=================================================================== RCS file: /home/cvs/OpenXM_contrib/gmp/mpn/generic/Attic/sqrtrem.c,v retrieving revision 1.1 retrieving revision 1.1.1.3 diff -u -p -r1.1 -r1.1.1.3 --- OpenXM_contrib/gmp/mpn/generic/Attic/sqrtrem.c 2000/01/10 15:35:24 1.1 +++ OpenXM_contrib/gmp/mpn/generic/Attic/sqrtrem.c 2003/08/25 16:06:20 1.1.1.3 @@ -1,498 +1,287 @@ -/* mpn_sqrtrem (root_ptr, rem_ptr, op_ptr, op_size) +/* mpn_sqrtrem -- square root and remainder */ - 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.) +/* +Copyright 1999, 2000, 2001, 2002 Free Software Foundation, Inc. - 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 +it under the terms of the GNU Lesser General Public License as published by +the Free Software Foundation; either version 2.1 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 +or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. -You should have received a copy of the GNU Library General Public License +You should have received a copy of the GNU Lesser 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. */ +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. */ +/* Contributed by Paul Zimmermann. + See "Karatsuba Square Root", reference in gmp.texi. */ + + +#include +#include + #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). +/* Square roots table. Generated by the following program: +#include "gmp.h" +main(){mpz_t x;int i;mpz_init(x);for(i=64;i<256;i++){mpz_set_ui(x,256*i); +mpz_sqrt(x,x);mpz_out_str(0,10,x);printf(",");if(i%16==15)printf("\n");}} +*/ +static const unsigned char approx_tab[192] = + { + 128,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142, + 143,144,144,145,146,147,148,149,150,150,151,152,153,154,155,155, + 156,157,158,159,160,160,161,162,163,163,164,165,166,167,167,168, + 169,170,170,171,172,173,173,174,175,176,176,177,178,178,179,180, + 181,181,182,183,183,184,185,185,186,187,187,188,189,189,190,191, + 192,192,193,193,194,195,195,196,197,197,198,199,199,200,201,201, + 202,203,203,204,204,205,206,206,207,208,208,209,209,210,211,211, + 212,212,213,214,214,215,215,216,217,217,218,218,219,219,220,221, + 221,222,222,223,224,224,225,225,226,226,227,227,228,229,229,230, + 230,231,231,232,232,233,234,234,235,235,236,236,237,237,238,238, + 239,240,240,241,241,242,242,243,243,244,244,245,245,246,246,247, + 247,248,248,249,249,250,250,251,251,252,252,253,253,254,254,255 + }; - 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 +#define HALF_NAIL (GMP_NAIL_BITS / 2) -#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] = +/* same as mpn_sqrtrem, but for size=1 and {np, 1} normalized */ +static mp_size_t +mpn_sqrtrem1 (mp_ptr sp, mp_ptr rp, mp_srcptr np) { - 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, -}; + mp_limb_t np0, s, r, q, u; + int prec; -/* 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 + ASSERT (np[0] >= GMP_NUMB_HIGHBIT / 2); + ASSERT (GMP_LIMB_BITS >= 16); - -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 */ + /* first compute a 8-bit approximation from the high 8-bits of np[0] */ + np0 = np[0] << GMP_NAIL_BITS; + q = np0 >> (GMP_LIMB_BITS - 8); + /* 2^6 = 64 <= q < 256 = 2^8 */ + s = approx_tab[q - 64]; /* 128 <= s < 255 */ + r = (np0 >> (GMP_LIMB_BITS - 16)) - s * s; /* r <= 2*s */ + if (r > 2 * s) + { + r -= 2 * s + 1; + s++; + } - /* 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) + prec = 8; + np0 <<= 2 * prec; + while (2 * prec < GMP_LIMB_BITS) { - cnt += BITS_PER_MP_LIMB; - tsize++; + /* invariant: s has prec bits, and r <= 2*s */ + r = (r << prec) + (np0 >> (GMP_LIMB_BITS - prec)); + np0 <<= prec; + u = 2 * s; + q = r / u; + u = r - q * u; + s = (s << prec) + q; + u = (u << prec) + (np0 >> (GMP_LIMB_BITS - prec)); + q = q * q; + r = u - q; + if (u < q) + { + r += 2 * s - 1; + s --; + } + np0 <<= prec; + prec = 2 * prec; } - rsize = tsize / 2; - rp = root_ptr; + ASSERT (2 * prec == GMP_LIMB_BITS); /* GMP_LIMB_BITS must be a power of 2 */ - TMP_MARK (marker); + /* normalize back, assuming GMP_NAIL_BITS is even */ + ASSERT (GMP_NAIL_BITS % 2 == 0); + sp[0] = s >> HALF_NAIL; + u = s - (sp[0] << HALF_NAIL); /* s mod 2^HALF_NAIL */ + r += u * ((sp[0] << (HALF_NAIL + 1)) + u); + r = r >> GMP_NAIL_BITS; - /* 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. + if (rp != NULL) + rp[0] = r; + return r != 0 ? 1 : 0; +} - 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. */ +#define Prec (GMP_NUMB_BITS >> 1) - tp = (mp_ptr) TMP_ALLOC (tsize * BYTES_PER_MP_LIMB); +/* same as mpn_sqrtrem, but for size=2 and {np, 2} normalized + return cc such that {np, 2} = sp[0]^2 + cc*2^GMP_NUMB_BITS + rp[0] */ +static mp_limb_t +mpn_sqrtrem2 (mp_ptr sp, mp_ptr rp, mp_srcptr np) +{ + mp_limb_t qhl, q, u, np0; + int cc; - 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); + ASSERT (np[1] >= GMP_NUMB_HIGHBIT / 2); - 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) + np0 = np[0]; + mpn_sqrtrem1 (sp, rp, np + 1); + qhl = 0; + while (rp[0] >= sp[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]; + qhl++; + rp[0] -= sp[0]; } - else + /* now rp[0] < sp[0] < 2^Prec */ + rp[0] = (rp[0] << Prec) + (np0 >> Prec); + u = 2 * sp[0]; + q = rp[0] / u; + u = rp[0] - q * u; + q += (qhl & 1) << (Prec - 1); + qhl >>= 1; /* if qhl=1, necessary q=0 as qhl*2^Prec + q <= 2^Prec */ + /* now we have (initial rp[0])<>Prec = (qhl<> Prec; + rp[0] = ((u << Prec) & GMP_NUMB_MASK) + (np0 & (((mp_limb_t) 1 << Prec) - 1)); + /* subtract q * q or qhl*2^(2*Prec) from rp */ + cc -= mpn_sub_1 (rp, rp, 1, q * q) + qhl; + /* now subtract 2*q*2^Prec + 2^(2*Prec) if qhl is set */ + if (cc < 0) { - /* 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]; + cc += sp[0] != 0 ? mpn_add_1 (rp, rp, 1, sp[0]) : 1; + cc += mpn_add_1 (rp, rp, 1, --sp[0]); } - 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; + return cc; +} - if (t_high0 >= initial_approx) - initial_approx = t_high0 + 1; +/* writes in {sp, n} the square root (rounded towards zero) of {np, 2n}, + and in {np, n} the low n limbs of the remainder, returns the high + limb of the remainder (which is 0 or 1). + Assumes {np, 2n} is normalized, i.e. np[2n-1] >= B/4 + where B=2^GMP_NUMB_BITS. */ +static mp_limb_t +mpn_dc_sqrtrem (mp_ptr sp, mp_ptr np, mp_size_t n) +{ + mp_limb_t q; /* carry out of {sp, n} */ + int c, b; /* carry out of remainder */ + mp_size_t l, h; - /* 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); + ASSERT (np[2 * n - 1] >= GMP_NUMB_HIGHBIT / 2); - /* 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; + if (n == 1) + c = mpn_sqrtrem2 (sp, np, np); + else + { + l = n / 2; + h = n - l; + q = mpn_dc_sqrtrem (sp + l, np + 2 * l, h); + if (q != 0) + mpn_sub_n (np + 2 * l, np + 2 * l, sp + l, h); + q += mpn_divrem (sp, 0, np + l, n, sp + l, h); + c = sp[0] & 1; + mpn_rshift (sp, sp, l, 1); + sp[l - 1] |= (q << (GMP_NUMB_BITS - 1)) & GMP_NUMB_MASK; + q >>= 1; + if (c != 0) + c = mpn_add_n (np + l, np + l, sp + l, h); + mpn_sqr_n (np + n, sp, l); + b = q + mpn_sub_n (np, np, np + n, 2 * l); + c -= (l == h) ? b : mpn_sub_1 (np + 2 * l, np + 2 * l, 1, b); + q = mpn_add_1 (sp + l, sp + l, h, q); - 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); - } + if (c < 0) + { + c += mpn_addmul_1 (np, sp, n, 2) + 2 * q; + c -= mpn_sub_1 (np, np, n, 1); + q -= mpn_sub_1 (sp, sp, n, 1); + } } -#endif - rp[0] = initial_approx; - rsize = 1; + return c; +} -#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). */ +mp_size_t +mpn_sqrtrem (mp_ptr sp, mp_ptr rp, mp_srcptr np, mp_size_t nn) +{ + mp_limb_t *tp, s0[1], cc, high, rl; + int c; + mp_size_t rn, tn; + TMP_DECL (marker); - xp = (mp_ptr) TMP_ALLOC (tsize * BYTES_PER_MP_LIMB); - ttp = (mp_ptr) TMP_ALLOC (tsize * BYTES_PER_MP_LIMB); + ASSERT (nn >= 0); - t_end_ptr = tp + tsize; + /* If OP is zero, both results are zero. */ + if (nn == 0) + return 0; - tmp = tsize / 2; - for (i = 0;; i++) - { - tsize = (tmp + 1) / 2; - if (tmp == tsize) - break; - tsizes[i] = tsize + tmp; - tmp = tsize; - } + ASSERT (np[nn - 1] != 0); + ASSERT (rp == NULL || MPN_SAME_OR_SEPARATE_P (np, rp, nn)); + ASSERT (rp == NULL || ! MPN_OVERLAP_P (sp, (nn + 1) / 2, rp, nn)); + ASSERT (! MPN_OVERLAP_P (sp, (nn + 1) / 2, np, nn)); - /* Main Newton iteration loop. For big arguments, most of the - time is spent here. */ + high = np[nn - 1]; + if (nn == 1 && (high & GMP_NUMB_HIGHBIT)) + return mpn_sqrtrem1 (sp, rp, np); + count_leading_zeros (c, high); + c -= GMP_NAIL_BITS; - /* 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. */ + c = c / 2; /* we have to shift left by 2c bits to normalize {np, nn} */ + tn = (nn + 1) / 2; /* 2*tn is the smallest even integer >= nn */ - /* 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) + TMP_MARK (marker); + if (nn % 2 != 0 || c > 0) + { + tp = TMP_ALLOC_LIMBS (2 * tn); + tp[0] = 0; /* needed only when 2*tn > nn, but saves a test */ + if (c != 0) + mpn_lshift (tp + 2 * tn - nn, np, nn, 2 * c); + else + MPN_COPY (tp + 2 * tn - nn, np, nn); + rl = mpn_dc_sqrtrem (sp, tp, tn); + /* We have 2^(2k)*N = S^2 + R where k = c + (2tn-nn)*GMP_NUMB_BITS/2, + thus 2^(2k)*N = (S-s0)^2 + 2*S*s0 - s0^2 + R where s0=S mod 2^k */ + c += (nn % 2) * GMP_NUMB_BITS / 2; /* c now represents k */ + s0[0] = sp[0] & (((mp_limb_t) 1 << c) - 1); /* S mod 2^k */ + rl += mpn_addmul_1 (tp, sp, tn, 2 * s0[0]); /* R = R + 2*s0*S */ + cc = mpn_submul_1 (tp, s0, 1, s0[0]); + rl -= (tn > 1) ? mpn_sub_1 (tp + 1, tp + 1, tn - 1, cc) : cc; + mpn_rshift (sp, sp, tn, c); + tp[tn] = rl; + if (rp == NULL) + rp = tp; + c = c << 1; + if (c < GMP_NUMB_BITS) + tn++; + else { - 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 + tp++; + c -= GMP_NUMB_BITS; } + if (c != 0) + mpn_rshift (rp, tp, tn, c); + else + MPN_COPY_INCR (rp, tp, tn); + rn = tn; } - -#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) + else { - mpn_rshift (rp, rp, rsize, cnt/2); - rsize -= rp[rsize - 1] == 0; + if (rp == NULL) + rp = TMP_ALLOC_LIMBS (nn); + if (rp != np) + MPN_COPY (rp, np, nn); + rn = tn + (rp[tn] = mpn_dc_sqrtrem (sp, rp, tn)); } - /* 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. */ + MPN_NORMALIZE (rp, rn); - /* 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; - } + TMP_FREE (marker); + return rn; }