1 /*- 2 * SPDX-License-Identifier: BSD-2-Clause 3 * 4 * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG> 5 * All rights reserved. 6 * 7 * Redistribution and use in source and binary forms, with or without 8 * modification, are permitted provided that the following conditions 9 * are met: 10 * 1. Redistributions of source code must retain the above copyright 11 * notice, this list of conditions and the following disclaimer. 12 * 2. Redistributions in binary form must reproduce the above copyright 13 * notice, this list of conditions and the following disclaimer in the 14 * documentation and/or other materials provided with the distribution. 15 * 16 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND 17 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 18 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 19 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE 20 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 21 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 22 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 23 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 24 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 25 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 26 * SUCH DAMAGE. 27 */ 28 29 #include <float.h> 30 31 #include "math.h" 32 #include "math_private.h" 33 34 #define TBLBITS 4 35 #define TBLSIZE (1 << TBLBITS) 36 37 static const float 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 45 huge = 0x1p100f, 46 twom100 = 0x1p-100f; 47 48 static const double exp2ft[TBLSIZE] = { 49 0x1.6a09e667f3bcdp-1, 50 0x1.7a11473eb0187p-1, 51 0x1.8ace5422aa0dbp-1, 52 0x1.9c49182a3f090p-1, 53 0x1.ae89f995ad3adp-1, 54 0x1.c199bdd85529cp-1, 55 0x1.d5818dcfba487p-1, 56 0x1.ea4afa2a490dap-1, 57 0x1.0000000000000p+0, 58 0x1.0b5586cf9890fp+0, 59 0x1.172b83c7d517bp+0, 60 0x1.2387a6e756238p+0, 61 0x1.306fe0a31b715p+0, 62 0x1.3dea64c123422p+0, 63 0x1.4bfdad5362a27p+0, 64 0x1.5ab07dd485429p+0, 65 }; 66 67 /* 68 * exp2f(x): compute the base 2 exponential of x 69 * 70 * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927. 71 * 72 * Method: (equally-spaced tables) 73 * 74 * Reduce x: 75 * x = 2**k + y, for integer k and |y| <= 1/2. 76 * Thus we have exp2f(x) = 2**k * exp2(y). 77 * 78 * Reduce y: 79 * y = i/TBLSIZE + z for integer i near y * TBLSIZE. 80 * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z), 81 * with |z| <= 2**-(TBLSIZE+1). 82 * 83 * We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a 84 * degree-4 minimax polynomial with maximum error under 1.4 * 2**-33. 85 * Using double precision for everything except the reduction makes 86 * roundoff error insignificant and simplifies the scaling step. 87 * 88 * This method is due to Tang, but I do not use his suggested parameters: 89 * 90 * Tang, P. Table-driven Implementation of the Exponential Function 91 * in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989). 92 */ 93 float 94 exp2f(float x) 95 { 96 double tv, twopk, u, z; 97 float t; 98 uint32_t hx, ix, i0; 99 int32_t k; 100 101 /* Filter out exceptional cases. */ 102 GET_FLOAT_WORD(hx, x); 103 ix = hx & 0x7fffffff; /* high word of |x| */ 104 if(ix >= 0x43000000) { /* |x| >= 128 */ 105 if(ix >= 0x7f800000) { 106 if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0) 107 return (x + x); /* x is NaN or +Inf */ 108 else 109 return (0.0); /* x is -Inf */ 110 } 111 if(x >= 0x1.0p7f) 112 return (huge * huge); /* overflow */ 113 if(x <= -0x1.2cp7f) 114 return (twom100 * twom100); /* underflow */ 115 } else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */ 116 return (1.0f + x); 117 } 118 119 /* Reduce x, computing z, i0, and k. */ 120 STRICT_ASSIGN(float, t, x + redux); 121 GET_FLOAT_WORD(i0, t); 122 i0 += TBLSIZE / 2; 123 k = (i0 >> TBLBITS) << 20; 124 i0 &= TBLSIZE - 1; 125 t -= redux; 126 z = x - t; 127 INSERT_WORDS(twopk, 0x3ff00000 + k, 0); 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