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Diffstat (limited to 'usr/src/libm/src/C/log1p.c')
| -rw-r--r-- | usr/src/libm/src/C/log1p.c | 201 |
1 files changed, 201 insertions, 0 deletions
diff --git a/usr/src/libm/src/C/log1p.c b/usr/src/libm/src/C/log1p.c new file mode 100644 index 0000000..845fd0e --- /dev/null +++ b/usr/src/libm/src/C/log1p.c @@ -0,0 +1,201 @@ +/* + * 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 2005 Sun Microsystems, Inc. All rights reserved. + * Use is subject to license terms. + */ + +#pragma ident "@(#)log1p.c 1.23 06/01/23 SMI" + +#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, 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; + } + } + 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); + 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)); +} |