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/* Double-precision floating point square root.
Copyright (C) 1997-2025 Free Software Foundation, Inc.
This file is part of the GNU C Library.
The GNU C Library is free software; you can redistribute it and/or
modify 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 C 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
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with the GNU C Library; if not, see
<https://www.gnu.org/licenses/>. */
#include <math.h>
#include <math_private.h>
#include <fenv_libc.h>
#include <libm-alias-finite.h>
#include <math-use-builtins.h>
double
__ieee754_sqrt (double x)
{
#if USE_SQRT_BUILTIN
return __builtin_sqrt (x);
#else
/* The method is based on a description in
Computation of elementary functions on the IBM RISC System/6000 processor,
P. W. Markstein, IBM J. Res. Develop, 34(1) 1990.
Basically, it consists of two interleaved Newton-Raphson approximations,
one to find the actual square root, and one to find its reciprocal
without the expense of a division operation. The tricky bit here
is the use of the POWER/PowerPC multiply-add operation to get the
required accuracy with high speed.
The argument reduction works by a combination of table lookup to
obtain the initial guesses, and some careful modification of the
generated guesses (which mostly runs on the integer unit, while the
Newton-Raphson is running on the FPU). */
extern const float __t_sqrt[1024];
if (x > 0)
{
/* schedule the EXTRACT_WORDS to get separation between the store
and the load. */
ieee_double_shape_type ew_u;
ieee_double_shape_type iw_u;
ew_u.value = (x);
if (x != INFINITY)
{
/* Variables named starting with 's' exist in the
argument-reduced space, so that 2 > sx >= 0.5,
1.41... > sg >= 0.70.., 0.70.. >= sy > 0.35... .
Variables named ending with 'i' are integer versions of
floating-point values. */
double sx; /* The value of which we're trying to find the
square root. */
double sg, g; /* Guess of the square root of x. */
double sd, d; /* Difference between the square of the guess and x. */
double sy; /* Estimate of 1/2g (overestimated by 1ulp). */
double sy2; /* 2*sy */
double e; /* Difference between y*g and 1/2 (se = e * fsy). */
double shx; /* == sx * fsg */
double fsg; /* sg*fsg == g. */
fenv_t fe; /* Saved floating-point environment (stores rounding
mode and whether the inexact exception is
enabled). */
uint32_t xi0, xi1, sxi, fsgi;
const float *t_sqrt;
fe = fegetenv_register ();
/* complete the EXTRACT_WORDS (xi0,xi1,x) operation. */
xi0 = ew_u.parts.msw;
xi1 = ew_u.parts.lsw;
relax_fenv_state ();
sxi = (xi0 & 0x3fffffff) | 0x3fe00000;
/* schedule the INSERT_WORDS (sx, sxi, xi1) to get separation
between the store and the load. */
iw_u.parts.msw = sxi;
iw_u.parts.lsw = xi1;
t_sqrt = __t_sqrt + (xi0 >> (52 - 32 - 8 - 1) & 0x3fe);
sg = t_sqrt[0];
sy = t_sqrt[1];
/* complete the INSERT_WORDS (sx, sxi, xi1) operation. */
sx = iw_u.value;
/* Here we have three Newton-Raphson iterations each of a
division and a square root and the remainder of the
argument reduction, all interleaved. */
sd = -__builtin_fma (sg, sg, -sx);
fsgi = (xi0 + 0x40000000) >> 1 & 0x7ff00000;
sy2 = sy + sy;
sg = __builtin_fma (sy, sd, sg); /* 16-bit approximation to
sqrt(sx). */
/* schedule the INSERT_WORDS (fsg, fsgi, 0) to get separation
between the store and the load. */
INSERT_WORDS (fsg, fsgi, 0);
iw_u.parts.msw = fsgi;
iw_u.parts.lsw = (0);
e = -__builtin_fma (sy, sg, -0x1.0000000000001p-1);
sd = -__builtin_fma (sg, sg, -sx);
if ((xi0 & 0x7ff00000) == 0)
goto denorm;
sy = __builtin_fma (e, sy2, sy);
sg = __builtin_fma (sy, sd, sg); /* 32-bit approximation to
sqrt(sx). */
sy2 = sy + sy;
/* complete the INSERT_WORDS (fsg, fsgi, 0) operation. */
fsg = iw_u.value;
e = -__builtin_fma (sy, sg, -0x1.0000000000001p-1);
sd = -__builtin_fma (sg, sg, -sx);
sy = __builtin_fma (e, sy2, sy);
shx = sx * fsg;
sg = __builtin_fma (sy, sd, sg); /* 64-bit approximation to
sqrt(sx), but perhaps
rounded incorrectly. */
sy2 = sy + sy;
g = sg * fsg;
e = -__builtin_fma (sy, sg, -0x1.0000000000001p-1);
d = -__builtin_fma (g, sg, -shx);
sy = __builtin_fma (e, sy2, sy);
fesetenv_register (fe);
return __builtin_fma (sy, d, g);
denorm:
/* For denormalised numbers, we normalise, calculate the
square root, and return an adjusted result. */
fesetenv_register (fe);
return __ieee754_sqrt (x * 0x1p+108f) * 0x1p-54f;
}
}
else if (x < 0)
{
/* For some reason, some PowerPC32 processors don't implement
FE_INVALID_SQRT. */
# ifdef FE_INVALID_SQRT
__feraiseexcept (FE_INVALID_SQRT);
fenv_union_t u = { .fenv = fegetenv_register () };
if ((u.l & FE_INVALID) == 0)
# endif
__feraiseexcept (FE_INVALID);
x = NAN;
}
return f_wash (x);
#endif /* USE_SQRT_BUILTIN */
}
libm_alias_finite (__ieee754_sqrt, __sqrt)
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