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 __FBSDID("$FreeBSD$"); 29 30 #include <float.h> 31 32 #include "math.h" 33 #include "math_private.h" 34 35 #define TBLBITS 4 36 #define TBLSIZE (1 << TBLBITS) 37 38 static const float 39 huge = 0x1p100f, 40 redux = 0x1.8p23f / TBLSIZE, 41 P1 = 0x1.62e430p-1f, 42 P2 = 0x1.ebfbe0p-3f, 43 P3 = 0x1.c6b348p-5f, 44 P4 = 0x1.3b2c9cp-7f; 45 46 static volatile float 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 in the final calculation avoids roundoff error. 86 * 87 * This method is due to Tang, but I do not use his suggested parameters: 88 * 89 * Tang, P. Table-driven Implementation of the Exponential Function 90 * in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989). 91 */ 92 float 93 exp2f(float x) 94 { 95 double tv, twopk; 96 float t, z; 97 uint32_t hx, htv, ix, i0; 98 int32_t k; 99 100 /* Filter out exceptional cases. */ 101 GET_FLOAT_WORD(hx, x); 102 ix = hx & 0x7fffffff; /* high word of |x| */ 103 if(ix >= 0x43000000) { /* |x| >= 128 */ 104 if(ix >= 0x7f800000) { 105 if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0) 106 return (x); /* x is NaN or +Inf */ 107 else 108 return (0.0); /* x is -Inf */ 109 } 110 if(x >= 0x1.0p7f) 111 return (huge * huge); /* overflow */ 112 if(x <= -0x1.2cp7f) 113 return (twom100 * twom100); /* underflow */ 114 } else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */ 115 return (1.0f + x); 116 } 117 118 /* Reduce x, computing z, i0, and k. */ 119 STRICT_ASSIGN(float, t, x + redux); 120 GET_FLOAT_WORD(i0, t); 121 i0 += TBLSIZE / 2; 122 k = (i0 >> TBLBITS) << 20; 123 i0 &= TBLSIZE - 1; 124 t -= redux; 125 z = x - t; 126 127 /* Compute r = exp2(y) = exp2ft[i0] * p(z). */ 128 tv = exp2ft[i0]; 129 tv = tv + tv * (z * (P1 + z * (P2 + z * (P3 + z * P4)))); 130 131 /* Scale by 2**(k>>20). */ 132 INSERT_WORDS(twopk, 0x3ff00000 + k, 0); 133 return (tv * twopk); 134 } 135