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Diffstat (limited to 'contrib/libgmp/mpn/generic/sqrtrem.c')
| -rw-r--r-- | contrib/libgmp/mpn/generic/sqrtrem.c | 498 |
1 files changed, 498 insertions, 0 deletions
diff --git a/contrib/libgmp/mpn/generic/sqrtrem.c b/contrib/libgmp/mpn/generic/sqrtrem.c new file mode 100644 index 0000000..539480d --- /dev/null +++ b/contrib/libgmp/mpn/generic/sqrtrem.c @@ -0,0 +1,498 @@ +/* mpn_sqrtrem (root_ptr, rem_ptr, op_ptr, op_size) + + Write the square root of {OP_PTR, OP_SIZE} at ROOT_PTR. + Write the remainder at REM_PTR, if REM_PTR != NULL. + Return the size of the remainder. + (The size of the root is always half of the size of the operand.) + + OP_PTR and ROOT_PTR may not point to the same object. + OP_PTR and REM_PTR may point to the same object. + + If REM_PTR is NULL, only the root is computed and the return value of + the function is 0 if OP is a perfect square, and *any* non-zero number + otherwise. + +Copyright (C) 1993, 1994, 1996 Free Software Foundation, Inc. + +This file is part of the GNU MP Library. + +The GNU MP Library is free software; you can redistribute it and/or modify +it under the terms of the GNU Library General Public License as published by +the Free Software Foundation; either version 2 of the License, or (at your +option) any later version. + +The GNU MP Library is distributed in the hope that it will be useful, but +WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY +or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Library General Public +License for more details. + +You should have received a copy of the GNU Library General Public License +along with the GNU MP Library; see the file COPYING.LIB. If not, write to +the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, +MA 02111-1307, USA. */ + +/* This code is just correct if "unsigned char" has at least 8 bits. It + doesn't help to use CHAR_BIT from limits.h, as the real problem is + the static arrays. */ + +#include "gmp.h" +#include "gmp-impl.h" +#include "longlong.h" + +/* Square root algorithm: + + 1. Shift OP (the input) to the left an even number of bits s.t. there + are an even number of words and either (or both) of the most + significant bits are set. This way, sqrt(OP) has exactly half as + many words as OP, and has its most significant bit set. + + 2. Get a 9-bit approximation to sqrt(OP) using the pre-computed tables. + This approximation is used for the first single-precision + iterations of Newton's method, yielding a full-word approximation + to sqrt(OP). + + 3. Perform multiple-precision Newton iteration until we have the + exact result. Only about half of the input operand is used in + this calculation, as the square root is perfectly determinable + from just the higher half of a number. */ + +/* Define this macro for IEEE P854 machines with a fast sqrt instruction. */ +#if defined __GNUC__ && ! defined __SOFT_FLOAT + +#if defined __sparc__ +#define SQRT(a) \ + ({ \ + double __sqrt_res; \ + asm ("fsqrtd %1,%0" : "=f" (__sqrt_res) : "f" (a)); \ + __sqrt_res; \ + }) +#endif + +#if defined __HAVE_68881__ +#define SQRT(a) \ + ({ \ + double __sqrt_res; \ + asm ("fsqrtx %1,%0" : "=f" (__sqrt_res) : "f" (a)); \ + __sqrt_res; \ + }) +#endif + +#if defined __hppa +#define SQRT(a) \ + ({ \ + double __sqrt_res; \ + asm ("fsqrt,dbl %1,%0" : "=fx" (__sqrt_res) : "fx" (a)); \ + __sqrt_res; \ + }) +#endif + +#if defined _ARCH_PWR2 +#define SQRT(a) \ + ({ \ + double __sqrt_res; \ + asm ("fsqrt %0,%1" : "=f" (__sqrt_res) : "f" (a)); \ + __sqrt_res; \ + }) +#endif + +#endif + +#ifndef SQRT + +/* Tables for initial approximation of the square root. These are + indexed with bits 1-8 of the operand for which the square root is + calculated, where bit 0 is the most significant non-zero bit. I.e. + the most significant one-bit is not used, since that per definition + is one. Likewise, the tables don't return the highest bit of the + result. That bit must be inserted by or:ing the returned value with + 0x100. This way, we get a 9-bit approximation from 8-bit tables! */ + +/* Table to be used for operands with an even total number of bits. + (Exactly as in the decimal system there are similarities between the + square root of numbers with the same initial digits and an even + difference in the total number of digits. Consider the square root + of 1, 10, 100, 1000, ...) */ +static unsigned char even_approx_tab[256] = +{ + 0x6a, 0x6a, 0x6b, 0x6c, 0x6c, 0x6d, 0x6e, 0x6e, + 0x6f, 0x70, 0x71, 0x71, 0x72, 0x73, 0x73, 0x74, + 0x75, 0x75, 0x76, 0x77, 0x77, 0x78, 0x79, 0x79, + 0x7a, 0x7b, 0x7b, 0x7c, 0x7d, 0x7d, 0x7e, 0x7f, + 0x80, 0x80, 0x81, 0x81, 0x82, 0x83, 0x83, 0x84, + 0x85, 0x85, 0x86, 0x87, 0x87, 0x88, 0x89, 0x89, + 0x8a, 0x8b, 0x8b, 0x8c, 0x8d, 0x8d, 0x8e, 0x8f, + 0x8f, 0x90, 0x90, 0x91, 0x92, 0x92, 0x93, 0x94, + 0x94, 0x95, 0x96, 0x96, 0x97, 0x97, 0x98, 0x99, + 0x99, 0x9a, 0x9b, 0x9b, 0x9c, 0x9c, 0x9d, 0x9e, + 0x9e, 0x9f, 0xa0, 0xa0, 0xa1, 0xa1, 0xa2, 0xa3, + 0xa3, 0xa4, 0xa4, 0xa5, 0xa6, 0xa6, 0xa7, 0xa7, + 0xa8, 0xa9, 0xa9, 0xaa, 0xaa, 0xab, 0xac, 0xac, + 0xad, 0xad, 0xae, 0xaf, 0xaf, 0xb0, 0xb0, 0xb1, + 0xb2, 0xb2, 0xb3, 0xb3, 0xb4, 0xb5, 0xb5, 0xb6, + 0xb6, 0xb7, 0xb7, 0xb8, 0xb9, 0xb9, 0xba, 0xba, + 0xbb, 0xbb, 0xbc, 0xbd, 0xbd, 0xbe, 0xbe, 0xbf, + 0xc0, 0xc0, 0xc1, 0xc1, 0xc2, 0xc2, 0xc3, 0xc3, + 0xc4, 0xc5, 0xc5, 0xc6, 0xc6, 0xc7, 0xc7, 0xc8, + 0xc9, 0xc9, 0xca, 0xca, 0xcb, 0xcb, 0xcc, 0xcc, + 0xcd, 0xce, 0xce, 0xcf, 0xcf, 0xd0, 0xd0, 0xd1, + 0xd1, 0xd2, 0xd3, 0xd3, 0xd4, 0xd4, 0xd5, 0xd5, + 0xd6, 0xd6, 0xd7, 0xd7, 0xd8, 0xd9, 0xd9, 0xda, + 0xda, 0xdb, 0xdb, 0xdc, 0xdc, 0xdd, 0xdd, 0xde, + 0xde, 0xdf, 0xe0, 0xe0, 0xe1, 0xe1, 0xe2, 0xe2, + 0xe3, 0xe3, 0xe4, 0xe4, 0xe5, 0xe5, 0xe6, 0xe6, + 0xe7, 0xe7, 0xe8, 0xe8, 0xe9, 0xea, 0xea, 0xeb, + 0xeb, 0xec, 0xec, 0xed, 0xed, 0xee, 0xee, 0xef, + 0xef, 0xf0, 0xf0, 0xf1, 0xf1, 0xf2, 0xf2, 0xf3, + 0xf3, 0xf4, 0xf4, 0xf5, 0xf5, 0xf6, 0xf6, 0xf7, + 0xf7, 0xf8, 0xf8, 0xf9, 0xf9, 0xfa, 0xfa, 0xfb, + 0xfb, 0xfc, 0xfc, 0xfd, 0xfd, 0xfe, 0xfe, 0xff, +}; + +/* Table to be used for operands with an odd total number of bits. + (Further comments before previous table.) */ +static unsigned char odd_approx_tab[256] = +{ + 0x00, 0x00, 0x00, 0x01, 0x01, 0x02, 0x02, 0x03, + 0x03, 0x04, 0x04, 0x05, 0x05, 0x06, 0x06, 0x07, + 0x07, 0x08, 0x08, 0x09, 0x09, 0x0a, 0x0a, 0x0b, + 0x0b, 0x0c, 0x0c, 0x0d, 0x0d, 0x0e, 0x0e, 0x0f, + 0x0f, 0x10, 0x10, 0x10, 0x11, 0x11, 0x12, 0x12, + 0x13, 0x13, 0x14, 0x14, 0x15, 0x15, 0x16, 0x16, + 0x16, 0x17, 0x17, 0x18, 0x18, 0x19, 0x19, 0x1a, + 0x1a, 0x1b, 0x1b, 0x1b, 0x1c, 0x1c, 0x1d, 0x1d, + 0x1e, 0x1e, 0x1f, 0x1f, 0x20, 0x20, 0x20, 0x21, + 0x21, 0x22, 0x22, 0x23, 0x23, 0x23, 0x24, 0x24, + 0x25, 0x25, 0x26, 0x26, 0x27, 0x27, 0x27, 0x28, + 0x28, 0x29, 0x29, 0x2a, 0x2a, 0x2a, 0x2b, 0x2b, + 0x2c, 0x2c, 0x2d, 0x2d, 0x2d, 0x2e, 0x2e, 0x2f, + 0x2f, 0x30, 0x30, 0x30, 0x31, 0x31, 0x32, 0x32, + 0x32, 0x33, 0x33, 0x34, 0x34, 0x35, 0x35, 0x35, + 0x36, 0x36, 0x37, 0x37, 0x37, 0x38, 0x38, 0x39, + 0x39, 0x39, 0x3a, 0x3a, 0x3b, 0x3b, 0x3b, 0x3c, + 0x3c, 0x3d, 0x3d, 0x3d, 0x3e, 0x3e, 0x3f, 0x3f, + 0x40, 0x40, 0x40, 0x41, 0x41, 0x41, 0x42, 0x42, + 0x43, 0x43, 0x43, 0x44, 0x44, 0x45, 0x45, 0x45, + 0x46, 0x46, 0x47, 0x47, 0x47, 0x48, 0x48, 0x49, + 0x49, 0x49, 0x4a, 0x4a, 0x4b, 0x4b, 0x4b, 0x4c, + 0x4c, 0x4c, 0x4d, 0x4d, 0x4e, 0x4e, 0x4e, 0x4f, + 0x4f, 0x50, 0x50, 0x50, 0x51, 0x51, 0x51, 0x52, + 0x52, 0x53, 0x53, 0x53, 0x54, 0x54, 0x54, 0x55, + 0x55, 0x56, 0x56, 0x56, 0x57, 0x57, 0x57, 0x58, + 0x58, 0x59, 0x59, 0x59, 0x5a, 0x5a, 0x5a, 0x5b, + 0x5b, 0x5b, 0x5c, 0x5c, 0x5d, 0x5d, 0x5d, 0x5e, + 0x5e, 0x5e, 0x5f, 0x5f, 0x60, 0x60, 0x60, 0x61, + 0x61, 0x61, 0x62, 0x62, 0x62, 0x63, 0x63, 0x63, + 0x64, 0x64, 0x65, 0x65, 0x65, 0x66, 0x66, 0x66, + 0x67, 0x67, 0x67, 0x68, 0x68, 0x68, 0x69, 0x69, +}; +#endif + + +mp_size_t +#if __STDC__ +mpn_sqrtrem (mp_ptr root_ptr, mp_ptr rem_ptr, mp_srcptr op_ptr, mp_size_t op_size) +#else +mpn_sqrtrem (root_ptr, rem_ptr, op_ptr, op_size) + mp_ptr root_ptr; + mp_ptr rem_ptr; + mp_srcptr op_ptr; + mp_size_t op_size; +#endif +{ + /* R (root result) */ + mp_ptr rp; /* Pointer to least significant word */ + mp_size_t rsize; /* The size in words */ + + /* T (OP shifted to the left a.k.a. normalized) */ + mp_ptr tp; /* Pointer to least significant word */ + mp_size_t tsize; /* The size in words */ + mp_ptr t_end_ptr; /* Pointer right beyond most sign. word */ + mp_limb_t t_high0, t_high1; /* The two most significant words */ + + /* TT (temporary for numerator/remainder) */ + mp_ptr ttp; /* Pointer to least significant word */ + + /* X (temporary for quotient in main loop) */ + mp_ptr xp; /* Pointer to least significant word */ + mp_size_t xsize; /* The size in words */ + + unsigned cnt; + mp_limb_t initial_approx; /* Initially made approximation */ + mp_size_t tsizes[BITS_PER_MP_LIMB]; /* Successive calculation precisions */ + mp_size_t tmp; + mp_size_t i; + + mp_limb_t cy_limb; + TMP_DECL (marker); + + /* If OP is zero, both results are zero. */ + if (op_size == 0) + return 0; + + count_leading_zeros (cnt, op_ptr[op_size - 1]); + tsize = op_size; + if ((tsize & 1) != 0) + { + cnt += BITS_PER_MP_LIMB; + tsize++; + } + + rsize = tsize / 2; + rp = root_ptr; + + TMP_MARK (marker); + + /* Shift OP an even number of bits into T, such that either the most or + the second most significant bit is set, and such that the number of + words in T becomes even. This way, the number of words in R=sqrt(OP) + is exactly half as many as in OP, and the most significant bit of R + is set. + + Also, the initial approximation is simplified by this up-shifted OP. + + Finally, the Newtonian iteration which is the main part of this + program performs division by R. The fast division routine expects + the divisor to be "normalized" in exactly the sense of having the + most significant bit set. */ + + tp = (mp_ptr) TMP_ALLOC (tsize * BYTES_PER_MP_LIMB); + + if ((cnt & ~1) % BITS_PER_MP_LIMB != 0) + t_high0 = mpn_lshift (tp + cnt / BITS_PER_MP_LIMB, op_ptr, op_size, + (cnt & ~1) % BITS_PER_MP_LIMB); + else + MPN_COPY (tp + cnt / BITS_PER_MP_LIMB, op_ptr, op_size); + + if (cnt >= BITS_PER_MP_LIMB) + tp[0] = 0; + + t_high0 = tp[tsize - 1]; + t_high1 = tp[tsize - 2]; /* Never stray. TSIZE is >= 2. */ + +/* Is there a fast sqrt instruction defined for this machine? */ +#ifdef SQRT + { + initial_approx = SQRT (t_high0 * 2.0 + * ((mp_limb_t) 1 << (BITS_PER_MP_LIMB - 1)) + + t_high1); + /* If t_high0,,t_high1 is big, the result in INITIAL_APPROX might have + become incorrect due to overflow in the conversion from double to + mp_limb_t above. It will typically be zero in that case, but might be + a small number on some machines. The most significant bit of + INITIAL_APPROX should be set, so that bit is a good overflow + indication. */ + if ((mp_limb_signed_t) initial_approx >= 0) + initial_approx = ~(mp_limb_t)0; + } +#else + /* Get a 9 bit approximation from the tables. The tables expect to + be indexed with the 8 high bits right below the highest bit. + Also, the highest result bit is not returned by the tables, and + must be or:ed into the result. The scheme gives 9 bits of start + approximation with just 256-entry 8 bit tables. */ + + if ((cnt & 1) == 0) + { + /* The most sign bit of t_high0 is set. */ + initial_approx = t_high0 >> (BITS_PER_MP_LIMB - 8 - 1); + initial_approx &= 0xff; + initial_approx = even_approx_tab[initial_approx]; + } + else + { + /* The most significant bit of T_HIGH0 is unset, + the second most significant is set. */ + initial_approx = t_high0 >> (BITS_PER_MP_LIMB - 8 - 2); + initial_approx &= 0xff; + initial_approx = odd_approx_tab[initial_approx]; + } + initial_approx |= 0x100; + initial_approx <<= BITS_PER_MP_LIMB - 8 - 1; + + /* Perform small precision Newtonian iterations to get a full word + approximation. For small operands, these iteration will make the + entire job. */ + if (t_high0 == ~(mp_limb_t)0) + initial_approx = t_high0; + else + { + mp_limb_t quot; + + if (t_high0 >= initial_approx) + initial_approx = t_high0 + 1; + + /* First get about 18 bits with pure C arithmetics. */ + quot = t_high0 / (initial_approx >> BITS_PER_MP_LIMB/2) << BITS_PER_MP_LIMB/2; + initial_approx = (initial_approx + quot) / 2; + initial_approx |= (mp_limb_t) 1 << (BITS_PER_MP_LIMB - 1); + + /* Now get a full word by one (or for > 36 bit machines) several + iterations. */ + for (i = 16; i < BITS_PER_MP_LIMB; i <<= 1) + { + mp_limb_t ignored_remainder; + + udiv_qrnnd (quot, ignored_remainder, + t_high0, t_high1, initial_approx); + initial_approx = (initial_approx + quot) / 2; + initial_approx |= (mp_limb_t) 1 << (BITS_PER_MP_LIMB - 1); + } + } +#endif + + rp[0] = initial_approx; + rsize = 1; + +#ifdef DEBUG + printf ("\n\nT = "); + mpn_dump (tp, tsize); +#endif + + if (tsize > 2) + { + /* Determine the successive precisions to use in the iteration. We + minimize the precisions, beginning with the highest (i.e. last + iteration) to the lowest (i.e. first iteration). */ + + xp = (mp_ptr) TMP_ALLOC (tsize * BYTES_PER_MP_LIMB); + ttp = (mp_ptr) TMP_ALLOC (tsize * BYTES_PER_MP_LIMB); + + t_end_ptr = tp + tsize; + + tmp = tsize / 2; + for (i = 0;; i++) + { + tsize = (tmp + 1) / 2; + if (tmp == tsize) + break; + tsizes[i] = tsize + tmp; + tmp = tsize; + } + + /* Main Newton iteration loop. For big arguments, most of the + time is spent here. */ + + /* It is possible to do a great optimization here. The successive + divisors in the mpn_divmod call below has more and more leading + words equal to its predecessor. Therefore the beginning of + each division will repeat the same work as did the last + division. If we could guarantee that the leading words of two + consecutive divisors are the same (i.e. in this case, a later + divisor has just more digits at the end) it would be a simple + matter of just using the old remainder of the last division in + a subsequent division, to take care of this optimization. This + idea would surely make a difference even for small arguments. */ + + /* Loop invariants: + + R <= shiftdown_to_same_size(floor(sqrt(OP))) < R + 1. + X - 1 < shiftdown_to_same_size(floor(sqrt(OP))) <= X. + R <= shiftdown_to_same_size(X). */ + + while (--i >= 0) + { + mp_limb_t cy; +#ifdef DEBUG + mp_limb_t old_least_sign_r = rp[0]; + mp_size_t old_rsize = rsize; + + printf ("R = "); + mpn_dump (rp, rsize); +#endif + tsize = tsizes[i]; + + /* Need to copy the numerator into temporary space, as + mpn_divmod overwrites its numerator argument with the + remainder (which we currently ignore). */ + MPN_COPY (ttp, t_end_ptr - tsize, tsize); + cy = mpn_divmod (xp, ttp, tsize, rp, rsize); + xsize = tsize - rsize; + +#ifdef DEBUG + printf ("X =%d ", cy); + mpn_dump (xp, xsize); +#endif + + /* Add X and R with the most significant limbs aligned, + temporarily ignoring at least one limb at the low end of X. */ + tmp = xsize - rsize; + cy += mpn_add_n (xp + tmp, rp, xp + tmp, rsize); + + /* If T begins with more than 2 x BITS_PER_MP_LIMB of ones, we get + intermediate roots that'd need an extra bit. We don't want to + handle that since it would make the subsequent divisor + non-normalized, so round such roots down to be only ones in the + current precision. */ + if (cy == 2) + { + mp_size_t j; + for (j = xsize; j >= 0; j--) + xp[j] = ~(mp_limb_t)0; + } + + /* Divide X by 2 and put the result in R. This is the new + approximation. Shift in the carry from the addition. */ + mpn_rshift (rp, xp, xsize, 1); + rp[xsize - 1] |= ((mp_limb_t) 1 << (BITS_PER_MP_LIMB - 1)); + rsize = xsize; +#ifdef DEBUG + if (old_least_sign_r != rp[rsize - old_rsize]) + printf (">>>>>>>> %d: %0*lX, %0*lX <<<<<<<<\n", + i, 2 * BYTES_PER_MP_LIMB, old_least_sign_r, + 2 * BYTES_PER_MP_LIMB, rp[rsize - old_rsize]); +#endif + } + } + +#ifdef DEBUG + printf ("(final) R = "); + mpn_dump (rp, rsize); +#endif + + /* We computed the square root of OP * 2**(2*floor(cnt/2)). + This has resulted in R being 2**floor(cnt/2) to large. + Shift it down here to fix that. */ + if (cnt / 2 != 0) + { + mpn_rshift (rp, rp, rsize, cnt/2); + rsize -= rp[rsize - 1] == 0; + } + + /* Calculate the remainder. */ + mpn_mul_n (tp, rp, rp, rsize); + tsize = rsize + rsize; + tsize -= tp[tsize - 1] == 0; + if (op_size < tsize + || (op_size == tsize && mpn_cmp (op_ptr, tp, op_size) < 0)) + { + /* R is too large. Decrement it. */ + + /* These operations can't overflow. */ + cy_limb = mpn_sub_n (tp, tp, rp, rsize); + cy_limb += mpn_sub_n (tp, tp, rp, rsize); + mpn_sub_1 (tp + rsize, tp + rsize, tsize - rsize, cy_limb); + mpn_add_1 (tp, tp, tsize, (mp_limb_t) 1); + + mpn_sub_1 (rp, rp, rsize, (mp_limb_t) 1); + +#ifdef DEBUG + printf ("(adjusted) R = "); + mpn_dump (rp, rsize); +#endif + } + + if (rem_ptr != NULL) + { + cy_limb = mpn_sub (rem_ptr, op_ptr, op_size, tp, tsize); + MPN_NORMALIZE (rem_ptr, op_size); + TMP_FREE (marker); + return op_size; + } + else + { + int res; + res = op_size != tsize || mpn_cmp (op_ptr, tp, op_size); + TMP_FREE (marker); + return res; + } +} |
