1/*- 2 * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG> 3 * All rights reserved. 4 * 5 * Redistribution and use in source and binary forms, with or without 6 * modification, are permitted provided that the following conditions 7 * are met: 8 * 1. Redistributions of source code must retain the above copyright 9 * notice, this list of conditions and the following disclaimer. 10 * 2. Redistributions in binary form must reproduce the above copyright 11 * notice, this list of conditions and the following disclaimer in the 12 * documentation and/or other materials provided with the distribution. 13 * 14 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND 15 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 16 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 17 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE 18 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 19 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 20 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 21 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 22 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 23 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 24 * SUCH DAMAGE. 25 */ 26 27#include <sys/cdefs.h> 28__RCSID("$NetBSD: n_exp2f.c,v 1.1 2014/03/06 10:55:57 martin Exp $"); 29#ifdef __FBSDID 30__FBSDID("$FreeBSD: src/lib/msun/src/s_exp2f.c,v 1.9 2008/02/22 02:27:34 das Exp $"); 31#endif 32 33#include <stdint.h> 34#include <float.h> 35#include <string.h> 36 37#include "math.h" 38 39#define TBLBITS 4 40#define TBLSIZE (1 << TBLBITS) 41 42static const float 43 huge = 0x1p100f, 44 redux = 0x1.8p23f / TBLSIZE, 45 P1 = 0x1.62e430p-1f, 46 P2 = 0x1.ebfbe0p-3f, 47 P3 = 0x1.c6b348p-5f, 48 P4 = 0x1.3b2c9cp-7f; 49 50static volatile float twom100 = 0x1p-100f; 51 52static const double exp2ft[TBLSIZE] = { 53 0x1.6a09e667f3bcdp-1, 54 0x1.7a11473eb0187p-1, 55 0x1.8ace5422aa0dbp-1, 56 0x1.9c49182a3f090p-1, 57 0x1.ae89f995ad3adp-1, 58 0x1.c199bdd85529cp-1, 59 0x1.d5818dcfba487p-1, 60 0x1.ea4afa2a490dap-1, 61 0x1.0000000000000p+0, 62 0x1.0b5586cf9890fp+0, 63 0x1.172b83c7d517bp+0, 64 0x1.2387a6e756238p+0, 65 0x1.306fe0a31b715p+0, 66 0x1.3dea64c123422p+0, 67 0x1.4bfdad5362a27p+0, 68 0x1.5ab07dd485429p+0, 69}; 70 71/* 72 * exp2f(x): compute the base 2 exponential of x 73 * 74 * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927. 75 * 76 * Method: (equally-spaced tables) 77 * 78 * Reduce x: 79 * x = 2**k + y, for integer k and |y| <= 1/2. 80 * Thus we have exp2f(x) = 2**k * exp2(y). 81 * 82 * Reduce y: 83 * y = i/TBLSIZE + z for integer i near y * TBLSIZE. 84 * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z), 85 * with |z| <= 2**-(TBLSIZE+1). 86 * 87 * We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a 88 * degree-4 minimax polynomial with maximum error under 1.4 * 2**-33. 89 * Using double precision for everything except the reduction makes 90 * roundoff error insignificant and simplifies the scaling step. 91 * 92 * This method is due to Tang, but I do not use his suggested parameters: 93 * 94 * Tang, P. Table-driven Implementation of the Exponential Function 95 * in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989). 96 */ 97float 98exp2f(float x) 99{ 100 double tv, twopk, u, z; 101 float t; 102 uint32_t hx, ix, i0; 103 int32_t k, temp; 104 105 /* Filter out exceptional cases. */ 106 memcpy(&hx, &x, sizeof(hx)); 107 ix = hx & 0x7fffffff; /* high word of |x| */ 108 if(ix >= 0x43000000) { /* |x| >= 128 */ 109 if(x >= 0x1.0p7f) 110 return (huge * huge); /* overflow */ 111 if(x <= -0x1.2cp7f) 112 return (twom100 * twom100); /* underflow */ 113 } else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */ 114 return (1.0f + x); 115 } 116 117 /* Reduce x, computing z, i0, and k. */ 118 i0 = x + redux; 119 memcpy(&t, &i0, sizeof(t)); 120 i0 += TBLSIZE / 2; 121 k = (i0 >> TBLBITS) << 20; 122 i0 &= TBLSIZE - 1; 123 t -= redux; 124 z = x - t; 125 temp = 0x3ff00000+k; 126 twopk = 0.0; 127 memcpy(&twopk, &temp, sizeof(temp)); 128 129 /* Compute r = exp2(y) = exp2ft[i0] * p(z). */ 130 tv = exp2ft[i0]; 131 u = tv * z; 132 tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4); 133 134 /* Scale by 2**(k>>20). */ 135 return (tv * twopk); 136} 137