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/* Single-precision pow function.
Copyright (C) 2017-2019 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-barriers.h>
#include <math-narrow-eval.h>
#include <stdint.h>
#include <shlib-compat.h>
#include <libm-alias-float.h>
#include "math_config.h"
/*
POWF_LOG2_POLY_ORDER = 5
EXP2F_TABLE_BITS = 5
ULP error: 0.82 (~ 0.5 + relerr*2^24)
relerr: 1.27 * 2^-26 (Relative error ~= 128*Ln2*relerr_log2 + relerr_exp2)
relerr_log2: 1.83 * 2^-33 (Relative error of logx.)
relerr_exp2: 1.69 * 2^-34 (Relative error of exp2(ylogx).)
*/
#define N (1 << POWF_LOG2_TABLE_BITS)
#define T __powf_log2_data.tab
#define A __powf_log2_data.poly
#define OFF 0x3f330000
/* Subnormal input is normalized so ix has negative biased exponent.
Output is multiplied by N (POWF_SCALE) if TOINT_INTRINICS is set. */
static inline double_t
log2_inline (uint32_t ix)
{
/* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
double_t z, r, r2, r4, p, q, y, y0, invc, logc;
uint32_t iz, top, tmp;
int k, i;
/* x = 2^k z; where z is in range [OFF,2*OFF] and exact.
The range is split into N subintervals.
The ith subinterval contains z and c is near its center. */
tmp = ix - OFF;
i = (tmp >> (23 - POWF_LOG2_TABLE_BITS)) % N;
top = tmp & 0xff800000;
iz = ix - top;
k = (int32_t) top >> (23 - POWF_SCALE_BITS); /* arithmetic shift */
invc = T[i].invc;
logc = T[i].logc;
z = (double_t) asfloat (iz);
/* log2(x) = log1p(z/c-1)/ln2 + log2(c) + k */
r = z * invc - 1;
y0 = logc + (double_t) k;
/* Pipelined polynomial evaluation to approximate log1p(r)/ln2. */
r2 = r * r;
y = A[0] * r + A[1];
p = A[2] * r + A[3];
r4 = r2 * r2;
q = A[4] * r + y0;
q = p * r2 + q;
y = y * r4 + q;
return y;
}
#undef N
#undef T
#define N (1 << EXP2F_TABLE_BITS)
#define T __exp2f_data.tab
#define SIGN_BIAS (1 << (EXP2F_TABLE_BITS + 11))
/* The output of log2 and thus the input of exp2 is either scaled by N
(in case of fast toint intrinsics) or not. The unscaled xd must be
in [-1021,1023], sign_bias sets the sign of the result. */
static inline double_t
exp2_inline (double_t xd, uint32_t sign_bias)
{
uint64_t ki, ski, t;
/* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
double_t kd, z, r, r2, y, s;
#if TOINT_INTRINSICS
# define C __exp2f_data.poly_scaled
/* N*x = k + r with r in [-1/2, 1/2] */
kd = roundtoint (xd); /* k */
ki = converttoint (xd);
#else
# define C __exp2f_data.poly
# define SHIFT __exp2f_data.shift_scaled
/* x = k/N + r with r in [-1/(2N), 1/(2N)] */
kd = (double) (xd + SHIFT); /* Rounding to double precision is required. */
ki = asuint64 (kd);
kd -= SHIFT; /* k/N */
#endif
r = xd - kd;
/* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */
t = T[ki % N];
ski = ki + sign_bias;
t += ski << (52 - EXP2F_TABLE_BITS);
s = asdouble (t);
z = C[0] * r + C[1];
r2 = r * r;
y = C[2] * r + 1;
y = z * r2 + y;
y = y * s;
return y;
}
/* Returns 0 if not int, 1 if odd int, 2 if even int. The argument is
the bit representation of a non-zero finite floating-point value. */
static inline int
checkint (uint32_t iy)
{
int e = iy >> 23 & 0xff;
if (e < 0x7f)
return 0;
if (e > 0x7f + 23)
return 2;
if (iy & ((1 << (0x7f + 23 - e)) - 1))
return 0;
if (iy & (1 << (0x7f + 23 - e)))
return 1;
return 2;
}
static inline int
zeroinfnan (uint32_t ix)
{
return 2 * ix - 1 >= 2u * 0x7f800000 - 1;
}
float
__powf (float x, float y)
{
uint32_t sign_bias = 0;
uint32_t ix, iy;
ix = asuint (x);
iy = asuint (y);
if (__glibc_unlikely (ix - 0x00800000 >= 0x7f800000 - 0x00800000
|| zeroinfnan (iy)))
{
/* Either (x < 0x1p-126 or inf or nan) or (y is 0 or inf or nan). */
if (__glibc_unlikely (zeroinfnan (iy)))
{
if (2 * iy == 0)
return issignalingf_inline (x) ? x + y : 1.0f;
if (ix == 0x3f800000)
return issignalingf_inline (y) ? x + y : 1.0f;
if (2 * ix > 2u * 0x7f800000 || 2 * iy > 2u * 0x7f800000)
return x + y;
if (2 * ix == 2 * 0x3f800000)
return 1.0f;
if ((2 * ix < 2 * 0x3f800000) == !(iy & 0x80000000))
return 0.0f; /* |x|<1 && y==inf or |x|>1 && y==-inf. */
return y * y;
}
if (__glibc_unlikely (zeroinfnan (ix)))
{
float_t x2 = x * x;
if (ix & 0x80000000 && checkint (iy) == 1)
{
x2 = -x2;
sign_bias = 1;
}
#if WANT_ERRNO
if (2 * ix == 0 && iy & 0x80000000)
return __math_divzerof (sign_bias);
#endif
return iy & 0x80000000 ? 1 / x2 : x2;
}
/* x and y are non-zero finite. */
if (ix & 0x80000000)
{
/* Finite x < 0. */
int yint = checkint (iy);
if (yint == 0)
return __math_invalidf (x);
if (yint == 1)
sign_bias = SIGN_BIAS;
ix &= 0x7fffffff;
}
if (ix < 0x00800000)
{
/* Normalize subnormal x so exponent becomes negative. */
ix = asuint (x * 0x1p23f);
ix &= 0x7fffffff;
ix -= 23 << 23;
}
}
double_t logx = log2_inline (ix);
double_t ylogx = y * logx; /* Note: cannot overflow, y is single prec. */
if (__glibc_unlikely ((asuint64 (ylogx) >> 47 & 0xffff)
>= asuint64 (126.0 * POWF_SCALE) >> 47))
{
/* |y*log(x)| >= 126. */
if (ylogx > 0x1.fffffffd1d571p+6 * POWF_SCALE)
/* |x^y| > 0x1.ffffffp127. */
return __math_oflowf (sign_bias);
if (WANT_ROUNDING && WANT_ERRNO
&& ylogx > 0x1.fffffffa3aae2p+6 * POWF_SCALE)
/* |x^y| > 0x1.fffffep127, check if we round away from 0. */
if ((!sign_bias
&& math_narrow_eval (1.0f + math_opt_barrier (0x1p-25f)) != 1.0f)
|| (sign_bias
&& math_narrow_eval (-1.0f - math_opt_barrier (0x1p-25f))
!= -1.0f))
return __math_oflowf (sign_bias);
if (ylogx <= -150.0 * POWF_SCALE)
return __math_uflowf (sign_bias);
#if WANT_ERRNO_UFLOW
if (ylogx < -149.0 * POWF_SCALE)
return __math_may_uflowf (sign_bias);
#endif
}
return (float) exp2_inline (ylogx, sign_bias);
}
#ifndef __powf
strong_alias (__powf, __ieee754_powf)
strong_alias (__powf, __powf_finite)
versioned_symbol (libm, __powf, powf, GLIBC_2_27);
libm_alias_float_other (__pow, pow)
#endif
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