1This is ../../gmp/doc/gmp.info, produced by makeinfo version 4.13 from 2../../gmp/doc/gmp.texi. 3 4This manual describes how to install and use the GNU multiple precision 5arithmetic library, version 5.0.5. 6 7 Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 82001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012 9Free Software Foundation, Inc. 10 11 Permission is granted to copy, distribute and/or modify this 12document under the terms of the GNU Free Documentation License, Version 131.3 or any later version published by the Free Software Foundation; 14with no Invariant Sections, with the Front-Cover Texts being "A GNU 15Manual", and with the Back-Cover Texts being "You have freedom to copy 16and modify this GNU Manual, like GNU software". A copy of the license 17is included in *note GNU Free Documentation License::. 18 19INFO-DIR-SECTION GNU libraries 20START-INFO-DIR-ENTRY 21* gmp: (gmp). GNU Multiple Precision Arithmetic Library. 22END-INFO-DIR-ENTRY 23 24 25File: gmp.info, Node: Subquadratic GCD, Next: Extended GCD, Prev: Lehmer's Algorithm, Up: Greatest Common Divisor Algorithms 26 2716.3.3 Subquadratic GCD 28----------------------- 29 30For inputs larger than `GCD_DC_THRESHOLD', GCD is computed via the HGCD 31(Half GCD) function, as a generalization to Lehmer's algorithm. 32 33 Let the inputs a,b be of size N limbs each. Put S = floor(N/2) + 1. 34Then HGCD(a,b) returns a transformation matrix T with non-negative 35elements, and reduced numbers (c;d) = T^-1 (a;b). The reduced numbers 36c,d must be larger than S limbs, while their difference abs(c-d) must 37fit in S limbs. The matrix elements will also be of size roughly N/2. 38 39 The HGCD base case uses Lehmer's algorithm, but with the above stop 40condition that returns reduced numbers and the corresponding 41transformation matrix half-way through. For inputs larger than 42`HGCD_THRESHOLD', HGCD is computed recursively, using the divide and 43conquer algorithm in "On Sch�nhage's algorithm and subquadratic integer 44GCD computation" by M�ller (*note References::). The recursive 45algorithm consists of these main steps. 46 47 * Call HGCD recursively, on the most significant N/2 limbs. Apply the 48 resulting matrix T_1 to the full numbers, reducing them to a size 49 just above 3N/2. 50 51 * Perform a small number of division or subtraction steps to reduce 52 the numbers to size below 3N/2. This is essential mainly for the 53 unlikely case of large quotients. 54 55 * Call HGCD recursively, on the most significant N/2 limbs of the 56 reduced numbers. Apply the resulting matrix T_2 to the full 57 numbers, reducing them to a size just above N/2. 58 59 * Compute T = T_1 T_2. 60 61 * Perform a small number of division and subtraction steps to 62 satisfy the requirements, and return. 63 64 GCD is then implemented as a loop around HGCD, similarly to Lehmer's 65algorithm. Where Lehmer repeatedly chops off the top two limbs, calls 66`mpn_hgcd2', and applies the resulting matrix to the full numbers, the 67subquadratic GCD chops off the most significant third of the limbs (the 68proportion is a tuning parameter, and 1/3 seems to be more efficient 69than, e.g, 1/2), calls `mpn_hgcd', and applies the resulting matrix. 70Once the input numbers are reduced to size below `GCD_DC_THRESHOLD', 71Lehmer's algorithm is used for the rest of the work. 72 73 The asymptotic running time of both HGCD and GCD is O(M(N)*log(N)), 74where M(N) is the time for multiplying two N-limb numbers. 75 76 77File: gmp.info, Node: Extended GCD, Next: Jacobi Symbol, Prev: Subquadratic GCD, Up: Greatest Common Divisor Algorithms 78 7916.3.4 Extended GCD 80------------------- 81 82The extended GCD function, or GCDEXT, calculates gcd(a,b) and also 83cofactors x and y satisfying a*x+b*y=gcd(a,b). All the algorithms used 84for plain GCD are extended to handle this case. The binary algorithm is 85used only for single-limb GCDEXT. Lehmer's algorithm is used for sizes 86up to `GCDEXT_DC_THRESHOLD'. Above this threshold, GCDEXT is 87implemented as a loop around HGCD, but with more book-keeping to keep 88track of the cofactors. This gives the same asymptotic running time as 89for GCD and HGCD, O(M(N)*log(N)) 90 91 One difference to plain GCD is that while the inputs a and b are 92reduced as the algorithm proceeds, the cofactors x and y grow in size. 93This makes the tuning of the chopping-point more difficult. The current 94code chops off the most significant half of the inputs for the call to 95HGCD in the first iteration, and the most significant two thirds for 96the remaining calls. This strategy could surely be improved. Also the 97stop condition for the loop, where Lehmer's algorithm is invoked once 98the inputs are reduced below `GCDEXT_DC_THRESHOLD', could maybe be 99improved by taking into account the current size of the cofactors. 100 101 102File: gmp.info, Node: Jacobi Symbol, Prev: Extended GCD, Up: Greatest Common Divisor Algorithms 103 10416.3.5 Jacobi Symbol 105-------------------- 106 107`mpz_jacobi' and `mpz_kronecker' are currently implemented with a 108simple binary algorithm similar to that described for the GCDs (*note 109Binary GCD::). They're not very fast when both inputs are large. 110Lehmer's multi-step improvement or a binary based multi-step algorithm 111is likely to be better. 112 113 When one operand fits a single limb, and that includes 114`mpz_kronecker_ui' and friends, an initial reduction is done with 115either `mpn_mod_1' or `mpn_modexact_1_odd', followed by the binary 116algorithm on a single limb. The binary algorithm is well suited to a 117single limb, and the whole calculation in this case is quite efficient. 118 119 In all the routines sign changes for the result are accumulated 120using some bit twiddling, avoiding table lookups or conditional jumps. 121 122 123File: gmp.info, Node: Powering Algorithms, Next: Root Extraction Algorithms, Prev: Greatest Common Divisor Algorithms, Up: Algorithms 124 12516.4 Powering Algorithms 126======================== 127 128* Menu: 129 130* Normal Powering Algorithm:: 131* Modular Powering Algorithm:: 132 133 134File: gmp.info, Node: Normal Powering Algorithm, Next: Modular Powering Algorithm, Prev: Powering Algorithms, Up: Powering Algorithms 135 13616.4.1 Normal Powering 137---------------------- 138 139Normal `mpz' or `mpf' powering uses a simple binary algorithm, 140successively squaring and then multiplying by the base when a 1 bit is 141seen in the exponent, as per Knuth section 4.6.3. The "left to right" 142variant described there is used rather than algorithm A, since it's 143just as easy and can be done with somewhat less temporary memory. 144 145 146File: gmp.info, Node: Modular Powering Algorithm, Prev: Normal Powering Algorithm, Up: Powering Algorithms 147 14816.4.2 Modular Powering 149----------------------- 150 151Modular powering is implemented using a 2^k-ary sliding window 152algorithm, as per "Handbook of Applied Cryptography" algorithm 14.85 153(*note References::). k is chosen according to the size of the 154exponent. Larger exponents use larger values of k, the choice being 155made to minimize the average number of multiplications that must 156supplement the squaring. 157 158 The modular multiplies and squares use either a simple division or 159the REDC method by Montgomery (*note References::). REDC is a little 160faster, essentially saving N single limb divisions in a fashion similar 161to an exact remainder (*note Exact Remainder::). 162 163 164File: gmp.info, Node: Root Extraction Algorithms, Next: Radix Conversion Algorithms, Prev: Powering Algorithms, Up: Algorithms 165 16616.5 Root Extraction Algorithms 167=============================== 168 169* Menu: 170 171* Square Root Algorithm:: 172* Nth Root Algorithm:: 173* Perfect Square Algorithm:: 174* Perfect Power Algorithm:: 175 176 177File: gmp.info, Node: Square Root Algorithm, Next: Nth Root Algorithm, Prev: Root Extraction Algorithms, Up: Root Extraction Algorithms 178 17916.5.1 Square Root 180------------------ 181 182Square roots are taken using the "Karatsuba Square Root" algorithm by 183Paul Zimmermann (*note References::). 184 185 An input n is split into four parts of k bits each, so with b=2^k we 186have n = a3*b^3 + a2*b^2 + a1*b + a0. Part a3 must be "normalized" so 187that either the high or second highest bit is set. In GMP, k is kept 188on a limb boundary and the input is left shifted (by an even number of 189bits) to normalize. 190 191 The square root of the high two parts is taken, by recursive 192application of the algorithm (bottoming out in a one-limb Newton's 193method), 194 195 s1,r1 = sqrtrem (a3*b + a2) 196 197 This is an approximation to the desired root and is extended by a 198division to give s,r, 199 200 q,u = divrem (r1*b + a1, 2*s1) 201 s = s1*b + q 202 r = u*b + a0 - q^2 203 204 The normalization requirement on a3 means at this point s is either 205correct or 1 too big. r is negative in the latter case, so 206 207 if r < 0 then 208 r = r + 2*s - 1 209 s = s - 1 210 211 The algorithm is expressed in a divide and conquer form, but as 212noted in the paper it can also be viewed as a discrete variant of 213Newton's method, or as a variation on the schoolboy method (no longer 214taught) for square roots two digits at a time. 215 216 If the remainder r is not required then usually only a few high limbs 217of r and u need to be calculated to determine whether an adjustment to 218s is required. This optimization is not currently implemented. 219 220 In the Karatsuba multiplication range this algorithm is 221O(1.5*M(N/2)), where M(n) is the time to multiply two numbers of n 222limbs. In the FFT multiplication range this grows to a bound of 223O(6*M(N/2)). In practice a factor of about 1.5 to 1.8 is found in the 224Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range. 225 226 The algorithm does all its calculations in integers and the resulting 227`mpn_sqrtrem' is used for both `mpz_sqrt' and `mpf_sqrt'. The extended 228precision given by `mpf_sqrt_ui' is obtained by padding with zero limbs. 229 230 231File: gmp.info, Node: Nth Root Algorithm, Next: Perfect Square Algorithm, Prev: Square Root Algorithm, Up: Root Extraction Algorithms 232 23316.5.2 Nth Root 234--------------- 235 236Integer Nth roots are taken using Newton's method with the following 237iteration, where A is the input and n is the root to be taken. 238 239 1 A 240 a[i+1] = - * ( --------- + (n-1)*a[i] ) 241 n a[i]^(n-1) 242 243 The initial approximation a[1] is generated bitwise by successively 244powering a trial root with or without new 1 bits, aiming to be just 245above the true root. The iteration converges quadratically when 246started from a good approximation. When n is large more initial bits 247are needed to get good convergence. The current implementation is not 248particularly well optimized. 249 250 251File: gmp.info, Node: Perfect Square Algorithm, Next: Perfect Power Algorithm, Prev: Nth Root Algorithm, Up: Root Extraction Algorithms 252 25316.5.3 Perfect Square 254--------------------- 255 256A significant fraction of non-squares can be quickly identified by 257checking whether the input is a quadratic residue modulo small integers. 258 259 `mpz_perfect_square_p' first tests the input mod 256, which means 260just examining the low byte. Only 44 different values occur for 261squares mod 256, so 82.8% of inputs can be immediately identified as 262non-squares. 263 264 On a 32-bit system similar tests are done mod 9, 5, 7, 13 and 17, 265for a total 99.25% of inputs identified as non-squares. On a 64-bit 266system 97 is tested too, for a total 99.62%. 267 268 These moduli are chosen because they're factors of 2^24-1 (or 2^48-1 269for 64-bits), and such a remainder can be quickly taken just using 270additions (see `mpn_mod_34lsub1'). 271 272 When nails are in use moduli are instead selected by the `gen-psqr.c' 273program and applied with an `mpn_mod_1'. The same 2^24-1 or 2^48-1 274could be done with nails using some extra bit shifts, but this is not 275currently implemented. 276 277 In any case each modulus is applied to the `mpn_mod_34lsub1' or 278`mpn_mod_1' remainder and a table lookup identifies non-squares. By 279using a "modexact" style calculation, and suitably permuted tables, 280just one multiply each is required, see the code for details. Moduli 281are also combined to save operations, so long as the lookup tables 282don't become too big. `gen-psqr.c' does all the pre-calculations. 283 284 A square root must still be taken for any value that passes these 285tests, to verify it's really a square and not one of the small fraction 286of non-squares that get through (i.e. a pseudo-square to all the tested 287bases). 288 289 Clearly more residue tests could be done, `mpz_perfect_square_p' only 290uses a compact and efficient set. Big inputs would probably benefit 291from more residue testing, small inputs might be better off with less. 292The assumed distribution of squares versus non-squares in the input 293would affect such considerations. 294 295 296File: gmp.info, Node: Perfect Power Algorithm, Prev: Perfect Square Algorithm, Up: Root Extraction Algorithms 297 29816.5.4 Perfect Power 299-------------------- 300 301Detecting perfect powers is required by some factorization algorithms. 302Currently `mpz_perfect_power_p' is implemented using repeated Nth root 303extractions, though naturally only prime roots need to be considered. 304(*Note Nth Root Algorithm::.) 305 306 If a prime divisor p with multiplicity e can be found, then only 307roots which are divisors of e need to be considered, much reducing the 308work necessary. To this end divisibility by a set of small primes is 309checked. 310 311 312File: gmp.info, Node: Radix Conversion Algorithms, Next: Other Algorithms, Prev: Root Extraction Algorithms, Up: Algorithms 313 31416.6 Radix Conversion 315===================== 316 317Radix conversions are less important than other algorithms. A program 318dominated by conversions should probably use a different data 319representation. 320 321* Menu: 322 323* Binary to Radix:: 324* Radix to Binary:: 325 326 327File: gmp.info, Node: Binary to Radix, Next: Radix to Binary, Prev: Radix Conversion Algorithms, Up: Radix Conversion Algorithms 328 32916.6.1 Binary to Radix 330---------------------- 331 332Conversions from binary to a power-of-2 radix use a simple and fast 333O(N) bit extraction algorithm. 334 335 Conversions from binary to other radices use one of two algorithms. 336Sizes below `GET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method. 337Repeated divisions by b^n are made, where b is the radix and n is the 338biggest power that fits in a limb. But instead of simply using the 339remainder r from such divisions, an extra divide step is done to give a 340fractional limb representing r/b^n. The digits of r can then be 341extracted using multiplications by b rather than divisions. Special 342case code is provided for decimal, allowing multiplications by 10 to 343optimize to shifts and adds. 344 345 Above `GET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is 346used. For an input t, powers b^(n*2^i) of the radix are calculated, 347until a power between t and sqrt(t) is reached. t is then divided by 348that largest power, giving a quotient which is the digits above that 349power, and a remainder which is those below. These two parts are in 350turn divided by the second highest power, and so on recursively. When 351a piece has been divided down to less than `GET_STR_DC_THRESHOLD' 352limbs, the basecase algorithm described above is used. 353 354 The advantage of this algorithm is that big divisions can make use 355of the sub-quadratic divide and conquer division (*note Divide and 356Conquer Division::), and big divisions tend to have less overheads than 357lots of separate single limb divisions anyway. But in any case the 358cost of calculating the powers b^(n*2^i) must first be overcome. 359 360 `GET_STR_PRECOMPUTE_THRESHOLD' and `GET_STR_DC_THRESHOLD' represent 361the same basic thing, the point where it becomes worth doing a big 362division to cut the input in half. `GET_STR_PRECOMPUTE_THRESHOLD' 363includes the cost of calculating the radix power required, whereas 364`GET_STR_DC_THRESHOLD' assumes that's already available, which is the 365case when recursing. 366 367 Since the base case produces digits from least to most significant 368but they want to be stored from most to least, it's necessary to 369calculate in advance how many digits there will be, or at least be sure 370not to underestimate that. For GMP the number of input bits is 371multiplied by `chars_per_bit_exactly' from `mp_bases', rounding up. 372The result is either correct or one too big. 373 374 Examining some of the high bits of the input could increase the 375chance of getting the exact number of digits, but an exact result every 376time would not be practical, since in general the difference between 377numbers 100... and 99... is only in the last few bits and the work to 378identify 99... might well be almost as much as a full conversion. 379 380 `mpf_get_str' doesn't currently use the algorithm described here, it 381multiplies or divides by a power of b to move the radix point to the 382just above the highest non-zero digit (or at worst one above that 383location), then multiplies by b^n to bring out digits. This is O(N^2) 384and is certainly not optimal. 385 386 The r/b^n scheme described above for using multiplications to bring 387out digits might be useful for more than a single limb. Some brief 388experiments with it on the base case when recursing didn't give a 389noticeable improvement, but perhaps that was only due to the 390implementation. Something similar would work for the sub-quadratic 391divisions too, though there would be the cost of calculating a bigger 392radix power. 393 394 Another possible improvement for the sub-quadratic part would be to 395arrange for radix powers that balanced the sizes of quotient and 396remainder produced, i.e. the highest power would be an b^(n*k) 397approximately equal to sqrt(t), not restricted to a 2^i factor. That 398ought to smooth out a graph of times against sizes, but may or may not 399be a net speedup. 400 401 402File: gmp.info, Node: Radix to Binary, Prev: Binary to Radix, Up: Radix Conversion Algorithms 403 40416.6.2 Radix to Binary 405---------------------- 406 407*This section needs to be rewritten, it currently describes the 408algorithms used before GMP 4.3.* 409 410 Conversions from a power-of-2 radix into binary use a simple and fast 411O(N) bitwise concatenation algorithm. 412 413 Conversions from other radices use one of two algorithms. Sizes 414below `SET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method. Groups 415of n digits are converted to limbs, where n is the biggest power of the 416base b which will fit in a limb, then those groups are accumulated into 417the result by multiplying by b^n and adding. This saves 418multi-precision operations, as per Knuth section 4.4 part E (*note 419References::). Some special case code is provided for decimal, giving 420the compiler a chance to optimize multiplications by 10. 421 422 Above `SET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is 423used. First groups of n digits are converted into limbs. Then adjacent 424limbs are combined into limb pairs with x*b^n+y, where x and y are the 425limbs. Adjacent limb pairs are combined into quads similarly with 426x*b^(2n)+y. This continues until a single block remains, that being 427the result. 428 429 The advantage of this method is that the multiplications for each x 430are big blocks, allowing Karatsuba and higher algorithms to be used. 431But the cost of calculating the powers b^(n*2^i) must be overcome. 432`SET_STR_PRECOMPUTE_THRESHOLD' usually ends up quite big, around 5000 433digits, and on some processors much bigger still. 434 435 `SET_STR_PRECOMPUTE_THRESHOLD' is based on the input digits (and 436tuned for decimal), though it might be better based on a limb count, so 437as to be independent of the base. But that sort of count isn't used by 438the base case and so would need some sort of initial calculation or 439estimate. 440 441 The main reason `SET_STR_PRECOMPUTE_THRESHOLD' is so much bigger 442than the corresponding `GET_STR_PRECOMPUTE_THRESHOLD' is that 443`mpn_mul_1' is much faster than `mpn_divrem_1' (often by a factor of 5, 444or more). 445 446 447File: gmp.info, Node: Other Algorithms, Next: Assembly Coding, Prev: Radix Conversion Algorithms, Up: Algorithms 448 44916.7 Other Algorithms 450===================== 451 452* Menu: 453 454* Prime Testing Algorithm:: 455* Factorial Algorithm:: 456* Binomial Coefficients Algorithm:: 457* Fibonacci Numbers Algorithm:: 458* Lucas Numbers Algorithm:: 459* Random Number Algorithms:: 460 461 462File: gmp.info, Node: Prime Testing Algorithm, Next: Factorial Algorithm, Prev: Other Algorithms, Up: Other Algorithms 463 46416.7.1 Prime Testing 465-------------------- 466 467The primality testing in `mpz_probab_prime_p' (*note Number Theoretic 468Functions::) first does some trial division by small factors and then 469uses the Miller-Rabin probabilistic primality testing algorithm, as 470described in Knuth section 4.5.4 algorithm P (*note References::). 471 472 For an odd input n, and with n = q*2^k+1 where q is odd, this 473algorithm selects a random base x and tests whether x^q mod n is 1 or 474-1, or an x^(q*2^j) mod n is 1, for 1<=j<=k. If so then n is probably 475prime, if not then n is definitely composite. 476 477 Any prime n will pass the test, but some composites do too. Such 478composites are known as strong pseudoprimes to base x. No n is a 479strong pseudoprime to more than 1/4 of all bases (see Knuth exercise 48022), hence with x chosen at random there's no more than a 1/4 chance a 481"probable prime" will in fact be composite. 482 483 In fact strong pseudoprimes are quite rare, making the test much more 484powerful than this analysis would suggest, but 1/4 is all that's proven 485for an arbitrary n. 486 487 488File: gmp.info, Node: Factorial Algorithm, Next: Binomial Coefficients Algorithm, Prev: Prime Testing Algorithm, Up: Other Algorithms 489 49016.7.2 Factorial 491---------------- 492 493Factorials are calculated by a combination of removal of twos, 494powering, and binary splitting. The procedure can be best illustrated 495with an example, 496 497 23! = 1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.18.19.20.21.22.23 498 499has factors of two removed, 500 501 23! = 2^19.1.1.3.1.5.3.7.1.9.5.11.3.13.7.15.1.17.9.19.5.21.11.23 502 503and the resulting terms collected up according to their multiplicity, 504 505 23! = 2^19.(3.5)^3.(7.9.11)^2.(13.15.17.19.21.23) 506 507 Each sequence such as 13.15.17.19.21.23 is evaluated by splitting 508into every second term, as for instance (13.17.21).(15.19.23), and the 509same recursively on each half. This is implemented iteratively using 510some bit twiddling. 511 512 Such splitting is more efficient than repeated Nx1 multiplies since 513it forms big multiplies, allowing Karatsuba and higher algorithms to be 514used. And even below the Karatsuba threshold a big block of work can 515be more efficient for the basecase algorithm. 516 517 Splitting into subsequences of every second term keeps the resulting 518products more nearly equal in size than would the simpler approach of 519say taking the first half and second half of the sequence. Nearly 520equal products are more efficient for the current multiply 521implementation. 522 523 524File: gmp.info, Node: Binomial Coefficients Algorithm, Next: Fibonacci Numbers Algorithm, Prev: Factorial Algorithm, Up: Other Algorithms 525 52616.7.3 Binomial Coefficients 527---------------------------- 528 529Binomial coefficients C(n,k) are calculated by first arranging k <= n/2 530using C(n,k) = C(n,n-k) if necessary, and then evaluating the following 531product simply from i=2 to i=k. 532 533 k (n-k+i) 534 C(n,k) = (n-k+1) * prod ------- 535 i=2 i 536 537 It's easy to show that each denominator i will divide the product so 538far, so the exact division algorithm is used (*note Exact Division::). 539 540 The numerators n-k+i and denominators i are first accumulated into 541as many fit a limb, to save multi-precision operations, though for 542`mpz_bin_ui' this applies only to the divisors, since n is an `mpz_t' 543and n-k+i in general won't fit in a limb at all. 544 545 546File: gmp.info, Node: Fibonacci Numbers Algorithm, Next: Lucas Numbers Algorithm, Prev: Binomial Coefficients Algorithm, Up: Other Algorithms 547 54816.7.4 Fibonacci Numbers 549------------------------ 550 551The Fibonacci functions `mpz_fib_ui' and `mpz_fib2_ui' are designed for 552calculating isolated F[n] or F[n],F[n-1] values efficiently. 553 554 For small n, a table of single limb values in `__gmp_fib_table' is 555used. On a 32-bit limb this goes up to F[47], or on a 64-bit limb up 556to F[93]. For convenience the table starts at F[-1]. 557 558 Beyond the table, values are generated with a binary powering 559algorithm, calculating a pair F[n] and F[n-1] working from high to low 560across the bits of n. The formulas used are 561 562 F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k 563 F[2k-1] = F[k]^2 + F[k-1]^2 564 565 F[2k] = F[2k+1] - F[2k-1] 566 567 At each step, k is the high b bits of n. If the next bit of n is 0 568then F[2k],F[2k-1] is used, or if it's a 1 then F[2k+1],F[2k] is used, 569and the process repeated until all bits of n are incorporated. Notice 570these formulas require just two squares per bit of n. 571 572 It'd be possible to handle the first few n above the single limb 573table with simple additions, using the defining Fibonacci recurrence 574F[k+1]=F[k]+F[k-1], but this is not done since it usually turns out to 575be faster for only about 10 or 20 values of n, and including a block of 576code for just those doesn't seem worthwhile. If they really mattered 577it'd be better to extend the data table. 578 579 Using a table avoids lots of calculations on small numbers, and 580makes small n go fast. A bigger table would make more small n go fast, 581it's just a question of balancing size against desired speed. For GMP 582the code is kept compact, with the emphasis primarily on a good 583powering algorithm. 584 585 `mpz_fib2_ui' returns both F[n] and F[n-1], but `mpz_fib_ui' is only 586interested in F[n]. In this case the last step of the algorithm can 587become one multiply instead of two squares. One of the following two 588formulas is used, according as n is odd or even. 589 590 F[2k] = F[k]*(F[k]+2F[k-1]) 591 592 F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k 593 594 F[2k+1] here is the same as above, just rearranged to be a multiply. 595For interest, the 2*(-1)^k term both here and above can be applied just 596to the low limb of the calculation, without a carry or borrow into 597further limbs, which saves some code size. See comments with 598`mpz_fib_ui' and the internal `mpn_fib2_ui' for how this is done. 599 600 601File: gmp.info, Node: Lucas Numbers Algorithm, Next: Random Number Algorithms, Prev: Fibonacci Numbers Algorithm, Up: Other Algorithms 602 60316.7.5 Lucas Numbers 604-------------------- 605 606`mpz_lucnum2_ui' derives a pair of Lucas numbers from a pair of 607Fibonacci numbers with the following simple formulas. 608 609 L[k] = F[k] + 2*F[k-1] 610 L[k-1] = 2*F[k] - F[k-1] 611 612 `mpz_lucnum_ui' is only interested in L[n], and some work can be 613saved. Trailing zero bits on n can be handled with a single square 614each. 615 616 L[2k] = L[k]^2 - 2*(-1)^k 617 618 And the lowest 1 bit can be handled with one multiply of a pair of 619Fibonacci numbers, similar to what `mpz_fib_ui' does. 620 621 L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k 622 623 624File: gmp.info, Node: Random Number Algorithms, Prev: Lucas Numbers Algorithm, Up: Other Algorithms 625 62616.7.6 Random Numbers 627--------------------- 628 629For the `urandomb' functions, random numbers are generated simply by 630concatenating bits produced by the generator. As long as the generator 631has good randomness properties this will produce well-distributed N bit 632numbers. 633 634 For the `urandomm' functions, random numbers in a range 0<=R<N are 635generated by taking values R of ceil(log2(N)) bits each until one 636satisfies R<N. This will normally require only one or two attempts, 637but the attempts are limited in case the generator is somehow 638degenerate and produces only 1 bits or similar. 639 640 The Mersenne Twister generator is by Matsumoto and Nishimura (*note 641References::). It has a non-repeating period of 2^19937-1, which is a 642Mersenne prime, hence the name of the generator. The state is 624 643words of 32-bits each, which is iterated with one XOR and shift for each 64432-bit word generated, making the algorithm very fast. Randomness 645properties are also very good and this is the default algorithm used by 646GMP. 647 648 Linear congruential generators are described in many text books, for 649instance Knuth volume 2 (*note References::). With a modulus M and 650parameters A and C, a integer state S is iterated by the formula S <- 651A*S+C mod M. At each step the new state is a linear function of the 652previous, mod M, hence the name of the generator. 653 654 In GMP only moduli of the form 2^N are supported, and the current 655implementation is not as well optimized as it could be. Overheads are 656significant when N is small, and when N is large clearly the multiply 657at each step will become slow. This is not a big concern, since the 658Mersenne Twister generator is better in every respect and is therefore 659recommended for all normal applications. 660 661 For both generators the current state can be deduced by observing 662enough output and applying some linear algebra (over GF(2) in the case 663of the Mersenne Twister). This generally means raw output is 664unsuitable for cryptographic applications without further hashing or 665the like. 666 667 668File: gmp.info, Node: Assembly Coding, Prev: Other Algorithms, Up: Algorithms 669 67016.8 Assembly Coding 671==================== 672 673The assembly subroutines in GMP are the most significant source of 674speed at small to moderate sizes. At larger sizes algorithm selection 675becomes more important, but of course speedups in low level routines 676will still speed up everything proportionally. 677 678 Carry handling and widening multiplies that are important for GMP 679can't be easily expressed in C. GCC `asm' blocks help a lot and are 680provided in `longlong.h', but hand coding low level routines invariably 681offers a speedup over generic C by a factor of anything from 2 to 10. 682 683* Menu: 684 685* Assembly Code Organisation:: 686* Assembly Basics:: 687* Assembly Carry Propagation:: 688* Assembly Cache Handling:: 689* Assembly Functional Units:: 690* Assembly Floating Point:: 691* Assembly SIMD Instructions:: 692* Assembly Software Pipelining:: 693* Assembly Loop Unrolling:: 694* Assembly Writing Guide:: 695 696 697File: gmp.info, Node: Assembly Code Organisation, Next: Assembly Basics, Prev: Assembly Coding, Up: Assembly Coding 698 69916.8.1 Code Organisation 700------------------------ 701 702The various `mpn' subdirectories contain machine-dependent code, written 703in C or assembly. The `mpn/generic' subdirectory contains default code, 704used when there's no machine-specific version of a particular file. 705 706 Each `mpn' subdirectory is for an ISA family. Generally 32-bit and 70764-bit variants in a family cannot share code and have separate 708directories. Within a family further subdirectories may exist for CPU 709variants. 710 711 In each directory a `nails' subdirectory may exist, holding code with 712nails support for that CPU variant. A `NAILS_SUPPORT' directive in each 713file indicates the nails values the code handles. Nails code only 714exists where it's faster, or promises to be faster, than plain code. 715There's no effort put into nails if they're not going to enhance a 716given CPU. 717 718 719File: gmp.info, Node: Assembly Basics, Next: Assembly Carry Propagation, Prev: Assembly Code Organisation, Up: Assembly Coding 720 72116.8.2 Assembly Basics 722---------------------- 723 724`mpn_addmul_1' and `mpn_submul_1' are the most important routines for 725overall GMP performance. All multiplications and divisions come down to 726repeated calls to these. `mpn_add_n', `mpn_sub_n', `mpn_lshift' and 727`mpn_rshift' are next most important. 728 729 On some CPUs assembly versions of the internal functions 730`mpn_mul_basecase' and `mpn_sqr_basecase' give significant speedups, 731mainly through avoiding function call overheads. They can also 732potentially make better use of a wide superscalar processor, as can 733bigger primitives like `mpn_addmul_2' or `mpn_addmul_4'. 734 735 The restrictions on overlaps between sources and destinations (*note 736Low-level Functions::) are designed to facilitate a variety of 737implementations. For example, knowing `mpn_add_n' won't have partly 738overlapping sources and destination means reading can be done far ahead 739of writing on superscalar processors, and loops can be vectorized on a 740vector processor, depending on the carry handling. 741 742 743File: gmp.info, Node: Assembly Carry Propagation, Next: Assembly Cache Handling, Prev: Assembly Basics, Up: Assembly Coding 744 74516.8.3 Carry Propagation 746------------------------ 747 748The problem that presents most challenges in GMP is propagating carries 749from one limb to the next. In functions like `mpn_addmul_1' and 750`mpn_add_n', carries are the only dependencies between limb operations. 751 752 On processors with carry flags, a straightforward CISC style `adc' is 753generally best. AMD K6 `mpn_addmul_1' however is an example of an 754unusual set of circumstances where a branch works out better. 755 756 On RISC processors generally an add and compare for overflow is 757used. This sort of thing can be seen in `mpn/generic/aors_n.c'. Some 758carry propagation schemes require 4 instructions, meaning at least 4 759cycles per limb, but other schemes may use just 1 or 2. On wide 760superscalar processors performance may be completely determined by the 761number of dependent instructions between carry-in and carry-out for 762each limb. 763 764 On vector processors good use can be made of the fact that a carry 765bit only very rarely propagates more than one limb. When adding a 766single bit to a limb, there's only a carry out if that limb was 767`0xFF...FF' which on random data will be only 1 in 2^mp_bits_per_limb. 768`mpn/cray/add_n.c' is an example of this, it adds all limbs in 769parallel, adds one set of carry bits in parallel and then only rarely 770needs to fall through to a loop propagating further carries. 771 772 On the x86s, GCC (as of version 2.95.2) doesn't generate 773particularly good code for the RISC style idioms that are necessary to 774handle carry bits in C. Often conditional jumps are generated where 775`adc' or `sbb' forms would be better. And so unfortunately almost any 776loop involving carry bits needs to be coded in assembly for best 777results. 778 779 780File: gmp.info, Node: Assembly Cache Handling, Next: Assembly Functional Units, Prev: Assembly Carry Propagation, Up: Assembly Coding 781 78216.8.4 Cache Handling 783--------------------- 784 785GMP aims to perform well both on operands that fit entirely in L1 cache 786and those which don't. 787 788 Basic routines like `mpn_add_n' or `mpn_lshift' are often used on 789large operands, so L2 and main memory performance is important for them. 790`mpn_mul_1' and `mpn_addmul_1' are mostly used for multiply and square 791basecases, so L1 performance matters most for them, unless assembly 792versions of `mpn_mul_basecase' and `mpn_sqr_basecase' exist, in which 793case the remaining uses are mostly for larger operands. 794 795 For L2 or main memory operands, memory access times will almost 796certainly be more than the calculation time. The aim therefore is to 797maximize memory throughput, by starting a load of the next cache line 798while processing the contents of the previous one. Clearly this is 799only possible if the chip has a lock-up free cache or some sort of 800prefetch instruction. Most current chips have both these features. 801 802 Prefetching sources combines well with loop unrolling, since a 803prefetch can be initiated once per unrolled loop (or more than once if 804the loop covers more than one cache line). 805 806 On CPUs without write-allocate caches, prefetching destinations will 807ensure individual stores don't go further down the cache hierarchy, 808limiting bandwidth. Of course for calculations which are slow anyway, 809like `mpn_divrem_1', write-throughs might be fine. 810 811 The distance ahead to prefetch will be determined by memory latency 812versus throughput. The aim of course is to have data arriving 813continuously, at peak throughput. Some CPUs have limits on the number 814of fetches or prefetches in progress. 815 816 If a special prefetch instruction doesn't exist then a plain load 817can be used, but in that case care must be taken not to attempt to read 818past the end of an operand, since that might produce a segmentation 819violation. 820 821 Some CPUs or systems have hardware that detects sequential memory 822accesses and initiates suitable cache movements automatically, making 823life easy. 824 825 826File: gmp.info, Node: Assembly Functional Units, Next: Assembly Floating Point, Prev: Assembly Cache Handling, Up: Assembly Coding 827 82816.8.5 Functional Units 829----------------------- 830 831When choosing an approach for an assembly loop, consideration is given 832to what operations can execute simultaneously and what throughput can 833thereby be achieved. In some cases an algorithm can be tweaked to 834accommodate available resources. 835 836 Loop control will generally require a counter and pointer updates, 837costing as much as 5 instructions, plus any delays a branch introduces. 838CPU addressing modes might reduce pointer updates, perhaps by allowing 839just one updating pointer and others expressed as offsets from it, or 840on CISC chips with all addressing done with the loop counter as a 841scaled index. 842 843 The final loop control cost can be amortised by processing several 844limbs in each iteration (*note Assembly Loop Unrolling::). This at 845least ensures loop control isn't a big fraction the work done. 846 847 Memory throughput is always a limit. If perhaps only one load or 848one store can be done per cycle then 3 cycles/limb will the top speed 849for "binary" operations like `mpn_add_n', and any code achieving that 850is optimal. 851 852 Integer resources can be freed up by having the loop counter in a 853float register, or by pressing the float units into use for some 854multiplying, perhaps doing every second limb on the float side (*note 855Assembly Floating Point::). 856 857 Float resources can be freed up by doing carry propagation on the 858integer side, or even by doing integer to float conversions in integers 859using bit twiddling. 860 861 862File: gmp.info, Node: Assembly Floating Point, Next: Assembly SIMD Instructions, Prev: Assembly Functional Units, Up: Assembly Coding 863 86416.8.6 Floating Point 865--------------------- 866 867Floating point arithmetic is used in GMP for multiplications on CPUs 868with poor integer multipliers. It's mostly useful for `mpn_mul_1', 869`mpn_addmul_1' and `mpn_submul_1' on 64-bit machines, and 870`mpn_mul_basecase' on both 32-bit and 64-bit machines. 871 872 With IEEE 53-bit double precision floats, integer multiplications 873producing up to 53 bits will give exact results. Breaking a 64x64 874multiplication into eight 16x32->48 bit pieces is convenient. With 875some care though six 21x32->53 bit products can be used, if one of the 876lower two 21-bit pieces also uses the sign bit. 877 878 For the `mpn_mul_1' family of functions on a 64-bit machine, the 879invariant single limb is split at the start, into 3 or 4 pieces. 880Inside the loop, the bignum operand is split into 32-bit pieces. Fast 881conversion of these unsigned 32-bit pieces to floating point is highly 882machine-dependent. In some cases, reading the data into the integer 883unit, zero-extending to 64-bits, then transferring to the floating 884point unit back via memory is the only option. 885 886 Converting partial products back to 64-bit limbs is usually best 887done as a signed conversion. Since all values are smaller than 2^53, 888signed and unsigned are the same, but most processors lack unsigned 889conversions. 890 891 892 893 Here is a diagram showing 16x32 bit products for an `mpn_mul_1' or 894`mpn_addmul_1' with a 64-bit limb. The single limb operand V is split 895into four 16-bit parts. The multi-limb operand U is split in the loop 896into two 32-bit parts. 897 898 +---+---+---+---+ 899 |v48|v32|v16|v00| V operand 900 +---+---+---+---+ 901 902 +-------+---+---+ 903 x | u32 | u00 | U operand (one limb) 904 +---------------+ 905 906 --------------------------------- 907 908 +-----------+ 909 | u00 x v00 | p00 48-bit products 910 +-----------+ 911 +-----------+ 912 | u00 x v16 | p16 913 +-----------+ 914 +-----------+ 915 | u00 x v32 | p32 916 +-----------+ 917 +-----------+ 918 | u00 x v48 | p48 919 +-----------+ 920 +-----------+ 921 | u32 x v00 | r32 922 +-----------+ 923 +-----------+ 924 | u32 x v16 | r48 925 +-----------+ 926 +-----------+ 927 | u32 x v32 | r64 928 +-----------+ 929 +-----------+ 930 | u32 x v48 | r80 931 +-----------+ 932 933 p32 and r32 can be summed using floating-point addition, and 934likewise p48 and r48. p00 and p16 can be summed with r64 and r80 from 935the previous iteration. 936 937 For each loop then, four 49-bit quantities are transferred to the 938integer unit, aligned as follows, 939 940 |-----64bits----|-----64bits----| 941 +------------+ 942 | p00 + r64' | i00 943 +------------+ 944 +------------+ 945 | p16 + r80' | i16 946 +------------+ 947 +------------+ 948 | p32 + r32 | i32 949 +------------+ 950 +------------+ 951 | p48 + r48 | i48 952 +------------+ 953 954 The challenge then is to sum these efficiently and add in a carry 955limb, generating a low 64-bit result limb and a high 33-bit carry limb 956(i48 extends 33 bits into the high half). 957 958 959File: gmp.info, Node: Assembly SIMD Instructions, Next: Assembly Software Pipelining, Prev: Assembly Floating Point, Up: Assembly Coding 960 96116.8.7 SIMD Instructions 962------------------------ 963 964The single-instruction multiple-data support in current microprocessors 965is aimed at signal processing algorithms where each data point can be 966treated more or less independently. There's generally not much support 967for propagating the sort of carries that arise in GMP. 968 969 SIMD multiplications of say four 16x16 bit multiplies only do as much 970work as one 32x32 from GMP's point of view, and need some shifts and 971adds besides. But of course if say the SIMD form is fully pipelined 972and uses less instruction decoding then it may still be worthwhile. 973 974 On the x86 chips, MMX has so far found a use in `mpn_rshift' and 975`mpn_lshift', and is used in a special case for 16-bit multipliers in 976the P55 `mpn_mul_1'. SSE2 is used for Pentium 4 `mpn_mul_1', 977`mpn_addmul_1', and `mpn_submul_1'. 978 979 980File: gmp.info, Node: Assembly Software Pipelining, Next: Assembly Loop Unrolling, Prev: Assembly SIMD Instructions, Up: Assembly Coding 981 98216.8.8 Software Pipelining 983-------------------------- 984 985Software pipelining consists of scheduling instructions around the 986branch point in a loop. For example a loop might issue a load not for 987use in the present iteration but the next, thereby allowing extra 988cycles for the data to arrive from memory. 989 990 Naturally this is wanted only when doing things like loads or 991multiplies that take several cycles to complete, and only where a CPU 992has multiple functional units so that other work can be done in the 993meantime. 994 995 A pipeline with several stages will have a data value in progress at 996each stage and each loop iteration moves them along one stage. This is 997like juggling. 998 999 If the latency of some instruction is greater than the loop time 1000then it will be necessary to unroll, so one register has a result ready 1001to use while another (or multiple others) are still in progress. 1002(*note Assembly Loop Unrolling::). 1003 1004 1005File: gmp.info, Node: Assembly Loop Unrolling, Next: Assembly Writing Guide, Prev: Assembly Software Pipelining, Up: Assembly Coding 1006 100716.8.9 Loop Unrolling 1008--------------------- 1009 1010Loop unrolling consists of replicating code so that several limbs are 1011processed in each loop. At a minimum this reduces loop overheads by a 1012corresponding factor, but it can also allow better register usage, for 1013example alternately using one register combination and then another. 1014Judicious use of `m4' macros can help avoid lots of duplication in the 1015source code. 1016 1017 Any amount of unrolling can be handled with a loop counter that's 1018decremented by N each time, stopping when the remaining count is less 1019than the further N the loop will process. Or by subtracting N at the 1020start, the termination condition becomes when the counter C is less 1021than 0 (and the count of remaining limbs is C+N). 1022 1023 Alternately for a power of 2 unroll the loop count and remainder can 1024be established with a shift and mask. This is convenient if also 1025making a computed jump into the middle of a large loop. 1026 1027 The limbs not a multiple of the unrolling can be handled in various 1028ways, for example 1029 1030 * A simple loop at the end (or the start) to process the excess. 1031 Care will be wanted that it isn't too much slower than the 1032 unrolled part. 1033 1034 * A set of binary tests, for example after an 8-limb unrolling, test 1035 for 4 more limbs to process, then a further 2 more or not, and 1036 finally 1 more or not. This will probably take more code space 1037 than a simple loop. 1038 1039 * A `switch' statement, providing separate code for each possible 1040 excess, for example an 8-limb unrolling would have separate code 1041 for 0 remaining, 1 remaining, etc, up to 7 remaining. This might 1042 take a lot of code, but may be the best way to optimize all cases 1043 in combination with a deep pipelined loop. 1044 1045 * A computed jump into the middle of the loop, thus making the first 1046 iteration handle the excess. This should make times smoothly 1047 increase with size, which is attractive, but setups for the jump 1048 and adjustments for pointers can be tricky and could become quite 1049 difficult in combination with deep pipelining. 1050 1051 1052File: gmp.info, Node: Assembly Writing Guide, Prev: Assembly Loop Unrolling, Up: Assembly Coding 1053 105416.8.10 Writing Guide 1055--------------------- 1056 1057This is a guide to writing software pipelined loops for processing limb 1058vectors in assembly. 1059 1060 First determine the algorithm and which instructions are needed. 1061Code it without unrolling or scheduling, to make sure it works. On a 10623-operand CPU try to write each new value to a new register, this will 1063greatly simplify later steps. 1064 1065 Then note for each instruction the functional unit and/or issue port 1066requirements. If an instruction can use either of two units, like U0 1067or U1 then make a category "U0/U1". Count the total using each unit 1068(or combined unit), and count all instructions. 1069 1070 Figure out from those counts the best possible loop time. The goal 1071will be to find a perfect schedule where instruction latencies are 1072completely hidden. The total instruction count might be the limiting 1073factor, or perhaps a particular functional unit. It might be possible 1074to tweak the instructions to help the limiting factor. 1075 1076 Suppose the loop time is N, then make N issue buckets, with the 1077final loop branch at the end of the last. Now fill the buckets with 1078dummy instructions using the functional units desired. Run this to 1079make sure the intended speed is reached. 1080 1081 Now replace the dummy instructions with the real instructions from 1082the slow but correct loop you started with. The first will typically 1083be a load instruction. Then the instruction using that value is placed 1084in a bucket an appropriate distance down. Run the loop again, to check 1085it still runs at target speed. 1086 1087 Keep placing instructions, frequently measuring the loop. After a 1088few you will need to wrap around from the last bucket back to the top 1089of the loop. If you used the new-register for new-value strategy above 1090then there will be no register conflicts. If not then take care not to 1091clobber something already in use. Changing registers at this time is 1092very error prone. 1093 1094 The loop will overlap two or more of the original loop iterations, 1095and the computation of one vector element result will be started in one 1096iteration of the new loop, and completed one or several iterations 1097later. 1098 1099 The final step is to create feed-in and wind-down code for the loop. 1100A good way to do this is to make a copy (or copies) of the loop at the 1101start and delete those instructions which don't have valid antecedents, 1102and at the end replicate and delete those whose results are unwanted 1103(including any further loads). 1104 1105 The loop will have a minimum number of limbs loaded and processed, 1106so the feed-in code must test if the request size is smaller and skip 1107either to a suitable part of the wind-down or to special code for small 1108sizes. 1109 1110 1111File: gmp.info, Node: Internals, Next: Contributors, Prev: Algorithms, Up: Top 1112 111317 Internals 1114************ 1115 1116*This chapter is provided only for informational purposes and the 1117various internals described here may change in future GMP releases. 1118Applications expecting to be compatible with future releases should use 1119only the documented interfaces described in previous chapters.* 1120 1121* Menu: 1122 1123* Integer Internals:: 1124* Rational Internals:: 1125* Float Internals:: 1126* Raw Output Internals:: 1127* C++ Interface Internals:: 1128 1129 1130File: gmp.info, Node: Integer Internals, Next: Rational Internals, Prev: Internals, Up: Internals 1131 113217.1 Integer Internals 1133====================== 1134 1135`mpz_t' variables represent integers using sign and magnitude, in space 1136dynamically allocated and reallocated. The fields are as follows. 1137 1138`_mp_size' 1139 The number of limbs, or the negative of that when representing a 1140 negative integer. Zero is represented by `_mp_size' set to zero, 1141 in which case the `_mp_d' data is unused. 1142 1143`_mp_d' 1144 A pointer to an array of limbs which is the magnitude. These are 1145 stored "little endian" as per the `mpn' functions, so `_mp_d[0]' 1146 is the least significant limb and `_mp_d[ABS(_mp_size)-1]' is the 1147 most significant. Whenever `_mp_size' is non-zero, the most 1148 significant limb is non-zero. 1149 1150 Currently there's always at least one limb allocated, so for 1151 instance `mpz_set_ui' never needs to reallocate, and `mpz_get_ui' 1152 can fetch `_mp_d[0]' unconditionally (though its value is then 1153 only wanted if `_mp_size' is non-zero). 1154 1155`_mp_alloc' 1156 `_mp_alloc' is the number of limbs currently allocated at `_mp_d', 1157 and naturally `_mp_alloc >= ABS(_mp_size)'. When an `mpz' routine 1158 is about to (or might be about to) increase `_mp_size', it checks 1159 `_mp_alloc' to see whether there's enough space, and reallocates 1160 if not. `MPZ_REALLOC' is generally used for this. 1161 1162 The various bitwise logical functions like `mpz_and' behave as if 1163negative values were twos complement. But sign and magnitude is always 1164used internally, and necessary adjustments are made during the 1165calculations. Sometimes this isn't pretty, but sign and magnitude are 1166best for other routines. 1167 1168 Some internal temporary variables are setup with `MPZ_TMP_INIT' and 1169these have `_mp_d' space obtained from `TMP_ALLOC' rather than the 1170memory allocation functions. Care is taken to ensure that these are 1171big enough that no reallocation is necessary (since it would have 1172unpredictable consequences). 1173 1174 `_mp_size' and `_mp_alloc' are `int', although `mp_size_t' is 1175usually a `long'. This is done to make the fields just 32 bits on some 117664 bits systems, thereby saving a few bytes of data space but still 1177providing plenty of range. 1178 1179 1180File: gmp.info, Node: Rational Internals, Next: Float Internals, Prev: Integer Internals, Up: Internals 1181 118217.2 Rational Internals 1183======================= 1184 1185`mpq_t' variables represent rationals using an `mpz_t' numerator and 1186denominator (*note Integer Internals::). 1187 1188 The canonical form adopted is denominator positive (and non-zero), 1189no common factors between numerator and denominator, and zero uniquely 1190represented as 0/1. 1191 1192 It's believed that casting out common factors at each stage of a 1193calculation is best in general. A GCD is an O(N^2) operation so it's 1194better to do a few small ones immediately than to delay and have to do 1195a big one later. Knowing the numerator and denominator have no common 1196factors can be used for example in `mpq_mul' to make only two cross 1197GCDs necessary, not four. 1198 1199 This general approach to common factors is badly sub-optimal in the 1200presence of simple factorizations or little prospect for cancellation, 1201but GMP has no way to know when this will occur. As per *note 1202Efficiency::, that's left to applications. The `mpq_t' framework might 1203still suit, with `mpq_numref' and `mpq_denref' for direct access to the 1204numerator and denominator, or of course `mpz_t' variables can be used 1205directly. 1206 1207 1208File: gmp.info, Node: Float Internals, Next: Raw Output Internals, Prev: Rational Internals, Up: Internals 1209 121017.3 Float Internals 1211==================== 1212 1213Efficient calculation is the primary aim of GMP floats and the use of 1214whole limbs and simple rounding facilitates this. 1215 1216 `mpf_t' floats have a variable precision mantissa and a single 1217machine word signed exponent. The mantissa is represented using sign 1218and magnitude. 1219 1220 most least 1221 significant significant 1222 limb limb 1223 1224 _mp_d 1225 |---- _mp_exp ---> | 1226 _____ _____ _____ _____ _____ 1227 |_____|_____|_____|_____|_____| 1228 . <------------ radix point 1229 1230 <-------- _mp_size ---------> 1231 1232The fields are as follows. 1233 1234`_mp_size' 1235 The number of limbs currently in use, or the negative of that when 1236 representing a negative value. Zero is represented by `_mp_size' 1237 and `_mp_exp' both set to zero, and in that case the `_mp_d' data 1238 is unused. (In the future `_mp_exp' might be undefined when 1239 representing zero.) 1240 1241`_mp_prec' 1242 The precision of the mantissa, in limbs. In any calculation the 1243 aim is to produce `_mp_prec' limbs of result (the most significant 1244 being non-zero). 1245 1246`_mp_d' 1247 A pointer to the array of limbs which is the absolute value of the 1248 mantissa. These are stored "little endian" as per the `mpn' 1249 functions, so `_mp_d[0]' is the least significant limb and 1250 `_mp_d[ABS(_mp_size)-1]' the most significant. 1251 1252 The most significant limb is always non-zero, but there are no 1253 other restrictions on its value, in particular the highest 1 bit 1254 can be anywhere within the limb. 1255 1256 `_mp_prec+1' limbs are allocated to `_mp_d', the extra limb being 1257 for convenience (see below). There are no reallocations during a 1258 calculation, only in a change of precision with `mpf_set_prec'. 1259 1260`_mp_exp' 1261 The exponent, in limbs, determining the location of the implied 1262 radix point. Zero means the radix point is just above the most 1263 significant limb. Positive values mean a radix point offset 1264 towards the lower limbs and hence a value >= 1, as for example in 1265 the diagram above. Negative exponents mean a radix point further 1266 above the highest limb. 1267 1268 Naturally the exponent can be any value, it doesn't have to fall 1269 within the limbs as the diagram shows, it can be a long way above 1270 or a long way below. Limbs other than those included in the 1271 `{_mp_d,_mp_size}' data are treated as zero. 1272 1273 The `_mp_size' and `_mp_prec' fields are `int', although the 1274`mp_size_t' type is usually a `long'. The `_mp_exp' field is usually 1275`long'. This is done to make some fields just 32 bits on some 64 bits 1276systems, thereby saving a few bytes of data space but still providing 1277plenty of precision and a very large range. 1278 1279 1280The following various points should be noted. 1281 1282Low Zeros 1283 The least significant limbs `_mp_d[0]' etc can be zero, though 1284 such low zeros can always be ignored. Routines likely to produce 1285 low zeros check and avoid them to save time in subsequent 1286 calculations, but for most routines they're quite unlikely and 1287 aren't checked. 1288 1289Mantissa Size Range 1290 The `_mp_size' count of limbs in use can be less than `_mp_prec' if 1291 the value can be represented in less. This means low precision 1292 values or small integers stored in a high precision `mpf_t' can 1293 still be operated on efficiently. 1294 1295 `_mp_size' can also be greater than `_mp_prec'. Firstly a value is 1296 allowed to use all of the `_mp_prec+1' limbs available at `_mp_d', 1297 and secondly when `mpf_set_prec_raw' lowers `_mp_prec' it leaves 1298 `_mp_size' unchanged and so the size can be arbitrarily bigger than 1299 `_mp_prec'. 1300 1301Rounding 1302 All rounding is done on limb boundaries. Calculating `_mp_prec' 1303 limbs with the high non-zero will ensure the application requested 1304 minimum precision is obtained. 1305 1306 The use of simple "trunc" rounding towards zero is efficient, 1307 since there's no need to examine extra limbs and increment or 1308 decrement. 1309 1310Bit Shifts 1311 Since the exponent is in limbs, there are no bit shifts in basic 1312 operations like `mpf_add' and `mpf_mul'. When differing exponents 1313 are encountered all that's needed is to adjust pointers to line up 1314 the relevant limbs. 1315 1316 Of course `mpf_mul_2exp' and `mpf_div_2exp' will require bit 1317 shifts, but the choice is between an exponent in limbs which 1318 requires shifts there, or one in bits which requires them almost 1319 everywhere else. 1320 1321Use of `_mp_prec+1' Limbs 1322 The extra limb on `_mp_d' (`_mp_prec+1' rather than just 1323 `_mp_prec') helps when an `mpf' routine might get a carry from its 1324 operation. `mpf_add' for instance will do an `mpn_add' of 1325 `_mp_prec' limbs. If there's no carry then that's the result, but 1326 if there is a carry then it's stored in the extra limb of space and 1327 `_mp_size' becomes `_mp_prec+1'. 1328 1329 Whenever `_mp_prec+1' limbs are held in a variable, the low limb 1330 is not needed for the intended precision, only the `_mp_prec' high 1331 limbs. But zeroing it out or moving the rest down is unnecessary. 1332 Subsequent routines reading the value will simply take the high 1333 limbs they need, and this will be `_mp_prec' if their target has 1334 that same precision. This is no more than a pointer adjustment, 1335 and must be checked anyway since the destination precision can be 1336 different from the sources. 1337 1338 Copy functions like `mpf_set' will retain a full `_mp_prec+1' limbs 1339 if available. This ensures that a variable which has `_mp_size' 1340 equal to `_mp_prec+1' will get its full exact value copied. 1341 Strictly speaking this is unnecessary since only `_mp_prec' limbs 1342 are needed for the application's requested precision, but it's 1343 considered that an `mpf_set' from one variable into another of the 1344 same precision ought to produce an exact copy. 1345 1346Application Precisions 1347 `__GMPF_BITS_TO_PREC' converts an application requested precision 1348 to an `_mp_prec'. The value in bits is rounded up to a whole limb 1349 then an extra limb is added since the most significant limb of 1350 `_mp_d' is only non-zero and therefore might contain only one bit. 1351 1352 `__GMPF_PREC_TO_BITS' does the reverse conversion, and removes the 1353 extra limb from `_mp_prec' before converting to bits. The net 1354 effect of reading back with `mpf_get_prec' is simply the precision 1355 rounded up to a multiple of `mp_bits_per_limb'. 1356 1357 Note that the extra limb added here for the high only being 1358 non-zero is in addition to the extra limb allocated to `_mp_d'. 1359 For example with a 32-bit limb, an application request for 250 1360 bits will be rounded up to 8 limbs, then an extra added for the 1361 high being only non-zero, giving an `_mp_prec' of 9. `_mp_d' then 1362 gets 10 limbs allocated. Reading back with `mpf_get_prec' will 1363 take `_mp_prec' subtract 1 limb and multiply by 32, giving 256 1364 bits. 1365 1366 Strictly speaking, the fact the high limb has at least one bit 1367 means that a float with, say, 3 limbs of 32-bits each will be 1368 holding at least 65 bits, but for the purposes of `mpf_t' it's 1369 considered simply to be 64 bits, a nice multiple of the limb size. 1370 1371 1372File: gmp.info, Node: Raw Output Internals, Next: C++ Interface Internals, Prev: Float Internals, Up: Internals 1373 137417.4 Raw Output Internals 1375========================= 1376 1377`mpz_out_raw' uses the following format. 1378 1379 +------+------------------------+ 1380 | size | data bytes | 1381 +------+------------------------+ 1382 1383 The size is 4 bytes written most significant byte first, being the 1384number of subsequent data bytes, or the twos complement negative of 1385that when a negative integer is represented. The data bytes are the 1386absolute value of the integer, written most significant byte first. 1387 1388 The most significant data byte is always non-zero, so the output is 1389the same on all systems, irrespective of limb size. 1390 1391 In GMP 1, leading zero bytes were written to pad the data bytes to a 1392multiple of the limb size. `mpz_inp_raw' will still accept this, for 1393compatibility. 1394 1395 The use of "big endian" for both the size and data fields is 1396deliberate, it makes the data easy to read in a hex dump of a file. 1397Unfortunately it also means that the limb data must be reversed when 1398reading or writing, so neither a big endian nor little endian system 1399can just read and write `_mp_d'. 1400 1401 1402File: gmp.info, Node: C++ Interface Internals, Prev: Raw Output Internals, Up: Internals 1403 140417.5 C++ Interface Internals 1405============================ 1406 1407A system of expression templates is used to ensure something like 1408`a=b+c' turns into a simple call to `mpz_add' etc. For `mpf_class' the 1409scheme also ensures the precision of the final destination is used for 1410any temporaries within a statement like `f=w*x+y*z'. These are 1411important features which a naive implementation cannot provide. 1412 1413 A simplified description of the scheme follows. The true scheme is 1414complicated by the fact that expressions have different return types. 1415For detailed information, refer to the source code. 1416 1417 To perform an operation, say, addition, we first define a "function 1418object" evaluating it, 1419 1420 struct __gmp_binary_plus 1421 { 1422 static void eval(mpf_t f, mpf_t g, mpf_t h) { mpf_add(f, g, h); } 1423 }; 1424 1425And an "additive expression" object, 1426 1427 __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> > 1428 operator+(const mpf_class &f, const mpf_class &g) 1429 { 1430 return __gmp_expr 1431 <__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >(f, g); 1432 } 1433 1434 The seemingly redundant `__gmp_expr<__gmp_binary_expr<...>>' is used 1435to encapsulate any possible kind of expression into a single template 1436type. In fact even `mpf_class' etc are `typedef' specializations of 1437`__gmp_expr'. 1438 1439 Next we define assignment of `__gmp_expr' to `mpf_class'. 1440 1441 template <class T> 1442 mpf_class & mpf_class::operator=(const __gmp_expr<T> &expr) 1443 { 1444 expr.eval(this->get_mpf_t(), this->precision()); 1445 return *this; 1446 } 1447 1448 template <class Op> 1449 void __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, Op> >::eval 1450 (mpf_t f, mp_bitcnt_t precision) 1451 { 1452 Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t()); 1453 } 1454 1455 where `expr.val1' and `expr.val2' are references to the expression's 1456operands (here `expr' is the `__gmp_binary_expr' stored within the 1457`__gmp_expr'). 1458 1459 This way, the expression is actually evaluated only at the time of 1460assignment, when the required precision (that of `f') is known. 1461Furthermore the target `mpf_t' is now available, thus we can call 1462`mpf_add' directly with `f' as the output argument. 1463 1464 Compound expressions are handled by defining operators taking 1465subexpressions as their arguments, like this: 1466 1467 template <class T, class U> 1468 __gmp_expr 1469 <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> > 1470 operator+(const __gmp_expr<T> &expr1, const __gmp_expr<U> &expr2) 1471 { 1472 return __gmp_expr 1473 <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> > 1474 (expr1, expr2); 1475 } 1476 1477 And the corresponding specializations of `__gmp_expr::eval': 1478 1479 template <class T, class U, class Op> 1480 void __gmp_expr 1481 <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, Op> >::eval 1482 (mpf_t f, mp_bitcnt_t precision) 1483 { 1484 // declare two temporaries 1485 mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision); 1486 Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t()); 1487 } 1488 1489 The expression is thus recursively evaluated to any level of 1490complexity and all subexpressions are evaluated to the precision of `f'. 1491 1492 1493File: gmp.info, Node: Contributors, Next: References, Prev: Internals, Up: Top 1494 1495Appendix A Contributors 1496*********************** 1497 1498Torbj�rn Granlund wrote the original GMP library and is still the main 1499developer. Code not explicitly attributed to others, was contributed by 1500Torbj�rn. Several other individuals and organizations have contributed 1501GMP. Here is a list in chronological order on first contribution: 1502 1503 Gunnar Sj�din and Hans Riesel helped with mathematical problems in 1504early versions of the library. 1505 1506 Richard Stallman helped with the interface design and revised the 1507first version of this manual. 1508 1509 Brian Beuning and Doug Lea helped with testing of early versions of 1510the library and made creative suggestions. 1511 1512 John Amanatides of York University in Canada contributed the function 1513`mpz_probab_prime_p'. 1514 1515 Paul Zimmermann wrote the REDC-based mpz_powm code, the 1516Sch�nhage-Strassen FFT multiply code, and the Karatsuba square root 1517code. He also improved the Toom3 code for GMP 4.2. Paul sparked the 1518development of GMP 2, with his comparisons between bignum packages. 1519The ECMNET project Paul is organizing was a driving force behind many 1520of the optimizations in GMP 3. Paul also wrote the new GMP 4.3 nth 1521root code (with Torbj�rn). 1522 1523 Ken Weber (Kent State University, Universidade Federal do Rio Grande 1524do Sul) contributed now defunct versions of `mpz_gcd', `mpz_divexact', 1525`mpn_gcd', and `mpn_bdivmod', partially supported by CNPq (Brazil) 1526grant 301314194-2. 1527 1528 Per Bothner of Cygnus Support helped to set up GMP to use Cygnus' 1529configure. He has also made valuable suggestions and tested numerous 1530intermediary releases. 1531 1532 Joachim Hollman was involved in the design of the `mpf' interface, 1533and in the `mpz' design revisions for version 2. 1534 1535 Bennet Yee contributed the initial versions of `mpz_jacobi' and 1536`mpz_legendre'. 1537 1538 Andreas Schwab contributed the files `mpn/m68k/lshift.S' and 1539`mpn/m68k/rshift.S' (now in `.asm' form). 1540 1541 Robert Harley of Inria, France and David Seal of ARM, England, 1542suggested clever improvements for population count. Robert also wrote 1543highly optimized Karatsuba and 3-way Toom multiplication functions for 1544GMP 3, and contributed the ARM assembly code. 1545 1546 Torsten Ekedahl of the Mathematical department of Stockholm 1547University provided significant inspiration during several phases of 1548the GMP development. His mathematical expertise helped improve several 1549algorithms. 1550 1551 Linus Nordberg wrote the new configure system based on autoconf and 1552implemented the new random functions. 1553 1554 Kevin Ryde worked on a large number of things: optimized x86 code, 1555m4 asm macros, parameter tuning, speed measuring, the configure system, 1556function inlining, divisibility tests, bit scanning, Jacobi symbols, 1557Fibonacci and Lucas number functions, printf and scanf functions, perl 1558interface, demo expression parser, the algorithms chapter in the 1559manual, `gmpasm-mode.el', and various miscellaneous improvements 1560elsewhere. 1561 1562 Kent Boortz made the Mac OS 9 port. 1563 1564 Steve Root helped write the optimized alpha 21264 assembly code. 1565 1566 Gerardo Ballabio wrote the `gmpxx.h' C++ class interface and the C++ 1567`istream' input routines. 1568 1569 Jason Moxham rewrote `mpz_fac_ui'. 1570 1571 Pedro Gimeno implemented the Mersenne Twister and made other random 1572number improvements. 1573 1574 Niels M�ller wrote the sub-quadratic GCD and extended GCD code, the 1575quadratic Hensel division code, and (with Torbj�rn) the new divide and 1576conquer division code for GMP 4.3. Niels also helped implement the new 1577Toom multiply code for GMP 4.3 and implemented helper functions to 1578simplify Toom evaluations for GMP 5.0. He wrote the original version 1579of mpn_mulmod_bnm1. 1580 1581 Alberto Zanoni and Marco Bodrato suggested the unbalanced multiply 1582strategy, and found the optimal strategies for evaluation and 1583interpolation in Toom multiplication. 1584 1585 Marco Bodrato helped implement the new Toom multiply code for GMP 15864.3 and implemented most of the new Toom multiply and squaring code for 15875.0. He is the main author of the current mpn_mulmod_bnm1 and 1588mpn_mullo_n. Marco also wrote the functions mpn_invert and 1589mpn_invertappr. 1590 1591 David Harvey suggested the internal function `mpn_bdiv_dbm1', 1592implementing division relevant to Toom multiplication. He also worked 1593on fast assembly sequences, in particular on a fast AMD64 1594`mpn_mul_basecase'. 1595 1596 Martin Boij wrote `mpn_perfect_power_p'. 1597 1598 (This list is chronological, not ordered after significance. If you 1599have contributed to GMP but are not listed above, please tell 1600<[email protected]> about the omission!) 1601 1602 The development of floating point functions of GNU MP 2, were 1603supported in part by the ESPRIT-BRA (Basic Research Activities) 6846 1604project POSSO (POlynomial System SOlving). 1605 1606 The development of GMP 2, 3, and 4 was supported in part by the IDA 1607Center for Computing Sciences. 1608 1609 Thanks go to Hans Thorsen for donating an SGI system for the GMP 1610test system environment. 1611 1612 1613File: gmp.info, Node: References, Next: GNU Free Documentation License, Prev: Contributors, Up: Top 1614 1615Appendix B References 1616********************* 1617 1618B.1 Books 1619========= 1620 1621 * Jonathan M. Borwein and Peter B. Borwein, "Pi and the AGM: A Study 1622 in Analytic Number Theory and Computational Complexity", Wiley, 1623 1998. 1624 1625 * Richard Crandall and Carl Pomerance, "Prime Numbers: A 1626 Computational Perspective", 2nd edition, Springer-Verlag, 2005. 1627 `http://www.math.dartmouth.edu/~carlp/' 1628 1629 * Henri Cohen, "A Course in Computational Algebraic Number Theory", 1630 Graduate Texts in Mathematics number 138, Springer-Verlag, 1993. 1631 `http://www.math.u-bordeaux.fr/~cohen/' 1632 1633 * Donald E. Knuth, "The Art of Computer Programming", volume 2, 1634 "Seminumerical Algorithms", 3rd edition, Addison-Wesley, 1998. 1635 `http://www-cs-faculty.stanford.edu/~knuth/taocp.html' 1636 1637 * John D. Lipson, "Elements of Algebra and Algebraic Computing", The 1638 Benjamin Cummings Publishing Company Inc, 1981. 1639 1640 * Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, 1641 "Handbook of Applied Cryptography", 1642 `http://www.cacr.math.uwaterloo.ca/hac/' 1643 1644 * Richard M. Stallman and the GCC Developer Community, "Using the 1645 GNU Compiler Collection", Free Software Foundation, 2008, 1646 available online `http://gcc.gnu.org/onlinedocs/', and in the GCC 1647 package `ftp://ftp.gnu.org/gnu/gcc/' 1648 1649B.2 Papers 1650========== 1651 1652 * Yves Bertot, Nicolas Magaud and Paul Zimmermann, "A Proof of GMP 1653 Square Root", Journal of Automated Reasoning, volume 29, 2002, pp. 1654 225-252. Also available online as INRIA Research Report 4475, 1655 June 2002, `http://hal.inria.fr/docs/00/07/21/13/PDF/RR-4475.pdf' 1656 1657 * Christoph Burnikel and Joachim Ziegler, "Fast Recursive Division", 1658 Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022, 1659 `http://data.mpi-sb.mpg.de/internet/reports.nsf/NumberView/1998-1-022' 1660 1661 * Torbj�rn Granlund and Peter L. Montgomery, "Division by Invariant 1662 Integers using Multiplication", in Proceedings of the SIGPLAN 1663 PLDI'94 Conference, June 1994. Also available 1664 `http://gmplib.org/~tege/divcnst-pldi94.pdf'. 1665 1666 * Niels M�ller and Torbj�rn Granlund, "Improved division by invariant 1667 integers", IEEE Transactions on Computers, 11 June 2010. 1668 `http://gmplib.org/~tege/division-paper.pdf' 1669 1670 * Torbj�rn Granlund and Niels M�ller, "Division of integers large and 1671 small", to appear. 1672 1673 * Tudor Jebelean, "An algorithm for exact division", Journal of 1674 Symbolic Computation, volume 15, 1993, pp. 169-180. Research 1675 report version available 1676 `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz' 1677 1678 * Tudor Jebelean, "Exact Division with Karatsuba Complexity - 1679 Extended Abstract", RISC-Linz technical report 96-31, 1680 `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz' 1681 1682 * Tudor Jebelean, "Practical Integer Division with Karatsuba 1683 Complexity", ISSAC 97, pp. 339-341. Technical report available 1684 `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz' 1685 1686 * Tudor Jebelean, "A Generalization of the Binary GCD Algorithm", 1687 ISSAC 93, pp. 111-116. Technical report version available 1688 `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz' 1689 1690 * Tudor Jebelean, "A Double-Digit Lehmer-Euclid Algorithm for 1691 Finding the GCD of Long Integers", Journal of Symbolic 1692 Computation, volume 19, 1995, pp. 145-157. Technical report 1693 version also available 1694 `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz' 1695 1696 * Werner Krandick and Tudor Jebelean, "Bidirectional Exact Integer 1697 Division", Journal of Symbolic Computation, volume 21, 1996, pp. 1698 441-455. Early technical report version also available 1699 `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz' 1700 1701 * Makoto Matsumoto and Takuji Nishimura, "Mersenne Twister: A 1702 623-dimensionally equidistributed uniform pseudorandom number 1703 generator", ACM Transactions on Modelling and Computer Simulation, 1704 volume 8, January 1998, pp. 3-30. Available online 1705 `http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/ARTICLES/mt.ps.gz' 1706 (or .pdf) 1707 1708 * R. Moenck and A. Borodin, "Fast Modular Transforms via Division", 1709 Proceedings of the 13th Annual IEEE Symposium on Switching and 1710 Automata Theory, October 1972, pp. 90-96. Reprinted as "Fast 1711 Modular Transforms", Journal of Computer and System Sciences, 1712 volume 8, number 3, June 1974, pp. 366-386. 1713 1714 * Niels M�ller, "On Sch�nhage's algorithm and subquadratic integer 1715 GCD computation", in Mathematics of Computation, volume 77, 1716 January 2008, pp. 589-607. 1717 1718 * Peter L. Montgomery, "Modular Multiplication Without Trial 1719 Division", in Mathematics of Computation, volume 44, number 170, 1720 April 1985. 1721 1722 * Arnold Sch�nhage and Volker Strassen, "Schnelle Multiplikation 1723 grosser Zahlen", Computing 7, 1971, pp. 281-292. 1724 1725 * Kenneth Weber, "The accelerated integer GCD algorithm", ACM 1726 Transactions on Mathematical Software, volume 21, number 1, March 1727 1995, pp. 111-122. 1728 1729 * Paul Zimmermann, "Karatsuba Square Root", INRIA Research Report 1730 3805, November 1999, 1731 `http://hal.inria.fr/inria-00072854/PDF/RR-3805.pdf' 1732 1733 * Paul Zimmermann, "A Proof of GMP Fast Division and Square Root 1734 Implementations", 1735 `http://www.loria.fr/~zimmerma/papers/proof-div-sqrt.ps.gz' 1736 1737 * Dan Zuras, "On Squaring and Multiplying Large Integers", ARITH-11: 1738 IEEE Symposium on Computer Arithmetic, 1993, pp. 260 to 271. 1739 Reprinted as "More on Multiplying and Squaring Large Integers", 1740 IEEE Transactions on Computers, volume 43, number 8, August 1994, 1741 pp. 899-908. 1742 1743 1744File: gmp.info, Node: GNU Free Documentation License, Next: Concept Index, Prev: References, Up: Top 1745 1746Appendix C GNU Free Documentation License 1747***************************************** 1748 1749 Version 1.3, 3 November 2008 1750 1751 Copyright (C) 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc. 1752 `http://fsf.org/' 1753 1754 Everyone is permitted to copy and distribute verbatim copies 1755 of this license document, but changing it is not allowed. 1756 1757 0. 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Preserve in that license notice the full lists of Invariant 1967 Sections and required Cover Texts given in the Document's 1968 license notice. 1969 1970 H. Include an unaltered copy of this License. 1971 1972 I. Preserve the section Entitled "History", Preserve its Title, 1973 and add to it an item stating at least the title, year, new 1974 authors, and publisher of the Modified Version as given on 1975 the Title Page. If there is no section Entitled "History" in 1976 the Document, create one stating the title, year, authors, 1977 and publisher of the Document as given on its Title Page, 1978 then add an item describing the Modified Version as stated in 1979 the previous sentence. 1980 1981 J. Preserve the network location, if any, given in the Document 1982 for public access to a Transparent copy of the Document, and 1983 likewise the network locations given in the Document for 1984 previous versions it was based on. These may be placed in 1985 the "History" section. You may omit a network location for a 1986 work that was published at least four years before the 1987 Document itself, or if the original publisher of the version 1988 it refers to gives permission. 1989 1990 K. For any section Entitled "Acknowledgements" or "Dedications", 1991 Preserve the Title of the section, and preserve in the 1992 section all the substance and tone of each of the contributor 1993 acknowledgements and/or dedications given therein. 1994 1995 L. Preserve all the Invariant Sections of the Document, 1996 unaltered in their text and in their titles. Section numbers 1997 or the equivalent are not considered part of the section 1998 titles. 1999 2000 M. Delete any section Entitled "Endorsements". Such a section 2001 may not be included in the Modified Version. 2002 2003 N. Do not retitle any existing section to be Entitled 2004 "Endorsements" or to conflict in title with any Invariant 2005 Section. 2006 2007 O. Preserve any Warranty Disclaimers. 2008 2009 If the Modified Version includes new front-matter sections or 2010 appendices that qualify as Secondary Sections and contain no 2011 material copied from the Document, you may at your option 2012 designate some or all of these sections as invariant. To do this, 2013 add their titles to the list of Invariant Sections in the Modified 2014 Version's license notice. These titles must be distinct from any 2015 other section titles. 2016 2017 You may add a section Entitled "Endorsements", provided it contains 2018 nothing but endorsements of your Modified Version by various 2019 parties--for example, statements of peer review or that the text 2020 has been approved by an organization as the authoritative 2021 definition of a standard. 2022 2023 You may add a passage of up to five words as a Front-Cover Text, 2024 and a passage of up to 25 words as a Back-Cover Text, to the end 2025 of the list of Cover Texts in the Modified Version. Only one 2026 passage of Front-Cover Text and one of Back-Cover Text may be 2027 added by (or through arrangements made by) any one entity. If the 2028 Document already includes a cover text for the same cover, 2029 previously added by you or by arrangement made by the same entity 2030 you are acting on behalf of, you may not add another; but you may 2031 replace the old one, on explicit permission from the previous 2032 publisher that added the old one. 2033 2034 The author(s) and publisher(s) of the Document do not by this 2035 License give permission to use their names for publicity for or to 2036 assert or imply endorsement of any Modified Version. 2037 2038 5. COMBINING DOCUMENTS 2039 2040 You may combine the Document with other documents released under 2041 this License, under the terms defined in section 4 above for 2042 modified versions, provided that you include in the combination 2043 all of the Invariant Sections of all of the original documents, 2044 unmodified, and list them all as Invariant Sections of your 2045 combined work in its license notice, and that you preserve all 2046 their Warranty Disclaimers. 2047 2048 The combined work need only contain one copy of this License, and 2049 multiple identical Invariant Sections may be replaced with a single 2050 copy. If there are multiple Invariant Sections with the same name 2051 but different contents, make the title of each such section unique 2052 by adding at the end of it, in parentheses, the name of the 2053 original author or publisher of that section if known, or else a 2054 unique number. Make the same adjustment to the section titles in 2055 the list of Invariant Sections in the license notice of the 2056 combined work. 2057 2058 In the combination, you must combine any sections Entitled 2059 "History" in the various original documents, forming one section 2060 Entitled "History"; likewise combine any sections Entitled 2061 "Acknowledgements", and any sections Entitled "Dedications". You 2062 must delete all sections Entitled "Endorsements." 2063 2064 6. COLLECTIONS OF DOCUMENTS 2065 2066 You may make a collection consisting of the Document and other 2067 documents released under this License, and replace the individual 2068 copies of this License in the various documents with a single copy 2069 that is included in the collection, provided that you follow the 2070 rules of this License for verbatim copying of each of the 2071 documents in all other respects. 2072 2073 You may extract a single document from such a collection, and 2074 distribute it individually under this License, provided you insert 2075 a copy of this License into the extracted document, and follow 2076 this License in all other respects regarding verbatim copying of 2077 that document. 2078 2079 7. AGGREGATION WITH INDEPENDENT WORKS 2080 2081 A compilation of the Document or its derivatives with other 2082 separate and independent documents or works, in or on a volume of 2083 a storage or distribution medium, is called an "aggregate" if the 2084 copyright resulting from the compilation is not used to limit the 2085 legal rights of the compilation's users beyond what the individual 2086 works permit. When the Document is included in an aggregate, this 2087 License does not apply to the other works in the aggregate which 2088 are not themselves derivative works of the Document. 2089 2090 If the Cover Text requirement of section 3 is applicable to these 2091 copies of the Document, then if the Document is less than one half 2092 of the entire aggregate, the Document's Cover Texts may be placed 2093 on covers that bracket the Document within the aggregate, or the 2094 electronic equivalent of covers if the Document is in electronic 2095 form. Otherwise they must appear on printed covers that bracket 2096 the whole aggregate. 2097 2098 8. TRANSLATION 2099 2100 Translation is considered a kind of modification, so you may 2101 distribute translations of the Document under the terms of section 2102 4. Replacing Invariant Sections with translations requires special 2103 permission from their copyright holders, but you may include 2104 translations of some or all Invariant Sections in addition to the 2105 original versions of these Invariant Sections. You may include a 2106 translation of this License, and all the license notices in the 2107 Document, and any Warranty Disclaimers, provided that you also 2108 include the original English version of this License and the 2109 original versions of those notices and disclaimers. In case of a 2110 disagreement between the translation and the original version of 2111 this License or a notice or disclaimer, the original version will 2112 prevail. 2113 2114 If a section in the Document is Entitled "Acknowledgements", 2115 "Dedications", or "History", the requirement (section 4) to 2116 Preserve its Title (section 1) will typically require changing the 2117 actual title. 2118 2119 9. TERMINATION 2120 2121 You may not copy, modify, sublicense, or distribute the Document 2122 except as expressly provided under this License. Any attempt 2123 otherwise to copy, modify, sublicense, or distribute it is void, 2124 and will automatically terminate your rights under this License. 2125 2126 However, if you cease all violation of this License, then your 2127 license from a particular copyright holder is reinstated (a) 2128 provisionally, unless and until the copyright holder explicitly 2129 and finally terminates your license, and (b) permanently, if the 2130 copyright holder fails to notify you of the violation by some 2131 reasonable means prior to 60 days after the cessation. 2132 2133 Moreover, your license from a particular copyright holder is 2134 reinstated permanently if the copyright holder notifies you of the 2135 violation by some reasonable means, this is the first time you have 2136 received notice of violation of this License (for any work) from 2137 that copyright holder, and you cure the violation prior to 30 days 2138 after your receipt of the notice. 2139 2140 Termination of your rights under this section does not terminate 2141 the licenses of parties who have received copies or rights from 2142 you under this License. If your rights have been terminated and 2143 not permanently reinstated, receipt of a copy of some or all of 2144 the same material does not give you any rights to use it. 2145 2146 10. FUTURE REVISIONS OF THIS LICENSE 2147 2148 The Free Software Foundation may publish new, revised versions of 2149 the GNU Free Documentation License from time to time. Such new 2150 versions will be similar in spirit to the present version, but may 2151 differ in detail to address new problems or concerns. See 2152 `http://www.gnu.org/copyleft/'. 2153 2154 Each version of the License is given a distinguishing version 2155 number. If the Document specifies that a particular numbered 2156 version of this License "or any later version" applies to it, you 2157 have the option of following the terms and conditions either of 2158 that specified version or of any later version that has been 2159 published (not as a draft) by the Free Software Foundation. If 2160 the Document does not specify a version number of this License, 2161 you may choose any version ever published (not as a draft) by the 2162 Free Software Foundation. If the Document specifies that a proxy 2163 can decide which future versions of this License can be used, that 2164 proxy's public statement of acceptance of a version permanently 2165 authorizes you to choose that version for the Document. 2166 2167 11. RELICENSING 2168 2169 "Massive Multiauthor Collaboration Site" (or "MMC Site") means any 2170 World Wide Web server that publishes copyrightable works and also 2171 provides prominent facilities for anybody to edit those works. A 2172 public wiki that anybody can edit is an example of such a server. 2173 A "Massive Multiauthor Collaboration" (or "MMC") contained in the 2174 site means any set of copyrightable works thus published on the MMC 2175 site. 2176 2177 "CC-BY-SA" means the Creative Commons Attribution-Share Alike 3.0 2178 license published by Creative Commons Corporation, a not-for-profit 2179 corporation with a principal place of business in San Francisco, 2180 California, as well as future copyleft versions of that license 2181 published by that same organization. 2182 2183 "Incorporate" means to publish or republish a Document, in whole or 2184 in part, as part of another Document. 2185 2186 An MMC is "eligible for relicensing" if it is licensed under this 2187 License, and if all works that were first published under this 2188 License somewhere other than this MMC, and subsequently 2189 incorporated in whole or in part into the MMC, (1) had no cover 2190 texts or invariant sections, and (2) were thus incorporated prior 2191 to November 1, 2008. 2192 2193 The operator of an MMC Site may republish an MMC contained in the 2194 site under CC-BY-SA on the same site at any time before August 1, 2195 2009, provided the MMC is eligible for relicensing. 2196 2197 2198ADDENDUM: How to use this License for your documents 2199==================================================== 2200 2201To use this License in a document you have written, include a copy of 2202the License in the document and put the following copyright and license 2203notices just after the title page: 2204 2205 Copyright (C) YEAR YOUR NAME. 2206 Permission is granted to copy, distribute and/or modify this document 2207 under the terms of the GNU Free Documentation License, Version 1.3 2208 or any later version published by the Free Software Foundation; 2209 with no Invariant Sections, no Front-Cover Texts, and no Back-Cover 2210 Texts. A copy of the license is included in the section entitled ``GNU 2211 Free Documentation License''. 2212 2213 If you have Invariant Sections, Front-Cover Texts and Back-Cover 2214Texts, replace the "with...Texts." line with this: 2215 2216 with the Invariant Sections being LIST THEIR TITLES, with 2217 the Front-Cover Texts being LIST, and with the Back-Cover Texts 2218 being LIST. 2219 2220 If you have Invariant Sections without Cover Texts, or some other 2221combination of the three, merge those two alternatives to suit the 2222situation. 2223 2224 If your document contains nontrivial examples of program code, we 2225recommend releasing these examples in parallel under your choice of 2226free software license, such as the GNU General Public License, to 2227permit their use in free software. 2228 2229 2230File: gmp.info, Node: Concept Index, Next: Function Index, Prev: GNU Free Documentation License, Up: Top 2231 2232Concept Index 2233************* 2234 2235�[index�] 2236* Menu: 2237 2238* #include: Headers and Libraries. 2239 (line 6) 2240* --build: Build Options. (line 52) 2241* --disable-fft: Build Options. (line 317) 2242* --disable-shared: Build Options. (line 45) 2243* --disable-static: Build Options. (line 45) 2244* --enable-alloca: Build Options. (line 278) 2245* --enable-assert: Build Options. (line 328) 2246* --enable-cxx: Build Options. (line 230) 2247* --enable-fat: Build Options. (line 164) 2248* --enable-mpbsd: Build Options. (line 323) 2249* --enable-profiling <1>: Profiling. (line 6) 2250* --enable-profiling: Build Options. (line 332) 2251* --exec-prefix: Build Options. (line 32) 2252* --host: Build Options. (line 66) 2253* --prefix: Build Options. (line 32) 2254* -finstrument-functions: Profiling. (line 66) 2255* 2exp functions: Efficiency. (line 43) 2256* 68000: Notes for Particular Systems. 2257 (line 80) 2258* 80x86: Notes for Particular Systems. 2259 (line 126) 2260* ABI <1>: Build Options. (line 171) 2261* ABI: ABI and ISA. (line 6) 2262* About this manual: Introduction to GMP. (line 58) 2263* AC_CHECK_LIB: Autoconf. (line 11) 2264* AIX <1>: ABI and ISA. (line 169) 2265* AIX: Notes for Particular Systems. 2266 (line 7) 2267* Algorithms: Algorithms. (line 6) 2268* alloca: Build Options. (line 278) 2269* Allocation of memory: Custom Allocation. (line 6) 2270* AMD64: ABI and ISA. (line 44) 2271* Anonymous FTP of latest version: Introduction to GMP. (line 38) 2272* Application Binary Interface: ABI and ISA. (line 6) 2273* Arithmetic functions <1>: Float Arithmetic. (line 6) 2274* Arithmetic functions <2>: Integer Arithmetic. (line 6) 2275* Arithmetic functions: Rational Arithmetic. (line 6) 2276* ARM: Notes for Particular Systems. 2277 (line 20) 2278* Assembly cache handling: Assembly Cache Handling. 2279 (line 6) 2280* Assembly carry propagation: Assembly Carry Propagation. 2281 (line 6) 2282* Assembly code organisation: Assembly Code Organisation. 2283 (line 6) 2284* Assembly coding: Assembly Coding. (line 6) 2285* Assembly floating Point: Assembly Floating Point. 2286 (line 6) 2287* Assembly loop unrolling: Assembly Loop Unrolling. 2288 (line 6) 2289* Assembly SIMD: Assembly SIMD Instructions. 2290 (line 6) 2291* Assembly software pipelining: Assembly Software Pipelining. 2292 (line 6) 2293* Assembly writing guide: Assembly Writing Guide. 2294 (line 6) 2295* Assertion checking <1>: Debugging. (line 79) 2296* Assertion checking: Build Options. (line 328) 2297* Assignment functions <1>: Assigning Integers. (line 6) 2298* Assignment functions <2>: Simultaneous Float Init & Assign. 2299 (line 6) 2300* Assignment functions <3>: Assigning Floats. (line 6) 2301* Assignment functions <4>: Initializing Rationals. 2302 (line 6) 2303* Assignment functions: Simultaneous Integer Init & Assign. 2304 (line 6) 2305* Autoconf: Autoconf. (line 6) 2306* Basics: GMP Basics. (line 6) 2307* Berkeley MP compatible functions <1>: Build Options. (line 323) 2308* Berkeley MP compatible functions: BSD Compatible Functions. 2309 (line 6) 2310* Binomial coefficient algorithm: Binomial Coefficients Algorithm. 2311 (line 6) 2312* Binomial coefficient functions: Number Theoretic Functions. 2313 (line 113) 2314* Binutils strip: Known Build Problems. 2315 (line 28) 2316* Bit manipulation functions: Integer Logic and Bit Fiddling. 2317 (line 6) 2318* Bit scanning functions: Integer Logic and Bit Fiddling. 2319 (line 38) 2320* Bit shift left: Integer Arithmetic. (line 35) 2321* Bit shift right: Integer Division. (line 53) 2322* Bits per limb: Useful Macros and Constants. 2323 (line 7) 2324* BSD MP compatible functions <1>: BSD Compatible Functions. 2325 (line 6) 2326* BSD MP compatible functions: Build Options. (line 323) 2327* Bug reporting: Reporting Bugs. (line 6) 2328* Build directory: Build Options. (line 19) 2329* Build notes for binary packaging: Notes for Package Builds. 2330 (line 6) 2331* Build notes for particular systems: Notes for Particular Systems. 2332 (line 6) 2333* Build options: Build Options. (line 6) 2334* Build problems known: Known Build Problems. 2335 (line 6) 2336* Build system: Build Options. (line 52) 2337* Building GMP: Installing GMP. (line 6) 2338* Bus error: Debugging. (line 7) 2339* C compiler: Build Options. (line 182) 2340* C++ compiler: Build Options. (line 254) 2341* C++ interface: C++ Class Interface. (line 6) 2342* C++ interface internals: C++ Interface Internals. 2343 (line 6) 2344* C++ istream input: C++ Formatted Input. (line 6) 2345* C++ ostream output: C++ Formatted Output. 2346 (line 6) 2347* C++ support: Build Options. (line 230) 2348* CC: Build Options. (line 182) 2349* CC_FOR_BUILD: Build Options. (line 217) 2350* CFLAGS: Build Options. (line 182) 2351* Checker: Debugging. (line 115) 2352* checkergcc: Debugging. (line 122) 2353* Code organisation: Assembly Code Organisation. 2354 (line 6) 2355* Compaq C++: Notes for Particular Systems. 2356 (line 25) 2357* Comparison functions <1>: Float Comparison. (line 6) 2358* Comparison functions <2>: Integer Comparisons. (line 6) 2359* Comparison functions: Comparing Rationals. (line 6) 2360* Compatibility with older versions: Compatibility with older versions. 2361 (line 6) 2362* Conditions for copying GNU MP: Copying. (line 6) 2363* Configuring GMP: Installing GMP. (line 6) 2364* Congruence algorithm: Exact Remainder. (line 29) 2365* Congruence functions: Integer Division. (line 124) 2366* Constants: Useful Macros and Constants. 2367 (line 6) 2368* Contributors: Contributors. (line 6) 2369* Conventions for parameters: Parameter Conventions. 2370 (line 6) 2371* Conventions for variables: Variable Conventions. 2372 (line 6) 2373* Conversion functions <1>: Rational Conversions. 2374 (line 6) 2375* Conversion functions <2>: Converting Integers. (line 6) 2376* Conversion functions: Converting Floats. (line 6) 2377* Copying conditions: Copying. (line 6) 2378* CPPFLAGS: Build Options. (line 208) 2379* CPU types <1>: Introduction to GMP. (line 24) 2380* CPU types: Build Options. (line 108) 2381* Cross compiling: Build Options. (line 66) 2382* Custom allocation: Custom Allocation. (line 6) 2383* CXX: Build Options. (line 254) 2384* CXXFLAGS: Build Options. (line 254) 2385* Cygwin: Notes for Particular Systems. 2386 (line 43) 2387* Darwin: Known Build Problems. 2388 (line 51) 2389* Debugging: Debugging. (line 6) 2390* Demonstration programs: Demonstration Programs. 2391 (line 6) 2392* Digits in an integer: Miscellaneous Integer Functions. 2393 (line 23) 2394* Divisibility algorithm: Exact Remainder. (line 29) 2395* Divisibility functions: Integer Division. (line 112) 2396* Divisibility testing: Efficiency. (line 91) 2397* Division algorithms: Division Algorithms. (line 6) 2398* Division functions <1>: Float Arithmetic. (line 33) 2399* Division functions <2>: Rational Arithmetic. (line 22) 2400* Division functions: Integer Division. (line 6) 2401* DJGPP <1>: Notes for Particular Systems. 2402 (line 43) 2403* DJGPP: Known Build Problems. 2404 (line 18) 2405* DLLs: Notes for Particular Systems. 2406 (line 56) 2407* DocBook: Build Options. (line 355) 2408* Documentation formats: Build Options. (line 348) 2409* Documentation license: GNU Free Documentation License. 2410 (line 6) 2411* DVI: Build Options. (line 351) 2412* Efficiency: Efficiency. (line 6) 2413* Emacs: Emacs. (line 6) 2414* Exact division functions: Integer Division. (line 102) 2415* Exact remainder: Exact Remainder. (line 6) 2416* Example programs: Demonstration Programs. 2417 (line 6) 2418* Exec prefix: Build Options. (line 32) 2419* Execution profiling <1>: Profiling. (line 6) 2420* Execution profiling: Build Options. (line 332) 2421* Exponentiation functions <1>: Integer Exponentiation. 2422 (line 6) 2423* Exponentiation functions: Float Arithmetic. (line 41) 2424* Export: Integer Import and Export. 2425 (line 45) 2426* Expression parsing demo: Demonstration Programs. 2427 (line 15) 2428* Extended GCD: Number Theoretic Functions. 2429 (line 47) 2430* Factor removal functions: Number Theoretic Functions. 2431 (line 103) 2432* Factorial algorithm: Factorial Algorithm. (line 6) 2433* Factorial functions: Number Theoretic Functions. 2434 (line 108) 2435* Factorization demo: Demonstration Programs. 2436 (line 25) 2437* Fast Fourier Transform: FFT Multiplication. (line 6) 2438* Fat binary: Build Options. (line 164) 2439* FFT multiplication <1>: Build Options. (line 317) 2440* FFT multiplication: FFT Multiplication. (line 6) 2441* Fibonacci number algorithm: Fibonacci Numbers Algorithm. 2442 (line 6) 2443* Fibonacci sequence functions: Number Theoretic Functions. 2444 (line 121) 2445* Float arithmetic functions: Float Arithmetic. (line 6) 2446* Float assignment functions <1>: Simultaneous Float Init & Assign. 2447 (line 6) 2448* Float assignment functions: Assigning Floats. (line 6) 2449* Float comparison functions: Float Comparison. (line 6) 2450* Float conversion functions: Converting Floats. (line 6) 2451* Float functions: Floating-point Functions. 2452 (line 6) 2453* Float initialization functions <1>: Simultaneous Float Init & Assign. 2454 (line 6) 2455* Float initialization functions: Initializing Floats. (line 6) 2456* Float input and output functions: I/O of Floats. (line 6) 2457* Float internals: Float Internals. (line 6) 2458* Float miscellaneous functions: Miscellaneous Float Functions. 2459 (line 6) 2460* Float random number functions: Miscellaneous Float Functions. 2461 (line 27) 2462* Float rounding functions: Miscellaneous Float Functions. 2463 (line 9) 2464* Float sign tests: Float Comparison. (line 33) 2465* Floating point mode: Notes for Particular Systems. 2466 (line 34) 2467* Floating-point functions: Floating-point Functions. 2468 (line 6) 2469* Floating-point number: Nomenclature and Types. 2470 (line 21) 2471* fnccheck: Profiling. (line 77) 2472* Formatted input: Formatted Input. (line 6) 2473* Formatted output: Formatted Output. (line 6) 2474* Free Documentation License: GNU Free Documentation License. 2475 (line 6) 2476* frexp <1>: Converting Floats. (line 23) 2477* frexp: Converting Integers. (line 42) 2478* FTP of latest version: Introduction to GMP. (line 38) 2479* Function classes: Function Classes. (line 6) 2480* FunctionCheck: Profiling. (line 77) 2481* GCC Checker: Debugging. (line 115) 2482* GCD algorithms: Greatest Common Divisor Algorithms. 2483 (line 6) 2484* GCD extended: Number Theoretic Functions. 2485 (line 47) 2486* GCD functions: Number Theoretic Functions. 2487 (line 30) 2488* GDB: Debugging. (line 58) 2489* Generic C: Build Options. (line 153) 2490* GMP Perl module: Demonstration Programs. 2491 (line 35) 2492* GMP version number: Useful Macros and Constants. 2493 (line 12) 2494* gmp.h: Headers and Libraries. 2495 (line 6) 2496* gmpxx.h: C++ Interface General. 2497 (line 8) 2498* GNU Debugger: Debugging. (line 58) 2499* GNU Free Documentation License: GNU Free Documentation License. 2500 (line 6) 2501* GNU strip: Known Build Problems. 2502 (line 28) 2503* gprof: Profiling. (line 41) 2504* Greatest common divisor algorithms: Greatest Common Divisor Algorithms. 2505 (line 6) 2506* Greatest common divisor functions: Number Theoretic Functions. 2507 (line 30) 2508* Hardware floating point mode: Notes for Particular Systems. 2509 (line 34) 2510* Headers: Headers and Libraries. 2511 (line 6) 2512* Heap problems: Debugging. (line 24) 2513* Home page: Introduction to GMP. (line 34) 2514* Host system: Build Options. (line 66) 2515* HP-UX: ABI and ISA. (line 107) 2516* HPPA: ABI and ISA. (line 68) 2517* I/O functions <1>: I/O of Floats. (line 6) 2518* I/O functions <2>: I/O of Integers. (line 6) 2519* I/O functions: I/O of Rationals. (line 6) 2520* i386: Notes for Particular Systems. 2521 (line 126) 2522* IA-64: ABI and ISA. (line 107) 2523* Import: Integer Import and Export. 2524 (line 11) 2525* In-place operations: Efficiency. (line 57) 2526* Include files: Headers and Libraries. 2527 (line 6) 2528* info-lookup-symbol: Emacs. (line 6) 2529* Initialization functions <1>: Initializing Integers. 2530 (line 6) 2531* Initialization functions <2>: Random State Initialization. 2532 (line 6) 2533* Initialization functions <3>: Initializing Rationals. 2534 (line 6) 2535* Initialization functions <4>: Initializing Floats. (line 6) 2536* Initialization functions <5>: Simultaneous Float Init & Assign. 2537 (line 6) 2538* Initialization functions: Simultaneous Integer Init & Assign. 2539 (line 6) 2540* Initializing and clearing: Efficiency. (line 21) 2541* Input functions <1>: I/O of Floats. (line 6) 2542* Input functions <2>: I/O of Rationals. (line 6) 2543* Input functions <3>: I/O of Integers. (line 6) 2544* Input functions: Formatted Input Functions. 2545 (line 6) 2546* Install prefix: Build Options. (line 32) 2547* Installing GMP: Installing GMP. (line 6) 2548* Instruction Set Architecture: ABI and ISA. (line 6) 2549* instrument-functions: Profiling. (line 66) 2550* Integer: Nomenclature and Types. 2551 (line 6) 2552* Integer arithmetic functions: Integer Arithmetic. (line 6) 2553* Integer assignment functions <1>: Assigning Integers. (line 6) 2554* Integer assignment functions: Simultaneous Integer Init & Assign. 2555 (line 6) 2556* Integer bit manipulation functions: Integer Logic and Bit Fiddling. 2557 (line 6) 2558* Integer comparison functions: Integer Comparisons. (line 6) 2559* Integer conversion functions: Converting Integers. (line 6) 2560* Integer division functions: Integer Division. (line 6) 2561* Integer exponentiation functions: Integer Exponentiation. 2562 (line 6) 2563* Integer export: Integer Import and Export. 2564 (line 45) 2565* Integer functions: Integer Functions. (line 6) 2566* Integer import: Integer Import and Export. 2567 (line 11) 2568* Integer initialization functions <1>: Initializing Integers. 2569 (line 6) 2570* Integer initialization functions: Simultaneous Integer Init & Assign. 2571 (line 6) 2572* Integer input and output functions: I/O of Integers. (line 6) 2573* Integer internals: Integer Internals. (line 6) 2574* Integer logical functions: Integer Logic and Bit Fiddling. 2575 (line 6) 2576* Integer miscellaneous functions: Miscellaneous Integer Functions. 2577 (line 6) 2578* Integer random number functions: Integer Random Numbers. 2579 (line 6) 2580* Integer root functions: Integer Roots. (line 6) 2581* Integer sign tests: Integer Comparisons. (line 28) 2582* Integer special functions: Integer Special Functions. 2583 (line 6) 2584* Interix: Notes for Particular Systems. 2585 (line 51) 2586* Internals: Internals. (line 6) 2587* Introduction: Introduction to GMP. (line 6) 2588* Inverse modulo functions: Number Theoretic Functions. 2589 (line 72) 2590* IRIX <1>: Known Build Problems. 2591 (line 38) 2592* IRIX: ABI and ISA. (line 132) 2593* ISA: ABI and ISA. (line 6) 2594* istream input: C++ Formatted Input. (line 6) 2595* Jacobi symbol algorithm: Jacobi Symbol. (line 6) 2596* Jacobi symbol functions: Number Theoretic Functions. 2597 (line 79) 2598* Karatsuba multiplication: Karatsuba Multiplication. 2599 (line 6) 2600* Karatsuba square root algorithm: Square Root Algorithm. 2601 (line 6) 2602* Kronecker symbol functions: Number Theoretic Functions. 2603 (line 91) 2604* Language bindings: Language Bindings. (line 6) 2605* Latest version of GMP: Introduction to GMP. (line 38) 2606* LCM functions: Number Theoretic Functions. 2607 (line 67) 2608* Least common multiple functions: Number Theoretic Functions. 2609 (line 67) 2610* Legendre symbol functions: Number Theoretic Functions. 2611 (line 82) 2612* libgmp: Headers and Libraries. 2613 (line 22) 2614* libgmpxx: Headers and Libraries. 2615 (line 27) 2616* Libraries: Headers and Libraries. 2617 (line 22) 2618* Libtool: Headers and Libraries. 2619 (line 33) 2620* Libtool versioning: Notes for Package Builds. 2621 (line 9) 2622* License conditions: Copying. (line 6) 2623* Limb: Nomenclature and Types. 2624 (line 31) 2625* Limb size: Useful Macros and Constants. 2626 (line 7) 2627* Linear congruential algorithm: Random Number Algorithms. 2628 (line 25) 2629* Linear congruential random numbers: Random State Initialization. 2630 (line 18) 2631* Linking: Headers and Libraries. 2632 (line 22) 2633* Logical functions: Integer Logic and Bit Fiddling. 2634 (line 6) 2635* Low-level functions: Low-level Functions. (line 6) 2636* Lucas number algorithm: Lucas Numbers Algorithm. 2637 (line 6) 2638* Lucas number functions: Number Theoretic Functions. 2639 (line 132) 2640* MacOS X: Known Build Problems. 2641 (line 51) 2642* Mailing lists: Introduction to GMP. (line 45) 2643* Malloc debugger: Debugging. (line 30) 2644* Malloc problems: Debugging. (line 24) 2645* Memory allocation: Custom Allocation. (line 6) 2646* Memory management: Memory Management. (line 6) 2647* Mersenne twister algorithm: Random Number Algorithms. 2648 (line 17) 2649* Mersenne twister random numbers: Random State Initialization. 2650 (line 13) 2651* MINGW: Notes for Particular Systems. 2652 (line 43) 2653* MIPS: ABI and ISA. (line 132) 2654* Miscellaneous float functions: Miscellaneous Float Functions. 2655 (line 6) 2656* Miscellaneous integer functions: Miscellaneous Integer Functions. 2657 (line 6) 2658* MMX: Notes for Particular Systems. 2659 (line 132) 2660* Modular inverse functions: Number Theoretic Functions. 2661 (line 72) 2662* Most significant bit: Miscellaneous Integer Functions. 2663 (line 34) 2664* mp.h: BSD Compatible Functions. 2665 (line 21) 2666* MPN_PATH: Build Options. (line 336) 2667* MS Windows: Notes for Particular Systems. 2668 (line 43) 2669* MS-DOS: Notes for Particular Systems. 2670 (line 43) 2671* Multi-threading: Reentrancy. (line 6) 2672* Multiplication algorithms: Multiplication Algorithms. 2673 (line 6) 2674* Nails: Low-level Functions. (line 485) 2675* Native compilation: Build Options. (line 52) 2676* NeXT: Known Build Problems. 2677 (line 57) 2678* Next prime function: Number Theoretic Functions. 2679 (line 23) 2680* Nomenclature: Nomenclature and Types. 2681 (line 6) 2682* Non-Unix systems: Build Options. (line 11) 2683* Nth root algorithm: Nth Root Algorithm. (line 6) 2684* Number sequences: Efficiency. (line 147) 2685* Number theoretic functions: Number Theoretic Functions. 2686 (line 6) 2687* Numerator and denominator: Applying Integer Functions. 2688 (line 6) 2689* obstack output: Formatted Output Functions. 2690 (line 81) 2691* OpenBSD: Notes for Particular Systems. 2692 (line 86) 2693* Optimizing performance: Performance optimization. 2694 (line 6) 2695* ostream output: C++ Formatted Output. 2696 (line 6) 2697* Other languages: Language Bindings. (line 6) 2698* Output functions <1>: I/O of Integers. (line 6) 2699* Output functions <2>: I/O of Rationals. (line 6) 2700* Output functions <3>: Formatted Output Functions. 2701 (line 6) 2702* Output functions: I/O of Floats. (line 6) 2703* Packaged builds: Notes for Package Builds. 2704 (line 6) 2705* Parameter conventions: Parameter Conventions. 2706 (line 6) 2707* Parsing expressions demo: Demonstration Programs. 2708 (line 21) 2709* Particular systems: Notes for Particular Systems. 2710 (line 6) 2711* Past GMP versions: Compatibility with older versions. 2712 (line 6) 2713* PDF: Build Options. (line 351) 2714* Perfect power algorithm: Perfect Power Algorithm. 2715 (line 6) 2716* Perfect power functions: Integer Roots. (line 27) 2717* Perfect square algorithm: Perfect Square Algorithm. 2718 (line 6) 2719* Perfect square functions: Integer Roots. (line 36) 2720* perl: Demonstration Programs. 2721 (line 35) 2722* Perl module: Demonstration Programs. 2723 (line 35) 2724* Postscript: Build Options. (line 351) 2725* Power/PowerPC <1>: Known Build Problems. 2726 (line 63) 2727* Power/PowerPC: Notes for Particular Systems. 2728 (line 92) 2729* Powering algorithms: Powering Algorithms. (line 6) 2730* Powering functions <1>: Float Arithmetic. (line 41) 2731* Powering functions: Integer Exponentiation. 2732 (line 6) 2733* PowerPC: ABI and ISA. (line 167) 2734* Precision of floats: Floating-point Functions. 2735 (line 6) 2736* Precision of hardware floating point: Notes for Particular Systems. 2737 (line 34) 2738* Prefix: Build Options. (line 32) 2739* Prime testing algorithms: Prime Testing Algorithm. 2740 (line 6) 2741* Prime testing functions: Number Theoretic Functions. 2742 (line 7) 2743* printf formatted output: Formatted Output. (line 6) 2744* Probable prime testing functions: Number Theoretic Functions. 2745 (line 7) 2746* prof: Profiling. (line 24) 2747* Profiling: Profiling. (line 6) 2748* Radix conversion algorithms: Radix Conversion Algorithms. 2749 (line 6) 2750* Random number algorithms: Random Number Algorithms. 2751 (line 6) 2752* Random number functions <1>: Random Number Functions. 2753 (line 6) 2754* Random number functions <2>: Miscellaneous Float Functions. 2755 (line 27) 2756* Random number functions: Integer Random Numbers. 2757 (line 6) 2758* Random number seeding: Random State Seeding. 2759 (line 6) 2760* Random number state: Random State Initialization. 2761 (line 6) 2762* Random state: Nomenclature and Types. 2763 (line 46) 2764* Rational arithmetic: Efficiency. (line 113) 2765* Rational arithmetic functions: Rational Arithmetic. (line 6) 2766* Rational assignment functions: Initializing Rationals. 2767 (line 6) 2768* Rational comparison functions: Comparing Rationals. (line 6) 2769* Rational conversion functions: Rational Conversions. 2770 (line 6) 2771* Rational initialization functions: Initializing Rationals. 2772 (line 6) 2773* Rational input and output functions: I/O of Rationals. (line 6) 2774* Rational internals: Rational Internals. (line 6) 2775* Rational number: Nomenclature and Types. 2776 (line 16) 2777* Rational number functions: Rational Number Functions. 2778 (line 6) 2779* Rational numerator and denominator: Applying Integer Functions. 2780 (line 6) 2781* Rational sign tests: Comparing Rationals. (line 27) 2782* Raw output internals: Raw Output Internals. 2783 (line 6) 2784* Reallocations: Efficiency. (line 30) 2785* Reentrancy: Reentrancy. (line 6) 2786* References: References. (line 6) 2787* Remove factor functions: Number Theoretic Functions. 2788 (line 103) 2789* Reporting bugs: Reporting Bugs. (line 6) 2790* Root extraction algorithm: Nth Root Algorithm. (line 6) 2791* Root extraction algorithms: Root Extraction Algorithms. 2792 (line 6) 2793* Root extraction functions <1>: Float Arithmetic. (line 37) 2794* Root extraction functions: Integer Roots. (line 6) 2795* Root testing functions: Integer Roots. (line 27) 2796* Rounding functions: Miscellaneous Float Functions. 2797 (line 9) 2798* Sample programs: Demonstration Programs. 2799 (line 6) 2800* Scan bit functions: Integer Logic and Bit Fiddling. 2801 (line 38) 2802* scanf formatted input: Formatted Input. (line 6) 2803* SCO: Known Build Problems. 2804 (line 38) 2805* Seeding random numbers: Random State Seeding. 2806 (line 6) 2807* Segmentation violation: Debugging. (line 7) 2808* Sequent Symmetry: Known Build Problems. 2809 (line 68) 2810* Services for Unix: Notes for Particular Systems. 2811 (line 51) 2812* Shared library versioning: Notes for Package Builds. 2813 (line 9) 2814* Sign tests <1>: Integer Comparisons. (line 28) 2815* Sign tests <2>: Comparing Rationals. (line 27) 2816* Sign tests: Float Comparison. (line 33) 2817* Size in digits: Miscellaneous Integer Functions. 2818 (line 23) 2819* Small operands: Efficiency. (line 7) 2820* Solaris <1>: Known Build Problems. 2821 (line 78) 2822* Solaris: ABI and ISA. (line 201) 2823* Sparc: Notes for Particular Systems. 2824 (line 103) 2825* Sparc V9: ABI and ISA. (line 201) 2826* Special integer functions: Integer Special Functions. 2827 (line 6) 2828* Square root algorithm: Square Root Algorithm. 2829 (line 6) 2830* SSE2: Notes for Particular Systems. 2831 (line 132) 2832* Stack backtrace: Debugging. (line 50) 2833* Stack overflow <1>: Build Options. (line 278) 2834* Stack overflow: Debugging. (line 7) 2835* Static linking: Efficiency. (line 14) 2836* stdarg.h: Headers and Libraries. 2837 (line 17) 2838* stdio.h: Headers and Libraries. 2839 (line 11) 2840* Stripped libraries: Known Build Problems. 2841 (line 28) 2842* Sun: ABI and ISA. (line 201) 2843* SunOS: Notes for Particular Systems. 2844 (line 120) 2845* Systems: Notes for Particular Systems. 2846 (line 6) 2847* Temporary memory: Build Options. (line 278) 2848* Texinfo: Build Options. (line 348) 2849* Text input/output: Efficiency. (line 153) 2850* Thread safety: Reentrancy. (line 6) 2851* Toom multiplication <1>: Other Multiplication. 2852 (line 6) 2853* Toom multiplication <2>: Toom 3-Way Multiplication. 2854 (line 6) 2855* Toom multiplication <3>: Toom 4-Way Multiplication. 2856 (line 6) 2857* Toom multiplication: Higher degree Toom'n'half. 2858 (line 6) 2859* Types: Nomenclature and Types. 2860 (line 6) 2861* ui and si functions: Efficiency. (line 50) 2862* Unbalanced multiplication: Unbalanced Multiplication. 2863 (line 6) 2864* Upward compatibility: Compatibility with older versions. 2865 (line 6) 2866* Useful macros and constants: Useful Macros and Constants. 2867 (line 6) 2868* User-defined precision: Floating-point Functions. 2869 (line 6) 2870* Valgrind: Debugging. (line 130) 2871* Variable conventions: Variable Conventions. 2872 (line 6) 2873* Version number: Useful Macros and Constants. 2874 (line 12) 2875* Web page: Introduction to GMP. (line 34) 2876* Windows: Notes for Particular Systems. 2877 (line 43) 2878* x86: Notes for Particular Systems. 2879 (line 126) 2880* x87: Notes for Particular Systems. 2881 (line 34) 2882* XML: Build Options. (line 355) 2883 2884 2885File: gmp.info, Node: Function Index, Prev: Concept Index, Up: Top 2886 2887Function and Type Index 2888*********************** 2889 2890�[index�] 2891* Menu: 2892 2893* __GMP_CC: Useful Macros and Constants. 2894 (line 23) 2895* __GMP_CFLAGS: Useful Macros and Constants. 2896 (line 24) 2897* __GNU_MP_VERSION: Useful Macros and Constants. 2898 (line 10) 2899* __GNU_MP_VERSION_MINOR: Useful Macros and Constants. 2900 (line 11) 2901* __GNU_MP_VERSION_PATCHLEVEL: Useful Macros and Constants. 2902 (line 12) 2903* _mpz_realloc: Integer Special Functions. 2904 (line 51) 2905* abs <1>: C++ Interface Floats. 2906 (line 79) 2907* abs <2>: C++ Interface Rationals. 2908 (line 43) 2909* abs: C++ Interface Integers. 2910 (line 42) 2911* ceil: C++ Interface Floats. 2912 (line 80) 2913* cmp <1>: C++ Interface Floats. 2914 (line 81) 2915* cmp <2>: C++ Interface Integers. 2916 (line 43) 2917* cmp <3>: C++ Interface Floats. 2918 (line 82) 2919* cmp <4>: C++ Interface Rationals. 2920 (line 45) 2921* cmp: C++ Interface Integers. 2922 (line 44) 2923* floor: C++ Interface Floats. 2924 (line 89) 2925* gcd: BSD Compatible Functions. 2926 (line 82) 2927* gmp_asprintf: Formatted Output Functions. 2928 (line 65) 2929* gmp_errno: Random State Initialization. 2930 (line 55) 2931* GMP_ERROR_INVALID_ARGUMENT: Random State Initialization. 2932 (line 55) 2933* GMP_ERROR_UNSUPPORTED_ARGUMENT: Random State Initialization. 2934 (line 55) 2935* gmp_fprintf: Formatted Output Functions. 2936 (line 29) 2937* gmp_fscanf: Formatted Input Functions. 2938 (line 25) 2939* GMP_LIMB_BITS: Low-level Functions. (line 515) 2940* GMP_NAIL_BITS: Low-level Functions. (line 513) 2941* GMP_NAIL_MASK: Low-level Functions. (line 523) 2942* GMP_NUMB_BITS: Low-level Functions. (line 514) 2943* GMP_NUMB_MASK: Low-level Functions. (line 524) 2944* GMP_NUMB_MAX: Low-level Functions. (line 532) 2945* gmp_obstack_printf: Formatted Output Functions. 2946 (line 79) 2947* gmp_obstack_vprintf: Formatted Output Functions. 2948 (line 81) 2949* gmp_printf: Formatted Output Functions. 2950 (line 24) 2951* GMP_RAND_ALG_DEFAULT: Random State Initialization. 2952 (line 49) 2953* GMP_RAND_ALG_LC: Random State Initialization. 2954 (line 49) 2955* gmp_randclass: C++ Interface Random Numbers. 2956 (line 7) 2957* gmp_randclass::get_f: C++ Interface Random Numbers. 2958 (line 45) 2959* gmp_randclass::get_z_bits: C++ Interface Random Numbers. 2960 (line 38) 2961* gmp_randclass::get_z_range: C++ Interface Random Numbers. 2962 (line 42) 2963* gmp_randclass::gmp_randclass: C++ Interface Random Numbers. 2964 (line 27) 2965* gmp_randclass::seed: C++ Interface Random Numbers. 2966 (line 34) 2967* gmp_randclear: Random State Initialization. 2968 (line 62) 2969* gmp_randinit: Random State Initialization. 2970 (line 47) 2971* gmp_randinit_default: Random State Initialization. 2972 (line 7) 2973* gmp_randinit_lc_2exp: Random State Initialization. 2974 (line 18) 2975* gmp_randinit_lc_2exp_size: Random State Initialization. 2976 (line 32) 2977* gmp_randinit_mt: Random State Initialization. 2978 (line 13) 2979* gmp_randinit_set: Random State Initialization. 2980 (line 43) 2981* gmp_randseed: Random State Seeding. 2982 (line 7) 2983* gmp_randseed_ui: Random State Seeding. 2984 (line 9) 2985* gmp_randstate_t: Nomenclature and Types. 2986 (line 46) 2987* gmp_scanf: Formatted Input Functions. 2988 (line 21) 2989* gmp_snprintf: Formatted Output Functions. 2990 (line 46) 2991* gmp_sprintf: Formatted Output Functions. 2992 (line 34) 2993* gmp_sscanf: Formatted Input Functions. 2994 (line 29) 2995* gmp_urandomb_ui: Random State Miscellaneous. 2996 (line 8) 2997* gmp_urandomm_ui: Random State Miscellaneous. 2998 (line 14) 2999* gmp_vasprintf: Formatted Output Functions. 3000 (line 66) 3001* gmp_version: Useful Macros and Constants. 3002 (line 18) 3003* gmp_vfprintf: Formatted Output Functions. 3004 (line 30) 3005* gmp_vfscanf: Formatted Input Functions. 3006 (line 26) 3007* gmp_vprintf: Formatted Output Functions. 3008 (line 25) 3009* gmp_vscanf: Formatted Input Functions. 3010 (line 22) 3011* gmp_vsnprintf: Formatted Output Functions. 3012 (line 48) 3013* gmp_vsprintf: Formatted Output Functions. 3014 (line 35) 3015* gmp_vsscanf: Formatted Input Functions. 3016 (line 31) 3017* hypot: C++ Interface Floats. 3018 (line 90) 3019* itom: BSD Compatible Functions. 3020 (line 29) 3021* madd: BSD Compatible Functions. 3022 (line 43) 3023* mcmp: BSD Compatible Functions. 3024 (line 85) 3025* mdiv: BSD Compatible Functions. 3026 (line 53) 3027* mfree: BSD Compatible Functions. 3028 (line 105) 3029* min: BSD Compatible Functions. 3030 (line 89) 3031* MINT: BSD Compatible Functions. 3032 (line 21) 3033* mout: BSD Compatible Functions. 3034 (line 94) 3035* move: BSD Compatible Functions. 3036 (line 39) 3037* mp_bitcnt_t: Nomenclature and Types. 3038 (line 42) 3039* mp_bits_per_limb: Useful Macros and Constants. 3040 (line 7) 3041* mp_exp_t: Nomenclature and Types. 3042 (line 27) 3043* mp_get_memory_functions: Custom Allocation. (line 93) 3044* mp_limb_t: Nomenclature and Types. 3045 (line 31) 3046* mp_set_memory_functions: Custom Allocation. (line 21) 3047* mp_size_t: Nomenclature and Types. 3048 (line 37) 3049* mpf_abs: Float Arithmetic. (line 47) 3050* mpf_add: Float Arithmetic. (line 7) 3051* mpf_add_ui: Float Arithmetic. (line 9) 3052* mpf_ceil: Miscellaneous Float Functions. 3053 (line 7) 3054* mpf_class: C++ Interface General. 3055 (line 20) 3056* mpf_class::fits_sint_p: C++ Interface Floats. 3057 (line 83) 3058* mpf_class::fits_slong_p: C++ Interface Floats. 3059 (line 84) 3060* mpf_class::fits_sshort_p: C++ Interface Floats. 3061 (line 85) 3062* mpf_class::fits_uint_p: C++ Interface Floats. 3063 (line 86) 3064* mpf_class::fits_ulong_p: C++ Interface Floats. 3065 (line 87) 3066* mpf_class::fits_ushort_p: C++ Interface Floats. 3067 (line 88) 3068* mpf_class::get_d: C++ Interface Floats. 3069 (line 91) 3070* mpf_class::get_mpf_t: C++ Interface General. 3071 (line 66) 3072* mpf_class::get_prec: C++ Interface Floats. 3073 (line 109) 3074* mpf_class::get_si: C++ Interface Floats. 3075 (line 92) 3076* mpf_class::get_str: C++ Interface Floats. 3077 (line 94) 3078* mpf_class::get_ui: C++ Interface Floats. 3079 (line 95) 3080* mpf_class::mpf_class: C++ Interface Floats. 3081 (line 12) 3082* mpf_class::operator=: C++ Interface Floats. 3083 (line 56) 3084* mpf_class::set_prec: C++ Interface Floats. 3085 (line 110) 3086* mpf_class::set_prec_raw: C++ Interface Floats. 3087 (line 111) 3088* mpf_class::set_str: C++ Interface Floats. 3089 (line 97) 3090* mpf_clear: Initializing Floats. (line 37) 3091* mpf_clears: Initializing Floats. (line 41) 3092* mpf_cmp: Float Comparison. (line 7) 3093* mpf_cmp_d: Float Comparison. (line 8) 3094* mpf_cmp_si: Float Comparison. (line 10) 3095* mpf_cmp_ui: Float Comparison. (line 9) 3096* mpf_div: Float Arithmetic. (line 29) 3097* mpf_div_2exp: Float Arithmetic. (line 53) 3098* mpf_div_ui: Float Arithmetic. (line 33) 3099* mpf_eq: Float Comparison. (line 17) 3100* mpf_fits_sint_p: Miscellaneous Float Functions. 3101 (line 20) 3102* mpf_fits_slong_p: Miscellaneous Float Functions. 3103 (line 18) 3104* mpf_fits_sshort_p: Miscellaneous Float Functions. 3105 (line 22) 3106* mpf_fits_uint_p: Miscellaneous Float Functions. 3107 (line 19) 3108* mpf_fits_ulong_p: Miscellaneous Float Functions. 3109 (line 17) 3110* mpf_fits_ushort_p: Miscellaneous Float Functions. 3111 (line 21) 3112* mpf_floor: Miscellaneous Float Functions. 3113 (line 8) 3114* mpf_get_d: Converting Floats. (line 7) 3115* mpf_get_d_2exp: Converting Floats. (line 16) 3116* mpf_get_default_prec: Initializing Floats. (line 12) 3117* mpf_get_prec: Initializing Floats. (line 62) 3118* mpf_get_si: Converting Floats. (line 27) 3119* mpf_get_str: Converting Floats. (line 37) 3120* mpf_get_ui: Converting Floats. (line 28) 3121* mpf_init: Initializing Floats. (line 19) 3122* mpf_init2: Initializing Floats. (line 26) 3123* mpf_init_set: Simultaneous Float Init & Assign. 3124 (line 16) 3125* mpf_init_set_d: Simultaneous Float Init & Assign. 3126 (line 19) 3127* mpf_init_set_si: Simultaneous Float Init & Assign. 3128 (line 18) 3129* mpf_init_set_str: Simultaneous Float Init & Assign. 3130 (line 25) 3131* mpf_init_set_ui: Simultaneous Float Init & Assign. 3132 (line 17) 3133* mpf_inits: Initializing Floats. (line 31) 3134* mpf_inp_str: I/O of Floats. (line 39) 3135* mpf_integer_p: Miscellaneous Float Functions. 3136 (line 14) 3137* mpf_mul: Float Arithmetic. (line 19) 3138* mpf_mul_2exp: Float Arithmetic. (line 50) 3139* mpf_mul_ui: Float Arithmetic. (line 21) 3140* mpf_neg: Float Arithmetic. (line 44) 3141* mpf_out_str: I/O of Floats. (line 19) 3142* mpf_pow_ui: Float Arithmetic. (line 41) 3143* mpf_random2: Miscellaneous Float Functions. 3144 (line 37) 3145* mpf_reldiff: Float Comparison. (line 29) 3146* mpf_set: Assigning Floats. (line 10) 3147* mpf_set_d: Assigning Floats. (line 13) 3148* mpf_set_default_prec: Initializing Floats. (line 7) 3149* mpf_set_prec: Initializing Floats. (line 65) 3150* mpf_set_prec_raw: Initializing Floats. (line 72) 3151* mpf_set_q: Assigning Floats. (line 15) 3152* mpf_set_si: Assigning Floats. (line 12) 3153* mpf_set_str: Assigning Floats. (line 18) 3154* mpf_set_ui: Assigning Floats. (line 11) 3155* mpf_set_z: Assigning Floats. (line 14) 3156* mpf_sgn: Float Comparison. (line 33) 3157* mpf_sqrt: Float Arithmetic. (line 36) 3158* mpf_sqrt_ui: Float Arithmetic. (line 37) 3159* mpf_sub: Float Arithmetic. (line 12) 3160* mpf_sub_ui: Float Arithmetic. (line 16) 3161* mpf_swap: Assigning Floats. (line 52) 3162* mpf_t: Nomenclature and Types. 3163 (line 21) 3164* mpf_trunc: Miscellaneous Float Functions. 3165 (line 9) 3166* mpf_ui_div: Float Arithmetic. (line 31) 3167* mpf_ui_sub: Float Arithmetic. (line 14) 3168* mpf_urandomb: Miscellaneous Float Functions. 3169 (line 27) 3170* mpn_add: Low-level Functions. (line 69) 3171* mpn_add_1: Low-level Functions. (line 64) 3172* mpn_add_n: Low-level Functions. (line 54) 3173* mpn_addmul_1: Low-level Functions. (line 148) 3174* mpn_and_n: Low-level Functions. (line 427) 3175* mpn_andn_n: Low-level Functions. (line 442) 3176* mpn_cmp: Low-level Functions. (line 284) 3177* mpn_com: Low-level Functions. (line 467) 3178* mpn_copyd: Low-level Functions. (line 476) 3179* mpn_copyi: Low-level Functions. (line 472) 3180* mpn_divexact_by3: Low-level Functions. (line 229) 3181* mpn_divexact_by3c: Low-level Functions. (line 231) 3182* mpn_divmod: Low-level Functions. (line 224) 3183* mpn_divmod_1: Low-level Functions. (line 208) 3184* mpn_divrem: Low-level Functions. (line 182) 3185* mpn_divrem_1: Low-level Functions. (line 206) 3186* mpn_gcd: Low-level Functions. (line 289) 3187* mpn_gcd_1: Low-level Functions. (line 299) 3188* mpn_gcdext: Low-level Functions. (line 305) 3189* mpn_get_str: Low-level Functions. (line 352) 3190* mpn_hamdist: Low-level Functions. (line 416) 3191* mpn_ior_n: Low-level Functions. (line 432) 3192* mpn_iorn_n: Low-level Functions. (line 447) 3193* mpn_lshift: Low-level Functions. (line 260) 3194* mpn_mod_1: Low-level Functions. (line 255) 3195* mpn_mul: Low-level Functions. (line 114) 3196* mpn_mul_1: Low-level Functions. (line 133) 3197* mpn_mul_n: Low-level Functions. (line 103) 3198* mpn_nand_n: Low-level Functions. (line 452) 3199* mpn_neg: Low-level Functions. (line 98) 3200* mpn_nior_n: Low-level Functions. (line 457) 3201* mpn_perfect_square_p: Low-level Functions. (line 422) 3202* mpn_popcount: Low-level Functions. (line 412) 3203* mpn_random: Low-level Functions. (line 401) 3204* mpn_random2: Low-level Functions. (line 402) 3205* mpn_rshift: Low-level Functions. (line 272) 3206* mpn_scan0: Low-level Functions. (line 386) 3207* mpn_scan1: Low-level Functions. (line 394) 3208* mpn_set_str: Low-level Functions. (line 367) 3209* mpn_sqr: Low-level Functions. (line 125) 3210* mpn_sqrtrem: Low-level Functions. (line 334) 3211* mpn_sub: Low-level Functions. (line 90) 3212* mpn_sub_1: Low-level Functions. (line 85) 3213* mpn_sub_n: Low-level Functions. (line 76) 3214* mpn_submul_1: Low-level Functions. (line 159) 3215* mpn_tdiv_qr: Low-level Functions. (line 171) 3216* mpn_xnor_n: Low-level Functions. (line 462) 3217* mpn_xor_n: Low-level Functions. (line 437) 3218* mpn_zero: Low-level Functions. (line 479) 3219* mpq_abs: Rational Arithmetic. (line 31) 3220* mpq_add: Rational Arithmetic. (line 7) 3221* mpq_canonicalize: Rational Number Functions. 3222 (line 22) 3223* mpq_class: C++ Interface General. 3224 (line 19) 3225* mpq_class::canonicalize: C++ Interface Rationals. 3226 (line 37) 3227* mpq_class::get_d: C++ Interface Rationals. 3228 (line 46) 3229* mpq_class::get_den: C++ Interface Rationals. 3230 (line 58) 3231* mpq_class::get_den_mpz_t: C++ Interface Rationals. 3232 (line 68) 3233* mpq_class::get_mpq_t: C++ Interface General. 3234 (line 65) 3235* mpq_class::get_num: C++ Interface Rationals. 3236 (line 57) 3237* mpq_class::get_num_mpz_t: C++ Interface Rationals. 3238 (line 67) 3239* mpq_class::get_str: C++ Interface Rationals. 3240 (line 47) 3241* mpq_class::mpq_class: C++ Interface Rationals. 3242 (line 30) 3243* mpq_class::set_str: C++ Interface Rationals. 3244 (line 48) 3245* mpq_clear: Initializing Rationals. 3246 (line 16) 3247* mpq_clears: Initializing Rationals. 3248 (line 20) 3249* mpq_cmp: Comparing Rationals. (line 7) 3250* mpq_cmp_si: Comparing Rationals. (line 17) 3251* mpq_cmp_ui: Comparing Rationals. (line 15) 3252* mpq_denref: Applying Integer Functions. 3253 (line 18) 3254* mpq_div: Rational Arithmetic. (line 22) 3255* mpq_div_2exp: Rational Arithmetic. (line 25) 3256* mpq_equal: Comparing Rationals. (line 33) 3257* mpq_get_d: Rational Conversions. 3258 (line 7) 3259* mpq_get_den: Applying Integer Functions. 3260 (line 24) 3261* mpq_get_num: Applying Integer Functions. 3262 (line 23) 3263* mpq_get_str: Rational Conversions. 3264 (line 22) 3265* mpq_init: Initializing Rationals. 3266 (line 7) 3267* mpq_inits: Initializing Rationals. 3268 (line 12) 3269* mpq_inp_str: I/O of Rationals. (line 26) 3270* mpq_inv: Rational Arithmetic. (line 34) 3271* mpq_mul: Rational Arithmetic. (line 15) 3272* mpq_mul_2exp: Rational Arithmetic. (line 18) 3273* mpq_neg: Rational Arithmetic. (line 28) 3274* mpq_numref: Applying Integer Functions. 3275 (line 17) 3276* mpq_out_str: I/O of Rationals. (line 18) 3277* mpq_set: Initializing Rationals. 3278 (line 24) 3279* mpq_set_d: Rational Conversions. 3280 (line 17) 3281* mpq_set_den: Applying Integer Functions. 3282 (line 26) 3283* mpq_set_f: Rational Conversions. 3284 (line 18) 3285* mpq_set_num: Applying Integer Functions. 3286 (line 25) 3287* mpq_set_si: Initializing Rationals. 3288 (line 31) 3289* mpq_set_str: Initializing Rationals. 3290 (line 36) 3291* mpq_set_ui: Initializing Rationals. 3292 (line 29) 3293* mpq_set_z: Initializing Rationals. 3294 (line 25) 3295* mpq_sgn: Comparing Rationals. (line 27) 3296* mpq_sub: Rational Arithmetic. (line 11) 3297* mpq_swap: Initializing Rationals. 3298 (line 56) 3299* mpq_t: Nomenclature and Types. 3300 (line 16) 3301* mpz_abs: Integer Arithmetic. (line 42) 3302* mpz_add: Integer Arithmetic. (line 7) 3303* mpz_add_ui: Integer Arithmetic. (line 9) 3304* mpz_addmul: Integer Arithmetic. (line 25) 3305* mpz_addmul_ui: Integer Arithmetic. (line 27) 3306* mpz_and: Integer Logic and Bit Fiddling. 3307 (line 11) 3308* mpz_array_init: Integer Special Functions. 3309 (line 11) 3310* mpz_bin_ui: Number Theoretic Functions. 3311 (line 111) 3312* mpz_bin_uiui: Number Theoretic Functions. 3313 (line 113) 3314* mpz_cdiv_q: Integer Division. (line 13) 3315* mpz_cdiv_q_2exp: Integer Division. (line 24) 3316* mpz_cdiv_q_ui: Integer Division. (line 17) 3317* mpz_cdiv_qr: Integer Division. (line 15) 3318* mpz_cdiv_qr_ui: Integer Division. (line 21) 3319* mpz_cdiv_r: Integer Division. (line 14) 3320* mpz_cdiv_r_2exp: Integer Division. (line 25) 3321* mpz_cdiv_r_ui: Integer Division. (line 19) 3322* mpz_cdiv_ui: Integer Division. (line 23) 3323* mpz_class: C++ Interface General. 3324 (line 18) 3325* mpz_class::fits_sint_p: C++ Interface Integers. 3326 (line 45) 3327* mpz_class::fits_slong_p: C++ Interface Integers. 3328 (line 46) 3329* mpz_class::fits_sshort_p: C++ Interface Integers. 3330 (line 47) 3331* mpz_class::fits_uint_p: C++ Interface Integers. 3332 (line 48) 3333* mpz_class::fits_ulong_p: C++ Interface Integers. 3334 (line 49) 3335* mpz_class::fits_ushort_p: C++ Interface Integers. 3336 (line 50) 3337* mpz_class::get_d: C++ Interface Integers. 3338 (line 51) 3339* mpz_class::get_mpz_t: C++ Interface General. 3340 (line 64) 3341* mpz_class::get_si: C++ Interface Integers. 3342 (line 52) 3343* mpz_class::get_str: C++ Interface Integers. 3344 (line 53) 3345* mpz_class::get_ui: C++ Interface Integers. 3346 (line 54) 3347* mpz_class::mpz_class: C++ Interface Integers. 3348 (line 20) 3349* mpz_class::set_str: C++ Interface Integers. 3350 (line 55) 3351* mpz_clear: Initializing Integers. 3352 (line 44) 3353* mpz_clears: Initializing Integers. 3354 (line 48) 3355* mpz_clrbit: Integer Logic and Bit Fiddling. 3356 (line 54) 3357* mpz_cmp: Integer Comparisons. (line 7) 3358* mpz_cmp_d: Integer Comparisons. (line 8) 3359* mpz_cmp_si: Integer Comparisons. (line 9) 3360* mpz_cmp_ui: Integer Comparisons. (line 10) 3361* mpz_cmpabs: Integer Comparisons. (line 18) 3362* mpz_cmpabs_d: Integer Comparisons. (line 19) 3363* mpz_cmpabs_ui: Integer Comparisons. (line 20) 3364* mpz_com: Integer Logic and Bit Fiddling. 3365 (line 20) 3366* mpz_combit: Integer Logic and Bit Fiddling. 3367 (line 57) 3368* mpz_congruent_2exp_p: Integer Division. (line 124) 3369* mpz_congruent_p: Integer Division. (line 121) 3370* mpz_congruent_ui_p: Integer Division. (line 123) 3371* mpz_divexact: Integer Division. (line 101) 3372* mpz_divexact_ui: Integer Division. (line 102) 3373* mpz_divisible_2exp_p: Integer Division. (line 112) 3374* mpz_divisible_p: Integer Division. (line 110) 3375* mpz_divisible_ui_p: Integer Division. (line 111) 3376* mpz_even_p: Miscellaneous Integer Functions. 3377 (line 18) 3378* mpz_export: Integer Import and Export. 3379 (line 45) 3380* mpz_fac_ui: Number Theoretic Functions. 3381 (line 108) 3382* mpz_fdiv_q: Integer Division. (line 27) 3383* mpz_fdiv_q_2exp: Integer Division. (line 38) 3384* mpz_fdiv_q_ui: Integer Division. (line 31) 3385* mpz_fdiv_qr: Integer Division. (line 29) 3386* mpz_fdiv_qr_ui: Integer Division. (line 35) 3387* mpz_fdiv_r: Integer Division. (line 28) 3388* mpz_fdiv_r_2exp: Integer Division. (line 39) 3389* mpz_fdiv_r_ui: Integer Division. (line 33) 3390* mpz_fdiv_ui: Integer Division. (line 37) 3391* mpz_fib2_ui: Number Theoretic Functions. 3392 (line 121) 3393* mpz_fib_ui: Number Theoretic Functions. 3394 (line 119) 3395* mpz_fits_sint_p: Miscellaneous Integer Functions. 3396 (line 10) 3397* mpz_fits_slong_p: Miscellaneous Integer Functions. 3398 (line 8) 3399* mpz_fits_sshort_p: Miscellaneous Integer Functions. 3400 (line 12) 3401* mpz_fits_uint_p: Miscellaneous Integer Functions. 3402 (line 9) 3403* mpz_fits_ulong_p: Miscellaneous Integer Functions. 3404 (line 7) 3405* mpz_fits_ushort_p: Miscellaneous Integer Functions. 3406 (line 11) 3407* mpz_gcd: Number Theoretic Functions. 3408 (line 30) 3409* mpz_gcd_ui: Number Theoretic Functions. 3410 (line 37) 3411* mpz_gcdext: Number Theoretic Functions. 3412 (line 47) 3413* mpz_get_d: Converting Integers. (line 27) 3414* mpz_get_d_2exp: Converting Integers. (line 35) 3415* mpz_get_si: Converting Integers. (line 18) 3416* mpz_get_str: Converting Integers. (line 46) 3417* mpz_get_ui: Converting Integers. (line 11) 3418* mpz_getlimbn: Integer Special Functions. 3419 (line 60) 3420* mpz_hamdist: Integer Logic and Bit Fiddling. 3421 (line 29) 3422* mpz_import: Integer Import and Export. 3423 (line 11) 3424* mpz_init: Initializing Integers. 3425 (line 26) 3426* mpz_init2: Initializing Integers. 3427 (line 33) 3428* mpz_init_set: Simultaneous Integer Init & Assign. 3429 (line 27) 3430* mpz_init_set_d: Simultaneous Integer Init & Assign. 3431 (line 30) 3432* mpz_init_set_si: Simultaneous Integer Init & Assign. 3433 (line 29) 3434* mpz_init_set_str: Simultaneous Integer Init & Assign. 3435 (line 34) 3436* mpz_init_set_ui: Simultaneous Integer Init & Assign. 3437 (line 28) 3438* mpz_inits: Initializing Integers. 3439 (line 29) 3440* mpz_inp_raw: I/O of Integers. (line 61) 3441* mpz_inp_str: I/O of Integers. (line 30) 3442* mpz_invert: Number Theoretic Functions. 3443 (line 72) 3444* mpz_ior: Integer Logic and Bit Fiddling. 3445 (line 14) 3446* mpz_jacobi: Number Theoretic Functions. 3447 (line 79) 3448* mpz_kronecker: Number Theoretic Functions. 3449 (line 87) 3450* mpz_kronecker_si: Number Theoretic Functions. 3451 (line 88) 3452* mpz_kronecker_ui: Number Theoretic Functions. 3453 (line 89) 3454* mpz_lcm: Number Theoretic Functions. 3455 (line 66) 3456* mpz_lcm_ui: Number Theoretic Functions. 3457 (line 67) 3458* mpz_legendre: Number Theoretic Functions. 3459 (line 82) 3460* mpz_lucnum2_ui: Number Theoretic Functions. 3461 (line 132) 3462* mpz_lucnum_ui: Number Theoretic Functions. 3463 (line 130) 3464* mpz_mod: Integer Division. (line 91) 3465* mpz_mod_ui: Integer Division. (line 93) 3466* mpz_mul: Integer Arithmetic. (line 19) 3467* mpz_mul_2exp: Integer Arithmetic. (line 35) 3468* mpz_mul_si: Integer Arithmetic. (line 20) 3469* mpz_mul_ui: Integer Arithmetic. (line 22) 3470* mpz_neg: Integer Arithmetic. (line 39) 3471* mpz_nextprime: Number Theoretic Functions. 3472 (line 23) 3473* mpz_odd_p: Miscellaneous Integer Functions. 3474 (line 17) 3475* mpz_out_raw: I/O of Integers. (line 45) 3476* mpz_out_str: I/O of Integers. (line 18) 3477* mpz_perfect_power_p: Integer Roots. (line 27) 3478* mpz_perfect_square_p: Integer Roots. (line 36) 3479* mpz_popcount: Integer Logic and Bit Fiddling. 3480 (line 23) 3481* mpz_pow_ui: Integer Exponentiation. 3482 (line 31) 3483* mpz_powm: Integer Exponentiation. 3484 (line 8) 3485* mpz_powm_sec: Integer Exponentiation. 3486 (line 18) 3487* mpz_powm_ui: Integer Exponentiation. 3488 (line 10) 3489* mpz_probab_prime_p: Number Theoretic Functions. 3490 (line 7) 3491* mpz_random: Integer Random Numbers. 3492 (line 42) 3493* mpz_random2: Integer Random Numbers. 3494 (line 51) 3495* mpz_realloc2: Initializing Integers. 3496 (line 52) 3497* mpz_remove: Number Theoretic Functions. 3498 (line 103) 3499* mpz_root: Integer Roots. (line 7) 3500* mpz_rootrem: Integer Roots. (line 13) 3501* mpz_rrandomb: Integer Random Numbers. 3502 (line 31) 3503* mpz_scan0: Integer Logic and Bit Fiddling. 3504 (line 37) 3505* mpz_scan1: Integer Logic and Bit Fiddling. 3506 (line 38) 3507* mpz_set: Assigning Integers. (line 10) 3508* mpz_set_d: Assigning Integers. (line 13) 3509* mpz_set_f: Assigning Integers. (line 15) 3510* mpz_set_q: Assigning Integers. (line 14) 3511* mpz_set_si: Assigning Integers. (line 12) 3512* mpz_set_str: Assigning Integers. (line 21) 3513* mpz_set_ui: Assigning Integers. (line 11) 3514* mpz_setbit: Integer Logic and Bit Fiddling. 3515 (line 51) 3516* mpz_sgn: Integer Comparisons. (line 28) 3517* mpz_si_kronecker: Number Theoretic Functions. 3518 (line 90) 3519* mpz_size: Integer Special Functions. 3520 (line 68) 3521* mpz_sizeinbase: Miscellaneous Integer Functions. 3522 (line 23) 3523* mpz_sqrt: Integer Roots. (line 17) 3524* mpz_sqrtrem: Integer Roots. (line 20) 3525* mpz_sub: Integer Arithmetic. (line 12) 3526* mpz_sub_ui: Integer Arithmetic. (line 14) 3527* mpz_submul: Integer Arithmetic. (line 30) 3528* mpz_submul_ui: Integer Arithmetic. (line 32) 3529* mpz_swap: Assigning Integers. (line 37) 3530* mpz_t: Nomenclature and Types. 3531 (line 6) 3532* mpz_tdiv_q: Integer Division. (line 41) 3533* mpz_tdiv_q_2exp: Integer Division. (line 52) 3534* mpz_tdiv_q_ui: Integer Division. (line 45) 3535* mpz_tdiv_qr: Integer Division. (line 43) 3536* mpz_tdiv_qr_ui: Integer Division. (line 49) 3537* mpz_tdiv_r: Integer Division. (line 42) 3538* mpz_tdiv_r_2exp: Integer Division. (line 53) 3539* mpz_tdiv_r_ui: Integer Division. (line 47) 3540* mpz_tdiv_ui: Integer Division. (line 51) 3541* mpz_tstbit: Integer Logic and Bit Fiddling. 3542 (line 60) 3543* mpz_ui_kronecker: Number Theoretic Functions. 3544 (line 91) 3545* mpz_ui_pow_ui: Integer Exponentiation. 3546 (line 33) 3547* mpz_ui_sub: Integer Arithmetic. (line 16) 3548* mpz_urandomb: Integer Random Numbers. 3549 (line 14) 3550* mpz_urandomm: Integer Random Numbers. 3551 (line 23) 3552* mpz_xor: Integer Logic and Bit Fiddling. 3553 (line 17) 3554* msqrt: BSD Compatible Functions. 3555 (line 63) 3556* msub: BSD Compatible Functions. 3557 (line 46) 3558* mtox: BSD Compatible Functions. 3559 (line 98) 3560* mult: BSD Compatible Functions. 3561 (line 49) 3562* operator%: C++ Interface Integers. 3563 (line 30) 3564* operator/: C++ Interface Integers. 3565 (line 29) 3566* operator<<: C++ Formatted Output. 3567 (line 11) 3568* operator>> <1>: C++ Formatted Input. (line 11) 3569* operator>> <2>: C++ Interface Rationals. 3570 (line 77) 3571* operator>>: C++ Formatted Input. (line 14) 3572* pow: BSD Compatible Functions. 3573 (line 71) 3574* rpow: BSD Compatible Functions. 3575 (line 79) 3576* sdiv: BSD Compatible Functions. 3577 (line 55) 3578* sgn <1>: C++ Interface Rationals. 3579 (line 50) 3580* sgn <2>: C++ Interface Integers. 3581 (line 57) 3582* sgn: C++ Interface Floats. 3583 (line 98) 3584* sqrt <1>: C++ Interface Floats. 3585 (line 99) 3586* sqrt: C++ Interface Integers. 3587 (line 58) 3588* trunc: C++ Interface Floats. 3589 (line 100) 3590* xtom: BSD Compatible Functions. 3591 (line 34) 3592 3593 3594 3595 3596Local Variables: 3597coding: iso-8859-1 3598End: 3599