1 /* $OpenBSD: s_exp2f.c,v 1.2 2012/12/05 23:20:04 deraadt Exp $ */ 2 /*- 3 * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG> 4 * All rights reserved. 5 * 6 * Redistribution and use in source and binary forms, with or without 7 * modification, are permitted provided that the following conditions 8 * are met: 9 * 1. Redistributions of source code must retain the above copyright 10 * notice, this list of conditions and the following disclaimer. 11 * 2. Redistributions in binary form must reproduce the above copyright 12 * notice, this list of conditions and the following disclaimer in the 13 * documentation and/or other materials provided with the distribution. 14 * 15 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND 16 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 17 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 18 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE 19 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 20 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 21 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 22 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 23 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 24 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 25 * SUCH DAMAGE. 26 */ 27 28 #include <float.h> 29 30 #include "math.h" 31 #include "math_private.h" 32 33 #define TBLBITS 4 34 #define TBLSIZE (1 << TBLBITS) 35 36 static const float 37 huge = 0x1p100f, 38 redux = 0x1.8p23f / TBLSIZE, 39 P1 = 0x1.62e430p-1f, 40 P2 = 0x1.ebfbe0p-3f, 41 P3 = 0x1.c6b348p-5f, 42 P4 = 0x1.3b2c9cp-7f; 43 44 static volatile float twom100 = 0x1p-100f; 45 46 static const double exp2ft[TBLSIZE] = { 47 0x1.6a09e667f3bcdp-1, 48 0x1.7a11473eb0187p-1, 49 0x1.8ace5422aa0dbp-1, 50 0x1.9c49182a3f090p-1, 51 0x1.ae89f995ad3adp-1, 52 0x1.c199bdd85529cp-1, 53 0x1.d5818dcfba487p-1, 54 0x1.ea4afa2a490dap-1, 55 0x1.0000000000000p+0, 56 0x1.0b5586cf9890fp+0, 57 0x1.172b83c7d517bp+0, 58 0x1.2387a6e756238p+0, 59 0x1.306fe0a31b715p+0, 60 0x1.3dea64c123422p+0, 61 0x1.4bfdad5362a27p+0, 62 0x1.5ab07dd485429p+0, 63 }; 64 65 /* 66 * exp2f(x): compute the base 2 exponential of x 67 * 68 * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927. 69 * 70 * Method: (equally-spaced tables) 71 * 72 * Reduce x: 73 * x = 2**k + y, for integer k and |y| <= 1/2. 74 * Thus we have exp2f(x) = 2**k * exp2(y). 75 * 76 * Reduce y: 77 * y = i/TBLSIZE + z for integer i near y * TBLSIZE. 78 * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z), 79 * with |z| <= 2**-(TBLSIZE+1). 80 * 81 * We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a 82 * degree-4 minimax polynomial with maximum error under 1.4 * 2**-33. 83 * Using double precision for everything except the reduction makes 84 * roundoff error insignificant and simplifies the scaling step. 85 * 86 * This method is due to Tang, but I do not use his suggested parameters: 87 * 88 * Tang, P. Table-driven Implementation of the Exponential Function 89 * in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989). 90 */ 91 float 92 exp2f(float x) 93 { 94 double tv, twopk, u, z; 95 float t; 96 uint32_t hx, ix, i0; 97 int32_t k; 98 99 /* Filter out exceptional cases. */ 100 GET_FLOAT_WORD(hx, x); 101 ix = hx & 0x7fffffff; /* high word of |x| */ 102 if(ix >= 0x43000000) { /* |x| >= 128 */ 103 if(ix >= 0x7f800000) { 104 if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0) 105 return (x + x); /* x is NaN or +Inf */ 106 else 107 return (0.0); /* x is -Inf */ 108 } 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 STRICT_ASSIGN(float, t, x + redux); 119 GET_FLOAT_WORD(i0, t); 120 i0 += TBLSIZE / 2; 121 k = (i0 >> TBLBITS) << 20; 122 i0 &= TBLSIZE - 1; 123 t -= redux; 124 z = x - t; 125 INSERT_WORDS(twopk, 0x3ff00000 + k, 0); 126 127 /* Compute r = exp2(y) = exp2ft[i0] * p(z). */ 128 tv = exp2ft[i0]; 129 u = tv * z; 130 tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4); 131 132 /* Scale by 2**(k>>20). */ 133 return (tv * twopk); 134 } 135