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 /* __FBSDID("$FreeBSD: src/lib/msun/src/s_exp2f.c,v 1.1 2005/04/05 02:57:15 das Exp $"); */ 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 twom100 = 0x1p-100f, 39 redux = 0x1.8p23f / TBLSIZE, 40 P1 = 0x1.62e430p-1f, 41 P2 = 0x1.ebfbe0p-3f, 42 P3 = 0x1.c6b348p-5f, 43 P4 = 0x1.3b2c9cp-7f; 44 45 static const double exp2ft[TBLSIZE] = { 46 0x1.6a09e667f3bcdp-1, 47 0x1.7a11473eb0187p-1, 48 0x1.8ace5422aa0dbp-1, 49 0x1.9c49182a3f090p-1, 50 0x1.ae89f995ad3adp-1, 51 0x1.c199bdd85529cp-1, 52 0x1.d5818dcfba487p-1, 53 0x1.ea4afa2a490dap-1, 54 0x1.0000000000000p+0, 55 0x1.0b5586cf9890fp+0, 56 0x1.172b83c7d517bp+0, 57 0x1.2387a6e756238p+0, 58 0x1.306fe0a31b715p+0, 59 0x1.3dea64c123422p+0, 60 0x1.4bfdad5362a27p+0, 61 0x1.5ab07dd485429p+0, 62 }; 63 64 /* 65 * exp2f(x): compute the base 2 exponential of x 66 * 67 * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927. 68 * 69 * Method: (equally-spaced tables) 70 * 71 * Reduce x: 72 * x = 2**k + y, for integer k and |y| <= 1/2. 73 * Thus we have exp2f(x) = 2**k * exp2(y). 74 * 75 * Reduce y: 76 * y = i/TBLSIZE + z for integer i near y * TBLSIZE. 77 * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z), 78 * with |z| <= 2**-(TBLSIZE+1). 79 * 80 * We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a 81 * degree-4 minimax polynomial with maximum error under 1.4 * 2**-33. 82 * Using double precision in the final calculation avoids roundoff error. 83 * 84 * This method is due to Tang, but I do not use his suggested parameters: 85 * 86 * Tang, P. Table-driven Implementation of the Exponential Function 87 * in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989). 88 */ 89 float 90 exp2f(float x) 91 { 92 double tv; 93 float r, z; 94 volatile float t; /* prevent gcc from using too much precision */ 95 uint32_t hx, hr, ix, i0; 96 int32_t k; 97 98 /* Filter out exceptional cases. */ 99 GET_FLOAT_WORD(hx,x); 100 ix = hx & 0x7fffffff; /* high word of |x| */ 101 if(ix >= 0x43000000) { /* |x| >= 128 */ 102 if(ix >= 0x7f800000) { 103 if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0) 104 return (x); /* x is NaN or +Inf */ 105 else 106 return (0.0); /* x is -Inf */ 107 } 108 if(x >= 0x1.0p7f) 109 return (huge * huge); /* overflow */ 110 if(x <= -0x1.2cp7f) 111 return (twom100 * twom100); /* underflow */ 112 } else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */ 113 return (1.0f + x); 114 } 115 116 /* Reduce x, computing z, i0, and k. */ 117 t = x + redux; 118 GET_FLOAT_WORD(i0, t); 119 i0 += TBLSIZE / 2; 120 k = (i0 >> TBLBITS) << 23; 121 i0 &= TBLSIZE - 1; 122 t -= redux; 123 z = x - t; 124 125 /* Compute r = exp2(y) = exp2ft[i0] * p(z). */ 126 tv = exp2ft[i0]; 127 r = tv + tv * (z * (P1 + z * (P2 + z * (P3 + z * P4)))); 128 129 /* Scale by 2**(k>>23). */ 130 if(k >= -125 << 23) { 131 if (k != 0) { 132 GET_FLOAT_WORD(hr, r); 133 SET_FLOAT_WORD(r, hr + k); 134 } 135 return (r); 136 } else { 137 GET_FLOAT_WORD(hr, r); 138 SET_FLOAT_WORD(r, hr + (k + (100 << 23))); 139 return (r * twom100); 140 } 141 } 142
