1 /* $OpenBSD: e_log10l.c,v 1.2 2013/11/12 20:35:19 martynas Exp $ */ 2 3 /* 4 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net> 5 * 6 * Permission to use, copy, modify, and distribute this software for any 7 * purpose with or without fee is hereby granted, provided that the above 8 * copyright notice and this permission notice appear in all copies. 9 * 10 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES 11 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF 12 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR 13 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES 14 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN 15 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF 16 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. 17 */ 18 19 /* log10l.c 20 * 21 * Common logarithm, long double precision 22 * 23 * 24 * 25 * SYNOPSIS: 26 * 27 * long double x, y, log10l(); 28 * 29 * y = log10l( x ); 30 * 31 * 32 * 33 * DESCRIPTION: 34 * 35 * Returns the base 10 logarithm of x. 36 * 37 * The argument is separated into its exponent and fractional 38 * parts. If the exponent is between -1 and +1, the logarithm 39 * of the fraction is approximated by 40 * 41 * log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x). 42 * 43 * Otherwise, setting z = 2(x-1)/x+1), 44 * 45 * log(x) = z + z**3 P(z)/Q(z). 46 * 47 * 48 * 49 * ACCURACY: 50 * 51 * Relative error: 52 * arithmetic domain # trials peak rms 53 * IEEE 0.5, 2.0 30000 9.0e-20 2.6e-20 54 * IEEE exp(+-10000) 30000 6.0e-20 2.3e-20 55 * 56 * In the tests over the interval exp(+-10000), the logarithms 57 * of the random arguments were uniformly distributed over 58 * [-10000, +10000]. 59 * 60 * ERROR MESSAGES: 61 * 62 * log singularity: x = 0; returns MINLOG 63 * log domain: x < 0; returns MINLOG 64 */ 65 66 #include <openlibm.h> 67 68 #include "math_private.h" 69 70 /* Coefficients for log(1+x) = x - x**2/2 + x**3 P(x)/Q(x) 71 * 1/sqrt(2) <= x < sqrt(2) 72 * Theoretical peak relative error = 6.2e-22 73 */ 74 static long double P[] = { 75 4.9962495940332550844739E-1L, 76 1.0767376367209449010438E1L, 77 7.7671073698359539859595E1L, 78 2.5620629828144409632571E2L, 79 4.2401812743503691187826E2L, 80 3.4258224542413922935104E2L, 81 1.0747524399916215149070E2L, 82 }; 83 static long double Q[] = { 84 /* 1.0000000000000000000000E0,*/ 85 2.3479774160285863271658E1L, 86 1.9444210022760132894510E2L, 87 7.7952888181207260646090E2L, 88 1.6911722418503949084863E3L, 89 2.0307734695595183428202E3L, 90 1.2695660352705325274404E3L, 91 3.2242573199748645407652E2L, 92 }; 93 94 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2), 95 * where z = 2(x-1)/(x+1) 96 * 1/sqrt(2) <= x < sqrt(2) 97 * Theoretical peak relative error = 6.16e-22 98 */ 99 100 static long double R[4] = { 101 1.9757429581415468984296E-3L, 102 -7.1990767473014147232598E-1L, 103 1.0777257190312272158094E1L, 104 -3.5717684488096787370998E1L, 105 }; 106 static long double S[4] = { 107 /* 1.00000000000000000000E0L,*/ 108 -2.6201045551331104417768E1L, 109 1.9361891836232102174846E2L, 110 -4.2861221385716144629696E2L, 111 }; 112 /* log10(2) */ 113 #define L102A 0.3125L 114 #define L102B -1.1470004336018804786261e-2L 115 /* log10(e) */ 116 #define L10EA 0.5L 117 #define L10EB -6.5705518096748172348871e-2L 118 119 #define SQRTH 0.70710678118654752440L 120 121 long double 122 log10l(long double x) 123 { 124 long double y; 125 volatile long double z; 126 int e; 127 128 if( isnan(x) ) 129 return(x); 130 /* Test for domain */ 131 if( x <= 0.0L ) 132 { 133 if( x == 0.0L ) 134 return (-1.0L / (x - x)); 135 else 136 return (x - x) / (x - x); 137 } 138 if( x == INFINITY ) 139 return(INFINITY); 140 /* separate mantissa from exponent */ 141 142 /* Note, frexp is used so that denormal numbers 143 * will be handled properly. 144 */ 145 x = frexpl( x, &e ); 146 147 148 /* logarithm using log(x) = z + z**3 P(z)/Q(z), 149 * where z = 2(x-1)/x+1) 150 */ 151 if( (e > 2) || (e < -2) ) 152 { 153 if( x < SQRTH ) 154 { /* 2( 2x-1 )/( 2x+1 ) */ 155 e -= 1; 156 z = x - 0.5L; 157 y = 0.5L * z + 0.5L; 158 } 159 else 160 { /* 2 (x-1)/(x+1) */ 161 z = x - 0.5L; 162 z -= 0.5L; 163 y = 0.5L * x + 0.5L; 164 } 165 x = z / y; 166 z = x*x; 167 y = x * ( z * __polevll( z, R, 3 ) / __p1evll( z, S, 3 ) ); 168 goto done; 169 } 170 171 172 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */ 173 174 if( x < SQRTH ) 175 { 176 e -= 1; 177 x = ldexpl( x, 1 ) - 1.0L; /* 2x - 1 */ 178 } 179 else 180 { 181 x = x - 1.0L; 182 } 183 z = x*x; 184 y = x * ( z * __polevll( x, P, 6 ) / __p1evll( x, Q, 7 ) ); 185 y = y - ldexpl( z, -1 ); /* -0.5x^2 + ... */ 186 187 done: 188 189 /* Multiply log of fraction by log10(e) 190 * and base 2 exponent by log10(2). 191 * 192 * ***CAUTION*** 193 * 194 * This sequence of operations is critical and it may 195 * be horribly defeated by some compiler optimizers. 196 */ 197 z = y * (L10EB); 198 z += x * (L10EB); 199 z += e * (L102B); 200 z += y * (L10EA); 201 z += x * (L10EA); 202 z += e * (L102A); 203 204 return( z ); 205 } 206