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/*
* 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));
}
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