/* * CDDL HEADER START * * The contents of this file are subject to the terms of the * Common Development and Distribution License (the "License"). * You may not use this file except in compliance with the License. * * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE * or http://www.opensolaris.org/os/licensing. * See the License for the specific language governing permissions * and limitations under the License. * * When distributing Covered Code, include this CDDL HEADER in each * file and include the License file at usr/src/OPENSOLARIS.LICENSE. * If applicable, add the following below this CDDL HEADER, with the * fields enclosed by brackets "[]" replaced with your own identifying * information: Portions Copyright [yyyy] [name of copyright owner] * * CDDL HEADER END */ /* * Copyright 2011 Nexenta Systems, Inc. All rights reserved. */ /* * Copyright 2005 Sun Microsystems, Inc. All rights reserved. * Use is subject to license terms. */ #pragma weak __log = log /* INDENT OFF */ /* * log(x) * Table look-up algorithm with product polynomial approximation. * By K.C. Ng, Oct 23, 2004. Updated Oct 18, 2005. * * (a). For x in [1-0.125, 1+0.1328125], using a special approximation: * Let f = x - 1 and z = f*f. * return f + ((a1*z) * * ((a2 + (a3*f)*(a4+f)) + (f*z)*(a5+f))) * * (((a6 + f*(a7+f)) + (f*z)*(a8+f)) * * ((a9 + (a10*f)*(a11+f)) + (f*z)*(a12+f))) * a1 -6.88821452420390473170286327331268694251775741577e-0002, * a2 1.97493380704769294631262255279580131173133850098e+0000, * a3 2.24963218866067560242072431719861924648284912109e+0000, * a4 -9.02975906958474405783476868236903101205825805664e-0001, * a5 -1.47391630715542865104339398385491222143173217773e+0000, * a6 1.86846544648220058704168877738993614912033081055e+0000, * a7 1.82277370459347465292410106485476717352867126465e+0000, * a8 1.25295479915214102994980294170090928673744201660e+0000, * a9 1.96709676945198275177517643896862864494323730469e+0000, * a10 -4.00127989749189894030934055990655906498432159424e-0001, * a11 3.01675528558798333733648178167641162872314453125e+0000, * a12 -9.52325445049240770778453679668018594384193420410e-0001, * * with remez error |(log(1+f) - P(f))/f| <= 2**-56.81 and * * (b). For 0.09375 <= x < 24 * Use an 8-bit table look-up (3-bit for exponent and 5 bit for * significand): * Let ix stands for the high part of x in IEEE double format. * Since 0.09375 <= x < 24, we have * 0x3fb80000 <= ix < 0x40380000. * Let j = (ix - 0x3fb80000) >> 15. Then 0 <= j < 256. Choose * a Y[j] such that HIWORD(Y[j]) ~ 0x3fb8400 + (j<<15) (the middle * number between 0x3fb80000 + (j<<15) and 3fb80000 + ((j+1)<<15)), * and at the same time 1/Y[j] as well as log(Y[j]) are very close * to 53-bits floating point numbers. * A table of Y[j], 1/Y[j], and log(Y[j]) are pre-computed and thus * log(x) = log(Y[j]) + log(1 + (x-Y[j])*(1/Y[j])) * = log(Y[j]) + log(1 + s) * where * s = (x-Y[j])*(1/Y[j]) * We compute max (x-Y[j])*(1/Y[j]) for the chosen Y[j] and obtain * |s| < 0.0154. By applying remez algorithm with Product Polynomial * Approximiation, we find the following approximated of log(1+s) * (b1*s)*(b2+s*(b3+s))*((b4+s*b5)+(s*s)*(b6+s))*(b7+s*(b8+s)) * with remez error |log(1+s) - P(s)| <= 2**-63.5 * * (c). Otherwise, get "n", the exponent of x, and then normalize x to * z in [1,2). Then similar to (b) find a Y[i] that matches z to 5.5 * significant bits. Then * log(x) = n*ln2 + log(Y[i]) + log(z/Y[i]). * * Special cases: * log(x) is NaN with signal if x < 0 (including -INF) ; * log(+INF) is +INF; log(0) is -INF with signal; * log(NaN) is that NaN with no signal. * * Maximum error observed: less than 0.90 ulp * * Constants: * The hexadecimal values are the intended ones for the following constants. * The decimal values may be used, provided that the compiler will convert * from decimal to binary accurately enough to produce the hexadecimal values * shown. */ /* INDENT ON */ #include "libm.h" extern const double _TBL_log[]; static const double P[] = { /* ONE */ 1.0, /* TWO52 */ 4503599627370496.0, /* LN2HI */ 6.93147180369123816490e-01, /* 3fe62e42, fee00000 */ /* LN2LO */ 1.90821492927058770002e-10, /* 3dea39ef, 35793c76 */ /* A1 */ -6.88821452420390473170286327331268694251775741577e-0002, /* A2 */ 1.97493380704769294631262255279580131173133850098e+0000, /* A3 */ 2.24963218866067560242072431719861924648284912109e+0000, /* A4 */ -9.02975906958474405783476868236903101205825805664e-0001, /* A5 */ -1.47391630715542865104339398385491222143173217773e+0000, /* A6 */ 1.86846544648220058704168877738993614912033081055e+0000, /* A7 */ 1.82277370459347465292410106485476717352867126465e+0000, /* A8 */ 1.25295479915214102994980294170090928673744201660e+0000, /* A9 */ 1.96709676945198275177517643896862864494323730469e+0000, /* A10 */ -4.00127989749189894030934055990655906498432159424e-0001, /* A11 */ 3.01675528558798333733648178167641162872314453125e+0000, /* A12 */ -9.52325445049240770778453679668018594384193420410e-0001, /* B1 */ -1.25041641589283658575482149899471551179885864258e-0001, /* B2 */ 1.87161713283355151891381127914642725337613123482e+0000, /* B3 */ -1.89082956295731507978530316904652863740921020508e+0000, /* B4 */ -2.50562891673640253387134180229622870683670043945e+0000, /* B5 */ 1.64822828085258366037635369139024987816810607910e+0000, /* B6 */ -1.24409107065868340669112512841820716857910156250e+0000, /* B7 */ 1.70534231658220414296067701798165217041969299316e+0000, /* B8 */ 1.99196833784655646937267192697618156671524047852e+0000, }; #define ONE P[0] #define TWO52 P[1] #define LN2HI P[2] #define LN2LO P[3] #define A1 P[4] #define A2 P[5] #define A3 P[6] #define A4 P[7] #define A5 P[8] #define A6 P[9] #define A7 P[10] #define A8 P[11] #define A9 P[12] #define A10 P[13] #define A11 P[14] #define A12 P[15] #define B1 P[16] #define B2 P[17] #define B3 P[18] #define B4 P[19] #define B5 P[20] #define B6 P[21] #define B7 P[22] #define B8 P[23] double log(double x) { double *tb, dn, dn1, s, z, r, w; int i, hx, ix, n, lx; n = 0; hx = ((int *)&x)[HIWORD]; ix = hx & 0x7fffffff; lx = ((int *)&x)[LOWORD]; /* subnormal,0,negative,inf,nan */ if ((hx + 0x100000) < 0x200000) { if (ix > 0x7ff00000 || (ix == 0x7ff00000 && lx != 0)) /* nan */ return (x * x); if (((hx << 1) | lx) == 0) /* zero */ return (_SVID_libm_err(x, x, 16)); if (hx < 0) /* negative */ return (_SVID_libm_err(x, x, 17)); if (((hx - 0x7ff00000) | lx) == 0) /* +inf */ return (x); /* x must be positive and subnormal */ x *= TWO52; n = -52; ix = ((int *)&x)[HIWORD]; lx = ((int *)&x)[LOWORD]; } i = ix >> 19; if (i >= 0x7f7 && i <= 0x806) { /* 0.09375 (0x3fb80000) <= x < 24 (0x40380000) */ if (ix >= 0x3fec0000 && ix < 0x3ff22000) { /* 0.875 <= x < 1.125 */ s = x - ONE; z = s * s; if (((ix - 0x3ff00000) | lx) == 0) /* x = 1 */ return (z); r = (A10 * s) * (A11 + s); w = z * s; return (s + ((A1 * z) * (A2 + ((A3 * s) * (A4 + s) + w * (A5 + s)))) * ((A6 + (s * (A7 + s) + w * (A8 + s))) * (A9 + (r + w * (A12 + s))))); } else { i = (ix - 0x3fb80000) >> 15; tb = (double *)_TBL_log + (i + i + i); s = (x - tb[0]) * tb[1]; return (tb[2] + ((B1 * s) * (B2 + s * (B3 + s))) * (((B4 + s * B5) + (s * s) * (B6 + s)) * (B7 + s * (B8 + s)))); } } else { dn = (double)(n + ((ix >> 20) - 0x3ff)); dn1 = dn * LN2HI; i = (ix & 0x000fffff) | 0x3ff00000; /* scale x to [1,2] */ ((int *)&x)[HIWORD] = i; i = (i - 0x3fb80000) >> 15; tb = (double *)_TBL_log + (i + i + i); s = (x - tb[0]) * tb[1]; dn = dn * LN2LO + tb[2]; return (dn1 + (dn + ((B1 * s) * (B2 + s * (B3 + s))) * (((B4 + s * B5) + (s * s) * (B6 + s)) * (B7 + s * (B8 + s))))); } }
