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/* 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
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
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; \
#if defined __HAVE_68881__
#define SQRT(a) \
({ \
double __sqrt_res; \
asm ("fsqrtx %1,%0" : "=f" (__sqrt_res) : "f" (a)); \
__sqrt_res; \
#if defined __hppa
#define SQRT(a) \
({ \
double __sqrt_res; \
asm ("fsqrt,dbl %1,%0" : "=fx" (__sqrt_res) : "fx" (a)); \
__sqrt_res; \
#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,
#ifdef __STDC__
mpn_sqrt (mp_ptr root_ptr, mp_ptr rem_ptr, mp_srcptr op_ptr, mp_size op_size)
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;
/* 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)
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;
/* 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];
/* 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;
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);
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);
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)
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);
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);
/* 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]);
#ifdef DEBUG
printf ("(final) R = ");
_mp_mout (rp, rsize);
/* 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);
if (rem_ptr != NULL)
mp_size retval = op_size + mpn_sub (rem_ptr, op_ptr, op_size, tp, tsize);
alloca (0);
return retval;
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 ("");