/* Implement powl for x86 using extra-precision log.
Copyright (C) 2012-2018 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
<http://www.gnu.org/licenses/>. */
#include <math.h>
#include <math_private.h>
#include <math-underflow.h>
#include <stdbool.h>
/* High parts and low parts of -log (k/16), for integer k from 12 to
24. */
static const long double powl_log_table[] =
{
0x4.9a58844d36e49e1p-4L, -0x1.0522624fd558f574p-68L,
0x3.527da7915b3c6de4p-4L, 0x1.7d4ef4b901b99b9ep-68L,
0x2.22f1d044fc8f7bc8p-4L, -0x1.8e97c071a42fc388p-68L,
0x1.08598b59e3a0688ap-4L, 0x3.fd9bf503372c12fcp-72L,
-0x0p+0L, 0x0p+0L,
-0xf.85186008b15330cp-8L, 0x1.9b47488a6687672cp-72L,
-0x1.e27076e2af2e5e9ep-4L, -0xa.87ffe1fe9e155dcp-72L,
-0x2.bfe60e14f27a791p-4L, 0x1.83bebf1bdb88a032p-68L,
-0x3.91fef8f353443584p-4L, -0xb.b03de5ff734495cp-72L,
-0x4.59d72aeae98380e8p-4L, 0xc.e0aa3be4747dc1p-72L,
-0x5.1862f08717b09f4p-4L, -0x2.decdeccf1cd10578p-68L,
-0x5.ce75fdaef401a738p-4L, -0x9.314feb4fbde5aaep-72L,
-0x6.7cc8fb2fe612fcbp-4L, 0x2.5ca2642feb779f98p-68L,
};
/* High 32 bits of log2 (e), and remainder rounded to 64 bits. */
static const long double log2e_hi = 0x1.71547652p+0L;
static const long double log2e_lo = 0xb.82fe1777d0ffda1p-36L;
/* Given a number with high part HI and low part LO, add the number X
to it and store the result in *RHI and *RLO. It is given that
either |X| < |0.7 * HI|, or HI == LO == 0, and that the values are
small enough that no overflow occurs. The result does not need to
be exact to 128 bits; 78-bit accuracy of the final accumulated
result suffices. */
static inline void
acc_split (long double *rhi, long double *rlo, long double hi, long double lo,
long double x)
{
long double thi = hi + x;
long double tlo = (hi - thi) + x + lo;
*rhi = thi + tlo;
*rlo = (thi - *rhi) + tlo;
}
extern long double __powl_helper (long double x, long double y);
libm_hidden_proto (__powl_helper)
/* Given X a value that is finite and nonzero, or a NaN, and Y a
finite nonzero value with 0x1p-79 <= |Y| <= 0x1p78, compute X to
the power Y. */
long double
__powl_helper (long double x, long double y)
{
if (isnan (x))
return __ieee754_expl (y * __ieee754_logl (x));
bool negate;
if (x < 0)
{
long double absy = fabsl (y);
if (absy >= 0x1p64L)
negate = false;
else
{
unsigned long long yll = absy;
if (yll != absy)
return __ieee754_expl (y * __ieee754_logl (x));
negate = (yll & 1) != 0;
}
x = fabsl (x);
}
else
negate = false;
/* We need to compute Y * log2 (X) to at least 64 bits after the
point for normal results (that is, to at least 78 bits
precision). */
int x_int_exponent;
long double x_frac;
x_frac = __frexpl (x, &x_int_exponent);
if (x_frac <= 0x0.aaaaaaaaaaaaaaaap0L) /* 2.0L / 3.0L, rounded down */
{
x_frac *= 2.0;
x_int_exponent--;
}
long double log_x_frac_hi, log_x_frac_lo;
/* Determine an initial approximation to log (X_FRAC) using
POWL_LOG_TABLE, and multiply by a value K/16 to reduce to an
interval (24/25, 26/25). */
int k = (int) ((16.0L / x_frac) + 0.5L);
log_x_frac_hi = powl_log_table[2 * k - 24];
log_x_frac_lo = powl_log_table[2 * k - 23];
long double x_frac_low;
if (k == 16)
x_frac_low = 0.0L;
else
{
/* Mask off low 5 bits of X_FRAC so the multiplication by K/16
is exact. These bits are small enough that they can be
corrected for by adding log2 (e) * X_FRAC_LOW to the final
result. */
int32_t se;
uint32_t i0, i1;
GET_LDOUBLE_WORDS (se, i0, i1, x_frac);
x_frac_low = x_frac;
i1 &= 0xffffffe0;
SET_LDOUBLE_WORDS (x_frac, se, i0, i1);
x_frac_low -= x_frac;
x_frac_low /= x_frac;
x_frac *= k / 16.0L;
}
/* Now compute log (X_FRAC) for X_FRAC in (24/25, 26/25). Separate
W = X_FRAC - 1 into high 16 bits and remaining bits, so that
multiplications for low-order power series terms are exact. The
remaining bits are small enough that adding a 64-bit value of
log2 (1 + W_LO / (1 + W_HI)) will be a sufficient correction for
them. */
long double w = x_frac - 1;
long double w_hi, w_lo;
int32_t se;
uint32_t i0, i1;
GET_LDOUBLE_WORDS (se, i0, i1, w);
i0 &= 0xffff0000;
i1 = 0;
SET_LDOUBLE_WORDS (w_hi, se, i0, i1);
w_lo = w - w_hi;
long double wp = w_hi;
acc_split (&log_x_frac_hi, &log_x_frac_lo, log_x_frac_hi, log_x_frac_lo, wp);
wp *= -w_hi;
acc_split (&log_x_frac_hi, &log_x_frac_lo, log_x_frac_hi, log_x_frac_lo,
wp / 2.0L);
wp *= -w_hi;
acc_split (&log_x_frac_hi, &log_x_frac_lo, log_x_frac_hi, log_x_frac_lo,
wp * 0x0.5555p0L); /* -W_HI**3 / 3, high part. */
acc_split (&log_x_frac_hi, &log_x_frac_lo, log_x_frac_hi, log_x_frac_lo,
wp * 0x0.5555555555555555p-16L); /* -W_HI**3 / 3, low part. */
wp *= -w_hi;
acc_split (&log_x_frac_hi, &log_x_frac_lo, log_x_frac_hi, log_x_frac_lo,
wp / 4.0L);
/* Subsequent terms are small enough that they only need be computed
to 64 bits. */
for (int i = 5; i <= 17; i++)
{
wp *= -w_hi;
acc_split (&log_x_frac_hi, &log_x_frac_lo, log_x_frac_hi, log_x_frac_lo,
wp / i);
}
/* Convert LOG_X_FRAC_HI + LOG_X_FRAC_LO to a base-2 logarithm. */
long double log2_x_frac_hi, log2_x_frac_lo;
long double log_x_frac_hi32, log_x_frac_lo64;
GET_LDOUBLE_WORDS (se, i0, i1, log_x_frac_hi);
i1 = 0;
SET_LDOUBLE_WORDS (log_x_frac_hi32, se, i0, i1);
log_x_frac_lo64 = (log_x_frac_hi - log_x_frac_hi32) + log_x_frac_lo;
long double log2_x_frac_hi1 = log_x_frac_hi32 * log2e_hi;
long double log2_x_frac_lo1
= log_x_frac_lo64 * log2e_hi + log_x_frac_hi * log2e_lo;
log2_x_frac_hi = log2_x_frac_hi1 + log2_x_frac_lo1;
log2_x_frac_lo = (log2_x_frac_hi1 - log2_x_frac_hi) + log2_x_frac_lo1;
/* Correct for the masking off of W_LO. */
long double log2_1p_w_lo;
asm ("fyl2xp1"
: "=t" (log2_1p_w_lo)
: "0" (w_lo / (1.0L + w_hi)), "u" (1.0L)
: "st(1)");
acc_split (&log2_x_frac_hi, &log2_x_frac_lo, log2_x_frac_hi, log2_x_frac_lo,
log2_1p_w_lo);
/* Correct for the masking off of X_FRAC_LOW. */
acc_split (&log2_x_frac_hi, &log2_x_frac_lo, log2_x_frac_hi, log2_x_frac_lo,
x_frac_low * M_LOG2El);
/* Add the integer and fractional parts of the base-2 logarithm. */
long double log2_x_hi, log2_x_lo;
log2_x_hi = x_int_exponent + log2_x_frac_hi;
log2_x_lo = ((x_int_exponent - log2_x_hi) + log2_x_frac_hi) + log2_x_frac_lo;
/* Compute the base-2 logarithm of the result. */
long double log2_res_hi, log2_res_lo;
long double log2_x_hi32, log2_x_lo64;
GET_LDOUBLE_WORDS (se, i0, i1, log2_x_hi);
i1 = 0;
SET_LDOUBLE_WORDS (log2_x_hi32, se, i0, i1);
log2_x_lo64 = (log2_x_hi - log2_x_hi32) + log2_x_lo;
long double y_hi32, y_lo32;
GET_LDOUBLE_WORDS (se, i0, i1, y);
i1 = 0;
SET_LDOUBLE_WORDS (y_hi32, se, i0, i1);
y_lo32 = y - y_hi32;
log2_res_hi = log2_x_hi32 * y_hi32;
log2_res_lo = log2_x_hi32 * y_lo32 + log2_x_lo64 * y;
/* Split the base-2 logarithm of the result into integer and
fractional parts. */
long double log2_res_int = __roundl (log2_res_hi);
long double log2_res_frac = log2_res_hi - log2_res_int + log2_res_lo;
/* If the integer part is very large, the computed fractional part
may be outside the valid range for f2xm1. */
if (fabsl (log2_res_int) > 16500)
log2_res_frac = 0;
/* Compute the final result. */
long double res;
asm ("f2xm1" : "=t" (res) : "0" (log2_res_frac));
res += 1.0L;
if (negate)
res = -res;
asm ("fscale" : "=t" (res) : "0" (res), "u" (log2_res_int));
math_check_force_underflow (res);
return res;
}
libm_hidden_def (__powl_helper)