/* * 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 __log1p = log1p /* INDENT OFF */ /* * Method : * 1. Argument Reduction: find k and f such that * 1+x = 2^k * (1+f), * where sqrt(2)/2 < 1+f < sqrt(2) . * * Note. If k=0, then f=x is exact. However, if k != 0, then f * may not be representable exactly. In that case, a correction * term is need. Let u=1+x rounded. Let c = (1+x)-u, then * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u), * and add back the correction term c/u. * (Note: when x > 2**53, one can simply return log(x)) * * 2. Approximation of log1p(f). * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) * = 2s + 2/3 s**3 + 2/5 s**5 + ....., * = 2s + s*R * We use a special Reme algorithm on [0,0.1716] to generate * a polynomial of degree 14 to approximate R The maximum error * of this polynomial approximation is bounded by 2**-58.45. In * other words, * 2 4 6 8 10 12 14 * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s * (the values of Lp1 to Lp7 are listed in the program) * and * | 2 14 | -58.45 * | Lp1*s +...+Lp7*s - R(z) | <= 2 * | | * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. * In order to guarantee error in log below 1ulp, we compute log * by * log1p(f) = f - (hfsq - s*(hfsq+R)). * * 3. Finally, log1p(x) = k*ln2 + log1p(f). * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) * Here ln2 is splitted into two floating point number: * ln2_hi + ln2_lo, * where n*ln2_hi is always exact for |n| < 2000. * * Special cases: * log1p(x) is NaN with signal if x < -1 (including -INF) ; * log1p(+INF) is +INF; log1p(-1) is -INF with signal; * log1p(NaN) is that NaN with no signal. * * Accuracy: * according to an error analysis, the error is always less than * 1 ulp (unit in the last place). * * 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. * * Note: Assuming log() return accurate answer, the following * algorithm can be used to compute log1p(x) to within a few ULP: * * u = 1+x; * if (u == 1.0) return x ; else * return log(u)*(x/(u-1.0)); * * See HP-15C Advanced Functions Handbook, p.193. */ /* INDENT ON */ #include "libm.h" static const double xxx[] = { /* ln2_hi */ 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ /* ln2_lo */ 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ /* two54 */ 1.80143985094819840000e+16, /* 43500000 00000000 */ /* Lp1 */ 6.666666666666735130e-01, /* 3FE55555 55555593 */ /* Lp2 */ 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ /* Lp3 */ 2.857142874366239149e-01, /* 3FD24924 94229359 */ /* Lp4 */ 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ /* Lp5 */ 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ /* Lp6 */ 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ /* Lp7 */ 1.479819860511658591e-01, /* 3FC2F112 DF3E5244 */ /* zero */ 0.0 }; #define ln2_hi xxx[0] #define ln2_lo xxx[1] #define two54 xxx[2] #define Lp1 xxx[3] #define Lp2 xxx[4] #define Lp3 xxx[5] #define Lp4 xxx[6] #define Lp5 xxx[7] #define Lp6 xxx[8] #define Lp7 xxx[9] #define zero xxx[10] double log1p(double x) { double hfsq, f, c = 0.0, s, z, R, u; int k, hx, hu, ax; hx = ((int *)&x)[HIWORD]; /* high word of x */ ax = hx & 0x7fffffff; if (ax >= 0x7ff00000) { /* x is inf or nan */ if (((hx - 0xfff00000) | ((int *)&x)[LOWORD]) == 0) /* -inf */ return (_SVID_libm_err(x, x, 44)); return (x * x); } k = 1; if (hx < 0x3FDA827A) { /* x < 0.41422 */ if (ax >= 0x3ff00000) /* x <= -1.0 */ return (_SVID_libm_err(x, x, x == -1.0 ? 43 : 44)); if (ax < 0x3e200000) { /* |x| < 2**-29 */ if (two54 + x > zero && /* raise inexact */ ax < 0x3c900000) /* |x| < 2**-54 */ return (x); else return (x - x * x * 0.5); } if (hx > 0 || hx <= (int)0xbfd2bec3) { /* -0.2929<x<0.41422 */ k = 0; f = x; hu = 1; } } /* We will initialize 'c' here. */ if (k != 0) { if (hx < 0x43400000) { u = 1.0 + x; hu = ((int *)&u)[HIWORD]; /* high word of u */ k = (hu >> 20) - 1023; /* * correction term */ c = k > 0 ? 1.0 - (u - x) : x - (u - 1.0); c /= u; } else { u = x; hu = ((int *)&u)[HIWORD]; /* high word of u */ k = (hu >> 20) - 1023; c = 0; } hu &= 0x000fffff; if (hu < 0x6a09e) { /* normalize u */ ((int *)&u)[HIWORD] = hu | 0x3ff00000; } else { /* normalize u/2 */ k += 1; ((int *)&u)[HIWORD] = hu | 0x3fe00000; hu = (0x00100000 - hu) >> 2; } f = u - 1.0; } hfsq = 0.5 * f * f; if (hu == 0) { /* |f| < 2**-20 */ if (f == zero) { if (k == 0) return (zero); /* We already initialized 'c' before, when (k != 0) */ c += k * ln2_lo; return (k * ln2_hi + c); } R = hfsq * (1.0 - 0.66666666666666666 * f); if (k == 0) return (f - R); return (k * ln2_hi - ((R - (k * ln2_lo + c)) - f)); } s = f / (2.0 + f); z = s * s; R = z * (Lp1 + z * (Lp2 + z * (Lp3 + z * (Lp4 + z * (Lp5 + z * (Lp6 + z * Lp7)))))); if (k == 0) return (f - (hfsq - s * (hfsq + R))); return (k * ln2_hi - ((hfsq - (s * (hfsq + R) + (k * ln2_lo + c))) - f)); }
