--- /dev/null
+/* 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. */
+\f
+/* 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
+
+\f
+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;
+ }
+}
projects / ghc-hetmet.git / blobdiff