1 /*- 2 * Copyright (c) 2005 David Schultz <das (at) 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: s_exp2f.c,v 1.2 2014/03/16 22:30:43 dsl 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 <float.h> 34 35 #include "math.h" 36 #include "math_private.h" 37 38 #define TBLBITS 4 39 #define TBLSIZE (1 << TBLBITS) 40 41 static const float 42 redux = 0x1.8p23f / TBLSIZE, 43 P1 = 0x1.62e430p-1f, 44 P2 = 0x1.ebfbe0p-3f, 45 P3 = 0x1.c6b348p-5f, 46 P4 = 0x1.3b2c9cp-7f; 47 48 /* 49 * For out of range values we need to generate the appropriate 50 * underflow or overflow trap as well as generating infinity or zero. 51 * This means we have to get the fpu to execute an instruction that 52 * will generate the trap (and not have the compiler optimise it away). 53 * This is normally done by calculating 'huge * huge' or 'tiny * tiny'. 54 * 55 * i386 is particularly problematic. 56 * The 'float' result is returned on the x87 stack, so is 'long double'. 57 * If we just multiply two 'float' values the caller will see 0x1p+/-200 58 * (not 0 or infinity). 59 * If we use 'double' the compiler does a store-load which will convert the 60 * value and generate the required exception. 61 */ 62 #ifdef __i386__ 63 static volatile double overflow = 0x1p+1000; 64 static volatile double underflow = 0x1p-1000; 65 #else 66 static volatile float huge = 0x1p+100; 67 static volatile float tiny = 0x1p-100; 68 #define overflow (huge * huge) 69 #define underflow (tiny * tiny) 70 #endif 71 72 static const double exp2ft[TBLSIZE] = { 73 0x1.6a09e667f3bcdp-1, 74 0x1.7a11473eb0187p-1, 75 0x1.8ace5422aa0dbp-1, 76 0x1.9c49182a3f090p-1, 77 0x1.ae89f995ad3adp-1, 78 0x1.c199bdd85529cp-1, 79 0x1.d5818dcfba487p-1, 80 0x1.ea4afa2a490dap-1, 81 0x1.0000000000000p+0, 82 0x1.0b5586cf9890fp+0, 83 0x1.172b83c7d517bp+0, 84 0x1.2387a6e756238p+0, 85 0x1.306fe0a31b715p+0, 86 0x1.3dea64c123422p+0, 87 0x1.4bfdad5362a27p+0, 88 0x1.5ab07dd485429p+0, 89 }; 90 91 /* 92 * exp2f(x): compute the base 2 exponential of x 93 * 94 * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927. 95 * 96 * Method: (equally-spaced tables) 97 * 98 * Reduce x: 99 * x = 2**k + y, for integer k and |y| <= 1/2. 100 * Thus we have exp2f(x) = 2**k * exp2(y). 101 * 102 * Reduce y: 103 * y = i/TBLSIZE + z for integer i near y * TBLSIZE. 104 * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z), 105 * with |z| <= 2**-(TBLSIZE+1). 106 * 107 * We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a 108 * degree-4 minimax polynomial with maximum error under 1.4 * 2**-33. 109 * Using double precision for everything except the reduction makes 110 * roundoff error insignificant and simplifies the scaling step. 111 * 112 * This method is due to Tang, but I do not use his suggested parameters: 113 * 114 * Tang, P. Table-driven Implementation of the Exponential Function 115 * in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989). 116 */ 117 float 118 exp2f(float x) 119 { 120 double tv, twopk, u, z; 121 float t; 122 uint32_t hx, ix, i0; 123 int32_t k; 124 125 /* Filter out exceptional cases. */ 126 GET_FLOAT_WORD(hx, x); 127 ix = hx & 0x7fffffff; /* high word of |x| */ 128 if(ix >= 0x43000000) { /* |x| >= 128 */ 129 if(ix >= 0x7f800000) { 130 if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0) 131 return (x + x); /* x is NaN or +Inf */ 132 else 133 return (0.0); /* x is -Inf */ 134 } 135 if(x >= 0x1.0p7f) 136 return overflow; /* +infinity with overflow */ 137 if(x <= -0x1.2cp7f) 138 return underflow; /* zero with underflow */ 139 } else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */ 140 return (1.0f + x); 141 } 142 143 /* Reduce x, computing z, i0, and k. */ 144 STRICT_ASSIGN(float, t, x + redux); 145 GET_FLOAT_WORD(i0, t); 146 i0 += TBLSIZE / 2; 147 k = (i0 >> TBLBITS) << 20; 148 i0 &= TBLSIZE - 1; 149 t -= redux; 150 z = x - t; 151 INSERT_WORDS(twopk, 0x3ff00000 + k, 0); 152 153 /* Compute r = exp2(y) = exp2ft[i0] * p(z). */ 154 tv = exp2ft[i0]; 155 u = tv * z; 156 tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4); 157 158 /* Scale by 2**(k>>20). */ 159 return (tv * twopk); 160 } 161
Indexes created Sun Oct 25 00:26:45 UTC 2026