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/* Copyright (C) 2007-2022 Free Software Foundation, Inc.
This file is part of GCC.
GCC 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 3, or (at your option) any later
version.
GCC 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.
Under Section 7 of GPL version 3, you are granted additional
permissions described in the GCC Runtime Library Exception, version
3.1, as published by the Free Software Foundation.
You should have received a copy of the GNU General Public License and
a copy of the GCC Runtime Library Exception along with this program;
see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
<http://www.gnu.org/licenses/>. */
#ifndef _SQRT_MACROS_H_
#define _SQRT_MACROS_H_
#define FENCE __fence
#if DOUBLE_EXTENDED_ON
extern BINARY80 SQRT80 (BINARY80);
__BID_INLINE__ UINT64
short_sqrt128 (UINT128 A10) {
BINARY80 lx, ly, l64;
int_float f64;
// 2^64
f64.i = 0x5f800000;
l64 = (BINARY80) f64.d;
lx = (BINARY80) A10.w[1] * l64 + (BINARY80) A10.w[0];
ly = SQRT80 (lx);
return (UINT64) ly;
}
__BID_INLINE__ void
long_sqrt128 (UINT128 * pCS, UINT256 C256) {
UINT256 C4;
UINT128 CS;
UINT64 X;
SINT64 SE;
BINARY80 l64, lm64, l128, lxL, lx, ly, lS, lSH, lSL, lE, l3, l2,
l1, l0, lp, lCl;
int_float fx, f64, fm64;
int *ple = (int *) &lx;
// 2^64
f64.i = 0x5f800000;
l64 = (BINARY80) f64.d;
l128 = l64 * l64;
lx = l3 = (BINARY80) C256.w[3] * l64 * l128;
l2 = (BINARY80) C256.w[2] * l128;
lx = FENCE (lx + l2);
l1 = (BINARY80) C256.w[1] * l64;
lx = FENCE (lx + l1);
l0 = (BINARY80) C256.w[0];
lx = FENCE (lx + l0);
// sqrt(C256)
lS = SQRT80 (lx);
// get coefficient
// 2^(-64)
fm64.i = 0x1f800000;
lm64 = (BINARY80) fm64.d;
CS.w[1] = (UINT64) (lS * lm64);
CS.w[0] = (UINT64) (lS - (BINARY80) CS.w[1] * l64);
///////////////////////////////////////
// CAUTION!
// little endian code only
// add solution for big endian
//////////////////////////////////////
lSH = lS;
*((UINT64 *) & lSH) &= 0xffffffff00000000ull;
// correction for C256 rounding
lCl = FENCE (l3 - lx);
lCl = FENCE (lCl + l2);
lCl = FENCE (lCl + l1);
lCl = FENCE (lCl + l0);
lSL = lS - lSH;
//////////////////////////////////////////
// Watch for compiler re-ordering
//
/////////////////////////////////////////
// C256-S^2
lxL = FENCE (lx - lSH * lSH);
lp = lSH * lSL;
lp += lp;
lxL = FENCE (lxL - lp);
lSL *= lSL;
lxL = FENCE (lxL - lSL);
lCl += lxL;
// correction term
lE = lCl / (lS + lS);
// get low part of coefficient
X = CS.w[0];
if (lCl >= 0) {
SE = (SINT64) (lE);
CS.w[0] += SE;
if (CS.w[0] < X)
CS.w[1]++;
} else {
SE = (SINT64) (-lE);
CS.w[0] -= SE;
if (CS.w[0] > X)
CS.w[1]--;
}
pCS->w[0] = CS.w[0];
pCS->w[1] = CS.w[1];
}
#else
extern double sqrt (double);
__BID_INLINE__ UINT64
short_sqrt128 (UINT128 A10) {
UINT256 ARS, ARS0, AE0, AE, S;
UINT64 MY, ES, CY;
double lx, l64;
int_double f64, ly;
int ey, k;
// 2^64
f64.i = 0x43f0000000000000ull;
l64 = f64.d;
lx = (double) A10.w[1] * l64 + (double) A10.w[0];
ly.d = 1.0 / sqrt (lx);
MY = (ly.i & 0x000fffffffffffffull) | 0x0010000000000000ull;
ey = 0x3ff - (ly.i >> 52);
// A10*RS^2
__mul_64x128_to_192 (ARS0, MY, A10);
__mul_64x192_to_256 (ARS, MY, ARS0);
// shr by 2*ey+40, to get a 64-bit value
k = (ey << 1) + 104 - 64;
if (k >= 128) {
if (k > 128)
ES = (ARS.w[2] >> (k - 128)) | (ARS.w[3] << (192 - k));
else
ES = ARS.w[2];
} else {
if (k >= 64) {
ARS.w[0] = ARS.w[1];
ARS.w[1] = ARS.w[2];
k -= 64;
}
if (k) {
__shr_128 (ARS, ARS, k);
}
ES = ARS.w[0];
}
ES = ((SINT64) ES) >> 1;
if (((SINT64) ES) < 0) {
ES = -ES;
// A*RS*eps (scaled by 2^64)
__mul_64x192_to_256 (AE0, ES, ARS0);
AE.w[0] = AE0.w[1];
AE.w[1] = AE0.w[2];
AE.w[2] = AE0.w[3];
__add_carry_out (S.w[0], CY, ARS0.w[0], AE.w[0]);
__add_carry_in_out (S.w[1], CY, ARS0.w[1], AE.w[1], CY);
S.w[2] = ARS0.w[2] + AE.w[2] + CY;
} else {
// A*RS*eps (scaled by 2^64)
__mul_64x192_to_256 (AE0, ES, ARS0);
AE.w[0] = AE0.w[1];
AE.w[1] = AE0.w[2];
AE.w[2] = AE0.w[3];
__sub_borrow_out (S.w[0], CY, ARS0.w[0], AE.w[0]);
__sub_borrow_in_out (S.w[1], CY, ARS0.w[1], AE.w[1], CY);
S.w[2] = ARS0.w[2] - AE.w[2] - CY;
}
k = ey + 51;
if (k >= 64) {
if (k >= 128) {
S.w[0] = S.w[2];
S.w[1] = 0;
k -= 128;
} else {
S.w[0] = S.w[1];
S.w[1] = S.w[2];
}
k -= 64;
}
if (k) {
__shr_128 (S, S, k);
}
return (UINT64) ((S.w[0] + 1) >> 1);
}
__BID_INLINE__ void
long_sqrt128 (UINT128 * pCS, UINT256 C256) {
UINT512 ARS0, ARS;
UINT256 ARS00, AE, AE2, S;
UINT128 ES, ES2, ARS1;
UINT64 ES32, CY, MY;
double l64, l128, lx, l2, l1, l0;
int_double f64, ly;
int ey, k, k2;
// 2^64
f64.i = 0x43f0000000000000ull;
l64 = f64.d;
l128 = l64 * l64;
lx = (double) C256.w[3] * l64 * l128;
l2 = (double) C256.w[2] * l128;
lx = FENCE (lx + l2);
l1 = (double) C256.w[1] * l64;
lx = FENCE (lx + l1);
l0 = (double) C256.w[0];
lx = FENCE (lx + l0);
// sqrt(C256)
ly.d = 1.0 / sqrt (lx);
MY = (ly.i & 0x000fffffffffffffull) | 0x0010000000000000ull;
ey = 0x3ff - (ly.i >> 52);
// A10*RS^2, scaled by 2^(2*ey+104)
__mul_64x256_to_320 (ARS0, MY, C256);
__mul_64x320_to_384 (ARS, MY, ARS0);
// shr by k=(2*ey+104)-128
// expect k is in the range (192, 256) if result in [10^33, 10^34)
// apply an additional signed shift by 1 at the same time (to get eps=eps0/2)
k = (ey << 1) + 104 - 128 - 192;
k2 = 64 - k;
ES.w[0] = (ARS.w[3] >> (k + 1)) | (ARS.w[4] << (k2 - 1));
ES.w[1] = (ARS.w[4] >> k) | (ARS.w[5] << k2);
ES.w[1] = ((SINT64) ES.w[1]) >> 1;
// A*RS >> 192 (for error term computation)
ARS1.w[0] = ARS0.w[3];
ARS1.w[1] = ARS0.w[4];
// A*RS>>64
ARS00.w[0] = ARS0.w[1];
ARS00.w[1] = ARS0.w[2];
ARS00.w[2] = ARS0.w[3];
ARS00.w[3] = ARS0.w[4];
if (((SINT64) ES.w[1]) < 0) {
ES.w[0] = -ES.w[0];
ES.w[1] = -ES.w[1];
if (ES.w[0])
ES.w[1]--;
// A*RS*eps
__mul_128x128_to_256 (AE, ES, ARS1);
__add_carry_out (S.w[0], CY, ARS00.w[0], AE.w[0]);
__add_carry_in_out (S.w[1], CY, ARS00.w[1], AE.w[1], CY);
__add_carry_in_out (S.w[2], CY, ARS00.w[2], AE.w[2], CY);
S.w[3] = ARS00.w[3] + AE.w[3] + CY;
} else {
// A*RS*eps
__mul_128x128_to_256 (AE, ES, ARS1);
__sub_borrow_out (S.w[0], CY, ARS00.w[0], AE.w[0]);
__sub_borrow_in_out (S.w[1], CY, ARS00.w[1], AE.w[1], CY);
__sub_borrow_in_out (S.w[2], CY, ARS00.w[2], AE.w[2], CY);
S.w[3] = ARS00.w[3] - AE.w[3] - CY;
}
// 3/2*eps^2, scaled by 2^128
ES32 = ES.w[1] + (ES.w[1] >> 1);
__mul_64x64_to_128 (ES2, ES32, ES.w[1]);
// A*RS*3/2*eps^2
__mul_128x128_to_256 (AE2, ES2, ARS1);
// result, scaled by 2^(ey+52-64)
__add_carry_out (S.w[0], CY, S.w[0], AE2.w[0]);
__add_carry_in_out (S.w[1], CY, S.w[1], AE2.w[1], CY);
__add_carry_in_out (S.w[2], CY, S.w[2], AE2.w[2], CY);
S.w[3] = S.w[3] + AE2.w[3] + CY;
// k in (0, 64)
k = ey + 51 - 128;
k2 = 64 - k;
S.w[0] = (S.w[1] >> k) | (S.w[2] << k2);
S.w[1] = (S.w[2] >> k) | (S.w[3] << k2);
// round to nearest
S.w[0]++;
if (!S.w[0])
S.w[1]++;
pCS->w[0] = (S.w[1] << 63) | (S.w[0] >> 1);
pCS->w[1] = S.w[1] >> 1;
}
#endif
#endif