1This is gmp.info, produced by makeinfo version 6.7 from gmp.texi. 2 3This manual describes how to install and use the GNU multiple precision 4arithmetic library, version 6.2.1. 5 6 Copyright 1991, 1993-2016, 2018-2020 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 305Jacobi symbol (A/B) 306 307 Initially if either operand fits in a single limb, a reduction is 308done with either 'mpn_mod_1' or 'mpn_modexact_1_odd', followed by the 309binary algorithm on a single limb. The binary algorithm is well suited 310to a single limb, and the whole calculation in this case is quite 311efficient. 312 313 For inputs larger than 'GCD_DC_THRESHOLD', 'mpz_jacobi', 314'mpz_legendre' and 'mpz_kronecker' are computed via the HGCD (Half GCD) 315function, as a generalization to Lehmer's algorithm. 316 317 Most GCD algorithms reduce a and b by repeatatily computing the 318quotient q = floor(a/b) and iteratively replacing 319 320 a, b = b, a - q * b 321 322 Different algorithms use different methods for calculating q, but the 323core algorithm is the same if we use *note Lehmer's Algorithm:: or *note 324HGCD: Subquadratic GCD. 325 326 At each step it is possible to compute if the reduction inverts the 327Jacobi symbol based on the two least significant bits of A and B. For 328more details see "Efficient computation of the Jacobi symbol" by M�ller 329(*note References::). 330 331 A small set of bits is thus used to track state 332 * current sign of result (1 bit) 333 334 * two least significant bits of A and B (4 bits) 335 336 * a pointer to which input is currently the denominator (1 bit) 337 338 In all the routines sign changes for the result are accumulated using 339fast bit twiddling which avoids conditional jumps. 340 341 The final result is calculated after verifying the inputs are coprime 342(GCD = 1) by raising (-1)^e 343 344 Much of the HGCD code is shared directly with the HGCD 345implementations, such as the 2x2 matrix calculation, *Note Lehmer's 346Algorithm:: basecase and 'GCD_DC_THRESHOLD'. 347 348 The asymptotic running time is O(M(N)*log(N)), where M(N) is the time 349for multiplying two N-limb numbers. 350 351 352File: gmp.info, Node: Powering Algorithms, Next: Root Extraction Algorithms, Prev: Greatest Common Divisor Algorithms, Up: Algorithms 353 35415.4 Powering Algorithms 355======================== 356 357* Menu: 358 359* Normal Powering Algorithm:: 360* Modular Powering Algorithm:: 361 362 363File: gmp.info, Node: Normal Powering Algorithm, Next: Modular Powering Algorithm, Prev: Powering Algorithms, Up: Powering Algorithms 364 36515.4.1 Normal Powering 366---------------------- 367 368Normal 'mpz' or 'mpf' powering uses a simple binary algorithm, 369successively squaring and then multiplying by the base when a 1 bit is 370seen in the exponent, as per Knuth section 4.6.3. The "left to right" 371variant described there is used rather than algorithm A, since it's just 372as easy and can be done with somewhat less temporary memory. 373 374 375File: gmp.info, Node: Modular Powering Algorithm, Prev: Normal Powering Algorithm, Up: Powering Algorithms 376 37715.4.2 Modular Powering 378----------------------- 379 380Modular powering is implemented using a 2^k-ary sliding window 381algorithm, as per "Handbook of Applied Cryptography" algorithm 14.85 382(*note References::). k is chosen according to the size of the 383exponent. Larger exponents use larger values of k, the choice being 384made to minimize the average number of multiplications that must 385supplement the squaring. 386 387 The modular multiplies and squarings use either a simple division or 388the REDC method by Montgomery (*note References::). REDC is a little 389faster, essentially saving N single limb divisions in a fashion similar 390to an exact remainder (*note Exact Remainder::). 391 392 393File: gmp.info, Node: Root Extraction Algorithms, Next: Radix Conversion Algorithms, Prev: Powering Algorithms, Up: Algorithms 394 39515.5 Root Extraction Algorithms 396=============================== 397 398* Menu: 399 400* Square Root Algorithm:: 401* Nth Root Algorithm:: 402* Perfect Square Algorithm:: 403* Perfect Power Algorithm:: 404 405 406File: gmp.info, Node: Square Root Algorithm, Next: Nth Root Algorithm, Prev: Root Extraction Algorithms, Up: Root Extraction Algorithms 407 40815.5.1 Square Root 409------------------ 410 411Square roots are taken using the "Karatsuba Square Root" algorithm by 412Paul Zimmermann (*note References::). 413 414 An input n is split into four parts of k bits each, so with b=2^k we 415have n = a3*b^3 + a2*b^2 + a1*b + a0. Part a3 must be "normalized" so 416that either the high or second highest bit is set. In GMP, k is kept on 417a limb boundary and the input is left shifted (by an even number of 418bits) to normalize. 419 420 The square root of the high two parts is taken, by recursive 421application of the algorithm (bottoming out in a one-limb Newton's 422method), 423 424 s1,r1 = sqrtrem (a3*b + a2) 425 426 This is an approximation to the desired root and is extended by a 427division to give s,r, 428 429 q,u = divrem (r1*b + a1, 2*s1) 430 s = s1*b + q 431 r = u*b + a0 - q^2 432 433 The normalization requirement on a3 means at this point s is either 434correct or 1 too big. r is negative in the latter case, so 435 436 if r < 0 then 437 r = r + 2*s - 1 438 s = s - 1 439 440 The algorithm is expressed in a divide and conquer form, but as noted 441in the paper it can also be viewed as a discrete variant of Newton's 442method, or as a variation on the schoolboy method (no longer taught) for 443square roots two digits at a time. 444 445 If the remainder r is not required then usually only a few high limbs 446of r and u need to be calculated to determine whether an adjustment to s 447is required. This optimization is not currently implemented. 448 449 In the Karatsuba multiplication range this algorithm is 450O(1.5*M(N/2)), where M(n) is the time to multiply two numbers of n 451limbs. In the FFT multiplication range this grows to a bound of 452O(6*M(N/2)). In practice a factor of about 1.5 to 1.8 is found in the 453Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range. 454 455 The algorithm does all its calculations in integers and the resulting 456'mpn_sqrtrem' is used for both 'mpz_sqrt' and 'mpf_sqrt'. The extended 457precision given by 'mpf_sqrt_ui' is obtained by padding with zero limbs. 458 459 460File: gmp.info, Node: Nth Root Algorithm, Next: Perfect Square Algorithm, Prev: Square Root Algorithm, Up: Root Extraction Algorithms 461 46215.5.2 Nth Root 463--------------- 464 465Integer Nth roots are taken using Newton's method with the following 466iteration, where A is the input and n is the root to be taken. 467 468 1 A 469 a[i+1] = - * ( --------- + (n-1)*a[i] ) 470 n a[i]^(n-1) 471 472 The initial approximation a[1] is generated bitwise by successively 473powering a trial root with or without new 1 bits, aiming to be just 474above the true root. The iteration converges quadratically when started 475from a good approximation. When n is large more initial bits are needed 476to get good convergence. The current implementation is not particularly 477well optimized. 478 479 480File: gmp.info, Node: Perfect Square Algorithm, Next: Perfect Power Algorithm, Prev: Nth Root Algorithm, Up: Root Extraction Algorithms 481 48215.5.3 Perfect Square 483--------------------- 484 485A significant fraction of non-squares can be quickly identified by 486checking whether the input is a quadratic residue modulo small integers. 487 488 'mpz_perfect_square_p' first tests the input mod 256, which means 489just examining the low byte. Only 44 different values occur for squares 490mod 256, so 82.8% of inputs can be immediately identified as 491non-squares. 492 493 On a 32-bit system similar tests are done mod 9, 5, 7, 13 and 17, for 494a total 99.25% of inputs identified as non-squares. On a 64-bit system 49597 is tested too, for a total 99.62%. 496 497 These moduli are chosen because they're factors of 2^24-1 (or 2^48-1 498for 64-bits), and such a remainder can be quickly taken just using 499additions (see 'mpn_mod_34lsub1'). 500 501 When nails are in use moduli are instead selected by the 'gen-psqr.c' 502program and applied with an 'mpn_mod_1'. The same 2^24-1 or 2^48-1 503could be done with nails using some extra bit shifts, but this is not 504currently implemented. 505 506 In any case each modulus is applied to the 'mpn_mod_34lsub1' or 507'mpn_mod_1' remainder and a table lookup identifies non-squares. By 508using a "modexact" style calculation, and suitably permuted tables, just 509one multiply each is required, see the code for details. Moduli are 510also combined to save operations, so long as the lookup tables don't 511become too big. 'gen-psqr.c' does all the pre-calculations. 512 513 A square root must still be taken for any value that passes these 514tests, to verify it's really a square and not one of the small fraction 515of non-squares that get through (i.e. a pseudo-square to all the tested 516bases). 517 518 Clearly more residue tests could be done, 'mpz_perfect_square_p' only 519uses a compact and efficient set. Big inputs would probably benefit 520from more residue testing, small inputs might be better off with less. 521The assumed distribution of squares versus non-squares in the input 522would affect such considerations. 523 524 525File: gmp.info, Node: Perfect Power Algorithm, Prev: Perfect Square Algorithm, Up: Root Extraction Algorithms 526 52715.5.4 Perfect Power 528-------------------- 529 530Detecting perfect powers is required by some factorization algorithms. 531Currently 'mpz_perfect_power_p' is implemented using repeated Nth root 532extractions, though naturally only prime roots need to be considered. 533(*Note Nth Root Algorithm::.) 534 535 If a prime divisor p with multiplicity e can be found, then only 536roots which are divisors of e need to be considered, much reducing the 537work necessary. To this end divisibility by a set of small primes is 538checked. 539 540 541File: gmp.info, Node: Radix Conversion Algorithms, Next: Other Algorithms, Prev: Root Extraction Algorithms, Up: Algorithms 542 54315.6 Radix Conversion 544===================== 545 546Radix conversions are less important than other algorithms. A program 547dominated by conversions should probably use a different data 548representation. 549 550* Menu: 551 552* Binary to Radix:: 553* Radix to Binary:: 554 555 556File: gmp.info, Node: Binary to Radix, Next: Radix to Binary, Prev: Radix Conversion Algorithms, Up: Radix Conversion Algorithms 557 55815.6.1 Binary to Radix 559---------------------- 560 561Conversions from binary to a power-of-2 radix use a simple and fast O(N) 562bit extraction algorithm. 563 564 Conversions from binary to other radices use one of two algorithms. 565Sizes below 'GET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method. 566Repeated divisions by b^n are made, where b is the radix and n is the 567biggest power that fits in a limb. But instead of simply using the 568remainder r from such divisions, an extra divide step is done to give a 569fractional limb representing r/b^n. The digits of r can then be 570extracted using multiplications by b rather than divisions. Special 571case code is provided for decimal, allowing multiplications by 10 to 572optimize to shifts and adds. 573 574 Above 'GET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is 575used. For an input t, powers b^(n*2^i) of the radix are calculated, 576until a power between t and sqrt(t) is reached. t is then divided by 577that largest power, giving a quotient which is the digits above that 578power, and a remainder which is those below. These two parts are in 579turn divided by the second highest power, and so on recursively. When a 580piece has been divided down to less than 'GET_STR_DC_THRESHOLD' limbs, 581the basecase algorithm described above is used. 582 583 The advantage of this algorithm is that big divisions can make use of 584the sub-quadratic divide and conquer division (*note Divide and Conquer 585Division::), and big divisions tend to have less overheads than lots of 586separate single limb divisions anyway. But in any case the cost of 587calculating the powers b^(n*2^i) must first be overcome. 588 589 'GET_STR_PRECOMPUTE_THRESHOLD' and 'GET_STR_DC_THRESHOLD' represent 590the same basic thing, the point where it becomes worth doing a big 591division to cut the input in half. 'GET_STR_PRECOMPUTE_THRESHOLD' 592includes the cost of calculating the radix power required, whereas 593'GET_STR_DC_THRESHOLD' assumes that's already available, which is the 594case when recursing. 595 596 Since the base case produces digits from least to most significant 597but they want to be stored from most to least, it's necessary to 598calculate in advance how many digits there will be, or at least be sure 599not to underestimate that. For GMP the number of input bits is 600multiplied by 'chars_per_bit_exactly' from 'mp_bases', rounding up. The 601result is either correct or one too big. 602 603 Examining some of the high bits of the input could increase the 604chance of getting the exact number of digits, but an exact result every 605time would not be practical, since in general the difference between 606numbers 100... and 99... is only in the last few bits and the work to 607identify 99... might well be almost as much as a full conversion. 608 609 The r/b^n scheme described above for using multiplications to bring 610out digits might be useful for more than a single limb. Some brief 611experiments with it on the base case when recursing didn't give a 612noticeable improvement, but perhaps that was only due to the 613implementation. Something similar would work for the sub-quadratic 614divisions too, though there would be the cost of calculating a bigger 615radix power. 616 617 Another possible improvement for the sub-quadratic part would be to 618arrange for radix powers that balanced the sizes of quotient and 619remainder produced, i.e. the highest power would be an b^(n*k) 620approximately equal to sqrt(t), not restricted to a 2^i factor. That 621ought to smooth out a graph of times against sizes, but may or may not 622be a net speedup. 623 624 625File: gmp.info, Node: Radix to Binary, Prev: Binary to Radix, Up: Radix Conversion Algorithms 626 62715.6.2 Radix to Binary 628---------------------- 629 630*This section needs to be rewritten, it currently describes the 631algorithms used before GMP 4.3.* 632 633 Conversions from a power-of-2 radix into binary use a simple and fast 634O(N) bitwise concatenation algorithm. 635 636 Conversions from other radices use one of two algorithms. Sizes 637below 'SET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method. Groups 638of n digits are converted to limbs, where n is the biggest power of the 639base b which will fit in a limb, then those groups are accumulated into 640the result by multiplying by b^n and adding. This saves multi-precision 641operations, as per Knuth section 4.4 part E (*note References::). Some 642special case code is provided for decimal, giving the compiler a chance 643to optimize multiplications by 10. 644 645 Above 'SET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is 646used. First groups of n digits are converted into limbs. Then adjacent 647limbs are combined into limb pairs with x*b^n+y, where x and y are the 648limbs. Adjacent limb pairs are combined into quads similarly with 649x*b^(2n)+y. This continues until a single block remains, that being the 650result. 651 652 The advantage of this method is that the multiplications for each x 653are big blocks, allowing Karatsuba and higher algorithms to be used. 654But the cost of calculating the powers b^(n*2^i) must be overcome. 655'SET_STR_PRECOMPUTE_THRESHOLD' usually ends up quite big, around 5000 656digits, and on some processors much bigger still. 657 658 'SET_STR_PRECOMPUTE_THRESHOLD' is based on the input digits (and 659tuned for decimal), though it might be better based on a limb count, so 660as to be independent of the base. But that sort of count isn't used by 661the base case and so would need some sort of initial calculation or 662estimate. 663 664 The main reason 'SET_STR_PRECOMPUTE_THRESHOLD' is so much bigger than 665the corresponding 'GET_STR_PRECOMPUTE_THRESHOLD' is that 'mpn_mul_1' is 666much faster than 'mpn_divrem_1' (often by a factor of 5, or more). 667 668 669File: gmp.info, Node: Other Algorithms, Next: Assembly Coding, Prev: Radix Conversion Algorithms, Up: Algorithms 670 67115.7 Other Algorithms 672===================== 673 674* Menu: 675 676* Prime Testing Algorithm:: 677* Factorial Algorithm:: 678* Binomial Coefficients Algorithm:: 679* Fibonacci Numbers Algorithm:: 680* Lucas Numbers Algorithm:: 681* Random Number Algorithms:: 682 683 684File: gmp.info, Node: Prime Testing Algorithm, Next: Factorial Algorithm, Prev: Other Algorithms, Up: Other Algorithms 685 68615.7.1 Prime Testing 687-------------------- 688 689The primality testing in 'mpz_probab_prime_p' (*note Number Theoretic 690Functions::) first does some trial division by small factors and then 691uses the Miller-Rabin probabilistic primality testing algorithm, as 692described in Knuth section 4.5.4 algorithm P (*note References::). 693 694 For an odd input n, and with n = q*2^k+1 where q is odd, this 695algorithm selects a random base x and tests whether x^q mod n is 1 or 696-1, or an x^(q*2^j) mod n is 1, for 1<=j<=k. If so then n is probably 697prime, if not then n is definitely composite. 698 699 Any prime n will pass the test, but some composites do too. Such 700composites are known as strong pseudoprimes to base x. No n is a strong 701pseudoprime to more than 1/4 of all bases (see Knuth exercise 22), hence 702with x chosen at random there's no more than a 1/4 chance a "probable 703prime" will in fact be composite. 704 705 In fact strong pseudoprimes are quite rare, making the test much more 706powerful than this analysis would suggest, but 1/4 is all that's proven 707for an arbitrary n. 708 709 710File: gmp.info, Node: Factorial Algorithm, Next: Binomial Coefficients Algorithm, Prev: Prime Testing Algorithm, Up: Other Algorithms 711 71215.7.2 Factorial 713---------------- 714 715Factorials are calculated by a combination of two algorithms. An idea 716is shared among them: to compute the odd part of the factorial; a final 717step takes account of the power of 2 term, by shifting. 718 719 For small n, the odd factor of n! is computed with the simple 720observation that it is equal to the product of all positive odd numbers 721smaller than n times the odd factor of [n/2]!, where [x] is the integer 722part of x, and so on recursively. The procedure can be best illustrated 723with an example, 724 725 23! = (23.21.19.17.15.13.11.9.7.5.3)(11.9.7.5.3)(5.3)2^{19} 726 727 Current code collects all the factors in a single list, with a loop 728and no recursion, and compute the product, with no special care for 729repeated chunks. 730 731 When n is larger, computation pass trough prime sieving. An helper 732function is used, as suggested by Peter Luschny: 733 734 n 735 ----- 736 n! | | L(p,n) 737 msf(n) = -------------- = | | p 738 [n/2]!^2.2^k p=3 739 740 Where p ranges on odd prime numbers. The exponent k is chosen to 741obtain an odd integer number: k is the number of 1 bits in the binary 742representation of [n/2]. The function L(p,n) can be defined as zero 743when p is composite, and, for any prime p, it is computed with: 744 745 --- 746 \ n 747 L(p,n) = / [---] mod 2 <= log (n) . 748 --- p^i p 749 i>0 750 751 With this helper function, we are able to compute the odd part of n! 752using the recursion implied by n!=[n/2]!^2*msf(n)*2^k. The recursion 753stops using the small-n algorithm on some [n/2^i]. 754 755 Both the above algorithms use binary splitting to compute the product 756of many small factors. At first as many products as possible are 757accumulated in a single register, generating a list of factors that fit 758in a machine word. This list is then split into halves, and the product 759is computed recursively. 760 761 Such splitting is more efficient than repeated Nx1 multiplies since 762it forms big multiplies, allowing Karatsuba and higher algorithms to be 763used. And even below the Karatsuba threshold a big block of work can be 764more efficient for the basecase algorithm. 765 766 767File: gmp.info, Node: Binomial Coefficients Algorithm, Next: Fibonacci Numbers Algorithm, Prev: Factorial Algorithm, Up: Other Algorithms 768 76915.7.3 Binomial Coefficients 770---------------------------- 771 772Binomial coefficients C(n,k) are calculated by first arranging k <= n/2 773using C(n,k) = C(n,n-k) if necessary, and then evaluating the following 774product simply from i=2 to i=k. 775 776 k (n-k+i) 777 C(n,k) = (n-k+1) * prod ------- 778 i=2 i 779 780 It's easy to show that each denominator i will divide the product so 781far, so the exact division algorithm is used (*note Exact Division::). 782 783 The numerators n-k+i and denominators i are first accumulated into as 784many fit a limb, to save multi-precision operations, though for 785'mpz_bin_ui' this applies only to the divisors, since n is an 'mpz_t' 786and n-k+i in general won't fit in a limb at all. 787 788 789File: gmp.info, Node: Fibonacci Numbers Algorithm, Next: Lucas Numbers Algorithm, Prev: Binomial Coefficients Algorithm, Up: Other Algorithms 790 79115.7.4 Fibonacci Numbers 792------------------------ 793 794The Fibonacci functions 'mpz_fib_ui' and 'mpz_fib2_ui' are designed for 795calculating isolated F[n] or F[n],F[n-1] values efficiently. 796 797 For small n, a table of single limb values in '__gmp_fib_table' is 798used. On a 32-bit limb this goes up to F[47], or on a 64-bit limb up to 799F[93]. For convenience the table starts at F[-1]. 800 801 Beyond the table, values are generated with a binary powering 802algorithm, calculating a pair F[n] and F[n-1] working from high to low 803across the bits of n. The formulas used are 804 805 F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k 806 F[2k-1] = F[k]^2 + F[k-1]^2 807 808 F[2k] = F[2k+1] - F[2k-1] 809 810 At each step, k is the high b bits of n. If the next bit of n is 0 811then F[2k],F[2k-1] is used, or if it's a 1 then F[2k+1],F[2k] is used, 812and the process repeated until all bits of n are incorporated. Notice 813these formulas require just two squares per bit of n. 814 815 It'd be possible to handle the first few n above the single limb 816table with simple additions, using the defining Fibonacci recurrence 817F[k+1]=F[k]+F[k-1], but this is not done since it usually turns out to 818be faster for only about 10 or 20 values of n, and including a block of 819code for just those doesn't seem worthwhile. If they really mattered 820it'd be better to extend the data table. 821 822 Using a table avoids lots of calculations on small numbers, and makes 823small n go fast. A bigger table would make more small n go fast, it's 824just a question of balancing size against desired speed. For GMP the 825code is kept compact, with the emphasis primarily on a good powering 826algorithm. 827 828 'mpz_fib2_ui' returns both F[n] and F[n-1], but 'mpz_fib_ui' is only 829interested in F[n]. In this case the last step of the algorithm can 830become one multiply instead of two squares. One of the following two 831formulas is used, according as n is odd or even. 832 833 F[2k] = F[k]*(F[k]+2F[k-1]) 834 835 F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k 836 837 F[2k+1] here is the same as above, just rearranged to be a multiply. 838For interest, the 2*(-1)^k term both here and above can be applied just 839to the low limb of the calculation, without a carry or borrow into 840further limbs, which saves some code size. See comments with 841'mpz_fib_ui' and the internal 'mpn_fib2_ui' for how this is done. 842 843 844File: gmp.info, Node: Lucas Numbers Algorithm, Next: Random Number Algorithms, Prev: Fibonacci Numbers Algorithm, Up: Other Algorithms 845 84615.7.5 Lucas Numbers 847-------------------- 848 849'mpz_lucnum2_ui' derives a pair of Lucas numbers from a pair of 850Fibonacci numbers with the following simple formulas. 851 852 L[k] = F[k] + 2*F[k-1] 853 L[k-1] = 2*F[k] - F[k-1] 854 855 'mpz_lucnum_ui' is only interested in L[n], and some work can be 856saved. Trailing zero bits on n can be handled with a single square 857each. 858 859 L[2k] = L[k]^2 - 2*(-1)^k 860 861 And the lowest 1 bit can be handled with one multiply of a pair of 862Fibonacci numbers, similar to what 'mpz_fib_ui' does. 863 864 L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k 865 866 867File: gmp.info, Node: Random Number Algorithms, Prev: Lucas Numbers Algorithm, Up: Other Algorithms 868 86915.7.6 Random Numbers 870--------------------- 871 872For the 'urandomb' functions, random numbers are generated simply by 873concatenating bits produced by the generator. As long as the generator 874has good randomness properties this will produce well-distributed N bit 875numbers. 876 877 For the 'urandomm' functions, random numbers in a range 0<=R<N are 878generated by taking values R of ceil(log2(N)) bits each until one 879satisfies R<N. This will normally require only one or two attempts, but 880the attempts are limited in case the generator is somehow degenerate and 881produces only 1 bits or similar. 882 883 The Mersenne Twister generator is by Matsumoto and Nishimura (*note 884References::). It has a non-repeating period of 2^19937-1, which is a 885Mersenne prime, hence the name of the generator. The state is 624 words 886of 32-bits each, which is iterated with one XOR and shift for each 88732-bit word generated, making the algorithm very fast. Randomness 888properties are also very good and this is the default algorithm used by 889GMP. 890 891 Linear congruential generators are described in many text books, for 892instance Knuth volume 2 (*note References::). With a modulus M and 893parameters A and C, an integer state S is iterated by the formula S <- 894A*S+C mod M. At each step the new state is a linear function of the 895previous, mod M, hence the name of the generator. 896 897 In GMP only moduli of the form 2^N are supported, and the current 898implementation is not as well optimized as it could be. Overheads are 899significant when N is small, and when N is large clearly the multiply at 900each step will become slow. This is not a big concern, since the 901Mersenne Twister generator is better in every respect and is therefore 902recommended for all normal applications. 903 904 For both generators the current state can be deduced by observing 905enough output and applying some linear algebra (over GF(2) in the case 906of the Mersenne Twister). This generally means raw output is unsuitable 907for cryptographic applications without further hashing or the like. 908 909 910File: gmp.info, Node: Assembly Coding, Prev: Other Algorithms, Up: Algorithms 911 91215.8 Assembly Coding 913==================== 914 915The assembly subroutines in GMP are the most significant source of speed 916at small to moderate sizes. At larger sizes algorithm selection becomes 917more important, but of course speedups in low level routines will still 918speed up everything proportionally. 919 920 Carry handling and widening multiplies that are important for GMP 921can't be easily expressed in C. GCC 'asm' blocks help a lot and are 922provided in 'longlong.h', but hand coding low level routines invariably 923offers a speedup over generic C by a factor of anything from 2 to 10. 924 925* Menu: 926 927* Assembly Code Organisation:: 928* Assembly Basics:: 929* Assembly Carry Propagation:: 930* Assembly Cache Handling:: 931* Assembly Functional Units:: 932* Assembly Floating Point:: 933* Assembly SIMD Instructions:: 934* Assembly Software Pipelining:: 935* Assembly Loop Unrolling:: 936* Assembly Writing Guide:: 937 938 939File: gmp.info, Node: Assembly Code Organisation, Next: Assembly Basics, Prev: Assembly Coding, Up: Assembly Coding 940 94115.8.1 Code Organisation 942------------------------ 943 944The various 'mpn' subdirectories contain machine-dependent code, written 945in C or assembly. The 'mpn/generic' subdirectory contains default code, 946used when there's no machine-specific version of a particular file. 947 948 Each 'mpn' subdirectory is for an ISA family. Generally 32-bit and 94964-bit variants in a family cannot share code and have separate 950directories. Within a family further subdirectories may exist for CPU 951variants. 952 953 In each directory a 'nails' subdirectory may exist, holding code with 954nails support for that CPU variant. A 'NAILS_SUPPORT' directive in each 955file indicates the nails values the code handles. Nails code only 956exists where it's faster, or promises to be faster, than plain code. 957There's no effort put into nails if they're not going to enhance a given 958CPU. 959 960 961File: gmp.info, Node: Assembly Basics, Next: Assembly Carry Propagation, Prev: Assembly Code Organisation, Up: Assembly Coding 962 96315.8.2 Assembly Basics 964---------------------- 965 966'mpn_addmul_1' and 'mpn_submul_1' are the most important routines for 967overall GMP performance. All multiplications and divisions come down to 968repeated calls to these. 'mpn_add_n', 'mpn_sub_n', 'mpn_lshift' and 969'mpn_rshift' are next most important. 970 971 On some CPUs assembly versions of the internal functions 972'mpn_mul_basecase' and 'mpn_sqr_basecase' give significant speedups, 973mainly through avoiding function call overheads. They can also 974potentially make better use of a wide superscalar processor, as can 975bigger primitives like 'mpn_addmul_2' or 'mpn_addmul_4'. 976 977 The restrictions on overlaps between sources and destinations (*note 978Low-level Functions::) are designed to facilitate a variety of 979implementations. For example, knowing 'mpn_add_n' won't have partly 980overlapping sources and destination means reading can be done far ahead 981of writing on superscalar processors, and loops can be vectorized on a 982vector processor, depending on the carry handling. 983 984 985File: gmp.info, Node: Assembly Carry Propagation, Next: Assembly Cache Handling, Prev: Assembly Basics, Up: Assembly Coding 986 98715.8.3 Carry Propagation 988------------------------ 989 990The problem that presents most challenges in GMP is propagating carries 991from one limb to the next. In functions like 'mpn_addmul_1' and 992'mpn_add_n', carries are the only dependencies between limb operations. 993 994 On processors with carry flags, a straightforward CISC style 'adc' is 995generally best. AMD K6 'mpn_addmul_1' however is an example of an 996unusual set of circumstances where a branch works out better. 997 998 On RISC processors generally an add and compare for overflow is used. 999This sort of thing can be seen in 'mpn/generic/aors_n.c'. Some carry 1000propagation schemes require 4 instructions, meaning at least 4 cycles 1001per limb, but other schemes may use just 1 or 2. On wide superscalar 1002processors performance may be completely determined by the number of 1003dependent instructions between carry-in and carry-out for each limb. 1004 1005 On vector processors good use can be made of the fact that a carry 1006bit only very rarely propagates more than one limb. When adding a 1007single bit to a limb, there's only a carry out if that limb was 1008'0xFF...FF' which on random data will be only 1 in 2^mp_bits_per_limb. 1009'mpn/cray/add_n.c' is an example of this, it adds all limbs in parallel, 1010adds one set of carry bits in parallel and then only rarely needs to 1011fall through to a loop propagating further carries. 1012 1013 On the x86s, GCC (as of version 2.95.2) doesn't generate particularly 1014good code for the RISC style idioms that are necessary to handle carry 1015bits in C. Often conditional jumps are generated where 'adc' or 'sbb' 1016forms would be better. And so unfortunately almost any loop involving 1017carry bits needs to be coded in assembly for best results. 1018 1019 1020File: gmp.info, Node: Assembly Cache Handling, Next: Assembly Functional Units, Prev: Assembly Carry Propagation, Up: Assembly Coding 1021 102215.8.4 Cache Handling 1023--------------------- 1024 1025GMP aims to perform well both on operands that fit entirely in L1 cache 1026and those which don't. 1027 1028 Basic routines like 'mpn_add_n' or 'mpn_lshift' are often used on 1029large operands, so L2 and main memory performance is important for them. 1030'mpn_mul_1' and 'mpn_addmul_1' are mostly used for multiply and square 1031basecases, so L1 performance matters most for them, unless assembly 1032versions of 'mpn_mul_basecase' and 'mpn_sqr_basecase' exist, in which 1033case the remaining uses are mostly for larger operands. 1034 1035 For L2 or main memory operands, memory access times will almost 1036certainly be more than the calculation time. The aim therefore is to 1037maximize memory throughput, by starting a load of the next cache line 1038while processing the contents of the previous one. Clearly this is only 1039possible if the chip has a lock-up free cache or some sort of prefetch 1040instruction. Most current chips have both these features. 1041 1042 Prefetching sources combines well with loop unrolling, since a 1043prefetch can be initiated once per unrolled loop (or more than once if 1044the loop covers more than one cache line). 1045 1046 On CPUs without write-allocate caches, prefetching destinations will 1047ensure individual stores don't go further down the cache hierarchy, 1048limiting bandwidth. Of course for calculations which are slow anyway, 1049like 'mpn_divrem_1', write-throughs might be fine. 1050 1051 The distance ahead to prefetch will be determined by memory latency 1052versus throughput. The aim of course is to have data arriving 1053continuously, at peak throughput. Some CPUs have limits on the number 1054of fetches or prefetches in progress. 1055 1056 If a special prefetch instruction doesn't exist then a plain load can 1057be used, but in that case care must be taken not to attempt to read past 1058the end of an operand, since that might produce a segmentation 1059violation. 1060 1061 Some CPUs or systems have hardware that detects sequential memory 1062accesses and initiates suitable cache movements automatically, making 1063life easy. 1064 1065 1066File: gmp.info, Node: Assembly Functional Units, Next: Assembly Floating Point, Prev: Assembly Cache Handling, Up: Assembly Coding 1067 106815.8.5 Functional Units 1069----------------------- 1070 1071When choosing an approach for an assembly loop, consideration is given 1072to what operations can execute simultaneously and what throughput can 1073thereby be achieved. In some cases an algorithm can be tweaked to 1074accommodate available resources. 1075 1076 Loop control will generally require a counter and pointer updates, 1077costing as much as 5 instructions, plus any delays a branch introduces. 1078CPU addressing modes might reduce pointer updates, perhaps by allowing 1079just one updating pointer and others expressed as offsets from it, or on 1080CISC chips with all addressing done with the loop counter as a scaled 1081index. 1082 1083 The final loop control cost can be amortised by processing several 1084limbs in each iteration (*note Assembly Loop Unrolling::). This at 1085least ensures loop control isn't a big fraction the work done. 1086 1087 Memory throughput is always a limit. If perhaps only one load or one 1088store can be done per cycle then 3 cycles/limb will the top speed for 1089"binary" operations like 'mpn_add_n', and any code achieving that is 1090optimal. 1091 1092 Integer resources can be freed up by having the loop counter in a 1093float register, or by pressing the float units into use for some 1094multiplying, perhaps doing every second limb on the float side (*note 1095Assembly Floating Point::). 1096 1097 Float resources can be freed up by doing carry propagation on the 1098integer side, or even by doing integer to float conversions in integers 1099using bit twiddling. 1100 1101 1102File: gmp.info, Node: Assembly Floating Point, Next: Assembly SIMD Instructions, Prev: Assembly Functional Units, Up: Assembly Coding 1103 110415.8.6 Floating Point 1105--------------------- 1106 1107Floating point arithmetic is used in GMP for multiplications on CPUs 1108with poor integer multipliers. It's mostly useful for 'mpn_mul_1', 1109'mpn_addmul_1' and 'mpn_submul_1' on 64-bit machines, and 1110'mpn_mul_basecase' on both 32-bit and 64-bit machines. 1111 1112 With IEEE 53-bit double precision floats, integer multiplications 1113producing up to 53 bits will give exact results. Breaking a 64x64 1114multiplication into eight 16x32->48 bit pieces is convenient. With some 1115care though six 21x32->53 bit products can be used, if one of the lower 1116two 21-bit pieces also uses the sign bit. 1117 1118 For the 'mpn_mul_1' family of functions on a 64-bit machine, the 1119invariant single limb is split at the start, into 3 or 4 pieces. Inside 1120the loop, the bignum operand is split into 32-bit pieces. Fast 1121conversion of these unsigned 32-bit pieces to floating point is highly 1122machine-dependent. In some cases, reading the data into the integer 1123unit, zero-extending to 64-bits, then transferring to the floating point 1124unit back via memory is the only option. 1125 1126 Converting partial products back to 64-bit limbs is usually best done 1127as a signed conversion. Since all values are smaller than 2^53, signed 1128and unsigned are the same, but most processors lack unsigned 1129conversions. 1130 1131 1132 1133 Here is a diagram showing 16x32 bit products for an 'mpn_mul_1' or 1134'mpn_addmul_1' with a 64-bit limb. The single limb operand V is split 1135into four 16-bit parts. The multi-limb operand U is split in the loop 1136into two 32-bit parts. 1137 1138 +---+---+---+---+ 1139 |v48|v32|v16|v00| V operand 1140 +---+---+---+---+ 1141 1142 +-------+---+---+ 1143 x | u32 | u00 | U operand (one limb) 1144 +---------------+ 1145 1146 --------------------------------- 1147 1148 +-----------+ 1149 | u00 x v00 | p00 48-bit products 1150 +-----------+ 1151 +-----------+ 1152 | u00 x v16 | p16 1153 +-----------+ 1154 +-----------+ 1155 | u00 x v32 | p32 1156 +-----------+ 1157 +-----------+ 1158 | u00 x v48 | p48 1159 +-----------+ 1160 +-----------+ 1161 | u32 x v00 | r32 1162 +-----------+ 1163 +-----------+ 1164 | u32 x v16 | r48 1165 +-----------+ 1166 +-----------+ 1167 | u32 x v32 | r64 1168 +-----------+ 1169 +-----------+ 1170 | u32 x v48 | r80 1171 +-----------+ 1172 1173 p32 and r32 can be summed using floating-point addition, and likewise 1174p48 and r48. p00 and p16 can be summed with r64 and r80 from the 1175previous iteration. 1176 1177 For each loop then, four 49-bit quantities are transferred to the 1178integer unit, aligned as follows, 1179 1180 |-----64bits----|-----64bits----| 1181 +------------+ 1182 | p00 + r64' | i00 1183 +------------+ 1184 +------------+ 1185 | p16 + r80' | i16 1186 +------------+ 1187 +------------+ 1188 | p32 + r32 | i32 1189 +------------+ 1190 +------------+ 1191 | p48 + r48 | i48 1192 +------------+ 1193 1194 The challenge then is to sum these efficiently and add in a carry 1195limb, generating a low 64-bit result limb and a high 33-bit carry limb 1196(i48 extends 33 bits into the high half). 1197 1198 1199File: gmp.info, Node: Assembly SIMD Instructions, Next: Assembly Software Pipelining, Prev: Assembly Floating Point, Up: Assembly Coding 1200 120115.8.7 SIMD Instructions 1202------------------------ 1203 1204The single-instruction multiple-data support in current microprocessors 1205is aimed at signal processing algorithms where each data point can be 1206treated more or less independently. There's generally not much support 1207for propagating the sort of carries that arise in GMP. 1208 1209 SIMD multiplications of say four 16x16 bit multiplies only do as much 1210work as one 32x32 from GMP's point of view, and need some shifts and 1211adds besides. But of course if say the SIMD form is fully pipelined and 1212uses less instruction decoding then it may still be worthwhile. 1213 1214 On the x86 chips, MMX has so far found a use in 'mpn_rshift' and 1215'mpn_lshift', and is used in a special case for 16-bit multipliers in 1216the P55 'mpn_mul_1'. SSE2 is used for Pentium 4 'mpn_mul_1', 1217'mpn_addmul_1', and 'mpn_submul_1'. 1218 1219 1220File: gmp.info, Node: Assembly Software Pipelining, Next: Assembly Loop Unrolling, Prev: Assembly SIMD Instructions, Up: Assembly Coding 1221 122215.8.8 Software Pipelining 1223-------------------------- 1224 1225Software pipelining consists of scheduling instructions around the 1226branch point in a loop. For example a loop might issue a load not for 1227use in the present iteration but the next, thereby allowing extra cycles 1228for the data to arrive from memory. 1229 1230 Naturally this is wanted only when doing things like loads or 1231multiplies that take several cycles to complete, and only where a CPU 1232has multiple functional units so that other work can be done in the 1233meantime. 1234 1235 A pipeline with several stages will have a data value in progress at 1236each stage and each loop iteration moves them along one stage. This is 1237like juggling. 1238 1239 If the latency of some instruction is greater than the loop time then 1240it will be necessary to unroll, so one register has a result ready to 1241use while another (or multiple others) are still in progress. (*note 1242Assembly Loop Unrolling::). 1243 1244 1245File: gmp.info, Node: Assembly Loop Unrolling, Next: Assembly Writing Guide, Prev: Assembly Software Pipelining, Up: Assembly Coding 1246 124715.8.9 Loop Unrolling 1248--------------------- 1249 1250Loop unrolling consists of replicating code so that several limbs are 1251processed in each loop. At a minimum this reduces loop overheads by a 1252corresponding factor, but it can also allow better register usage, for 1253example alternately using one register combination and then another. 1254Judicious use of 'm4' macros can help avoid lots of duplication in the 1255source code. 1256 1257 Any amount of unrolling can be handled with a loop counter that's 1258decremented by N each time, stopping when the remaining count is less 1259than the further N the loop will process. Or by subtracting N at the 1260start, the termination condition becomes when the counter C is less than 12610 (and the count of remaining limbs is C+N). 1262 1263 Alternately for a power of 2 unroll the loop count and remainder can 1264be established with a shift and mask. This is convenient if also making 1265a computed jump into the middle of a large loop. 1266 1267 The limbs not a multiple of the unrolling can be handled in various 1268ways, for example 1269 1270 * A simple loop at the end (or the start) to process the excess. 1271 Care will be wanted that it isn't too much slower than the unrolled 1272 part. 1273 1274 * A set of binary tests, for example after an 8-limb unrolling, test 1275 for 4 more limbs to process, then a further 2 more or not, and 1276 finally 1 more or not. This will probably take more code space 1277 than a simple loop. 1278 1279 * A 'switch' statement, providing separate code for each possible 1280 excess, for example an 8-limb unrolling would have separate code 1281 for 0 remaining, 1 remaining, etc, up to 7 remaining. This might 1282 take a lot of code, but may be the best way to optimize all cases 1283 in combination with a deep pipelined loop. 1284 1285 * A computed jump into the middle of the loop, thus making the first 1286 iteration handle the excess. This should make times smoothly 1287 increase with size, which is attractive, but setups for the jump 1288 and adjustments for pointers can be tricky and could become quite 1289 difficult in combination with deep pipelining. 1290 1291 1292File: gmp.info, Node: Assembly Writing Guide, Prev: Assembly Loop Unrolling, Up: Assembly Coding 1293 129415.8.10 Writing Guide 1295--------------------- 1296 1297This is a guide to writing software pipelined loops for processing limb 1298vectors in assembly. 1299 1300 First determine the algorithm and which instructions are needed. 1301Code it without unrolling or scheduling, to make sure it works. On a 13023-operand CPU try to write each new value to a new register, this will 1303greatly simplify later steps. 1304 1305 Then note for each instruction the functional unit and/or issue port 1306requirements. If an instruction can use either of two units, like U0 or 1307U1 then make a category "U0/U1". Count the total using each unit (or 1308combined unit), and count all instructions. 1309 1310 Figure out from those counts the best possible loop time. The goal 1311will be to find a perfect schedule where instruction latencies are 1312completely hidden. The total instruction count might be the limiting 1313factor, or perhaps a particular functional unit. It might be possible 1314to tweak the instructions to help the limiting factor. 1315 1316 Suppose the loop time is N, then make N issue buckets, with the final 1317loop branch at the end of the last. Now fill the buckets with dummy 1318instructions using the functional units desired. Run this to make sure 1319the intended speed is reached. 1320 1321 Now replace the dummy instructions with the real instructions from 1322the slow but correct loop you started with. The first will typically be 1323a load instruction. Then the instruction using that value is placed in 1324a bucket an appropriate distance down. Run the loop again, to check it 1325still runs at target speed. 1326 1327 Keep placing instructions, frequently measuring the loop. After a 1328few you will need to wrap around from the last bucket back to the top of 1329the loop. If you used the new-register for new-value strategy above 1330then there will be no register conflicts. If not then take care not to 1331clobber something already in use. Changing registers at this time is 1332very error prone. 1333 1334 The loop will overlap two or more of the original loop iterations, 1335and the computation of one vector element result will be started in one 1336iteration of the new loop, and completed one or several iterations 1337later. 1338 1339 The final step is to create feed-in and wind-down code for the loop. 1340A good way to do this is to make a copy (or copies) of the loop at the 1341start and delete those instructions which don't have valid antecedents, 1342and at the end replicate and delete those whose results are unwanted 1343(including any further loads). 1344 1345 The loop will have a minimum number of limbs loaded and processed, so 1346the feed-in code must test if the request size is smaller and skip 1347either to a suitable part of the wind-down or to special code for small 1348sizes. 1349 1350 1351File: gmp.info, Node: Internals, Next: Contributors, Prev: Algorithms, Up: Top 1352 135316 Internals 1354************ 1355 1356*This chapter is provided only for informational purposes and the 1357various internals described here may change in future GMP releases. 1358Applications expecting to be compatible with future releases should use 1359only the documented interfaces described in previous chapters.* 1360 1361* Menu: 1362 1363* Integer Internals:: 1364* Rational Internals:: 1365* Float Internals:: 1366* Raw Output Internals:: 1367* C++ Interface Internals:: 1368 1369 1370File: gmp.info, Node: Integer Internals, Next: Rational Internals, Prev: Internals, Up: Internals 1371 137216.1 Integer Internals 1373====================== 1374 1375'mpz_t' variables represent integers using sign and magnitude, in space 1376dynamically allocated and reallocated. The fields are as follows. 1377 1378'_mp_size' 1379 The number of limbs, or the negative of that when representing a 1380 negative integer. Zero is represented by '_mp_size' set to zero, 1381 in which case the '_mp_d' data is undefined. 1382 1383'_mp_d' 1384 A pointer to an array of limbs which is the magnitude. These are 1385 stored "little endian" as per the 'mpn' functions, so '_mp_d[0]' is 1386 the least significant limb and '_mp_d[ABS(_mp_size)-1]' is the most 1387 significant. Whenever '_mp_size' is non-zero, the most significant 1388 limb is non-zero. 1389 1390 Currently there's always at least one readable limb, so for 1391 instance 'mpz_get_ui' can fetch '_mp_d[0]' unconditionally (though 1392 its value is undefined if '_mp_size' is zero). 1393 1394'_mp_alloc' 1395 '_mp_alloc' is the number of limbs currently allocated at '_mp_d', 1396 and normally '_mp_alloc >= ABS(_mp_size)'. When an 'mpz' routine 1397 is about to (or might be about to) increase '_mp_size', it checks 1398 '_mp_alloc' to see whether there's enough space, and reallocates if 1399 not. 'MPZ_REALLOC' is generally used for this. 1400 1401 'mpz_t' variables initialised with the 'mpz_roinit_n' function or 1402 the 'MPZ_ROINIT_N' macro have '_mp_alloc = 0' but can have a 1403 non-zero '_mp_size'. They can only be used as read-only constants. 1404 See *note Integer Special Functions:: for details. 1405 1406 The various bitwise logical functions like 'mpz_and' behave as if 1407negative values were twos complement. But sign and magnitude is always 1408used internally, and necessary adjustments are made during the 1409calculations. Sometimes this isn't pretty, but sign and magnitude are 1410best for other routines. 1411 1412 Some internal temporary variables are setup with 'MPZ_TMP_INIT' and 1413these have '_mp_d' space obtained from 'TMP_ALLOC' rather than the 1414memory allocation functions. Care is taken to ensure that these are big 1415enough that no reallocation is necessary (since it would have 1416unpredictable consequences). 1417 1418 '_mp_size' and '_mp_alloc' are 'int', although 'mp_size_t' is usually 1419a 'long'. This is done to make the fields just 32 bits on some 64 bits 1420systems, thereby saving a few bytes of data space but still providing 1421plenty of range. 1422 1423 1424File: gmp.info, Node: Rational Internals, Next: Float Internals, Prev: Integer Internals, Up: Internals 1425 142616.2 Rational Internals 1427======================= 1428 1429'mpq_t' variables represent rationals using an 'mpz_t' numerator and 1430denominator (*note Integer Internals::). 1431 1432 The canonical form adopted is denominator positive (and non-zero), no 1433common factors between numerator and denominator, and zero uniquely 1434represented as 0/1. 1435 1436 It's believed that casting out common factors at each stage of a 1437calculation is best in general. A GCD is an O(N^2) operation so it's 1438better to do a few small ones immediately than to delay and have to do a 1439big one later. Knowing the numerator and denominator have no common 1440factors can be used for example in 'mpq_mul' to make only two cross GCDs 1441necessary, not four. 1442 1443 This general approach to common factors is badly sub-optimal in the 1444presence of simple factorizations or little prospect for cancellation, 1445but GMP has no way to know when this will occur. As per *note 1446Efficiency::, that's left to applications. The 'mpq_t' framework might 1447still suit, with 'mpq_numref' and 'mpq_denref' for direct access to the 1448numerator and denominator, or of course 'mpz_t' variables can be used 1449directly. 1450 1451 1452File: gmp.info, Node: Float Internals, Next: Raw Output Internals, Prev: Rational Internals, Up: Internals 1453 145416.3 Float Internals 1455==================== 1456 1457Efficient calculation is the primary aim of GMP floats and the use of 1458whole limbs and simple rounding facilitates this. 1459 1460 'mpf_t' floats have a variable precision mantissa and a single 1461machine word signed exponent. The mantissa is represented using sign 1462and magnitude. 1463 1464 most least 1465 significant significant 1466 limb limb 1467 1468 _mp_d 1469 |---- _mp_exp ---> | 1470 _____ _____ _____ _____ _____ 1471 |_____|_____|_____|_____|_____| 1472 . <------------ radix point 1473 1474 <-------- _mp_size ---------> 1475 1476 1477The fields are as follows. 1478 1479'_mp_size' 1480 The number of limbs currently in use, or the negative of that when 1481 representing a negative value. Zero is represented by '_mp_size' 1482 and '_mp_exp' both set to zero, and in that case the '_mp_d' data 1483 is unused. (In the future '_mp_exp' might be undefined when 1484 representing zero.) 1485 1486'_mp_prec' 1487 The precision of the mantissa, in limbs. In any calculation the 1488 aim is to produce '_mp_prec' limbs of result (the most significant 1489 being non-zero). 1490 1491'_mp_d' 1492 A pointer to the array of limbs which is the absolute value of the 1493 mantissa. These are stored "little endian" as per the 'mpn' 1494 functions, so '_mp_d[0]' is the least significant limb and 1495 '_mp_d[ABS(_mp_size)-1]' the most significant. 1496 1497 The most significant limb is always non-zero, but there are no 1498 other restrictions on its value, in particular the highest 1 bit 1499 can be anywhere within the limb. 1500 1501 '_mp_prec+1' limbs are allocated to '_mp_d', the extra limb being 1502 for convenience (see below). There are no reallocations during a 1503 calculation, only in a change of precision with 'mpf_set_prec'. 1504 1505'_mp_exp' 1506 The exponent, in limbs, determining the location of the implied 1507 radix point. Zero means the radix point is just above the most 1508 significant limb. Positive values mean a radix point offset 1509 towards the lower limbs and hence a value >= 1, as for example in 1510 the diagram above. Negative exponents mean a radix point further 1511 above the highest limb. 1512 1513 Naturally the exponent can be any value, it doesn't have to fall 1514 within the limbs as the diagram shows, it can be a long way above 1515 or a long way below. Limbs other than those included in the 1516 '{_mp_d,_mp_size}' data are treated as zero. 1517 1518 The '_mp_size' and '_mp_prec' fields are 'int', although the 1519'mp_size_t' type is usually a 'long'. The '_mp_exp' field is usually 1520'long'. This is done to make some fields just 32 bits on some 64 bits 1521systems, thereby saving a few bytes of data space but still providing 1522plenty of precision and a very large range. 1523 1524 1525The following various points should be noted. 1526 1527Low Zeros 1528 The least significant limbs '_mp_d[0]' etc can be zero, though such 1529 low zeros can always be ignored. Routines likely to produce low 1530 zeros check and avoid them to save time in subsequent calculations, 1531 but for most routines they're quite unlikely and aren't checked. 1532 1533Mantissa Size Range 1534 The '_mp_size' count of limbs in use can be less than '_mp_prec' if 1535 the value can be represented in less. This means low precision 1536 values or small integers stored in a high precision 'mpf_t' can 1537 still be operated on efficiently. 1538 1539 '_mp_size' can also be greater than '_mp_prec'. Firstly a value is 1540 allowed to use all of the '_mp_prec+1' limbs available at '_mp_d', 1541 and secondly when 'mpf_set_prec_raw' lowers '_mp_prec' it leaves 1542 '_mp_size' unchanged and so the size can be arbitrarily bigger than 1543 '_mp_prec'. 1544 1545Rounding 1546 All rounding is done on limb boundaries. Calculating '_mp_prec' 1547 limbs with the high non-zero will ensure the application requested 1548 minimum precision is obtained. 1549 1550 The use of simple "trunc" rounding towards zero is efficient, since 1551 there's no need to examine extra limbs and increment or decrement. 1552 1553Bit Shifts 1554 Since the exponent is in limbs, there are no bit shifts in basic 1555 operations like 'mpf_add' and 'mpf_mul'. When differing exponents 1556 are encountered all that's needed is to adjust pointers to line up 1557 the relevant limbs. 1558 1559 Of course 'mpf_mul_2exp' and 'mpf_div_2exp' will require bit 1560 shifts, but the choice is between an exponent in limbs which 1561 requires shifts there, or one in bits which requires them almost 1562 everywhere else. 1563 1564Use of '_mp_prec+1' Limbs 1565 The extra limb on '_mp_d' ('_mp_prec+1' rather than just 1566 '_mp_prec') helps when an 'mpf' routine might get a carry from its 1567 operation. 'mpf_add' for instance will do an 'mpn_add' of 1568 '_mp_prec' limbs. If there's no carry then that's the result, but 1569 if there is a carry then it's stored in the extra limb of space and 1570 '_mp_size' becomes '_mp_prec+1'. 1571 1572 Whenever '_mp_prec+1' limbs are held in a variable, the low limb is 1573 not needed for the intended precision, only the '_mp_prec' high 1574 limbs. But zeroing it out or moving the rest down is unnecessary. 1575 Subsequent routines reading the value will simply take the high 1576 limbs they need, and this will be '_mp_prec' if their target has 1577 that same precision. This is no more than a pointer adjustment, 1578 and must be checked anyway since the destination precision can be 1579 different from the sources. 1580 1581 Copy functions like 'mpf_set' will retain a full '_mp_prec+1' limbs 1582 if available. This ensures that a variable which has '_mp_size' 1583 equal to '_mp_prec+1' will get its full exact value copied. 1584 Strictly speaking this is unnecessary since only '_mp_prec' limbs 1585 are needed for the application's requested precision, but it's 1586 considered that an 'mpf_set' from one variable into another of the 1587 same precision ought to produce an exact copy. 1588 1589Application Precisions 1590 '__GMPF_BITS_TO_PREC' converts an application requested precision 1591 to an '_mp_prec'. The value in bits is rounded up to a whole limb 1592 then an extra limb is added since the most significant limb of 1593 '_mp_d' is only non-zero and therefore might contain only one bit. 1594 1595 '__GMPF_PREC_TO_BITS' does the reverse conversion, and removes the 1596 extra limb from '_mp_prec' before converting to bits. The net 1597 effect of reading back with 'mpf_get_prec' is simply the precision 1598 rounded up to a multiple of 'mp_bits_per_limb'. 1599 1600 Note that the extra limb added here for the high only being 1601 non-zero is in addition to the extra limb allocated to '_mp_d'. 1602 For example with a 32-bit limb, an application request for 250 bits 1603 will be rounded up to 8 limbs, then an extra added for the high 1604 being only non-zero, giving an '_mp_prec' of 9. '_mp_d' then gets 1605 10 limbs allocated. Reading back with 'mpf_get_prec' will take 1606 '_mp_prec' subtract 1 limb and multiply by 32, giving 256 bits. 1607 1608 Strictly speaking, the fact the high limb has at least one bit 1609 means that a float with, say, 3 limbs of 32-bits each will be 1610 holding at least 65 bits, but for the purposes of 'mpf_t' it's 1611 considered simply to be 64 bits, a nice multiple of the limb size. 1612 1613 1614File: gmp.info, Node: Raw Output Internals, Next: C++ Interface Internals, Prev: Float Internals, Up: Internals 1615 161616.4 Raw Output Internals 1617========================= 1618 1619'mpz_out_raw' uses the following format. 1620 1621 +------+------------------------+ 1622 | size | data bytes | 1623 +------+------------------------+ 1624 1625 The size is 4 bytes written most significant byte first, being the 1626number of subsequent data bytes, or the twos complement negative of that 1627when a negative integer is represented. The data bytes are the absolute 1628value of the integer, written most significant byte first. 1629 1630 The most significant data byte is always non-zero, so the output is 1631the same on all systems, irrespective of limb size. 1632 1633 In GMP 1, leading zero bytes were written to pad the data bytes to a 1634multiple of the limb size. 'mpz_inp_raw' will still accept this, for 1635compatibility. 1636 1637 The use of "big endian" for both the size and data fields is 1638deliberate, it makes the data easy to read in a hex dump of a file. 1639Unfortunately it also means that the limb data must be reversed when 1640reading or writing, so neither a big endian nor little endian system can 1641just read and write '_mp_d'. 1642 1643 1644File: gmp.info, Node: C++ Interface Internals, Prev: Raw Output Internals, Up: Internals 1645 164616.5 C++ Interface Internals 1647============================ 1648 1649A system of expression templates is used to ensure something like 1650'a=b+c' turns into a simple call to 'mpz_add' etc. For 'mpf_class' the 1651scheme also ensures the precision of the final destination is used for 1652any temporaries within a statement like 'f=w*x+y*z'. These are 1653important features which a naive implementation cannot provide. 1654 1655 A simplified description of the scheme follows. The true scheme is 1656complicated by the fact that expressions have different return types. 1657For detailed information, refer to the source code. 1658 1659 To perform an operation, say, addition, we first define a "function 1660object" evaluating it, 1661 1662 struct __gmp_binary_plus 1663 { 1664 static void eval(mpf_t f, const mpf_t g, const mpf_t h) 1665 { 1666 mpf_add(f, g, h); 1667 } 1668 }; 1669 1670And an "additive expression" object, 1671 1672 __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> > 1673 operator+(const mpf_class &f, const mpf_class &g) 1674 { 1675 return __gmp_expr 1676 <__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >(f, g); 1677 } 1678 1679 The seemingly redundant '__gmp_expr<__gmp_binary_expr<...>>' is used 1680to encapsulate any possible kind of expression into a single template 1681type. In fact even 'mpf_class' etc are 'typedef' specializations of 1682'__gmp_expr'. 1683 1684 Next we define assignment of '__gmp_expr' to 'mpf_class'. 1685 1686 template <class T> 1687 mpf_class & mpf_class::operator=(const __gmp_expr<T> &expr) 1688 { 1689 expr.eval(this->get_mpf_t(), this->precision()); 1690 return *this; 1691 } 1692 1693 template <class Op> 1694 void __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, Op> >::eval 1695 (mpf_t f, mp_bitcnt_t precision) 1696 { 1697 Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t()); 1698 } 1699 1700 where 'expr.val1' and 'expr.val2' are references to the expression's 1701operands (here 'expr' is the '__gmp_binary_expr' stored within the 1702'__gmp_expr'). 1703 1704 This way, the expression is actually evaluated only at the time of 1705assignment, when the required precision (that of 'f') is known. 1706Furthermore the target 'mpf_t' is now available, thus we can call 1707'mpf_add' directly with 'f' as the output argument. 1708 1709 Compound expressions are handled by defining operators taking 1710subexpressions as their arguments, like this: 1711 1712 template <class T, class U> 1713 __gmp_expr 1714 <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> > 1715 operator+(const __gmp_expr<T> &expr1, const __gmp_expr<U> &expr2) 1716 { 1717 return __gmp_expr 1718 <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> > 1719 (expr1, expr2); 1720 } 1721 1722 And the corresponding specializations of '__gmp_expr::eval': 1723 1724 template <class T, class U, class Op> 1725 void __gmp_expr 1726 <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, Op> >::eval 1727 (mpf_t f, mp_bitcnt_t precision) 1728 { 1729 // declare two temporaries 1730 mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision); 1731 Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t()); 1732 } 1733 1734 The expression is thus recursively evaluated to any level of 1735complexity and all subexpressions are evaluated to the precision of 'f'. 1736 1737 1738File: gmp.info, Node: Contributors, Next: References, Prev: Internals, Up: Top 1739 1740Appendix A Contributors 1741*********************** 1742 1743Torbj�rn Granlund wrote the original GMP library and is still the main 1744developer. Code not explicitly attributed to others, was contributed by 1745Torbj�rn. Several other individuals and organizations have contributed 1746GMP. Here is a list in chronological order on first contribution: 1747 1748 Gunnar Sj�din and Hans Riesel helped with mathematical problems in 1749early versions of the library. 1750 1751 Richard Stallman helped with the interface design and revised the 1752first version of this manual. 1753 1754 Brian Beuning and Doug Lea helped with testing of early versions of 1755the library and made creative suggestions. 1756 1757 John Amanatides of York University in Canada contributed the function 1758'mpz_probab_prime_p'. 1759 1760 Paul Zimmermann wrote the REDC-based mpz_powm code, the 1761Sch�nhage-Strassen FFT multiply code, and the Karatsuba square root 1762code. He also improved the Toom3 code for GMP 4.2. Paul sparked the 1763development of GMP 2, with his comparisons between bignum packages. The 1764ECMNET project Paul is organizing was a driving force behind many of the 1765optimizations in GMP 3. Paul also wrote the new GMP 4.3 nth root code 1766(with Torbj�rn). 1767 1768 Ken Weber (Kent State University, Universidade Federal do Rio Grande 1769do Sul) contributed now defunct versions of 'mpz_gcd', 'mpz_divexact', 1770'mpn_gcd', and 'mpn_bdivmod', partially supported by CNPq (Brazil) grant 1771301314194-2. 1772 1773 Per Bothner of Cygnus Support helped to set up GMP to use Cygnus' 1774configure. He has also made valuable suggestions and tested numerous 1775intermediary releases. 1776 1777 Joachim Hollman was involved in the design of the 'mpf' interface, 1778and in the 'mpz' design revisions for version 2. 1779 1780 Bennet Yee contributed the initial versions of 'mpz_jacobi' and 1781'mpz_legendre'. 1782 1783 Andreas Schwab contributed the files 'mpn/m68k/lshift.S' and 1784'mpn/m68k/rshift.S' (now in '.asm' form). 1785 1786 Robert Harley of Inria, France and David Seal of ARM, England, 1787suggested clever improvements for population count. Robert also wrote 1788highly optimized Karatsuba and 3-way Toom multiplication functions for 1789GMP 3, and contributed the ARM assembly code. 1790 1791 Torsten Ekedahl of the Mathematical department of Stockholm 1792University provided significant inspiration during several phases of the 1793GMP development. His mathematical expertise helped improve several 1794algorithms. 1795 1796 Linus Nordberg wrote the new configure system based on autoconf and 1797implemented the new random functions. 1798 1799 Kevin Ryde worked on a large number of things: optimized x86 code, m4 1800asm macros, parameter tuning, speed measuring, the configure system, 1801function inlining, divisibility tests, bit scanning, Jacobi symbols, 1802Fibonacci and Lucas number functions, printf and scanf functions, perl 1803interface, demo expression parser, the algorithms chapter in the manual, 1804'gmpasm-mode.el', and various miscellaneous improvements elsewhere. 1805 1806 Kent Boortz made the Mac OS 9 port. 1807 1808 Steve Root helped write the optimized alpha 21264 assembly code. 1809 1810 Gerardo Ballabio wrote the 'gmpxx.h' C++ class interface and the C++ 1811'istream' input routines. 1812 1813 Jason Moxham rewrote 'mpz_fac_ui'. 1814 1815 Pedro Gimeno implemented the Mersenne Twister and made other random 1816number improvements. 1817 1818 Niels M�ller wrote the sub-quadratic GCD, extended GCD and jacobi 1819code, the quadratic Hensel division code, and (with Torbj�rn) the new 1820divide and conquer division code for GMP 4.3. Niels also helped 1821implement the new Toom multiply code for GMP 4.3 and implemented helper 1822functions to simplify Toom evaluations for GMP 5.0. He wrote the 1823original version of mpn_mulmod_bnm1, and he is the main author of the 1824mini-gmp package used for gmp bootstrapping. 1825 1826 Alberto Zanoni and Marco Bodrato suggested the unbalanced multiply 1827strategy, and found the optimal strategies for evaluation and 1828interpolation in Toom multiplication. 1829 1830 Marco Bodrato helped implement the new Toom multiply code for GMP 4.3 1831and implemented most of the new Toom multiply and squaring code for 5.0. 1832He is the main author of the current mpn_mulmod_bnm1, mpn_mullo_n, and 1833mpn_sqrlo. Marco also wrote the functions mpn_invert and 1834mpn_invertappr, and improved the speed of integer root extraction. He 1835is the author of mini-mpq, an additional layer to mini-gmp; of most of 1836the combinatorial functions and the BPSW primality testing 1837implementation, for both the main library and the mini-gmp package. 1838 1839 David Harvey suggested the internal function 'mpn_bdiv_dbm1', 1840implementing division relevant to Toom multiplication. He also worked 1841on fast assembly sequences, in particular on a fast AMD64 1842'mpn_mul_basecase'. He wrote the internal middle product functions 1843'mpn_mulmid_basecase', 'mpn_toom42_mulmid', 'mpn_mulmid_n' and related 1844helper routines. 1845 1846 Martin Boij wrote 'mpn_perfect_power_p'. 1847 1848 Marc Glisse improved 'gmpxx.h': use fewer temporaries (faster), 1849specializations of 'numeric_limits' and 'common_type', C++11 features 1850(move constructors, explicit bool conversion, UDL), make the conversion 1851from 'mpq_class' to 'mpz_class' explicit, optimize operations where one 1852argument is a small compile-time constant, replace some heap allocations 1853by stack allocations. He also fixed the eofbit handling of C++ streams, 1854and removed one division from 'mpq/aors.c'. 1855 1856 David S Miller wrote assembly code for SPARC T3 and T4. 1857 1858 Mark Sofroniou cleaned up the types of mul_fft.c, letting it work for 1859huge operands. 1860 1861 Ulrich Weigand ported GMP to the powerpc64le ABI. 1862 1863 (This list is chronological, not ordered after significance. If you 1864have contributed to GMP but are not listed above, please tell 1865<gmp-devel@gmplib.org> about the omission!) 1866 1867 The development of floating point functions of GNU MP 2, were 1868supported in part by the ESPRIT-BRA (Basic Research Activities) 6846 1869project POSSO (POlynomial System SOlving). 1870 1871 The development of GMP 2, 3, and 4.0 was supported in part by the IDA 1872Center for Computing Sciences. 1873 1874 The development of GMP 4.3, 5.0, and 5.1 was supported in part by the 1875Swedish Foundation for Strategic Research. 1876 1877 Thanks go to Hans Thorsen for donating an SGI system for the GMP test 1878system environment. 1879 1880 1881File: gmp.info, Node: References, Next: GNU Free Documentation License, Prev: Contributors, Up: Top 1882 1883Appendix B References 1884********************* 1885 1886B.1 Books 1887========= 1888 1889 * Jonathan M. Borwein and Peter B. Borwein, "Pi and the AGM: A Study 1890 in Analytic Number Theory and Computational Complexity", Wiley, 1891 1998. 1892 1893 * Richard Crandall and Carl Pomerance, "Prime Numbers: A 1894 Computational Perspective", 2nd edition, Springer-Verlag, 2005. 1895 <https://www.math.dartmouth.edu/~carlp/> 1896 1897 * Henri Cohen, "A Course in Computational Algebraic Number Theory", 1898 Graduate Texts in Mathematics number 138, Springer-Verlag, 1993. 1899 <https://www.math.u-bordeaux.fr/~cohen/> 1900 1901 * Donald E. Knuth, "The Art of Computer Programming", volume 2, 1902 "Seminumerical Algorithms", 3rd edition, Addison-Wesley, 1998. 1903 <https://www-cs-faculty.stanford.edu/~knuth/taocp.html> 1904 1905 * John D. Lipson, "Elements of Algebra and Algebraic Computing", The 1906 Benjamin Cummings Publishing Company Inc, 1981. 1907 1908 * Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, 1909 "Handbook of Applied Cryptography", 1910 <http://www.cacr.math.uwaterloo.ca/hac/> 1911 1912 * Richard M. Stallman and the GCC Developer Community, "Using the GNU 1913 Compiler Collection", Free Software Foundation, 2008, available 1914 online <https://gcc.gnu.org/onlinedocs/>, and in the GCC package 1915 <https://ftp.gnu.org/gnu/gcc/> 1916 1917B.2 Papers 1918========== 1919 1920 * Yves Bertot, Nicolas Magaud and Paul Zimmermann, "A Proof of GMP 1921 Square Root", Journal of Automated Reasoning, volume 29, 2002, pp. 1922 225-252. Also available online as INRIA Research Report 4475, June 1923 2002, <https://hal.inria.fr/docs/00/07/21/13/PDF/RR-4475.pdf> 1924 1925 * Christoph Burnikel and Joachim Ziegler, "Fast Recursive Division", 1926 Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022, 1927 <https://www.mpi-inf.mpg.de/~ziegler/TechRep.ps.gz> 1928 1929 * Torbj�rn Granlund and Peter L. Montgomery, "Division by Invariant 1930 Integers using Multiplication", in Proceedings of the SIGPLAN 1931 PLDI'94 Conference, June 1994. Also available 1932 <https://gmplib.org/~tege/divcnst-pldi94.pdf>. 1933 1934 * Niels M�ller and Torbj�rn Granlund, "Improved division by invariant 1935 integers", IEEE Transactions on Computers, 11 June 2010. 1936 <https://gmplib.org/~tege/division-paper.pdf> 1937 1938 * Torbj�rn Granlund and Niels M�ller, "Division of integers large and 1939 small", to appear. 1940 1941 * Tudor Jebelean, "An algorithm for exact division", Journal of 1942 Symbolic Computation, volume 15, 1993, pp. 169-180. Research 1943 report version available 1944 <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz> 1945 1946 * Tudor Jebelean, "Exact Division with Karatsuba Complexity - 1947 Extended Abstract", RISC-Linz technical report 96-31, 1948 <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz> 1949 1950 * Tudor Jebelean, "Practical Integer Division with Karatsuba 1951 Complexity", ISSAC 97, pp. 339-341. Technical report available 1952 <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz> 1953 1954 * Tudor Jebelean, "A Generalization of the Binary GCD Algorithm", 1955 ISSAC 93, pp. 111-116. Technical report version available 1956 <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz> 1957 1958 * Tudor Jebelean, "A Double-Digit Lehmer-Euclid Algorithm for Finding 1959 the GCD of Long Integers", Journal of Symbolic Computation, volume 1960 19, 1995, pp. 145-157. Technical report version also available 1961 <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz> 1962 1963 * Werner Krandick and Tudor Jebelean, "Bidirectional Exact Integer 1964 Division", Journal of Symbolic Computation, volume 21, 1996, pp. 1965 441-455. Early technical report version also available 1966 <ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz> 1967 1968 * Makoto Matsumoto and Takuji Nishimura, "Mersenne Twister: A 1969 623-dimensionally equidistributed uniform pseudorandom number 1970 generator", ACM Transactions on Modelling and Computer Simulation, 1971 volume 8, January 1998, pp. 3-30. Available online 1972 <http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/ARTICLES/mt.pdf> 1973 1974 * R. Moenck and A. Borodin, "Fast Modular Transforms via Division", 1975 Proceedings of the 13th Annual IEEE Symposium on Switching and 1976 Automata Theory, October 1972, pp. 90-96. Reprinted as "Fast 1977 Modular Transforms", Journal of Computer and System Sciences, 1978 volume 8, number 3, June 1974, pp. 366-386. 1979 1980 * Niels M�ller, "On Sch�nhage's algorithm and subquadratic integer 1981 GCD computation", in Mathematics of Computation, volume 77, January 1982 2008, pp. 589-607, 1983 <https://www.ams.org/journals/mcom/2008-77-261/S0025-5718-07-02017-0/home.html> 1984 1985 * Peter L. Montgomery, "Modular Multiplication Without Trial 1986 Division", in Mathematics of Computation, volume 44, number 170, 1987 April 1985. 1988 1989 * Arnold Sch�nhage and Volker Strassen, "Schnelle Multiplikation 1990 grosser Zahlen", Computing 7, 1971, pp. 281-292. 1991 1992 * Kenneth Weber, "The accelerated integer GCD algorithm", ACM 1993 Transactions on Mathematical Software, volume 21, number 1, March 1994 1995, pp. 111-122. 1995 1996 * Paul Zimmermann, "Karatsuba Square Root", INRIA Research Report 1997 3805, November 1999, 1998 <https://hal.inria.fr/inria-00072854/PDF/RR-3805.pdf> 1999 2000 * Paul Zimmermann, "A Proof of GMP Fast Division and Square Root 2001 Implementations", 2002 <https://homepages.loria.fr/PZimmermann/papers/proof-div-sqrt.ps.gz> 2003 2004 * Dan Zuras, "On Squaring and Multiplying Large Integers", ARITH-11: 2005 IEEE Symposium on Computer Arithmetic, 1993, pp. 260 to 271. 2006 Reprinted as "More on Multiplying and Squaring Large Integers", 2007 IEEE Transactions on Computers, volume 43, number 8, August 1994, 2008 pp. 899-908. 2009 2010 * Niels M�ller, "Efficient computation of the Jacobi symbol", 2011 <https://arxiv.org/abs/1907.07795> 2012 2013 2014File: gmp.info, Node: GNU Free Documentation License, Next: Concept Index, Prev: References, Up: Top 2015 2016Appendix C GNU Free Documentation License 2017***************************************** 2018 2019 Version 1.3, 3 November 2008 2020 2021 Copyright � 2000-2002, 2007, 2008 Free Software Foundation, Inc. 2022 <http://fsf.org/> 2023 2024 Everyone is permitted to copy and distribute verbatim copies 2025 of this license document, but changing it is not allowed. 2026 2027 0. PREAMBLE 2028 2029 The purpose of this License is to make a manual, textbook, or other 2030 functional and useful document "free" in the sense of freedom: to 2031 assure everyone the effective freedom to copy and redistribute it, 2032 with or without modifying it, either commercially or 2033 noncommercially. Secondarily, this License preserves for the 2034 author and publisher a way to get credit for their work, while not 2035 being considered responsible for modifications made by others. 2036 2037 This License is a kind of "copyleft", which means that derivative 2038 works of the document must themselves be free in the same sense. 2039 It complements the GNU General Public License, which is a copyleft 2040 license designed for free software. 2041 2042 We have designed this License in order to use it for manuals for 2043 free software, because free software needs free documentation: a 2044 free program should come with manuals providing the same freedoms 2045 that the software does. 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A copy of the license is included in the section entitled ``GNU 2477 Free Documentation License''. 2478 2479 If you have Invariant Sections, Front-Cover Texts and Back-Cover 2480Texts, replace the "with...Texts." line with this: 2481 2482 with the Invariant Sections being LIST THEIR TITLES, with 2483 the Front-Cover Texts being LIST, and with the Back-Cover Texts 2484 being LIST. 2485 2486 If you have Invariant Sections without Cover Texts, or some other 2487combination of the three, merge those two alternatives to suit the 2488situation. 2489 2490 If your document contains nontrivial examples of program code, we 2491recommend releasing these examples in parallel under your choice of free 2492software license, such as the GNU General Public License, to permit 2493their use in free software. 2494 2495 2496File: gmp.info, Node: Concept Index, Next: Function Index, Prev: GNU Free Documentation License, Up: Top 2497 2498Concept Index 2499************* 2500 2501�[index�] 2502* Menu: 2503 2504* #include: Headers and Libraries. 2505 (line 6) 2506* --build: Build Options. (line 51) 2507* --disable-fft: Build Options. (line 307) 2508* --disable-shared: Build Options. (line 44) 2509* --disable-static: Build Options. (line 44) 2510* --enable-alloca: Build Options. (line 273) 2511* --enable-assert: Build Options. (line 313) 2512* --enable-cxx: Build Options. (line 225) 2513* --enable-fat: Build Options. (line 160) 2514* --enable-profiling: Build Options. (line 317) 2515* --enable-profiling <1>: Profiling. (line 6) 2516* --exec-prefix: Build Options. (line 32) 2517* --host: Build Options. (line 65) 2518* --prefix: Build Options. (line 32) 2519* -finstrument-functions: Profiling. (line 66) 2520* 2exp functions: Efficiency. (line 43) 2521* 68000: Notes for Particular Systems. 2522 (line 94) 2523* 80x86: Notes for Particular Systems. 2524 (line 150) 2525* ABI: Build Options. (line 167) 2526* ABI <1>: ABI and ISA. (line 6) 2527* About this manual: Introduction to GMP. (line 57) 2528* AC_CHECK_LIB: Autoconf. (line 11) 2529* AIX: ABI and ISA. (line 174) 2530* AIX <1>: Notes for Particular Systems. 2531 (line 7) 2532* Algorithms: Algorithms. (line 6) 2533* alloca: Build Options. (line 273) 2534* Allocation of memory: Custom Allocation. (line 6) 2535* AMD64: ABI and ISA. (line 44) 2536* Anonymous FTP of latest version: Introduction to GMP. (line 37) 2537* Application Binary Interface: ABI and ISA. (line 6) 2538* Arithmetic functions: Integer Arithmetic. (line 6) 2539* Arithmetic functions <1>: Rational Arithmetic. (line 6) 2540* Arithmetic functions <2>: Float Arithmetic. (line 6) 2541* ARM: Notes for Particular Systems. 2542 (line 20) 2543* Assembly cache handling: Assembly Cache Handling. 2544 (line 6) 2545* Assembly carry propagation: Assembly Carry Propagation. 2546 (line 6) 2547* Assembly code organisation: Assembly Code Organisation. 2548 (line 6) 2549* Assembly coding: Assembly Coding. (line 6) 2550* Assembly floating Point: Assembly Floating Point. 2551 (line 6) 2552* Assembly loop unrolling: Assembly Loop Unrolling. 2553 (line 6) 2554* Assembly SIMD: Assembly SIMD Instructions. 2555 (line 6) 2556* Assembly software pipelining: Assembly Software Pipelining. 2557 (line 6) 2558* Assembly writing guide: Assembly Writing Guide. 2559 (line 6) 2560* Assertion checking: Build Options. (line 313) 2561* Assertion checking <1>: Debugging. (line 74) 2562* Assignment functions: Assigning Integers. (line 6) 2563* Assignment functions <1>: Simultaneous Integer Init & Assign. 2564 (line 6) 2565* Assignment functions <2>: Initializing Rationals. 2566 (line 6) 2567* Assignment functions <3>: Assigning Floats. (line 6) 2568* Assignment functions <4>: Simultaneous Float Init & Assign. 2569 (line 6) 2570* Autoconf: Autoconf. (line 6) 2571* Basics: GMP Basics. (line 6) 2572* Binomial coefficient algorithm: Binomial Coefficients Algorithm. 2573 (line 6) 2574* Binomial coefficient functions: Number Theoretic Functions. 2575 (line 128) 2576* Binutils strip: Known Build Problems. 2577 (line 28) 2578* Bit manipulation functions: Integer Logic and Bit Fiddling. 2579 (line 6) 2580* Bit scanning functions: Integer Logic and Bit Fiddling. 2581 (line 39) 2582* Bit shift left: Integer Arithmetic. (line 38) 2583* Bit shift right: Integer Division. (line 74) 2584* Bits per limb: Useful Macros and Constants. 2585 (line 7) 2586* Bug reporting: Reporting Bugs. (line 6) 2587* Build directory: Build Options. (line 19) 2588* Build notes for binary packaging: Notes for Package Builds. 2589 (line 6) 2590* Build notes for particular systems: Notes for Particular Systems. 2591 (line 6) 2592* Build options: Build Options. (line 6) 2593* Build problems known: Known Build Problems. 2594 (line 6) 2595* Build system: Build Options. (line 51) 2596* Building GMP: Installing GMP. (line 6) 2597* Bus error: Debugging. (line 7) 2598* C compiler: Build Options. (line 178) 2599* C++ compiler: Build Options. (line 249) 2600* C++ interface: C++ Class Interface. (line 6) 2601* C++ interface internals: C++ Interface Internals. 2602 (line 6) 2603* C++ istream input: C++ Formatted Input. (line 6) 2604* C++ ostream output: C++ Formatted Output. 2605 (line 6) 2606* C++ support: Build Options. (line 225) 2607* CC: Build Options. (line 178) 2608* CC_FOR_BUILD: Build Options. (line 212) 2609* CFLAGS: Build Options. (line 178) 2610* Checker: Debugging. (line 110) 2611* checkergcc: Debugging. (line 117) 2612* Code organisation: Assembly Code Organisation. 2613 (line 6) 2614* Compaq C++: Notes for Particular Systems. 2615 (line 25) 2616* Comparison functions: Integer Comparisons. (line 6) 2617* Comparison functions <1>: Comparing Rationals. (line 6) 2618* Comparison functions <2>: Float Comparison. (line 6) 2619* Compatibility with older versions: Compatibility with older versions. 2620 (line 6) 2621* Conditions for copying GNU MP: Copying. (line 6) 2622* Configuring GMP: Installing GMP. (line 6) 2623* Congruence algorithm: Exact Remainder. (line 30) 2624* Congruence functions: Integer Division. (line 150) 2625* Constants: Useful Macros and Constants. 2626 (line 6) 2627* Contributors: Contributors. (line 6) 2628* Conventions for parameters: Parameter Conventions. 2629 (line 6) 2630* Conventions for variables: Variable Conventions. 2631 (line 6) 2632* Conversion functions: Converting Integers. (line 6) 2633* Conversion functions <1>: Rational Conversions. 2634 (line 6) 2635* Conversion functions <2>: Converting Floats. (line 6) 2636* Copying conditions: Copying. (line 6) 2637* CPPFLAGS: Build Options. (line 204) 2638* CPU types: Introduction to GMP. (line 24) 2639* CPU types <1>: Build Options. (line 107) 2640* Cross compiling: Build Options. (line 65) 2641* Cryptography functions, low-level: Low-level Functions. (line 507) 2642* Custom allocation: Custom Allocation. (line 6) 2643* CXX: Build Options. (line 249) 2644* CXXFLAGS: Build Options. (line 249) 2645* Cygwin: Notes for Particular Systems. 2646 (line 57) 2647* Darwin: Known Build Problems. 2648 (line 51) 2649* Debugging: Debugging. (line 6) 2650* Demonstration programs: Demonstration Programs. 2651 (line 6) 2652* Digits in an integer: Miscellaneous Integer Functions. 2653 (line 23) 2654* Divisibility algorithm: Exact Remainder. (line 30) 2655* Divisibility functions: Integer Division. (line 136) 2656* Divisibility functions <1>: Integer Division. (line 150) 2657* Divisibility testing: Efficiency. (line 91) 2658* Division algorithms: Division Algorithms. (line 6) 2659* Division functions: Integer Division. (line 6) 2660* Division functions <1>: Rational Arithmetic. (line 24) 2661* Division functions <2>: Float Arithmetic. (line 33) 2662* DJGPP: Notes for Particular Systems. 2663 (line 57) 2664* DJGPP <1>: Known Build Problems. 2665 (line 18) 2666* DLLs: Notes for Particular Systems. 2667 (line 70) 2668* DocBook: Build Options. (line 340) 2669* Documentation formats: Build Options. (line 333) 2670* Documentation license: GNU Free Documentation License. 2671 (line 6) 2672* DVI: Build Options. (line 336) 2673* Efficiency: Efficiency. (line 6) 2674* Emacs: Emacs. (line 6) 2675* Exact division functions: Integer Division. (line 125) 2676* Exact remainder: Exact Remainder. (line 6) 2677* Example programs: Demonstration Programs. 2678 (line 6) 2679* Exec prefix: Build Options. (line 32) 2680* Execution profiling: Build Options. (line 317) 2681* Execution profiling <1>: Profiling. (line 6) 2682* Exponentiation functions: Integer Exponentiation. 2683 (line 6) 2684* Exponentiation functions <1>: Float Arithmetic. (line 41) 2685* Export: Integer Import and Export. 2686 (line 45) 2687* Expression parsing demo: Demonstration Programs. 2688 (line 15) 2689* Expression parsing demo <1>: Demonstration Programs. 2690 (line 17) 2691* Expression parsing demo <2>: Demonstration Programs. 2692 (line 19) 2693* Extended GCD: Number Theoretic Functions. 2694 (line 47) 2695* Factor removal functions: Number Theoretic Functions. 2696 (line 108) 2697* Factorial algorithm: Factorial Algorithm. (line 6) 2698* Factorial functions: Number Theoretic Functions. 2699 (line 116) 2700* Factorization demo: Demonstration Programs. 2701 (line 22) 2702* Fast Fourier Transform: FFT Multiplication. (line 6) 2703* Fat binary: Build Options. (line 160) 2704* FFT multiplication: Build Options. (line 307) 2705* FFT multiplication <1>: FFT Multiplication. (line 6) 2706* Fibonacci number algorithm: Fibonacci Numbers Algorithm. 2707 (line 6) 2708* Fibonacci sequence functions: Number Theoretic Functions. 2709 (line 136) 2710* Float arithmetic functions: Float Arithmetic. (line 6) 2711* Float assignment functions: Assigning Floats. (line 6) 2712* Float assignment functions <1>: Simultaneous Float Init & Assign. 2713 (line 6) 2714* Float comparison functions: Float Comparison. (line 6) 2715* Float conversion functions: Converting Floats. (line 6) 2716* Float functions: Floating-point Functions. 2717 (line 6) 2718* Float initialization functions: Initializing Floats. (line 6) 2719* Float initialization functions <1>: Simultaneous Float Init & Assign. 2720 (line 6) 2721* Float input and output functions: I/O of Floats. (line 6) 2722* Float internals: Float Internals. (line 6) 2723* Float miscellaneous functions: Miscellaneous Float Functions. 2724 (line 6) 2725* Float random number functions: Miscellaneous Float Functions. 2726 (line 27) 2727* Float rounding functions: Miscellaneous Float Functions. 2728 (line 9) 2729* Float sign tests: Float Comparison. (line 34) 2730* Floating point mode: Notes for Particular Systems. 2731 (line 34) 2732* Floating-point functions: Floating-point Functions. 2733 (line 6) 2734* Floating-point number: Nomenclature and Types. 2735 (line 21) 2736* fnccheck: Profiling. (line 77) 2737* Formatted input: Formatted Input. (line 6) 2738* Formatted output: Formatted Output. (line 6) 2739* Free Documentation License: GNU Free Documentation License. 2740 (line 6) 2741* FreeBSD: Notes for Particular Systems. 2742 (line 43) 2743* FreeBSD <1>: Notes for Particular Systems. 2744 (line 52) 2745* frexp: Converting Integers. (line 43) 2746* frexp <1>: Converting Floats. (line 24) 2747* FTP of latest version: Introduction to GMP. (line 37) 2748* Function classes: Function Classes. (line 6) 2749* FunctionCheck: Profiling. (line 77) 2750* GCC Checker: Debugging. (line 110) 2751* GCD algorithms: Greatest Common Divisor Algorithms. 2752 (line 6) 2753* GCD extended: Number Theoretic Functions. 2754 (line 47) 2755* GCD functions: Number Theoretic Functions. 2756 (line 30) 2757* GDB: Debugging. (line 53) 2758* Generic C: Build Options. (line 151) 2759* GMP Perl module: Demonstration Programs. 2760 (line 28) 2761* GMP version number: Useful Macros and Constants. 2762 (line 12) 2763* gmp.h: Headers and Libraries. 2764 (line 6) 2765* gmpxx.h: C++ Interface General. 2766 (line 8) 2767* GNU Debugger: Debugging. (line 53) 2768* GNU Free Documentation License: GNU Free Documentation License. 2769 (line 6) 2770* GNU strip: Known Build Problems. 2771 (line 28) 2772* gprof: Profiling. (line 41) 2773* Greatest common divisor algorithms: Greatest Common Divisor Algorithms. 2774 (line 6) 2775* Greatest common divisor functions: Number Theoretic Functions. 2776 (line 30) 2777* Hardware floating point mode: Notes for Particular Systems. 2778 (line 34) 2779* Headers: Headers and Libraries. 2780 (line 6) 2781* Heap problems: Debugging. (line 23) 2782* Home page: Introduction to GMP. (line 33) 2783* Host system: Build Options. (line 65) 2784* HP-UX: ABI and ISA. (line 76) 2785* HP-UX <1>: ABI and ISA. (line 114) 2786* HPPA: ABI and ISA. (line 76) 2787* I/O functions: I/O of Integers. (line 6) 2788* I/O functions <1>: I/O of Rationals. (line 6) 2789* I/O functions <2>: I/O of Floats. (line 6) 2790* i386: Notes for Particular Systems. 2791 (line 150) 2792* IA-64: ABI and ISA. (line 114) 2793* Import: Integer Import and Export. 2794 (line 11) 2795* In-place operations: Efficiency. (line 57) 2796* Include files: Headers and Libraries. 2797 (line 6) 2798* info-lookup-symbol: Emacs. (line 6) 2799* Initialization functions: Initializing Integers. 2800 (line 6) 2801* Initialization functions <1>: Simultaneous Integer Init & Assign. 2802 (line 6) 2803* Initialization functions <2>: Initializing Rationals. 2804 (line 6) 2805* Initialization functions <3>: Initializing Floats. (line 6) 2806* Initialization functions <4>: Simultaneous Float Init & Assign. 2807 (line 6) 2808* Initialization functions <5>: Random State Initialization. 2809 (line 6) 2810* Initializing and clearing: Efficiency. (line 21) 2811* Input functions: I/O of Integers. (line 6) 2812* Input functions <1>: I/O of Rationals. (line 6) 2813* Input functions <2>: I/O of Floats. (line 6) 2814* Input functions <3>: Formatted Input Functions. 2815 (line 6) 2816* Install prefix: Build Options. (line 32) 2817* Installing GMP: Installing GMP. (line 6) 2818* Instruction Set Architecture: ABI and ISA. (line 6) 2819* instrument-functions: Profiling. (line 66) 2820* Integer: Nomenclature and Types. 2821 (line 6) 2822* Integer arithmetic functions: Integer Arithmetic. (line 6) 2823* Integer assignment functions: Assigning Integers. (line 6) 2824* Integer assignment functions <1>: Simultaneous Integer Init & Assign. 2825 (line 6) 2826* Integer bit manipulation functions: Integer Logic and Bit Fiddling. 2827 (line 6) 2828* Integer comparison functions: Integer Comparisons. (line 6) 2829* Integer conversion functions: Converting Integers. (line 6) 2830* Integer division functions: Integer Division. (line 6) 2831* Integer exponentiation functions: Integer Exponentiation. 2832 (line 6) 2833* Integer export: Integer Import and Export. 2834 (line 45) 2835* Integer functions: Integer Functions. (line 6) 2836* Integer import: Integer Import and Export. 2837 (line 11) 2838* Integer initialization functions: Initializing Integers. 2839 (line 6) 2840* Integer initialization functions <1>: Simultaneous Integer Init & Assign. 2841 (line 6) 2842* Integer input and output functions: I/O of Integers. (line 6) 2843* Integer internals: Integer Internals. (line 6) 2844* Integer logical functions: Integer Logic and Bit Fiddling. 2845 (line 6) 2846* Integer miscellaneous functions: Miscellaneous Integer Functions. 2847 (line 6) 2848* Integer random number functions: Integer Random Numbers. 2849 (line 6) 2850* Integer root functions: Integer Roots. (line 6) 2851* Integer sign tests: Integer Comparisons. (line 28) 2852* Integer special functions: Integer Special Functions. 2853 (line 6) 2854* Interix: Notes for Particular Systems. 2855 (line 65) 2856* Internals: Internals. (line 6) 2857* Introduction: Introduction to GMP. (line 6) 2858* Inverse modulo functions: Number Theoretic Functions. 2859 (line 74) 2860* IRIX: ABI and ISA. (line 139) 2861* IRIX <1>: Known Build Problems. 2862 (line 38) 2863* ISA: ABI and ISA. (line 6) 2864* istream input: C++ Formatted Input. (line 6) 2865* Jacobi symbol algorithm: Jacobi Symbol. (line 6) 2866* Jacobi symbol functions: Number Theoretic Functions. 2867 (line 83) 2868* Karatsuba multiplication: Karatsuba Multiplication. 2869 (line 6) 2870* Karatsuba square root algorithm: Square Root Algorithm. 2871 (line 6) 2872* Kronecker symbol functions: Number Theoretic Functions. 2873 (line 95) 2874* Language bindings: Language Bindings. (line 6) 2875* Latest version of GMP: Introduction to GMP. (line 37) 2876* LCM functions: Number Theoretic Functions. 2877 (line 68) 2878* Least common multiple functions: Number Theoretic Functions. 2879 (line 68) 2880* Legendre symbol functions: Number Theoretic Functions. 2881 (line 86) 2882* libgmp: Headers and Libraries. 2883 (line 22) 2884* libgmpxx: Headers and Libraries. 2885 (line 27) 2886* Libraries: Headers and Libraries. 2887 (line 22) 2888* Libtool: Headers and Libraries. 2889 (line 33) 2890* Libtool versioning: Notes for Package Builds. 2891 (line 9) 2892* License conditions: Copying. (line 6) 2893* Limb: Nomenclature and Types. 2894 (line 31) 2895* Limb size: Useful Macros and Constants. 2896 (line 7) 2897* Linear congruential algorithm: Random Number Algorithms. 2898 (line 25) 2899* Linear congruential random numbers: Random State Initialization. 2900 (line 18) 2901* Linear congruential random numbers <1>: Random State Initialization. 2902 (line 32) 2903* Linking: Headers and Libraries. 2904 (line 22) 2905* Logical functions: Integer Logic and Bit Fiddling. 2906 (line 6) 2907* Low-level functions: Low-level Functions. (line 6) 2908* Low-level functions for cryptography: Low-level Functions. (line 507) 2909* Lucas number algorithm: Lucas Numbers Algorithm. 2910 (line 6) 2911* Lucas number functions: Number Theoretic Functions. 2912 (line 147) 2913* MacOS X: Known Build Problems. 2914 (line 51) 2915* Mailing lists: Introduction to GMP. (line 44) 2916* Malloc debugger: Debugging. (line 29) 2917* Malloc problems: Debugging. (line 23) 2918* Memory allocation: Custom Allocation. (line 6) 2919* Memory management: Memory Management. (line 6) 2920* Mersenne twister algorithm: Random Number Algorithms. 2921 (line 17) 2922* Mersenne twister random numbers: Random State Initialization. 2923 (line 13) 2924* MINGW: Notes for Particular Systems. 2925 (line 57) 2926* MIPS: ABI and ISA. (line 139) 2927* Miscellaneous float functions: Miscellaneous Float Functions. 2928 (line 6) 2929* Miscellaneous integer functions: Miscellaneous Integer Functions. 2930 (line 6) 2931* MMX: Notes for Particular Systems. 2932 (line 156) 2933* Modular inverse functions: Number Theoretic Functions. 2934 (line 74) 2935* Most significant bit: Miscellaneous Integer Functions. 2936 (line 34) 2937* MPN_PATH: Build Options. (line 321) 2938* MS Windows: Notes for Particular Systems. 2939 (line 57) 2940* MS Windows <1>: Notes for Particular Systems. 2941 (line 70) 2942* MS-DOS: Notes for Particular Systems. 2943 (line 57) 2944* Multi-threading: Reentrancy. (line 6) 2945* Multiplication algorithms: Multiplication Algorithms. 2946 (line 6) 2947* Nails: Low-level Functions. (line 686) 2948* Native compilation: Build Options. (line 51) 2949* NetBSD: Notes for Particular Systems. 2950 (line 100) 2951* NeXT: Known Build Problems. 2952 (line 57) 2953* Next prime function: Number Theoretic Functions. 2954 (line 23) 2955* Nomenclature: Nomenclature and Types. 2956 (line 6) 2957* Non-Unix systems: Build Options. (line 11) 2958* Nth root algorithm: Nth Root Algorithm. (line 6) 2959* Number sequences: Efficiency. (line 145) 2960* Number theoretic functions: Number Theoretic Functions. 2961 (line 6) 2962* Numerator and denominator: Applying Integer Functions. 2963 (line 6) 2964* obstack output: Formatted Output Functions. 2965 (line 79) 2966* OpenBSD: Notes for Particular Systems. 2967 (line 109) 2968* Optimizing performance: Performance optimization. 2969 (line 6) 2970* ostream output: C++ Formatted Output. 2971 (line 6) 2972* Other languages: Language Bindings. (line 6) 2973* Output functions: I/O of Integers. (line 6) 2974* Output functions <1>: I/O of Rationals. (line 6) 2975* Output functions <2>: I/O of Floats. (line 6) 2976* Output functions <3>: Formatted Output Functions. 2977 (line 6) 2978* Packaged builds: Notes for Package Builds. 2979 (line 6) 2980* Parameter conventions: Parameter Conventions. 2981 (line 6) 2982* Parsing expressions demo: Demonstration Programs. 2983 (line 15) 2984* Parsing expressions demo <1>: Demonstration Programs. 2985 (line 17) 2986* Parsing expressions demo <2>: Demonstration Programs. 2987 (line 19) 2988* Particular systems: Notes for Particular Systems. 2989 (line 6) 2990* Past GMP versions: Compatibility with older versions. 2991 (line 6) 2992* PDF: Build Options. (line 336) 2993* Perfect power algorithm: Perfect Power Algorithm. 2994 (line 6) 2995* Perfect power functions: Integer Roots. (line 28) 2996* Perfect square algorithm: Perfect Square Algorithm. 2997 (line 6) 2998* Perfect square functions: Integer Roots. (line 37) 2999* perl: Demonstration Programs. 3000 (line 28) 3001* Perl module: Demonstration Programs. 3002 (line 28) 3003* Postscript: Build Options. (line 336) 3004* Power/PowerPC: Notes for Particular Systems. 3005 (line 115) 3006* Power/PowerPC <1>: Known Build Problems. 3007 (line 63) 3008* Powering algorithms: Powering Algorithms. (line 6) 3009* Powering functions: Integer Exponentiation. 3010 (line 6) 3011* Powering functions <1>: Float Arithmetic. (line 41) 3012* PowerPC: ABI and ISA. (line 173) 3013* Precision of floats: Floating-point Functions. 3014 (line 6) 3015* Precision of hardware floating point: Notes for Particular Systems. 3016 (line 34) 3017* Prefix: Build Options. (line 32) 3018* Prime testing algorithms: Prime Testing Algorithm. 3019 (line 6) 3020* Prime testing functions: Number Theoretic Functions. 3021 (line 7) 3022* Primorial functions: Number Theoretic Functions. 3023 (line 121) 3024* printf formatted output: Formatted Output. (line 6) 3025* Probable prime testing functions: Number Theoretic Functions. 3026 (line 7) 3027* prof: Profiling. (line 24) 3028* Profiling: Profiling. (line 6) 3029* Radix conversion algorithms: Radix Conversion Algorithms. 3030 (line 6) 3031* Random number algorithms: Random Number Algorithms. 3032 (line 6) 3033* Random number functions: Integer Random Numbers. 3034 (line 6) 3035* Random number functions <1>: Miscellaneous Float Functions. 3036 (line 27) 3037* Random number functions <2>: Random Number Functions. 3038 (line 6) 3039* Random number seeding: Random State Seeding. 3040 (line 6) 3041* Random number state: Random State Initialization. 3042 (line 6) 3043* Random state: Nomenclature and Types. 3044 (line 46) 3045* Rational arithmetic: Efficiency. (line 111) 3046* Rational arithmetic functions: Rational Arithmetic. (line 6) 3047* Rational assignment functions: Initializing Rationals. 3048 (line 6) 3049* Rational comparison functions: Comparing Rationals. (line 6) 3050* Rational conversion functions: Rational Conversions. 3051 (line 6) 3052* Rational initialization functions: Initializing Rationals. 3053 (line 6) 3054* Rational input and output functions: I/O of Rationals. (line 6) 3055* Rational internals: Rational Internals. (line 6) 3056* Rational number: Nomenclature and Types. 3057 (line 16) 3058* Rational number functions: Rational Number Functions. 3059 (line 6) 3060* Rational numerator and denominator: Applying Integer Functions. 3061 (line 6) 3062* Rational sign tests: Comparing Rationals. (line 28) 3063* Raw output internals: Raw Output Internals. 3064 (line 6) 3065* Reallocations: Efficiency. (line 30) 3066* Reentrancy: Reentrancy. (line 6) 3067* References: References. (line 5) 3068* Remove factor functions: Number Theoretic Functions. 3069 (line 108) 3070* Reporting bugs: Reporting Bugs. (line 6) 3071* Root extraction algorithm: Nth Root Algorithm. (line 6) 3072* Root extraction algorithms: Root Extraction Algorithms. 3073 (line 6) 3074* Root extraction functions: Integer Roots. (line 6) 3075* Root extraction functions <1>: Float Arithmetic. (line 37) 3076* Root testing functions: Integer Roots. (line 28) 3077* Root testing functions <1>: Integer Roots. (line 37) 3078* Rounding functions: Miscellaneous Float Functions. 3079 (line 9) 3080* Sample programs: Demonstration Programs. 3081 (line 6) 3082* Scan bit functions: Integer Logic and Bit Fiddling. 3083 (line 39) 3084* scanf formatted input: Formatted Input. (line 6) 3085* SCO: Known Build Problems. 3086 (line 38) 3087* Seeding random numbers: Random State Seeding. 3088 (line 6) 3089* Segmentation violation: Debugging. (line 7) 3090* Sequent Symmetry: Known Build Problems. 3091 (line 68) 3092* Services for Unix: Notes for Particular Systems. 3093 (line 65) 3094* Shared library versioning: Notes for Package Builds. 3095 (line 9) 3096* Sign tests: Integer Comparisons. (line 28) 3097* Sign tests <1>: Comparing Rationals. (line 28) 3098* Sign tests <2>: Float Comparison. (line 34) 3099* Size in digits: Miscellaneous Integer Functions. 3100 (line 23) 3101* Small operands: Efficiency. (line 7) 3102* Solaris: ABI and ISA. (line 204) 3103* Solaris <1>: Known Build Problems. 3104 (line 72) 3105* Solaris <2>: Known Build Problems. 3106 (line 77) 3107* Sparc: Notes for Particular Systems. 3108 (line 127) 3109* Sparc <1>: Notes for Particular Systems. 3110 (line 132) 3111* Sparc V9: ABI and ISA. (line 204) 3112* Special integer functions: Integer Special Functions. 3113 (line 6) 3114* Square root algorithm: Square Root Algorithm. 3115 (line 6) 3116* SSE2: Notes for Particular Systems. 3117 (line 156) 3118* Stack backtrace: Debugging. (line 45) 3119* Stack overflow: Build Options. (line 273) 3120* Stack overflow <1>: Debugging. (line 7) 3121* Static linking: Efficiency. (line 14) 3122* stdarg.h: Headers and Libraries. 3123 (line 17) 3124* stdio.h: Headers and Libraries. 3125 (line 11) 3126* Stripped libraries: Known Build Problems. 3127 (line 28) 3128* Sun: ABI and ISA. (line 204) 3129* SunOS: Notes for Particular Systems. 3130 (line 144) 3131* Systems: Notes for Particular Systems. 3132 (line 6) 3133* Temporary memory: Build Options. (line 273) 3134* Texinfo: Build Options. (line 333) 3135* Text input/output: Efficiency. (line 151) 3136* Thread safety: Reentrancy. (line 6) 3137* Toom multiplication: Toom 3-Way Multiplication. 3138 (line 6) 3139* Toom multiplication <1>: Toom 4-Way Multiplication. 3140 (line 6) 3141* Toom multiplication <2>: Higher degree Toom'n'half. 3142 (line 6) 3143* Toom multiplication <3>: Other Multiplication. 3144 (line 6) 3145* Types: Nomenclature and Types. 3146 (line 6) 3147* ui and si functions: Efficiency. (line 50) 3148* Unbalanced multiplication: Unbalanced Multiplication. 3149 (line 6) 3150* Upward compatibility: Compatibility with older versions. 3151 (line 6) 3152* Useful macros and constants: Useful Macros and Constants. 3153 (line 6) 3154* User-defined precision: Floating-point Functions. 3155 (line 6) 3156* Valgrind: Debugging. (line 125) 3157* Variable conventions: Variable Conventions. 3158 (line 6) 3159* Version number: Useful Macros and Constants. 3160 (line 12) 3161* Web page: Introduction to GMP. (line 33) 3162* Windows: Notes for Particular Systems. 3163 (line 57) 3164* Windows <1>: Notes for Particular Systems. 3165 (line 70) 3166* x86: Notes for Particular Systems. 3167 (line 150) 3168* x87: Notes for Particular Systems. 3169 (line 34) 3170* XML: Build Options. (line 340) 3171 3172 3173File: gmp.info, Node: Function Index, Prev: Concept Index, Up: Top 3174 3175Function and Type Index 3176*********************** 3177 3178�[index�] 3179* Menu: 3180 3181* _mpz_realloc: Integer Special Functions. 3182 (line 13) 3183* __GMP_CC: Useful Macros and Constants. 3184 (line 22) 3185* __GMP_CFLAGS: Useful Macros and Constants. 3186 (line 23) 3187* __GNU_MP_VERSION: Useful Macros and Constants. 3188 (line 9) 3189* __GNU_MP_VERSION_MINOR: Useful Macros and Constants. 3190 (line 10) 3191* __GNU_MP_VERSION_PATCHLEVEL: Useful Macros and Constants. 3192 (line 11) 3193* abs: C++ Interface Integers. 3194 (line 46) 3195* abs <1>: C++ Interface Rationals. 3196 (line 47) 3197* abs <2>: C++ Interface Floats. 3198 (line 82) 3199* ceil: C++ Interface Floats. 3200 (line 83) 3201* cmp: C++ Interface Integers. 3202 (line 47) 3203* cmp <1>: C++ Interface Integers. 3204 (line 48) 3205* cmp <2>: C++ Interface Rationals. 3206 (line 48) 3207* cmp <3>: C++ Interface Rationals. 3208 (line 49) 3209* cmp <4>: C++ Interface Floats. 3210 (line 84) 3211* cmp <5>: C++ Interface Floats. 3212 (line 85) 3213* factorial: C++ Interface Integers. 3214 (line 71) 3215* fibonacci: C++ Interface Integers. 3216 (line 75) 3217* floor: C++ Interface Floats. 3218 (line 95) 3219* gcd: C++ Interface Integers. 3220 (line 68) 3221* gmp_asprintf: Formatted Output Functions. 3222 (line 63) 3223* gmp_errno: Random State Initialization. 3224 (line 56) 3225* GMP_ERROR_INVALID_ARGUMENT: Random State Initialization. 3226 (line 56) 3227* GMP_ERROR_UNSUPPORTED_ARGUMENT: Random State Initialization. 3228 (line 56) 3229* gmp_fprintf: Formatted Output Functions. 3230 (line 28) 3231* gmp_fscanf: Formatted Input Functions. 3232 (line 24) 3233* GMP_LIMB_BITS: Low-level Functions. (line 714) 3234* GMP_NAIL_BITS: Low-level Functions. (line 712) 3235* GMP_NAIL_MASK: Low-level Functions. (line 722) 3236* GMP_NUMB_BITS: Low-level Functions. (line 713) 3237* GMP_NUMB_MASK: Low-level Functions. (line 723) 3238* GMP_NUMB_MAX: Low-level Functions. (line 731) 3239* gmp_obstack_printf: Formatted Output Functions. 3240 (line 75) 3241* gmp_obstack_vprintf: Formatted Output Functions. 3242 (line 77) 3243* gmp_printf: Formatted Output Functions. 3244 (line 23) 3245* gmp_randclass: C++ Interface Random Numbers. 3246 (line 6) 3247* gmp_randclass::get_f: C++ Interface Random Numbers. 3248 (line 44) 3249* gmp_randclass::get_f <1>: C++ Interface Random Numbers. 3250 (line 45) 3251* gmp_randclass::get_z_bits: C++ Interface Random Numbers. 3252 (line 37) 3253* gmp_randclass::get_z_bits <1>: C++ Interface Random Numbers. 3254 (line 38) 3255* gmp_randclass::get_z_range: C++ Interface Random Numbers. 3256 (line 41) 3257* gmp_randclass::gmp_randclass: C++ Interface Random Numbers. 3258 (line 11) 3259* gmp_randclass::gmp_randclass <1>: C++ Interface Random Numbers. 3260 (line 26) 3261* gmp_randclass::seed: C++ Interface Random Numbers. 3262 (line 32) 3263* gmp_randclass::seed <1>: C++ Interface Random Numbers. 3264 (line 33) 3265* gmp_randclear: Random State Initialization. 3266 (line 62) 3267* gmp_randinit: Random State Initialization. 3268 (line 45) 3269* gmp_randinit_default: Random State Initialization. 3270 (line 6) 3271* gmp_randinit_lc_2exp: Random State Initialization. 3272 (line 16) 3273* gmp_randinit_lc_2exp_size: Random State Initialization. 3274 (line 30) 3275* gmp_randinit_mt: Random State Initialization. 3276 (line 12) 3277* gmp_randinit_set: Random State Initialization. 3278 (line 41) 3279* gmp_randseed: Random State Seeding. 3280 (line 6) 3281* gmp_randseed_ui: Random State Seeding. 3282 (line 8) 3283* gmp_randstate_t: Nomenclature and Types. 3284 (line 46) 3285* GMP_RAND_ALG_DEFAULT: Random State Initialization. 3286 (line 50) 3287* GMP_RAND_ALG_LC: Random State Initialization. 3288 (line 50) 3289* gmp_scanf: Formatted Input Functions. 3290 (line 20) 3291* gmp_snprintf: Formatted Output Functions. 3292 (line 44) 3293* gmp_sprintf: Formatted Output Functions. 3294 (line 33) 3295* gmp_sscanf: Formatted Input Functions. 3296 (line 28) 3297* gmp_urandomb_ui: Random State Miscellaneous. 3298 (line 6) 3299* gmp_urandomm_ui: Random State Miscellaneous. 3300 (line 12) 3301* gmp_vasprintf: Formatted Output Functions. 3302 (line 64) 3303* gmp_version: Useful Macros and Constants. 3304 (line 18) 3305* gmp_vfprintf: Formatted Output Functions. 3306 (line 29) 3307* gmp_vfscanf: Formatted Input Functions. 3308 (line 25) 3309* gmp_vprintf: Formatted Output Functions. 3310 (line 24) 3311* gmp_vscanf: Formatted Input Functions. 3312 (line 21) 3313* gmp_vsnprintf: Formatted Output Functions. 3314 (line 46) 3315* gmp_vsprintf: Formatted Output Functions. 3316 (line 34) 3317* gmp_vsscanf: Formatted Input Functions. 3318 (line 29) 3319* hypot: C++ Interface Floats. 3320 (line 96) 3321* lcm: C++ Interface Integers. 3322 (line 69) 3323* mpf_abs: Float Arithmetic. (line 46) 3324* mpf_add: Float Arithmetic. (line 6) 3325* mpf_add_ui: Float Arithmetic. (line 7) 3326* mpf_ceil: Miscellaneous Float Functions. 3327 (line 6) 3328* mpf_class: C++ Interface General. 3329 (line 19) 3330* mpf_class::fits_sint_p: C++ Interface Floats. 3331 (line 87) 3332* mpf_class::fits_slong_p: C++ Interface Floats. 3333 (line 88) 3334* mpf_class::fits_sshort_p: C++ Interface Floats. 3335 (line 89) 3336* mpf_class::fits_uint_p: C++ Interface Floats. 3337 (line 91) 3338* mpf_class::fits_ulong_p: C++ Interface Floats. 3339 (line 92) 3340* mpf_class::fits_ushort_p: C++ Interface Floats. 3341 (line 93) 3342* mpf_class::get_d: C++ Interface Floats. 3343 (line 98) 3344* mpf_class::get_mpf_t: C++ Interface General. 3345 (line 65) 3346* mpf_class::get_prec: C++ Interface Floats. 3347 (line 120) 3348* mpf_class::get_si: C++ Interface Floats. 3349 (line 99) 3350* mpf_class::get_str: C++ Interface Floats. 3351 (line 100) 3352* mpf_class::get_ui: C++ Interface Floats. 3353 (line 102) 3354* mpf_class::mpf_class: C++ Interface Floats. 3355 (line 11) 3356* mpf_class::mpf_class <1>: C++ Interface Floats. 3357 (line 12) 3358* mpf_class::mpf_class <2>: C++ Interface Floats. 3359 (line 32) 3360* mpf_class::mpf_class <3>: C++ Interface Floats. 3361 (line 33) 3362* mpf_class::mpf_class <4>: C++ Interface Floats. 3363 (line 41) 3364* mpf_class::mpf_class <5>: C++ Interface Floats. 3365 (line 42) 3366* mpf_class::mpf_class <6>: C++ Interface Floats. 3367 (line 44) 3368* mpf_class::mpf_class <7>: C++ Interface Floats. 3369 (line 45) 3370* mpf_class::operator=: C++ Interface Floats. 3371 (line 59) 3372* mpf_class::set_prec: C++ Interface Floats. 3373 (line 121) 3374* mpf_class::set_prec_raw: C++ Interface Floats. 3375 (line 122) 3376* mpf_class::set_str: C++ Interface Floats. 3377 (line 104) 3378* mpf_class::set_str <1>: C++ Interface Floats. 3379 (line 105) 3380* mpf_class::swap: C++ Interface Floats. 3381 (line 109) 3382* mpf_clear: Initializing Floats. (line 36) 3383* mpf_clears: Initializing Floats. (line 40) 3384* mpf_cmp: Float Comparison. (line 6) 3385* mpf_cmp_d: Float Comparison. (line 8) 3386* mpf_cmp_si: Float Comparison. (line 10) 3387* mpf_cmp_ui: Float Comparison. (line 9) 3388* mpf_cmp_z: Float Comparison. (line 7) 3389* mpf_div: Float Arithmetic. (line 28) 3390* mpf_div_2exp: Float Arithmetic. (line 53) 3391* mpf_div_ui: Float Arithmetic. (line 31) 3392* mpf_eq: Float Comparison. (line 17) 3393* mpf_fits_sint_p: Miscellaneous Float Functions. 3394 (line 19) 3395* mpf_fits_slong_p: Miscellaneous Float Functions. 3396 (line 17) 3397* mpf_fits_sshort_p: Miscellaneous Float Functions. 3398 (line 21) 3399* mpf_fits_uint_p: Miscellaneous Float Functions. 3400 (line 18) 3401* mpf_fits_ulong_p: Miscellaneous Float Functions. 3402 (line 16) 3403* mpf_fits_ushort_p: Miscellaneous Float Functions. 3404 (line 20) 3405* mpf_floor: Miscellaneous Float Functions. 3406 (line 7) 3407* mpf_get_d: Converting Floats. (line 6) 3408* mpf_get_default_prec: Initializing Floats. (line 11) 3409* mpf_get_d_2exp: Converting Floats. (line 15) 3410* mpf_get_prec: Initializing Floats. (line 61) 3411* mpf_get_si: Converting Floats. (line 27) 3412* mpf_get_str: Converting Floats. (line 36) 3413* mpf_get_ui: Converting Floats. (line 28) 3414* mpf_init: Initializing Floats. (line 18) 3415* mpf_init2: Initializing Floats. (line 25) 3416* mpf_inits: Initializing Floats. (line 30) 3417* mpf_init_set: Simultaneous Float Init & Assign. 3418 (line 15) 3419* mpf_init_set_d: Simultaneous Float Init & Assign. 3420 (line 18) 3421* mpf_init_set_si: Simultaneous Float Init & Assign. 3422 (line 17) 3423* mpf_init_set_str: Simultaneous Float Init & Assign. 3424 (line 24) 3425* mpf_init_set_ui: Simultaneous Float Init & Assign. 3426 (line 16) 3427* mpf_inp_str: I/O of Floats. (line 38) 3428* mpf_integer_p: Miscellaneous Float Functions. 3429 (line 13) 3430* mpf_mul: Float Arithmetic. (line 18) 3431* mpf_mul_2exp: Float Arithmetic. (line 49) 3432* mpf_mul_ui: Float Arithmetic. (line 19) 3433* mpf_neg: Float Arithmetic. (line 43) 3434* mpf_out_str: I/O of Floats. (line 17) 3435* mpf_pow_ui: Float Arithmetic. (line 39) 3436* mpf_random2: Miscellaneous Float Functions. 3437 (line 35) 3438* mpf_reldiff: Float Comparison. (line 28) 3439* mpf_set: Assigning Floats. (line 9) 3440* mpf_set_d: Assigning Floats. (line 12) 3441* mpf_set_default_prec: Initializing Floats. (line 6) 3442* mpf_set_prec: Initializing Floats. (line 64) 3443* mpf_set_prec_raw: Initializing Floats. (line 71) 3444* mpf_set_q: Assigning Floats. (line 14) 3445* mpf_set_si: Assigning Floats. (line 11) 3446* mpf_set_str: Assigning Floats. (line 17) 3447* mpf_set_ui: Assigning Floats. (line 10) 3448* mpf_set_z: Assigning Floats. (line 13) 3449* mpf_sgn: Float Comparison. (line 33) 3450* mpf_sqrt: Float Arithmetic. (line 35) 3451* mpf_sqrt_ui: Float Arithmetic. (line 36) 3452* mpf_sub: Float Arithmetic. (line 11) 3453* mpf_sub_ui: Float Arithmetic. (line 14) 3454* mpf_swap: Assigning Floats. (line 50) 3455* mpf_t: Nomenclature and Types. 3456 (line 21) 3457* mpf_trunc: Miscellaneous Float Functions. 3458 (line 8) 3459* mpf_ui_div: Float Arithmetic. (line 29) 3460* mpf_ui_sub: Float Arithmetic. (line 12) 3461* mpf_urandomb: Miscellaneous Float Functions. 3462 (line 25) 3463* mpn_add: Low-level Functions. (line 67) 3464* mpn_addmul_1: Low-level Functions. (line 148) 3465* mpn_add_1: Low-level Functions. (line 62) 3466* mpn_add_n: Low-level Functions. (line 52) 3467* mpn_andn_n: Low-level Functions. (line 462) 3468* mpn_and_n: Low-level Functions. (line 447) 3469* mpn_cmp: Low-level Functions. (line 293) 3470* mpn_cnd_add_n: Low-level Functions. (line 540) 3471* mpn_cnd_sub_n: Low-level Functions. (line 542) 3472* mpn_cnd_swap: Low-level Functions. (line 567) 3473* mpn_com: Low-level Functions. (line 487) 3474* mpn_copyd: Low-level Functions. (line 496) 3475* mpn_copyi: Low-level Functions. (line 492) 3476* mpn_divexact_1: Low-level Functions. (line 231) 3477* mpn_divexact_by3: Low-level Functions. (line 238) 3478* mpn_divexact_by3c: Low-level Functions. (line 240) 3479* mpn_divmod: Low-level Functions. (line 226) 3480* mpn_divmod_1: Low-level Functions. (line 210) 3481* mpn_divrem: Low-level Functions. (line 183) 3482* mpn_divrem_1: Low-level Functions. (line 208) 3483* mpn_gcd: Low-level Functions. (line 301) 3484* mpn_gcdext: Low-level Functions. (line 316) 3485* mpn_gcd_1: Low-level Functions. (line 311) 3486* mpn_get_str: Low-level Functions. (line 371) 3487* mpn_hamdist: Low-level Functions. (line 436) 3488* mpn_iorn_n: Low-level Functions. (line 467) 3489* mpn_ior_n: Low-level Functions. (line 452) 3490* mpn_lshift: Low-level Functions. (line 269) 3491* mpn_mod_1: Low-level Functions. (line 264) 3492* mpn_mul: Low-level Functions. (line 114) 3493* mpn_mul_1: Low-level Functions. (line 133) 3494* mpn_mul_n: Low-level Functions. (line 103) 3495* mpn_nand_n: Low-level Functions. (line 472) 3496* mpn_neg: Low-level Functions. (line 96) 3497* mpn_nior_n: Low-level Functions. (line 477) 3498* mpn_perfect_square_p: Low-level Functions. (line 442) 3499* mpn_popcount: Low-level Functions. (line 432) 3500* mpn_random: Low-level Functions. (line 422) 3501* mpn_random2: Low-level Functions. (line 423) 3502* mpn_rshift: Low-level Functions. (line 281) 3503* mpn_scan0: Low-level Functions. (line 406) 3504* mpn_scan1: Low-level Functions. (line 414) 3505* mpn_sec_add_1: Low-level Functions. (line 553) 3506* mpn_sec_div_qr: Low-level Functions. (line 630) 3507* mpn_sec_div_qr_itch: Low-level Functions. (line 633) 3508* mpn_sec_div_r: Low-level Functions. (line 649) 3509* mpn_sec_div_r_itch: Low-level Functions. (line 651) 3510* mpn_sec_invert: Low-level Functions. (line 665) 3511* mpn_sec_invert_itch: Low-level Functions. (line 667) 3512* mpn_sec_mul: Low-level Functions. (line 574) 3513* mpn_sec_mul_itch: Low-level Functions. (line 577) 3514* mpn_sec_powm: Low-level Functions. (line 604) 3515* mpn_sec_powm_itch: Low-level Functions. (line 607) 3516* mpn_sec_sqr: Low-level Functions. (line 590) 3517* mpn_sec_sqr_itch: Low-level Functions. (line 592) 3518* mpn_sec_sub_1: Low-level Functions. (line 555) 3519* mpn_sec_tabselect: Low-level Functions. (line 622) 3520* mpn_set_str: Low-level Functions. (line 386) 3521* mpn_sizeinbase: Low-level Functions. (line 364) 3522* mpn_sqr: Low-level Functions. (line 125) 3523* mpn_sqrtrem: Low-level Functions. (line 346) 3524* mpn_sub: Low-level Functions. (line 88) 3525* mpn_submul_1: Low-level Functions. (line 160) 3526* mpn_sub_1: Low-level Functions. (line 83) 3527* mpn_sub_n: Low-level Functions. (line 74) 3528* mpn_tdiv_qr: Low-level Functions. (line 172) 3529* mpn_xnor_n: Low-level Functions. (line 482) 3530* mpn_xor_n: Low-level Functions. (line 457) 3531* mpn_zero: Low-level Functions. (line 500) 3532* mpn_zero_p: Low-level Functions. (line 298) 3533* mpq_abs: Rational Arithmetic. (line 33) 3534* mpq_add: Rational Arithmetic. (line 6) 3535* mpq_canonicalize: Rational Number Functions. 3536 (line 21) 3537* mpq_class: C++ Interface General. 3538 (line 18) 3539* mpq_class::canonicalize: C++ Interface Rationals. 3540 (line 41) 3541* mpq_class::get_d: C++ Interface Rationals. 3542 (line 51) 3543* mpq_class::get_den: C++ Interface Rationals. 3544 (line 67) 3545* mpq_class::get_den_mpz_t: C++ Interface Rationals. 3546 (line 77) 3547* mpq_class::get_mpq_t: C++ Interface General. 3548 (line 64) 3549* mpq_class::get_num: C++ Interface Rationals. 3550 (line 66) 3551* mpq_class::get_num_mpz_t: C++ Interface Rationals. 3552 (line 76) 3553* mpq_class::get_str: C++ Interface Rationals. 3554 (line 52) 3555* mpq_class::mpq_class: C++ Interface Rationals. 3556 (line 9) 3557* mpq_class::mpq_class <1>: C++ Interface Rationals. 3558 (line 10) 3559* mpq_class::mpq_class <2>: C++ Interface Rationals. 3560 (line 21) 3561* mpq_class::mpq_class <3>: C++ Interface Rationals. 3562 (line 26) 3563* mpq_class::mpq_class <4>: C++ Interface Rationals. 3564 (line 28) 3565* mpq_class::set_str: C++ Interface Rationals. 3566 (line 54) 3567* mpq_class::set_str <1>: C++ Interface Rationals. 3568 (line 55) 3569* mpq_class::swap: C++ Interface Rationals. 3570 (line 58) 3571* mpq_clear: Initializing Rationals. 3572 (line 15) 3573* mpq_clears: Initializing Rationals. 3574 (line 19) 3575* mpq_cmp: Comparing Rationals. (line 6) 3576* mpq_cmp_si: Comparing Rationals. (line 16) 3577* mpq_cmp_ui: Comparing Rationals. (line 14) 3578* mpq_cmp_z: Comparing Rationals. (line 7) 3579* mpq_denref: Applying Integer Functions. 3580 (line 16) 3581* mpq_div: Rational Arithmetic. (line 22) 3582* mpq_div_2exp: Rational Arithmetic. (line 26) 3583* mpq_equal: Comparing Rationals. (line 33) 3584* mpq_get_d: Rational Conversions. 3585 (line 6) 3586* mpq_get_den: Applying Integer Functions. 3587 (line 22) 3588* mpq_get_num: Applying Integer Functions. 3589 (line 21) 3590* mpq_get_str: Rational Conversions. 3591 (line 21) 3592* mpq_init: Initializing Rationals. 3593 (line 6) 3594* mpq_inits: Initializing Rationals. 3595 (line 11) 3596* mpq_inp_str: I/O of Rationals. (line 32) 3597* mpq_inv: Rational Arithmetic. (line 36) 3598* mpq_mul: Rational Arithmetic. (line 14) 3599* mpq_mul_2exp: Rational Arithmetic. (line 18) 3600* mpq_neg: Rational Arithmetic. (line 30) 3601* mpq_numref: Applying Integer Functions. 3602 (line 15) 3603* mpq_out_str: I/O of Rationals. (line 17) 3604* mpq_set: Initializing Rationals. 3605 (line 23) 3606* mpq_set_d: Rational Conversions. 3607 (line 16) 3608* mpq_set_den: Applying Integer Functions. 3609 (line 24) 3610* mpq_set_f: Rational Conversions. 3611 (line 17) 3612* mpq_set_num: Applying Integer Functions. 3613 (line 23) 3614* mpq_set_si: Initializing Rationals. 3615 (line 29) 3616* mpq_set_str: Initializing Rationals. 3617 (line 35) 3618* mpq_set_ui: Initializing Rationals. 3619 (line 27) 3620* mpq_set_z: Initializing Rationals. 3621 (line 24) 3622* mpq_sgn: Comparing Rationals. (line 27) 3623* mpq_sub: Rational Arithmetic. (line 10) 3624* mpq_swap: Initializing Rationals. 3625 (line 54) 3626* mpq_t: Nomenclature and Types. 3627 (line 16) 3628* mpz_2fac_ui: Number Theoretic Functions. 3629 (line 113) 3630* mpz_abs: Integer Arithmetic. (line 44) 3631* mpz_add: Integer Arithmetic. (line 6) 3632* mpz_addmul: Integer Arithmetic. (line 24) 3633* mpz_addmul_ui: Integer Arithmetic. (line 26) 3634* mpz_add_ui: Integer Arithmetic. (line 7) 3635* mpz_and: Integer Logic and Bit Fiddling. 3636 (line 10) 3637* mpz_array_init: Integer Special Functions. 3638 (line 9) 3639* mpz_bin_ui: Number Theoretic Functions. 3640 (line 124) 3641* mpz_bin_uiui: Number Theoretic Functions. 3642 (line 126) 3643* mpz_cdiv_q: Integer Division. (line 12) 3644* mpz_cdiv_qr: Integer Division. (line 14) 3645* mpz_cdiv_qr_ui: Integer Division. (line 21) 3646* mpz_cdiv_q_2exp: Integer Division. (line 26) 3647* mpz_cdiv_q_ui: Integer Division. (line 17) 3648* mpz_cdiv_r: Integer Division. (line 13) 3649* mpz_cdiv_r_2exp: Integer Division. (line 29) 3650* mpz_cdiv_r_ui: Integer Division. (line 19) 3651* mpz_cdiv_ui: Integer Division. (line 23) 3652* mpz_class: C++ Interface General. 3653 (line 17) 3654* mpz_class::factorial: C++ Interface Integers. 3655 (line 70) 3656* mpz_class::fibonacci: C++ Interface Integers. 3657 (line 74) 3658* mpz_class::fits_sint_p: C++ Interface Integers. 3659 (line 50) 3660* mpz_class::fits_slong_p: C++ Interface Integers. 3661 (line 51) 3662* mpz_class::fits_sshort_p: C++ Interface Integers. 3663 (line 52) 3664* mpz_class::fits_uint_p: C++ Interface Integers. 3665 (line 54) 3666* mpz_class::fits_ulong_p: C++ Interface Integers. 3667 (line 55) 3668* mpz_class::fits_ushort_p: C++ Interface Integers. 3669 (line 56) 3670* mpz_class::get_d: C++ Interface Integers. 3671 (line 58) 3672* mpz_class::get_mpz_t: C++ Interface General. 3673 (line 63) 3674* mpz_class::get_si: C++ Interface Integers. 3675 (line 59) 3676* mpz_class::get_str: C++ Interface Integers. 3677 (line 60) 3678* mpz_class::get_ui: C++ Interface Integers. 3679 (line 61) 3680* mpz_class::mpz_class: C++ Interface Integers. 3681 (line 6) 3682* mpz_class::mpz_class <1>: C++ Interface Integers. 3683 (line 14) 3684* mpz_class::mpz_class <2>: C++ Interface Integers. 3685 (line 19) 3686* mpz_class::mpz_class <3>: C++ Interface Integers. 3687 (line 21) 3688* mpz_class::primorial: C++ Interface Integers. 3689 (line 72) 3690* mpz_class::set_str: C++ Interface Integers. 3691 (line 63) 3692* mpz_class::set_str <1>: C++ Interface Integers. 3693 (line 64) 3694* mpz_class::swap: C++ Interface Integers. 3695 (line 77) 3696* mpz_clear: Initializing Integers. 3697 (line 48) 3698* mpz_clears: Initializing Integers. 3699 (line 52) 3700* mpz_clrbit: Integer Logic and Bit Fiddling. 3701 (line 54) 3702* mpz_cmp: Integer Comparisons. (line 6) 3703* mpz_cmpabs: Integer Comparisons. (line 17) 3704* mpz_cmpabs_d: Integer Comparisons. (line 18) 3705* mpz_cmpabs_ui: Integer Comparisons. (line 19) 3706* mpz_cmp_d: Integer Comparisons. (line 7) 3707* mpz_cmp_si: Integer Comparisons. (line 8) 3708* mpz_cmp_ui: Integer Comparisons. (line 9) 3709* mpz_com: Integer Logic and Bit Fiddling. 3710 (line 19) 3711* mpz_combit: Integer Logic and Bit Fiddling. 3712 (line 57) 3713* mpz_congruent_2exp_p: Integer Division. (line 148) 3714* mpz_congruent_p: Integer Division. (line 144) 3715* mpz_congruent_ui_p: Integer Division. (line 146) 3716* mpz_divexact: Integer Division. (line 122) 3717* mpz_divexact_ui: Integer Division. (line 123) 3718* mpz_divisible_2exp_p: Integer Division. (line 135) 3719* mpz_divisible_p: Integer Division. (line 132) 3720* mpz_divisible_ui_p: Integer Division. (line 133) 3721* mpz_even_p: Miscellaneous Integer Functions. 3722 (line 17) 3723* mpz_export: Integer Import and Export. 3724 (line 43) 3725* mpz_fac_ui: Number Theoretic Functions. 3726 (line 112) 3727* mpz_fdiv_q: Integer Division. (line 33) 3728* mpz_fdiv_qr: Integer Division. (line 35) 3729* mpz_fdiv_qr_ui: Integer Division. (line 42) 3730* mpz_fdiv_q_2exp: Integer Division. (line 47) 3731* mpz_fdiv_q_ui: Integer Division. (line 38) 3732* mpz_fdiv_r: Integer Division. (line 34) 3733* mpz_fdiv_r_2exp: Integer Division. (line 50) 3734* mpz_fdiv_r_ui: Integer Division. (line 40) 3735* mpz_fdiv_ui: Integer Division. (line 44) 3736* mpz_fib2_ui: Number Theoretic Functions. 3737 (line 134) 3738* mpz_fib_ui: Number Theoretic Functions. 3739 (line 133) 3740* mpz_fits_sint_p: Miscellaneous Integer Functions. 3741 (line 9) 3742* mpz_fits_slong_p: Miscellaneous Integer Functions. 3743 (line 7) 3744* mpz_fits_sshort_p: Miscellaneous Integer Functions. 3745 (line 11) 3746* mpz_fits_uint_p: Miscellaneous Integer Functions. 3747 (line 8) 3748* mpz_fits_ulong_p: Miscellaneous Integer Functions. 3749 (line 6) 3750* mpz_fits_ushort_p: Miscellaneous Integer Functions. 3751 (line 10) 3752* mpz_gcd: Number Theoretic Functions. 3753 (line 29) 3754* mpz_gcdext: Number Theoretic Functions. 3755 (line 45) 3756* mpz_gcd_ui: Number Theoretic Functions. 3757 (line 35) 3758* mpz_getlimbn: Integer Special Functions. 3759 (line 22) 3760* mpz_get_d: Converting Integers. (line 26) 3761* mpz_get_d_2exp: Converting Integers. (line 34) 3762* mpz_get_si: Converting Integers. (line 17) 3763* mpz_get_str: Converting Integers. (line 46) 3764* mpz_get_ui: Converting Integers. (line 10) 3765* mpz_hamdist: Integer Logic and Bit Fiddling. 3766 (line 28) 3767* mpz_import: Integer Import and Export. 3768 (line 9) 3769* mpz_init: Initializing Integers. 3770 (line 25) 3771* mpz_init2: Initializing Integers. 3772 (line 32) 3773* mpz_inits: Initializing Integers. 3774 (line 28) 3775* mpz_init_set: Simultaneous Integer Init & Assign. 3776 (line 26) 3777* mpz_init_set_d: Simultaneous Integer Init & Assign. 3778 (line 29) 3779* mpz_init_set_si: Simultaneous Integer Init & Assign. 3780 (line 28) 3781* mpz_init_set_str: Simultaneous Integer Init & Assign. 3782 (line 33) 3783* mpz_init_set_ui: Simultaneous Integer Init & Assign. 3784 (line 27) 3785* mpz_inp_raw: I/O of Integers. (line 61) 3786* mpz_inp_str: I/O of Integers. (line 30) 3787* mpz_invert: Number Theoretic Functions. 3788 (line 72) 3789* mpz_ior: Integer Logic and Bit Fiddling. 3790 (line 13) 3791* mpz_jacobi: Number Theoretic Functions. 3792 (line 82) 3793* mpz_kronecker: Number Theoretic Functions. 3794 (line 90) 3795* mpz_kronecker_si: Number Theoretic Functions. 3796 (line 91) 3797* mpz_kronecker_ui: Number Theoretic Functions. 3798 (line 92) 3799* mpz_lcm: Number Theoretic Functions. 3800 (line 65) 3801* mpz_lcm_ui: Number Theoretic Functions. 3802 (line 66) 3803* mpz_legendre: Number Theoretic Functions. 3804 (line 85) 3805* mpz_limbs_finish: Integer Special Functions. 3806 (line 47) 3807* mpz_limbs_modify: Integer Special Functions. 3808 (line 40) 3809* mpz_limbs_read: Integer Special Functions. 3810 (line 34) 3811* mpz_limbs_write: Integer Special Functions. 3812 (line 39) 3813* mpz_lucnum2_ui: Number Theoretic Functions. 3814 (line 145) 3815* mpz_lucnum_ui: Number Theoretic Functions. 3816 (line 144) 3817* mpz_mfac_uiui: Number Theoretic Functions. 3818 (line 114) 3819* mpz_mod: Integer Division. (line 112) 3820* mpz_mod_ui: Integer Division. (line 113) 3821* mpz_mul: Integer Arithmetic. (line 18) 3822* mpz_mul_2exp: Integer Arithmetic. (line 36) 3823* mpz_mul_si: Integer Arithmetic. (line 19) 3824* mpz_mul_ui: Integer Arithmetic. (line 20) 3825* mpz_neg: Integer Arithmetic. (line 41) 3826* mpz_nextprime: Number Theoretic Functions. 3827 (line 22) 3828* mpz_odd_p: Miscellaneous Integer Functions. 3829 (line 16) 3830* mpz_out_raw: I/O of Integers. (line 45) 3831* mpz_out_str: I/O of Integers. (line 17) 3832* mpz_perfect_power_p: Integer Roots. (line 27) 3833* mpz_perfect_square_p: Integer Roots. (line 36) 3834* mpz_popcount: Integer Logic and Bit Fiddling. 3835 (line 22) 3836* mpz_powm: Integer Exponentiation. 3837 (line 6) 3838* mpz_powm_sec: Integer Exponentiation. 3839 (line 16) 3840* mpz_powm_ui: Integer Exponentiation. 3841 (line 8) 3842* mpz_pow_ui: Integer Exponentiation. 3843 (line 29) 3844* mpz_primorial_ui: Number Theoretic Functions. 3845 (line 120) 3846* mpz_probab_prime_p: Number Theoretic Functions. 3847 (line 6) 3848* mpz_random: Integer Random Numbers. 3849 (line 41) 3850* mpz_random2: Integer Random Numbers. 3851 (line 50) 3852* mpz_realloc2: Initializing Integers. 3853 (line 56) 3854* mpz_remove: Number Theoretic Functions. 3855 (line 106) 3856* mpz_roinit_n: Integer Special Functions. 3857 (line 67) 3858* MPZ_ROINIT_N: Integer Special Functions. 3859 (line 83) 3860* mpz_root: Integer Roots. (line 6) 3861* mpz_rootrem: Integer Roots. (line 12) 3862* mpz_rrandomb: Integer Random Numbers. 3863 (line 29) 3864* mpz_scan0: Integer Logic and Bit Fiddling. 3865 (line 35) 3866* mpz_scan1: Integer Logic and Bit Fiddling. 3867 (line 37) 3868* mpz_set: Assigning Integers. (line 9) 3869* mpz_setbit: Integer Logic and Bit Fiddling. 3870 (line 51) 3871* mpz_set_d: Assigning Integers. (line 12) 3872* mpz_set_f: Assigning Integers. (line 14) 3873* mpz_set_q: Assigning Integers. (line 13) 3874* mpz_set_si: Assigning Integers. (line 11) 3875* mpz_set_str: Assigning Integers. (line 20) 3876* mpz_set_ui: Assigning Integers. (line 10) 3877* mpz_sgn: Integer Comparisons. (line 27) 3878* mpz_size: Integer Special Functions. 3879 (line 30) 3880* mpz_sizeinbase: Miscellaneous Integer Functions. 3881 (line 22) 3882* mpz_si_kronecker: Number Theoretic Functions. 3883 (line 93) 3884* mpz_sqrt: Integer Roots. (line 17) 3885* mpz_sqrtrem: Integer Roots. (line 20) 3886* mpz_sub: Integer Arithmetic. (line 11) 3887* mpz_submul: Integer Arithmetic. (line 30) 3888* mpz_submul_ui: Integer Arithmetic. (line 32) 3889* mpz_sub_ui: Integer Arithmetic. (line 12) 3890* mpz_swap: Assigning Integers. (line 36) 3891* mpz_t: Nomenclature and Types. 3892 (line 6) 3893* mpz_tdiv_q: Integer Division. (line 54) 3894* mpz_tdiv_qr: Integer Division. (line 56) 3895* mpz_tdiv_qr_ui: Integer Division. (line 63) 3896* mpz_tdiv_q_2exp: Integer Division. (line 68) 3897* mpz_tdiv_q_ui: Integer Division. (line 59) 3898* mpz_tdiv_r: Integer Division. (line 55) 3899* mpz_tdiv_r_2exp: Integer Division. (line 71) 3900* mpz_tdiv_r_ui: Integer Division. (line 61) 3901* mpz_tdiv_ui: Integer Division. (line 65) 3902* mpz_tstbit: Integer Logic and Bit Fiddling. 3903 (line 60) 3904* mpz_ui_kronecker: Number Theoretic Functions. 3905 (line 94) 3906* mpz_ui_pow_ui: Integer Exponentiation. 3907 (line 31) 3908* mpz_ui_sub: Integer Arithmetic. (line 14) 3909* mpz_urandomb: Integer Random Numbers. 3910 (line 12) 3911* mpz_urandomm: Integer Random Numbers. 3912 (line 21) 3913* mpz_xor: Integer Logic and Bit Fiddling. 3914 (line 16) 3915* mp_bitcnt_t: Nomenclature and Types. 3916 (line 42) 3917* mp_bits_per_limb: Useful Macros and Constants. 3918 (line 7) 3919* mp_exp_t: Nomenclature and Types. 3920 (line 27) 3921* mp_get_memory_functions: Custom Allocation. (line 86) 3922* mp_limb_t: Nomenclature and Types. 3923 (line 31) 3924* mp_set_memory_functions: Custom Allocation. (line 14) 3925* mp_size_t: Nomenclature and Types. 3926 (line 37) 3927* operator"": C++ Interface Integers. 3928 (line 29) 3929* operator"" <1>: C++ Interface Rationals. 3930 (line 36) 3931* operator"" <2>: C++ Interface Floats. 3932 (line 55) 3933* operator%: C++ Interface Integers. 3934 (line 34) 3935* operator/: C++ Interface Integers. 3936 (line 33) 3937* operator<<: C++ Formatted Output. 3938 (line 10) 3939* operator<< <1>: C++ Formatted Output. 3940 (line 19) 3941* operator<< <2>: C++ Formatted Output. 3942 (line 32) 3943* operator>>: C++ Formatted Input. (line 10) 3944* operator>> <1>: C++ Formatted Input. (line 13) 3945* operator>> <2>: C++ Formatted Input. (line 24) 3946* operator>> <3>: C++ Interface Rationals. 3947 (line 86) 3948* primorial: C++ Interface Integers. 3949 (line 73) 3950* sgn: C++ Interface Integers. 3951 (line 65) 3952* sgn <1>: C++ Interface Rationals. 3953 (line 56) 3954* sgn <2>: C++ Interface Floats. 3955 (line 106) 3956* sqrt: C++ Interface Integers. 3957 (line 66) 3958* sqrt <1>: C++ Interface Floats. 3959 (line 107) 3960* swap: C++ Interface Integers. 3961 (line 78) 3962* swap <1>: C++ Interface Rationals. 3963 (line 59) 3964* swap <2>: C++ Interface Floats. 3965 (line 110) 3966* trunc: C++ Interface Floats. 3967 (line 111) 3968 3969