1 /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_log10l.c */ 2 /* 3 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net> 4 * 5 * Permission to use, copy, modify, and distribute this software for any 6 * purpose with or without fee is hereby granted, provided that the above 7 * copyright notice and this permission notice appear in all copies. 8 * 9 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES 10 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF 11 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR 12 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES 13 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN 14 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF 15 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. 16 */ 17 /* 18 * Common logarithm, long double precision 19 * 20 * 21 * SYNOPSIS: 22 * 23 * long double x, y, log10l(); 24 * 25 * y = log10l( x ); 26 * 27 * 28 * DESCRIPTION: 29 * 30 * Returns the base 10 logarithm of x. 31 * 32 * The argument is separated into its exponent and fractional 33 * parts. If the exponent is between -1 and +1, the logarithm 34 * of the fraction is approximated by 35 * 36 * log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x). 37 * 38 * Otherwise, setting z = 2(x-1)/x+1), 39 * 40 * log(x) = z + z**3 P(z)/Q(z). 41 * 42 * 43 * ACCURACY: 44 * 45 * Relative error: 46 * arithmetic domain # trials peak rms 47 * IEEE 0.5, 2.0 30000 9.0e-20 2.6e-20 48 * IEEE exp(+-10000) 30000 6.0e-20 2.3e-20 49 * 50 * In the tests over the interval exp(+-10000), the logarithms 51 * of the random arguments were uniformly distributed over 52 * [-10000, +10000]. 53 * 54 * ERROR MESSAGES: 55 * 56 * log singularity: x = 0; returns MINLOG 57 * log domain: x < 0; returns MINLOG 58 */ 59 60 #include "libm.h" 61 62 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024 63 long double log10l(long double x) 64 { 65 return log10(x); 66 } 67 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384 68 /* Coefficients for log(1+x) = x - x**2/2 + x**3 P(x)/Q(x) 69 * 1/sqrt(2) <= x < sqrt(2) 70 * Theoretical peak relative error = 6.2e-22 71 */ 72 static const long double P[] = { 73 4.9962495940332550844739E-1L, 74 1.0767376367209449010438E1L, 75 7.7671073698359539859595E1L, 76 2.5620629828144409632571E2L, 77 4.2401812743503691187826E2L, 78 3.4258224542413922935104E2L, 79 1.0747524399916215149070E2L, 80 }; 81 static const long double Q[] = { 82 /* 1.0000000000000000000000E0,*/ 83 2.3479774160285863271658E1L, 84 1.9444210022760132894510E2L, 85 7.7952888181207260646090E2L, 86 1.6911722418503949084863E3L, 87 2.0307734695595183428202E3L, 88 1.2695660352705325274404E3L, 89 3.2242573199748645407652E2L, 90 }; 91 92 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2), 93 * where z = 2(x-1)/(x+1) 94 * 1/sqrt(2) <= x < sqrt(2) 95 * Theoretical peak relative error = 6.16e-22 96 */ 97 static const long double R[4] = { 98 1.9757429581415468984296E-3L, 99 -7.1990767473014147232598E-1L, 100 1.0777257190312272158094E1L, 101 -3.5717684488096787370998E1L, 102 }; 103 static const long double S[4] = { 104 /* 1.00000000000000000000E0L,*/ 105 -2.6201045551331104417768E1L, 106 1.9361891836232102174846E2L, 107 -4.2861221385716144629696E2L, 108 }; 109 /* log10(2) */ 110 #define L102A 0.3125L 111 #define L102B -1.1470004336018804786261e-2L 112 /* log10(e) */ 113 #define L10EA 0.5L 114 #define L10EB -6.5705518096748172348871e-2L 115 116 #define SQRTH 0.70710678118654752440L 117 118 long double log10l(long double x) 119 { 120 long double y, z; 121 int e; 122 123 if (isnan(x)) 124 return x; 125 if(x <= 0.0) { 126 if(x == 0.0) 127 return -1.0 / (x*x); 128 return (x - x) / 0.0; 129 } 130 if (x == INFINITY) 131 return INFINITY; 132 /* separate mantissa from exponent */ 133 /* Note, frexp is used so that denormal numbers 134 * will be handled properly. 135 */ 136 x = frexpl(x, &e); 137 138 /* logarithm using log(x) = z + z**3 P(z)/Q(z), 139 * where z = 2(x-1)/x+1) 140 */ 141 if (e > 2 || e < -2) { 142 if (x < SQRTH) { /* 2(2x-1)/(2x+1) */ 143 e -= 1; 144 z = x - 0.5; 145 y = 0.5 * z + 0.5; 146 } else { /* 2 (x-1)/(x+1) */ 147 z = x - 0.5; 148 z -= 0.5; 149 y = 0.5 * x + 0.5; 150 } 151 x = z / y; 152 z = x*x; 153 y = x * (z * __polevll(z, R, 3) / __p1evll(z, S, 3)); 154 goto done; 155 } 156 157 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */ 158 if (x < SQRTH) { 159 e -= 1; 160 x = 2.0*x - 1.0; 161 } else { 162 x = x - 1.0; 163 } 164 z = x*x; 165 y = x * (z * __polevll(x, P, 6) / __p1evll(x, Q, 7)); 166 y = y - 0.5*z; 167 168 done: 169 /* Multiply log of fraction by log10(e) 170 * and base 2 exponent by log10(2). 171 * 172 * ***CAUTION*** 173 * 174 * This sequence of operations is critical and it may 175 * be horribly defeated by some compiler optimizers. 176 */ 177 z = y * (L10EB); 178 z += x * (L10EB); 179 z += e * (L102B); 180 z += y * (L10EA); 181 z += x * (L10EA); 182 z += e * (L102A); 183 return z; 184 } 185 #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384 186 // TODO: broken implementation to make things compile 187 long double log10l(long double x) 188 { 189 return log10(x); 190 } 191 #endif 192