 3664 (line 19) 3665* mpz_combit: Integer Logic and Bit Fiddling. 3666 (line 57) 3667* mpz_congruent_2exp_p: Integer Division. (line 148) 3668* mpz_congruent_p: Integer Division. (line 144) 3669* mpz_congruent_ui_p: Integer Division. (line 146) 3670* mpz_divexact: Integer Division. (line 122) 3671* mpz_divexact_ui: Integer Division. (line 123) 3672* mpz_divisible_2exp_p: Integer Division. (line 135) 3673* mpz_divisible_p: Integer Division. (line 132) 3674* mpz_divisible_ui_p: Integer Division. (line 133) 3675* mpz_even_p: Miscellaneous Integer Functions. 3676 (line 17) 3677* mpz_export: Integer Import and Export. 3678 (line 43) 3679* mpz_fac_ui: Number Theoretic Functions. 3680 (line 108) 3681* mpz_fdiv_q: Integer Division. (line 33) 3682* mpz_fdiv_qr: Integer Division. (line 35) 3683* mpz_fdiv_qr_ui: Integer Division. (line 42) 3684* mpz_fdiv_q_2exp: Integer Division. (line 47) 3685* mpz_fdiv_q_ui: Integer Division. (line 38) 3686* mpz_fdiv_r: Integer Division. (line 34) 3687* mpz_fdiv_r_2exp: Integer Division. (line 50) 3688* mpz_fdiv_r_ui: Integer Division. (line 40) 3689* mpz_fdiv_ui: Integer Division. (line 44) 3690* mpz_fib2_ui: Number Theoretic Functions. 3691 (line 130) 3692* mpz_fib_ui: Number Theoretic Functions. 3693 (line 129) 3694* mpz_fits_sint_p: Miscellaneous Integer Functions. 3695 (line 9) 3696* mpz_fits_slong_p: Miscellaneous Integer Functions. 3697 (line 7) 3698* mpz_fits_sshort_p: Miscellaneous Integer Functions. 3699 (line 11) 3700* mpz_fits_uint_p: Miscellaneous Integer Functions. 3701 (line 8) 3702* mpz_fits_ulong_p: Miscellaneous Integer Functions. 3703 (line 6) 3704* mpz_fits_ushort_p: Miscellaneous Integer Functions. 3705 (line 10) 3706* mpz_gcd: Number Theoretic Functions. 3707 (line 25) 3708* mpz_gcdext: Number Theoretic Functions. 3709 (line 41) 3710* mpz_gcd_ui: Number Theoretic Functions. 3711 (line 31) 3712* mpz_getlimbn: Integer Special Functions. 3713 (line 22) 3714* mpz_get_d: Converting Integers. (line 26) 3715* mpz_get_d_2exp: Converting Integers. (line 34) 3716* mpz_get_si: Converting Integers. (line 17) 3717* mpz_get_str: Converting Integers. (line 46) 3718* mpz_get_ui: Converting Integers. (line 10) 3719* mpz_hamdist: Integer Logic and Bit Fiddling. 3720 (line 28) 3721* mpz_import: Integer Import and Export. 3722 (line 9) 3723* mpz_init: Initializing Integers. 3724 (line 25) 3725* mpz_init2: Initializing Integers. 3726 (line 32) 3727* mpz_inits: Initializing Integers. 3728 (line 28) 3729* mpz_init_set: Simultaneous Integer Init & Assign. 3730 (line 26) 3731* mpz_init_set_d: Simultaneous Integer Init & Assign. 3732 (line 29) 3733* mpz_init_set_si: Simultaneous Integer Init & Assign. 3734 (line 28) 3735* mpz_init_set_str: Simultaneous Integer Init & Assign. 3736 (line 33) 3737* mpz_init_set_ui: Simultaneous Integer Init & Assign. 3738 (line 27) 3739* mpz_inp_raw: I/O of Integers. (line 61) 3740* mpz_inp_str: I/O of Integers. (line 30) 3741* mpz_invert: Number Theoretic Functions. 3742 (line 68) 3743* mpz_ior: Integer Logic and Bit Fiddling. 3744 (line 13) 3745* mpz_jacobi: Number Theoretic Functions. 3746 (line 78) 3747* mpz_kronecker: Number Theoretic Functions. 3748 (line 86) 3749* mpz_kronecker_si: Number Theoretic Functions. 3750 (line 87) 3751* mpz_kronecker_ui: Number Theoretic Functions. 3752 (line 88) 3753* mpz_lcm: Number Theoretic Functions. 3754 (line 61) 3755* mpz_lcm_ui: Number Theoretic Functions. 3756 (line 62) 3757* mpz_legendre: Number Theoretic Functions. 3758 (line 81) 3759* mpz_limbs_finish: Integer Special Functions. 3760 (line 47) 3761* mpz_limbs_modify: Integer Special Functions. 3762 (line 40) 3763* mpz_limbs_read: Integer Special Functions. 3764 (line 34) 3765* mpz_limbs_write: Integer Special Functions. 3766 (line 39) 3767* mpz_lucnum2_ui: Number Theoretic Functions. 3768 (line 141) 3769* mpz_lucnum_ui: Number Theoretic Functions. 3770 (line 140) 3771* mpz_mfac_uiui: Number Theoretic Functions. 3772 (line 110) 3773* mpz_mod: Integer Division. (line 112) 3774* mpz_mod_ui: Integer Division. (line 113) 3775* mpz_mul: Integer Arithmetic. (line 18) 3776* mpz_mul_2exp: Integer Arithmetic. (line 36) 3777* mpz_mul_si: Integer Arithmetic. (line 19) 3778* mpz_mul_ui: Integer Arithmetic. (line 20) 3779* mpz_neg: Integer Arithmetic. (line 41) 3780* mpz_nextprime: Number Theoretic Functions. 3781 (line 18) 3782* mpz_odd_p: Miscellaneous Integer Functions. 3783 (line 16) 3784* mpz_out_raw: I/O of Integers. (line 45) 3785* mpz_out_str: I/O of Integers. (line 17) 3786* mpz_perfect_power_p: Integer Roots. (line 27) 3787* mpz_perfect_square_p: Integer Roots. (line 36) 3788* mpz_popcount: Integer Logic and Bit Fiddling. 3789 (line 22) 3790* mpz_powm: Integer Exponentiation. 3791 (line 6) 3792* mpz_powm_sec: Integer Exponentiation. 3793 (line 16) 3794* mpz_powm_ui: Integer Exponentiation. 3795 (line 8) 3796* mpz_pow_ui: Integer Exponentiation. 3797 (line 29) 3798* mpz_primorial_ui: Number Theoretic Functions. 3799 (line 116) 3800* mpz_probab_prime_p: Number Theoretic Functions. 3801 (line 6) 3802* mpz_random: Integer Random Numbers. 3803 (line 41) 3804* mpz_random2: Integer Random Numbers. 3805 (line 50) 3806* mpz_realloc2: Initializing Integers. 3807 (line 56) 3808* mpz_remove: Number Theoretic Functions. 3809 (line 102) 3810* mpz_roinit_n: Integer Special Functions. 3811 (line 67) 3812* MPZ_ROINIT_N: Integer Special Functions. 3813 (line 83) 3814* mpz_root: Integer Roots. (line 6) 3815* mpz_rootrem: Integer Roots. (line 12) 3816* mpz_rrandomb: Integer Random Numbers. 3817 (line 29) 3818* mpz_scan0: Integer Logic and Bit Fiddling. 3819 (line 35) 3820* mpz_scan1: Integer Logic and Bit Fiddling. 3821 (line 37) 3822* mpz_set: Assigning Integers. (line 9) 3823* mpz_setbit: Integer Logic and Bit Fiddling. 3824 (line 51) 3825* mpz_set_d: Assigning Integers. (line 12) 3826* mpz_set_f: Assigning Integers. (line 14) 3827* mpz_set_q: Assigning Integers. (line 13) 3828* mpz_set_si: Assigning Integers. (line 11) 3829* mpz_set_str: Assigning Integers. (line 20) 3830* mpz_set_ui: Assigning Integers. (line 10) 3831* mpz_sgn: Integer Comparisons. (line 27) 3832* mpz_size: Integer Special Functions. 3833 (line 30) 3834* mpz_sizeinbase: Miscellaneous Integer Functions. 3835 (line 22) 3836* mpz_si_kronecker: Number Theoretic Functions. 3837 (line 89) 3838* mpz_sqrt: Integer Roots. (line 17) 3839* mpz_sqrtrem: Integer Roots. (line 20) 3840* mpz_sub: Integer Arithmetic. (line 11) 3841* mpz_submul: Integer Arithmetic. (line 30) 3842* mpz_submul_ui: Integer Arithmetic. (line 32) 3843* mpz_sub_ui: Integer Arithmetic. (line 12) 3844* mpz_swap: Assigning Integers. (line 36) 3845* mpz_t: Nomenclature and Types. 3846 (line 6) 3847* mpz_tdiv_q: Integer Division. (line 54) 3848* mpz_tdiv_qr: Integer Division. (line 56) 3849* mpz_tdiv_qr_ui: Integer Division. (line 63) 3850* mpz_tdiv_q_2exp: Integer Division. (line 68) 3851* mpz_tdiv_q_ui: Integer Division. (line 59) 3852* mpz_tdiv_r: Integer Division. (line 55) 3853* mpz_tdiv_r_2exp: Integer Division. (line 71) 3854* mpz_tdiv_r_ui: Integer Division. (line 61) 3855* mpz_tdiv_ui: Integer Division. (line 65) 3856* mpz_tstbit: Integer Logic and Bit Fiddling. 3857 (line 60) 3858* mpz_ui_kronecker: Number Theoretic Functions. 3859 (line 90) 3860* mpz_ui_pow_ui: Integer Exponentiation. 3861 (line 31) 3862* mpz_ui_sub: Integer Arithmetic. (line 14) 3863* mpz_urandomb: Integer Random Numbers. 3864 (line 12) 3865* mpz_urandomm: Integer Random Numbers. 3866 (line 21) 3867* mpz_xor: Integer Logic and Bit Fiddling. 3868 (line 16) 3869* mp_bitcnt_t: Nomenclature and Types. 3870 (line 42) 3871* mp_bits_per_limb: Useful Macros and Constants. 3872 (line 7) 3873* mp_exp_t: Nomenclature and Types. 3874 (line 27) 3875* mp_get_memory_functions: Custom Allocation. (line 86) 3876* mp_limb_t: Nomenclature and Types. 3877 (line 31) 3878* mp_set_memory_functions: Custom Allocation. (line 14) 3879* mp_size_t: Nomenclature and Types. 3880 (line 37) 3881* operator"": C++ Interface Integers. 3882 (line 29) 3883* operator"" <1>: C++ Interface Rationals. 3884 (line 36) 3885* operator"" <2>: C++ Interface Floats. 3886 (line 55) 3887* operator%: C++ Interface Integers. 3888 (line 34) 3889* operator/: C++ Interface Integers. 3890 (line 33) 3891* operator<<: C++ Formatted Output. 3892 (line 10) 3893* operator<< <1>: C++ Formatted Output. 3894 (line 19) 3895* operator<< <2>: C++ Formatted Output. 3896 (line 32) 3897* operator>>: C++ Formatted Input. (line 10) 3898* operator>> <1>: C++ Formatted Input. (line 13) 3899* operator>> <2>: C++ Formatted Input. (line 24) 3900* operator>> <3>: C++ Interface Rationals. 3901 (line 86) 3902* sgn: C++ Interface Integers. 3903 (line 65) 3904* sgn <1>: C++ Interface Rationals. 3905 (line 56) 3906* sgn <2>: C++ Interface Floats. 3907 (line 106) 3908* sqrt: C++ Interface Integers. 3909 (line 66) 3910* sqrt <1>: C++ Interface Floats. 3911 (line 107) 3912* swap: C++ Interface Integers. 3913 (line 72) 3914* swap <1>: C++ Interface Rationals. 3915 (line 59) 3916* swap <2>: C++ Interface Floats. 3917 (line 110) 3918* trunc: C++ Interface Floats. 3919 (line 111) 3920 3921