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/*-
* 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 <sys/cdefs.h>
__RCSID("$NetBSD: s_exp2f.c,v 1.2 2014/03/16 22:30:43 dsl Exp $");
#ifdef __FBSDID
__FBSDID("$FreeBSD: src/lib/msun/src/s_exp2f.c,v 1.9 2008/02/22 02:27:34 das Exp $");
#endif
#include <float.h>
#include "math.h"
#include "math_private.h"
#define TBLBITS 4
#define TBLSIZE (1 << TBLBITS)
static const float
redux = 0x1.8p23f / TBLSIZE,
P1 = 0x1.62e430p-1f,
P2 = 0x1.ebfbe0p-3f,
P3 = 0x1.c6b348p-5f,
P4 = 0x1.3b2c9cp-7f;
/*
* For out of range values we need to generate the appropriate
* underflow or overflow trap as well as generating infinity or zero.
* This means we have to get the fpu to execute an instruction that
* will generate the trap (and not have the compiler optimise it away).
* This is normally done by calculating 'huge * huge' or 'tiny * tiny'.
*
* i386 is particularly problematic.
* The 'float' result is returned on the x87 stack, so is 'long double'.
* If we just multiply two 'float' values the caller will see 0x1p+/-200
* (not 0 or infinity).
* If we use 'double' the compiler does a store-load which will convert the
* value and generate the required exception.
*/
#ifdef __i386__
static volatile double overflow = 0x1p+1000;
static volatile double underflow = 0x1p-1000;
#else
static volatile float huge = 0x1p+100;
static volatile float tiny = 0x1p-100;
#define overflow (huge * huge)
#define underflow (tiny * tiny)
#endif
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 = 2**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 tv, twopk, u, z;
float t;
uint32_t hx, ix, i0;
int32_t k;
/* Filter out exceptional cases. */
GET_FLOAT_WORD(hx, x);
ix = hx & 0x7fffffff; /* high word of |x| */
if(ix >= 0x43000000) { /* |x| >= 128 */
if(ix >= 0x7f800000) {
if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0)
return (x + x); /* x is NaN or +Inf */
else
return (0.0); /* x is -Inf */
}
if(x >= 0x1.0p7f)
return overflow; /* +infinity with overflow */
if(x <= -0x1.2cp7f)
return underflow; /* zero with underflow */
} else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */
return (1.0f + x);
}
/* Reduce x, computing z, i0, and k. */
STRICT_ASSIGN(float, t, x + redux);
GET_FLOAT_WORD(i0, t);
i0 += TBLSIZE / 2;
k = (i0 >> TBLBITS) << 20;
i0 &= TBLSIZE - 1;
t -= redux;
z = x - t;
INSERT_WORDS(twopk, 0x3ff00000 + k, 0);
/* Compute r = exp2(y) = exp2ft[i0] * p(z). */
tv = exp2ft[i0];
u = tv * z;
tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4);
/* Scale by 2**(k>>20). */
return (tv * twopk);
}
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