mirror of https://gitlab.com/rnger/amath
239 lines
7.3 KiB
C
239 lines
7.3 KiB
C
/*-
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* Copyright (c) 2014-2017 Carsten Sonne Larsen <cs@innolan.net>
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* All rights reserved.
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions
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* are met:
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* 1. Redistributions of source code must retain the above copyright
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* notice, this list of conditions and the following disclaimer.
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* 2. Redistributions in binary form must reproduce the above copyright
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* notice, this list of conditions and the following disclaimer in the
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* documentation and/or other materials provided with the distribution.
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*
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* THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
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* IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
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* OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
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* IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
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* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
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* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
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* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
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* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
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* THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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*
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* Project homepage:
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* https://amath.innolan.net
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*
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* The original source code can be obtained from:
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* http://www.netlib.org/fdlibm/e_exp.c
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*
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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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* Developed at SunSoft, a Sun Microsystems, Inc. business.
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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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*/
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#include "prim.h"
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static const double
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one = 1.0,
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halF[2] = {
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0.5, -0.5,
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},
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huge = 1.0e+300, twom1000 = 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
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o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
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u_threshold = -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
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ln2HI[2] = {
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6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
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-6.93147180369123816490e-01, /* 0xbfe62e42, 0xfee00000 */
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},
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ln2LO[2] = {
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1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
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-1.90821492927058770002e-10, /* 0xbdea39ef, 0x35793c76 */
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},
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invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
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P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
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P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
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P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
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P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
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P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
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/**
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* @brief Returns the exponential of x
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* @details
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* <pre>
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* Method
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*
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* 1. Argument reduction:
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* Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
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* Given x, find r and integer k such that
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*
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* x = k*ln2 + r, |r| <= 0.5*ln2.
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*
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* Here r will be represented as r = hi-lo for better
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* accuracy.
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*
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* 2. Approximation of exp(r) by a special rational function on
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* the interval [0,0.34658]:
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*
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* Write
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* R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
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* We use a special Remes algorithm on [0,0.34658] to generate
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* a polynomial of degree 5 to approximate R. The maximum error
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* of this polynomial approximation is bounded by 2**-59. In
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* other words,
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* R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
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* (where z=r*r, and the values of P1 to P5 are listed below)
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* and
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*
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* | 5 | -59
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* | 2.0+P1*z+...+P5*z - R(z) | <= 2
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* | |
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*
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* The computation of exp(r) thus becomes
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* 2*r
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* exp(r) = 1 + -------
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* R - r
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* r*R1(r)
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* = 1 + r + ----------- (for better accuracy)
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* 2 - R1(r)
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* where
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* 2 4 10
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* R1(r) = r - (P1*r + P2*r + ... + P5*r ).
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*
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* 3. Scale back to obtain exp(x):
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* From step 1, we have
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* exp(x) = 2^k * exp(r)
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*
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* Special cases:
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*
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* exp(INF) is INF, exp(NaN) is NaN;
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* exp(-INF) is 0, and
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* for finite argument, only exp(0)=1 is exact.
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*
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* Accuracy:
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*
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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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* Misc. info:
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*
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* For IEEE double
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* if x > 7.09782712893383973096e+02 then exp(x) overflow
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* if x < -7.45133219101941108420e+02 then exp(x) underflow
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*
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* Constants:
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*
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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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* </pre>
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*/
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double exp(double x)
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{
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double y, hi, lo, c, t;
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int32_t k, xsb;
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uint32_t hx, hy;
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lo = 0.0;
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hi = 0.0;
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GET_HIGH_WORD(hx, x); // high word of x
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xsb = (hx >> 31) & 1; // sign bit of x
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hx &= 0x7FFFFFFF; // high word of |x|
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// filter out non-finite argument
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// if |x|>=709.78...
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if (hx >= 0x40862E42)
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{
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if (hx >= 0x7FF00000)
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{
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uint32_t lx;
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GET_LOW_WORD(lx, x);
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if (((hx & 0xFFFFF) | lx) != 0)
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{
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return NAN;
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}
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// exp(+-inf)={inf,0}
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return (xsb == 0) ? x : 0.0;
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}
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if (x > o_threshold)
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{
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// overflow
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return huge * huge;
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}
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if (x < u_threshold)
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{
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// underflow
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return twom1000 * twom1000;
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}
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}
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// argument reduction
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// |x| > 0.5 ln2
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if (hx > 0x3FD62E42)
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{
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// |x| < 1.5 ln2
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if (hx < 0x3FF0A2B2)
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{
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hi = x - ln2HI[xsb];
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lo = ln2LO[xsb];
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k = 1 - xsb - xsb;
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}
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else
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{
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k = (int32_t)(invln2 * x + halF[xsb]);
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t = k;
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hi = x - t * ln2HI[0]; // t*ln2HI is exact here
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lo = t * ln2LO[0];
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}
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x = hi - lo;
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}
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// when |x|<2**-28
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else if (hx < 0x3E300000)
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{
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if (huge + x > one)
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{
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return one + x; // trigger inexact
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}
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else
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{
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k = 0;
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}
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}
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else
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{
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k = 0;
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}
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// x is now in primary range
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t = x * x;
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c = x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
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if (k == 0)
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{
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return one - ((x * c) / (c - 2.0) - x);
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}
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y = one - ((lo - (x * c) / (2.0 - c)) - hi);
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if (k >= -1021)
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{
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GET_HIGH_WORD(hy, y);
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SET_HIGH_WORD(y, hy + (k << 20)); // add k to y's exponent
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return y;
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}
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GET_HIGH_WORD(hy, y);
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SET_HIGH_WORD(y, hy + ((k + 1000) << 20)); // add k to y's exponent
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return y * twom1000;
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}
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