forked from xuos/xiuos
246 lines
6.7 KiB
C
246 lines
6.7 KiB
C
/* Copyright JS Foundation and other contributors, http://js.foundation
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*
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* Licensed under the Apache License, Version 2.0 (the "License");
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* you may not use this file except in compliance with the License.
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* You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*
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* This file is based on work under the following copyright and permission
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* notice:
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*
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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*
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* @(#)s_log1p.c 5.1 93/09/24
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*/
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#include "jerry-math-internal.h"
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/* log1p(x)
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* Method :
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* 1. Argument Reduction: find k and f such that
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* 1+x = 2^k * (1+f),
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* where sqrt(2)/2 < 1+f < sqrt(2) .
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*
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* Note. If k=0, then f=x is exact. However, if k!=0, then f
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* may not be representable exactly. In that case, a correction
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* term is need. Let u=1+x rounded. Let c = (1+x)-u, then
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* log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
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* and add back the correction term c/u.
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* (Note: when x > 2**53, one can simply return log(x))
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*
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* 2. Approximation of log1p(f).
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* Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
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* = 2s + 2/3 s**3 + 2/5 s**5 + .....,
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* = 2s + s*R
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* We use a special Reme algorithm on [0,0.1716] to generate
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* a polynomial of degree 14 to approximate R The maximum error
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* of this polynomial approximation is bounded by 2**-58.45. In
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* other words,
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* 2 4 6 8 10 12 14
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* R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
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* (the values of Lp1 to Lp7 are listed in the program)
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* and
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* | 2 14 | -58.45
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* | Lp1*s +...+Lp7*s - R(z) | <= 2
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* | |
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* Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
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* In order to guarantee error in log below 1ulp, we compute log
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* by
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* log1p(f) = f - (hfsq - s*(hfsq+R)).
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*
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* 3. Finally, log1p(x) = k*ln2 + log1p(f).
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* = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
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* Here ln2 is split into two floating point number:
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* ln2_hi + ln2_lo,
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* where n*ln2_hi is always exact for |n| < 2000.
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*
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* Special cases:
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* log1p(x) is NaN with signal if x < -1 (including -INF) ;
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* log1p(+INF) is +INF; log1p(-1) is -INF with signal;
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* log1p(NaN) is that NaN with no signal.
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*
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* Accuracy:
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* according to an error analysis, the error is always less than
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* 1 ulp (unit in the last place).
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*
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* Constants:
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* The hexadecimal values are the intended ones for the following
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* constants. The decimal values may be used, provided that the
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* compiler will convert from decimal to binary accurately enough
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* to produce the hexadecimal values shown.
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*
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* Note: Assuming log() return accurate answer, the following
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* algorithm can be used to compute log1p(x) to within a few ULP:
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*
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* u = 1+x;
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* if(u==1.0) return x ; else
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* return log(u)*(x/(u-1.0));
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*
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* See HP-15C Advanced Functions Handbook, p.193.
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*/
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#define zero 0.0
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#define ln2_hi 6.93147180369123816490e-01 /* 3fe62e42 fee00000 */
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#define ln2_lo 1.90821492927058770002e-10 /* 3dea39ef 35793c76 */
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#define two54 1.80143985094819840000e+16 /* 43500000 00000000 */
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#define Lp1 6.666666666666735130e-01 /* 3FE55555 55555593 */
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#define Lp2 3.999999999940941908e-01 /* 3FD99999 9997FA04 */
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#define Lp3 2.857142874366239149e-01 /* 3FD24924 94229359 */
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#define Lp4 2.222219843214978396e-01 /* 3FCC71C5 1D8E78AF */
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#define Lp5 1.818357216161805012e-01 /* 3FC74664 96CB03DE */
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#define Lp6 1.531383769920937332e-01 /* 3FC39A09 D078C69F */
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#define Lp7 1.479819860511658591e-01 /* 3FC2F112 DF3E5244 */
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double
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log1p (double x)
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{
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double hfsq, f, c, s, z, R;
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double_accessor u;
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int k, hx, hu, ax;
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hx = __HI (x);
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ax = hx & 0x7fffffff;
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c = 0;
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k = 1;
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if (hx < 0x3FDA827A)
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{
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/* 1+x < sqrt(2)+ */
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if (ax >= 0x3ff00000)
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{
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/* x <= -1.0 */
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if (x == -1.0)
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{
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/* log1p(-1) = -inf */
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return -two54 / zero;
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}
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else
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{
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/* log1p(x<-1) = NaN */
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return NAN;
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}
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}
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if (ax < 0x3e200000)
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{ /* |x| < 2**-29 */
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if ((two54 + x > zero) /* raise inexact */
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&& (ax < 0x3c900000)) /* |x| < 2**-54 */
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{
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return x;
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}
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else
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{
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return x - x * x * 0.5;
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}
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}
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if ((hx > 0) || hx <= ((int) 0xbfd2bec4))
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{
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/* sqrt(2)/2- <= 1+x < sqrt(2)+ */
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k = 0;
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f = x;
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hu = 1;
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}
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}
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if (hx >= 0x7ff00000)
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{
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return x + x;
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}
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if (k != 0)
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{
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if (hx < 0x43400000)
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{
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u.dbl = 1.0 + x;
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hu = u.as_int.hi;
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k = (hu >> 20) - 1023;
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c = (k > 0) ? 1.0 - (u.dbl - x) : x - (u.dbl - 1.0); /* correction term */
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c /= u.dbl;
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}
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else
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{
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u.dbl = x;
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hu = u.as_int.hi;
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k = (hu >> 20) - 1023;
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c = 0;
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}
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hu &= 0x000fffff;
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/*
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* The approximation to sqrt(2) used in thresholds is not
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* critical. However, the ones used above must give less
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* strict bounds than the one here so that the k==0 case is
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* never reached from here, since here we have committed to
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* using the correction term but don't use it if k==0.
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*/
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if (hu < 0x6a09e)
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{
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/* u ~< sqrt(2) */
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u.as_int.hi = hu | 0x3ff00000; /* normalize u */
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}
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else
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{
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k += 1;
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u.as_int.hi = hu | 0x3fe00000; /* normalize u/2 */
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hu = (0x00100000 - hu) >> 2;
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}
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f = u.dbl - 1.0;
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}
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hfsq = 0.5 * f * f;
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if (hu == 0)
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{
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/* |f| < 2**-20 */
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if (f == zero)
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{
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if (k == 0)
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{
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return zero;
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}
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else
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{
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c += k * ln2_lo;
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return k * ln2_hi + c;
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}
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}
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R = hfsq * (1.0 - 0.66666666666666666 * f);
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if (k == 0)
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{
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return f - R;
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}
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else
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{
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return k * ln2_hi - ((R - (k * ln2_lo + c)) - f);
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}
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}
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s = f / (2.0 + f);
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z = s * s;
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R = z * (Lp1 +
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z * (Lp2 + z * (Lp3 + z * (Lp4 + z * (Lp5 + z * (Lp6 + z * Lp7))))));
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if (k == 0)
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{
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return f - (hfsq - s * (hfsq + R));
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}
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else
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{
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return k * ln2_hi - ((hfsq - (s * (hfsq + R) + (k * ln2_lo + c))) - f);
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}
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} /* log1p */
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#undef zero
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#undef ln2_hi
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#undef ln2_lo
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#undef two54
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#undef Lp1
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#undef Lp2
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#undef Lp3
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#undef Lp4
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#undef Lp5
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#undef Lp6
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#undef Lp7
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