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| | /* origin: FreeBSD /usr/src/lib/msun/src/s_exp2f.c */
/*-
* Copyright (c) 2005 David Schultz <das@FreeBSD.ORG>
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
#include "libm.h"
#define TBLSIZE 16
static const float
redux = 0x1.8p23f / TBLSIZE,
P1 = 0x1.62e430p-1f,
P2 = 0x1.ebfbe0p-3f,
P3 = 0x1.c6b348p-5f,
P4 = 0x1.3b2c9cp-7f;
static const double exp2ft[TBLSIZE] = {
0x1.6a09e667f3bcdp-1,
0x1.7a11473eb0187p-1,
0x1.8ace5422aa0dbp-1,
0x1.9c49182a3f090p-1,
0x1.ae89f995ad3adp-1,
0x1.c199bdd85529cp-1,
0x1.d5818dcfba487p-1,
0x1.ea4afa2a490dap-1,
0x1.0000000000000p+0,
0x1.0b5586cf9890fp+0,
0x1.172b83c7d517bp+0,
0x1.2387a6e756238p+0,
0x1.306fe0a31b715p+0,
0x1.3dea64c123422p+0,
0x1.4bfdad5362a27p+0,
0x1.5ab07dd485429p+0,
};
/*
* exp2f(x): compute the base 2 exponential of x
*
* Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
*
* Method: (equally-spaced tables)
*
* Reduce x:
* x = k + y, for integer k and |y| <= 1/2.
* Thus we have exp2f(x) = 2**k * exp2(y).
*
* Reduce y:
* y = i/TBLSIZE + z for integer i near y * TBLSIZE.
* Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
* with |z| <= 2**-(TBLSIZE+1).
*
* We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
* degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
* Using double precision for everything except the reduction makes
* roundoff error insignificant and simplifies the scaling step.
*
* This method is due to Tang, but I do not use his suggested parameters:
*
* Tang, P. Table-driven Implementation of the Exponential Function
* in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989).
*/
float exp2f(float x)
{
double_t t, r, z;
union {float f; uint32_t i;} u = {x};
union {double f; uint64_t i;} uk;
uint32_t ix, i0, k;
/* Filter out exceptional cases. */
ix = u.i & 0x7fffffff;
if (ix > 0x42fc0000) { /* |x| > 126 */
if (ix > 0x7f800000) /* NaN */
return x;
if (u.i >= 0x43000000 && u.i < 0x80000000) { /* x >= 128 */
x *= 0x1p127f;
return x;
}
if (u.i >= 0x80000000) { /* x < -126 */
if (u.i >= 0xc3160000 || (u.i & 0x0000ffff))
FORCE_EVAL(-0x1p-149f/x);
if (u.i >= 0xc3160000) /* x <= -150 */
return 0;
}
} else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */
return 1.0f + x;
}
/* Reduce x, computing z, i0, and k. */
u.f = x + redux;
i0 = u.i;
i0 += TBLSIZE / 2;
k = i0 / TBLSIZE;
uk.i = (uint64_t)(0x3ff + k)<<52;
i0 &= TBLSIZE - 1;
u.f -= redux;
z = x - u.f;
/* Compute r = exp2(y) = exp2ft[i0] * p(z). */
r = exp2ft[i0];
t = r * z;
r = r + t * (P1 + z * P2) + t * (z * z) * (P3 + z * P4);
/* Scale by 2**k */
return r * uk.f;
}
|