1 /* 2 * CDDL HEADER START 3 * 4 * The contents of this file are subject to the terms of the 5 * Common Development and Distribution License (the "License"). 6 * You may not use this file except in compliance with the License. 7 * 8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE 9 * or http://www.opensolaris.org/os/licensing. 10 * See the License for the specific language governing permissions 11 * and limitations under the License. 12 * 13 * When distributing Covered Code, include this CDDL HEADER in each 14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE. 15 * If applicable, add the following below this CDDL HEADER, with the 16 * fields enclosed by brackets "[]" replaced with your own identifying 17 * information: Portions Copyright [yyyy] [name of copyright owner] 18 * 19 * CDDL HEADER END 20 */ 21 /* 22 * Copyright 2011 Nexenta Systems, Inc. All rights reserved. 23 */ 24 /* 25 * Copyright 2005 Sun Microsystems, Inc. All rights reserved. 26 * Use is subject to license terms. 27 */ 28 29 #pragma weak log = __log 30 31 /* INDENT OFF */ 32 /* 33 * log(x) 34 * Table look-up algorithm with product polynomial approximation. 35 * By K.C. Ng, Oct 23, 2004. Updated Oct 18, 2005. 36 * 37 * (a). For x in [1-0.125, 1+0.1328125], using a special approximation: 38 * Let f = x - 1 and z = f*f. 39 * return f + ((a1*z) * 40 * ((a2 + (a3*f)*(a4+f)) + (f*z)*(a5+f))) * 41 * (((a6 + f*(a7+f)) + (f*z)*(a8+f)) * 42 * ((a9 + (a10*f)*(a11+f)) + (f*z)*(a12+f))) 43 * a1 -6.88821452420390473170286327331268694251775741577e-0002, 44 * a2 1.97493380704769294631262255279580131173133850098e+0000, 45 * a3 2.24963218866067560242072431719861924648284912109e+0000, 46 * a4 -9.02975906958474405783476868236903101205825805664e-0001, 47 * a5 -1.47391630715542865104339398385491222143173217773e+0000, 48 * a6 1.86846544648220058704168877738993614912033081055e+0000, 49 * a7 1.82277370459347465292410106485476717352867126465e+0000, 50 * a8 1.25295479915214102994980294170090928673744201660e+0000, 51 * a9 1.96709676945198275177517643896862864494323730469e+0000, 52 * a10 -4.00127989749189894030934055990655906498432159424e-0001, 53 * a11 3.01675528558798333733648178167641162872314453125e+0000, 54 * a12 -9.52325445049240770778453679668018594384193420410e-0001, 55 * 56 * with remez error |(log(1+f) - P(f))/f| <= 2**-56.81 and 57 * 58 * (b). For 0.09375 <= x < 24 59 * Use an 8-bit table look-up (3-bit for exponent and 5 bit for 60 * significand): 61 * Let ix stands for the high part of x in IEEE double format. 62 * Since 0.09375 <= x < 24, we have 63 * 0x3fb80000 <= ix < 0x40380000. 64 * Let j = (ix - 0x3fb80000) >> 15. Then 0 <= j < 256. Choose 65 * a Y[j] such that HIWORD(Y[j]) ~ 0x3fb8400 + (j<<15) (the middle 66 * number between 0x3fb80000 + (j<<15) and 3fb80000 + ((j+1)<<15)), 67 * and at the same time 1/Y[j] as well as log(Y[j]) are very close 68 * to 53-bits floating point numbers. 69 * A table of Y[j], 1/Y[j], and log(Y[j]) are pre-computed and thus 70 * log(x) = log(Y[j]) + log(1 + (x-Y[j])*(1/Y[j])) 71 * = log(Y[j]) + log(1 + s) 72 * where 73 * s = (x-Y[j])*(1/Y[j]) 74 * We compute max (x-Y[j])*(1/Y[j]) for the chosen Y[j] and obtain 75 * |s| < 0.0154. By applying remez algorithm with Product Polynomial 76 * Approximiation, we find the following approximated of log(1+s) 77 * (b1*s)*(b2+s*(b3+s))*((b4+s*b5)+(s*s)*(b6+s))*(b7+s*(b8+s)) 78 * with remez error |log(1+s) - P(s)| <= 2**-63.5 79 * 80 * (c). Otherwise, get "n", the exponent of x, and then normalize x to 81 * z in [1,2). Then similar to (b) find a Y[i] that matches z to 5.5 82 * significant bits. Then 83 * log(x) = n*ln2 + log(Y[i]) + log(z/Y[i]). 84 * 85 * Special cases: 86 * log(x) is NaN with signal if x < 0 (including -INF) ; 87 * log(+INF) is +INF; log(0) is -INF with signal; 88 * log(NaN) is that NaN with no signal. 89 * 90 * Maximum error observed: less than 0.90 ulp 91 * 92 * Constants: 93 * The hexadecimal values are the intended ones for the following constants. 94 * The decimal values may be used, provided that the compiler will convert 95 * from decimal to binary accurately enough to produce the hexadecimal values 96 * shown. 97 */ 98 /* INDENT ON */ 99 100 #include "libm.h" 101 102 extern const double _TBL_log[]; 103 104 static const double P[] = { 105 /* ONE */ 1.0, 106 /* TWO52 */ 4503599627370496.0, 107 /* LN2HI */ 6.93147180369123816490e-01, /* 3fe62e42, fee00000 */ 108 /* LN2LO */ 1.90821492927058770002e-10, /* 3dea39ef, 35793c76 */ 109 /* A1 */ -6.88821452420390473170286327331268694251775741577e-0002, 110 /* A2 */ 1.97493380704769294631262255279580131173133850098e+0000, 111 /* A3 */ 2.24963218866067560242072431719861924648284912109e+0000, 112 /* A4 */ -9.02975906958474405783476868236903101205825805664e-0001, 113 /* A5 */ -1.47391630715542865104339398385491222143173217773e+0000, 114 /* A6 */ 1.86846544648220058704168877738993614912033081055e+0000, 115 /* A7 */ 1.82277370459347465292410106485476717352867126465e+0000, 116 /* A8 */ 1.25295479915214102994980294170090928673744201660e+0000, 117 /* A9 */ 1.96709676945198275177517643896862864494323730469e+0000, 118 /* A10 */ -4.00127989749189894030934055990655906498432159424e-0001, 119 /* A11 */ 3.01675528558798333733648178167641162872314453125e+0000, 120 /* A12 */ -9.52325445049240770778453679668018594384193420410e-0001, 121 /* B1 */ -1.25041641589283658575482149899471551179885864258e-0001, 122 /* B2 */ 1.87161713283355151891381127914642725337613123482e+0000, 123 /* B3 */ -1.89082956295731507978530316904652863740921020508e+0000, 124 /* B4 */ -2.50562891673640253387134180229622870683670043945e+0000, 125 /* B5 */ 1.64822828085258366037635369139024987816810607910e+0000, 126 /* B6 */ -1.24409107065868340669112512841820716857910156250e+0000, 127 /* B7 */ 1.70534231658220414296067701798165217041969299316e+0000, 128 /* B8 */ 1.99196833784655646937267192697618156671524047852e+0000, 129 }; 130 131 #define ONE P[0] 132 #define TWO52 P[1] 133 #define LN2HI P[2] 134 #define LN2LO P[3] 135 #define A1 P[4] 136 #define A2 P[5] 137 #define A3 P[6] 138 #define A4 P[7] 139 #define A5 P[8] 140 #define A6 P[9] 141 #define A7 P[10] 142 #define A8 P[11] 143 #define A9 P[12] 144 #define A10 P[13] 145 #define A11 P[14] 146 #define A12 P[15] 147 #define B1 P[16] 148 #define B2 P[17] 149 #define B3 P[18] 150 #define B4 P[19] 151 #define B5 P[20] 152 #define B6 P[21] 153 #define B7 P[22] 154 #define B8 P[23] 155 156 double 157 log(double x) { 158 double *tb, dn, dn1, s, z, r, w; 159 int i, hx, ix, n, lx; 160 161 n = 0; 162 hx = ((int *)&x)[HIWORD]; 163 ix = hx & 0x7fffffff; 164 lx = ((int *)&x)[LOWORD]; 165 166 /* subnormal,0,negative,inf,nan */ 167 if ((hx + 0x100000) < 0x200000) { 168 if (ix > 0x7ff00000 || (ix == 0x7ff00000 && lx != 0)) /* nan */ 169 return (x * x); 170 if (((hx << 1) | lx) == 0) /* zero */ 171 return (_SVID_libm_err(x, x, 16)); 172 if (hx < 0) /* negative */ 173 return (_SVID_libm_err(x, x, 17)); 174 if (((hx - 0x7ff00000) | lx) == 0) /* +inf */ 175 return (x); 176 177 /* x must be positive and subnormal */ 178 x *= TWO52; 179 n = -52; 180 ix = ((int *)&x)[HIWORD]; 181 lx = ((int *)&x)[LOWORD]; 182 } 183 184 i = ix >> 19; 185 if (i >= 0x7f7 && i <= 0x806) { 186 /* 0.09375 (0x3fb80000) <= x < 24 (0x40380000) */ 187 if (ix >= 0x3fec0000 && ix < 0x3ff22000) { 188 /* 0.875 <= x < 1.125 */ 189 s = x - ONE; 190 z = s * s; 191 if (((ix - 0x3ff00000) | lx) == 0) /* x = 1 */ 192 return (z); 193 r = (A10 * s) * (A11 + s); 194 w = z * s; 195 return (s + ((A1 * z) * 196 (A2 + ((A3 * s) * (A4 + s) + w * (A5 + s)))) * 197 ((A6 + (s * (A7 + s) + w * (A8 + s))) * 198 (A9 + (r + w * (A12 + s))))); 199 } else { 200 i = (ix - 0x3fb80000) >> 15; 201 tb = (double *)_TBL_log + (i + i + i); 202 s = (x - tb[0]) * tb[1]; 203 return (tb[2] + ((B1 * s) * (B2 + s * (B3 + s))) * 204 (((B4 + s * B5) + (s * s) * (B6 + s)) * 205 (B7 + s * (B8 + s)))); 206 } 207 } else { 208 dn = (double)(n + ((ix >> 20) - 0x3ff)); 209 dn1 = dn * LN2HI; 210 i = (ix & 0x000fffff) | 0x3ff00000; /* scale x to [1,2] */ 211 ((int *)&x)[HIWORD] = i; 212 i = (i - 0x3fb80000) >> 15; 213 tb = (double *)_TBL_log + (i + i + i); 214 s = (x - tb[0]) * tb[1]; 215 dn = dn * LN2LO + tb[2]; 216 return (dn1 + (dn + ((B1 * s) * (B2 + s * (B3 + s))) * 217 (((B4 + s * B5) + (s * s) * (B6 + s)) * 218 (B7 + s * (B8 + s))))); 219 } 220 } 221