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Diffstat (limited to 'gnu/lib/libgmp/mpn_sqrt.c')
| -rw-r--r-- | gnu/lib/libgmp/mpn_sqrt.c | 479 |
1 files changed, 0 insertions, 479 deletions
diff --git a/gnu/lib/libgmp/mpn_sqrt.c b/gnu/lib/libgmp/mpn_sqrt.c deleted file mode 100644 index 7dda9e4..0000000 --- a/gnu/lib/libgmp/mpn_sqrt.c +++ /dev/null @@ -1,479 +0,0 @@ -/* mpn_sqrt(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) 1991, 1993 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 General Public License as published by -the Free Software Foundation; either version 2, 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 General Public License for more details. - -You should have received a copy of the GNU General Public License -along with the GNU MP Library; see the file COPYING. If not, write to -the Free Software Foundation, 675 Mass Ave, Cambridge, MA 02139, 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__ - -#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 - -#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 -#ifdef __STDC__ -mpn_sqrt (mp_ptr root_ptr, mp_ptr rem_ptr, mp_srcptr op_ptr, mp_size op_size) -#else -mpn_sqrt (root_ptr, rem_ptr, op_ptr, op_size) - mp_ptr root_ptr; - mp_ptr rem_ptr; - mp_srcptr op_ptr; - mp_size op_size; -#endif -{ - /* R (root result) */ - mp_ptr rp; /* Pointer to least significant word */ - mp_size 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 tsize; /* The size in words */ - mp_ptr t_end_ptr; /* Pointer right beyond most sign. word */ - mp_limb 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 xsize; /* The size in words */ - - unsigned cnt; - mp_limb initial_approx; /* Initially made approximation */ - mp_size tsizes[BITS_PER_MP_LIMB]; /* Successive calculation precisions */ - mp_size tmp; - mp_size i; - - /* 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; - - /* 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) alloca (tsize * BYTES_PER_MP_LIMB); - - t_high0 = mpn_lshift (tp + cnt / BITS_PER_MP_LIMB, op_ptr, op_size, - (cnt & ~1) % BITS_PER_MP_LIMB); - 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) 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 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) initial_approx >= 0) - initial_approx = ~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 == ~0) - initial_approx = t_high0; - else - { - mp_limb 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) 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 ignored_remainder; - - udiv_qrnnd (quot, ignored_remainder, - t_high0, t_high1, initial_approx); - initial_approx = (initial_approx + quot) / 2; - initial_approx |= (mp_limb) 1 << (BITS_PER_MP_LIMB - 1); - } - } -#endif - - rp[0] = initial_approx; - rsize = 1; - - xp = (mp_ptr) alloca (tsize * BYTES_PER_MP_LIMB); - ttp = (mp_ptr) alloca (tsize * BYTES_PER_MP_LIMB); - - t_end_ptr = tp + tsize; - -#ifdef DEBUG - printf ("\n\nT = "); - _mp_mout (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). */ - - 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_div 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 cy; -#ifdef DEBUG - mp_limb old_least_sign_r = rp[0]; - mp_size old_rsize = rsize; - - printf ("R = "); - _mp_mout (rp, rsize); -#endif - tsize = tsizes[i]; - - /* Need to copy the numerator into temporary space, as - mpn_div overwrites its numerator argument with the - remainder (which we currently ignore). */ - MPN_COPY (ttp, t_end_ptr - tsize, tsize); - cy = mpn_div (xp, ttp, tsize, rp, rsize); - xsize = tsize - rsize; - cy = cy ? xp[xsize] : 0; - -#ifdef DEBUG - printf ("X =%d", cy); - _mp_mout (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 (xp + tmp, rp, rsize, 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 j; - for (j = xsize; j >= 0; j--) - xp[j] = ~(mp_limb)0; - } - - /* Divide X by 2 and put the result in R. This is the new - approximation. Shift in the carry from the addition. */ - rsize = mpn_rshiftci (rp, xp, xsize, 1, (mp_limb) 1); -#ifdef DEBUG - if (old_least_sign_r != rp[rsize - old_rsize]) - printf (">>>>>>>> %d: %08x, %08x <<<<<<<<\n", - i, old_least_sign_r, rp[rsize - old_rsize]); -#endif - } - } - -#ifdef DEBUG - printf ("(final) R = "); - _mp_mout (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. */ - rsize = mpn_rshift (rp, rp, rsize, cnt/2); - - /* Calculate the remainder. */ - tsize = mpn_mul (tp, rp, rsize, rp, rsize); - if (op_size < tsize - || (op_size == tsize && mpn_cmp (op_ptr, tp, op_size) < 0)) - { - /* R is too large. Decrement it. */ - mp_limb one = 1; - - tsize = tsize + mpn_sub (tp, tp, tsize, rp, rsize); - tsize = tsize + mpn_sub (tp, tp, tsize, rp, rsize); - tsize = tsize + mpn_add (tp, tp, tsize, &one, 1); - - (void) mpn_sub (rp, rp, rsize, &one, 1); - -#ifdef DEBUG - printf ("(adjusted) R = "); - _mp_mout (rp, rsize); -#endif - } - - if (rem_ptr != NULL) - { - mp_size retval = op_size + mpn_sub (rem_ptr, op_ptr, op_size, tp, tsize); - alloca (0); - return retval; - } - else - { - mp_size retval = (op_size != tsize || mpn_cmp (op_ptr, tp, op_size)); - alloca (0); - return retval; - } -} - -#ifdef DEBUG -_mp_mout (mp_srcptr p, mp_size size) -{ - mp_size ii; - for (ii = size - 1; ii >= 0; ii--) - printf ("%08X", p[ii]); - - puts (""); -} -#endif |
