1 /* $OpenBSD: e_log10l.c,v 1.1 2011/07/06 00:02:42 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, 128-bit 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 2.3e-34 4.9e-35 54 * IEEE exp(+-10000) 30000 1.0e-34 4.1e-35 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 */ 61 62 #include <openlibm.h> 63 64 #include "math_private.h" 65 66 /* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x) 67 * 1/sqrt(2) <= x < sqrt(2) 68 * Theoretical peak relative error = 5.3e-37, 69 * relative peak error spread = 2.3e-14 70 */ 71 static const long double P[13] = 72 { 73 1.313572404063446165910279910527789794488E4L, 74 7.771154681358524243729929227226708890930E4L, 75 2.014652742082537582487669938141683759923E5L, 76 3.007007295140399532324943111654767187848E5L, 77 2.854829159639697837788887080758954924001E5L, 78 1.797628303815655343403735250238293741397E5L, 79 7.594356839258970405033155585486712125861E4L, 80 2.128857716871515081352991964243375186031E4L, 81 3.824952356185897735160588078446136783779E3L, 82 4.114517881637811823002128927449878962058E2L, 83 2.321125933898420063925789532045674660756E1L, 84 4.998469661968096229986658302195402690910E-1L, 85 1.538612243596254322971797716843006400388E-6L 86 }; 87 static const long double Q[12] = 88 { 89 3.940717212190338497730839731583397586124E4L, 90 2.626900195321832660448791748036714883242E5L, 91 7.777690340007566932935753241556479363645E5L, 92 1.347518538384329112529391120390701166528E6L, 93 1.514882452993549494932585972882995548426E6L, 94 1.158019977462989115839826904108208787040E6L, 95 6.132189329546557743179177159925690841200E5L, 96 2.248234257620569139969141618556349415120E5L, 97 5.605842085972455027590989944010492125825E4L, 98 9.147150349299596453976674231612674085381E3L, 99 9.104928120962988414618126155557301584078E2L, 100 4.839208193348159620282142911143429644326E1L 101 /* 1.000000000000000000000000000000000000000E0L, */ 102 }; 103 104 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2), 105 * where z = 2(x-1)/(x+1) 106 * 1/sqrt(2) <= x < sqrt(2) 107 * Theoretical peak relative error = 1.1e-35, 108 * relative peak error spread 1.1e-9 109 */ 110 static const long double R[6] = 111 { 112 1.418134209872192732479751274970992665513E5L, 113 -8.977257995689735303686582344659576526998E4L, 114 2.048819892795278657810231591630928516206E4L, 115 -2.024301798136027039250415126250455056397E3L, 116 8.057002716646055371965756206836056074715E1L, 117 -8.828896441624934385266096344596648080902E-1L 118 }; 119 static const long double S[6] = 120 { 121 1.701761051846631278975701529965589676574E6L, 122 -1.332535117259762928288745111081235577029E6L, 123 4.001557694070773974936904547424676279307E5L, 124 -5.748542087379434595104154610899551484314E4L, 125 3.998526750980007367835804959888064681098E3L, 126 -1.186359407982897997337150403816839480438E2L 127 /* 1.000000000000000000000000000000000000000E0L, */ 128 }; 129 130 static const long double 131 /* log10(2) */ 132 L102A = 0.3125L, 133 L102B = -1.14700043360188047862611052755069732318101185E-2L, 134 /* log10(e) */ 135 L10EA = 0.5L, 136 L10EB = -6.570551809674817234887108108339491770560299E-2L, 137 /* sqrt(2)/2 */ 138 SQRTH = 7.071067811865475244008443621048490392848359E-1L; 139 140 141 142 /* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */ 143 144 static long double 145 neval (long double x, const long double *p, int n) 146 { 147 long double y; 148 149 p += n; 150 y = *p--; 151 do 152 { 153 y = y * x + *p--; 154 } 155 while (--n > 0); 156 return y; 157 } 158 159 160 /* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */ 161 162 static long double 163 deval (long double x, const long double *p, int n) 164 { 165 long double y; 166 167 p += n; 168 y = x + *p--; 169 do 170 { 171 y = y * x + *p--; 172 } 173 while (--n > 0); 174 return y; 175 } 176 177 178 179 long double 180 log10l(long double x) 181 { 182 long double z; 183 long double y; 184 int e; 185 int64_t hx, lx; 186 187 /* Test for domain */ 188 GET_LDOUBLE_WORDS64 (hx, lx, x); 189 if (((hx & 0x7fffffffffffffffLL) | lx) == 0) 190 return (-1.0L / (x - x)); 191 if (hx < 0) 192 return (x - x) / (x - x); 193 if (hx >= 0x7fff000000000000LL) 194 return (x + x); 195 196 /* separate mantissa from exponent */ 197 198 /* Note, frexp is used so that denormal numbers 199 * will be handled properly. 200 */ 201 x = frexpl (x, &e); 202 203 204 /* logarithm using log(x) = z + z**3 P(z)/Q(z), 205 * where z = 2(x-1)/x+1) 206 */ 207 if ((e > 2) || (e < -2)) 208 { 209 if (x < SQRTH) 210 { /* 2( 2x-1 )/( 2x+1 ) */ 211 e -= 1; 212 z = x - 0.5L; 213 y = 0.5L * z + 0.5L; 214 } 215 else 216 { /* 2 (x-1)/(x+1) */ 217 z = x - 0.5L; 218 z -= 0.5L; 219 y = 0.5L * x + 0.5L; 220 } 221 x = z / y; 222 z = x * x; 223 y = x * (z * neval (z, R, 5) / deval (z, S, 5)); 224 goto done; 225 } 226 227 228 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */ 229 230 if (x < SQRTH) 231 { 232 e -= 1; 233 x = 2.0 * x - 1.0L; /* 2x - 1 */ 234 } 235 else 236 { 237 x = x - 1.0L; 238 } 239 z = x * x; 240 y = x * (z * neval (x, P, 12) / deval (x, Q, 11)); 241 y = y - 0.5 * z; 242 243 done: 244 245 /* Multiply log of fraction by log10(e) 246 * and base 2 exponent by log10(2). 247 */ 248 z = y * L10EB; 249 z += x * L10EB; 250 z += e * L102B; 251 z += y * L10EA; 252 z += x * L10EA; 253 z += e * L102A; 254 return (z); 255 } 256