418 lines
11 KiB
C
418 lines
11 KiB
C
/*
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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 SunPro, 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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/* Expansions and modifications for 128-bit long double are
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Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
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and are incorporated herein by permission of the author. The author
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reserves the right to distribute this material elsewhere under different
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copying permissions. These modifications are distributed here under
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the following terms:
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This library is free software; you can redistribute it and/or
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modify it under the terms of the GNU Lesser General Public
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License as published by the Free Software Foundation; either
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version 2.1 of the License, or (at your option) any later version.
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This library is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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Lesser General Public License for more details.
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You should have received a copy of the GNU Lesser General Public
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License along with this library; if not, see
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<https://www.gnu.org/licenses/>. */
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/* __ieee754_powl(x,y) return x**y
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*
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* n
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* Method: Let x = 2 * (1+f)
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* 1. Compute and return log2(x) in two pieces:
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* log2(x) = w1 + w2,
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* where w1 has 113-53 = 60 bit trailing zeros.
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* 2. Perform y*log2(x) = n+y' by simulating muti-precision
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* arithmetic, where |y'|<=0.5.
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* 3. Return x**y = 2**n*exp(y'*log2)
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*
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* Special cases:
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* 1. (anything) ** 0 is 1
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* 2. (anything) ** 1 is itself
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* 3. (anything) ** NAN is NAN
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* 4. NAN ** (anything except 0) is NAN
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* 5. +-(|x| > 1) ** +INF is +INF
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* 6. +-(|x| > 1) ** -INF is +0
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* 7. +-(|x| < 1) ** +INF is +0
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* 8. +-(|x| < 1) ** -INF is +INF
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* 9. +-1 ** +-INF is NAN
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* 10. +0 ** (+anything except 0, NAN) is +0
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* 11. -0 ** (+anything except 0, NAN, odd integer) is +0
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* 12. +0 ** (-anything except 0, NAN) is +INF
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* 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
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* 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
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* 15. +INF ** (+anything except 0,NAN) is +INF
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* 16. +INF ** (-anything except 0,NAN) is +0
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* 17. -INF ** (anything) = -0 ** (-anything)
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* 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
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* 19. (-anything except 0 and inf) ** (non-integer) is NAN
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*
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*/
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#include <math.h>
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#include <math_private.h>
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#include <math-underflow.h>
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#include <libm-alias-finite.h>
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static const long double bp[] = {
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1.0L,
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1.5L,
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};
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/* log_2(1.5) */
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static const long double dp_h[] = {
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0.0,
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5.8496250072115607565592654282227158546448E-1L
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};
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/* Low part of log_2(1.5) */
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static const long double dp_l[] = {
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0.0,
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1.0579781240112554492329533686862998106046E-16L
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};
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static const long double zero = 0.0L,
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one = 1.0L,
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two = 2.0L,
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two113 = 1.0384593717069655257060992658440192E34L,
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huge = 1.0e300L,
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tiny = 1.0e-300L;
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/* 3/2 log x = 3 z + z^3 + z^3 (z^2 R(z^2))
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z = (x-1)/(x+1)
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1 <= x <= 1.25
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Peak relative error 2.3e-37 */
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static const long double LN[] =
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{
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-3.0779177200290054398792536829702930623200E1L,
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6.5135778082209159921251824580292116201640E1L,
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-4.6312921812152436921591152809994014413540E1L,
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1.2510208195629420304615674658258363295208E1L,
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-9.9266909031921425609179910128531667336670E-1L
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};
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static const long double LD[] =
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{
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-5.129862866715009066465422805058933131960E1L,
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1.452015077564081884387441590064272782044E2L,
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-1.524043275549860505277434040464085593165E2L,
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7.236063513651544224319663428634139768808E1L,
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-1.494198912340228235853027849917095580053E1L
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/* 1.0E0 */
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};
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/* exp(x) = 1 + x - x / (1 - 2 / (x - x^2 R(x^2)))
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0 <= x <= 0.5
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Peak relative error 5.7e-38 */
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static const long double PN[] =
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{
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5.081801691915377692446852383385968225675E8L,
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9.360895299872484512023336636427675327355E6L,
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4.213701282274196030811629773097579432957E4L,
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5.201006511142748908655720086041570288182E1L,
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9.088368420359444263703202925095675982530E-3L,
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};
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static const long double PD[] =
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{
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3.049081015149226615468111430031590411682E9L,
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1.069833887183886839966085436512368982758E8L,
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8.259257717868875207333991924545445705394E5L,
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1.872583833284143212651746812884298360922E3L,
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/* 1.0E0 */
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};
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static const long double
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/* ln 2 */
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lg2 = 6.9314718055994530941723212145817656807550E-1L,
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lg2_h = 6.9314718055994528622676398299518041312695E-1L,
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lg2_l = 2.3190468138462996154948554638754786504121E-17L,
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ovt = 8.0085662595372944372e-0017L,
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/* 2/(3*log(2)) */
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cp = 9.6179669392597560490661645400126142495110E-1L,
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cp_h = 9.6179669392597555432899980587535537779331E-1L,
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cp_l = 5.0577616648125906047157785230014751039424E-17L;
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long double
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__ieee754_powl (long double x, long double y)
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{
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long double z, ax, z_h, z_l, p_h, p_l;
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long double y1, t1, t2, r, s, sgn, t, u, v, w;
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long double s2, s_h, s_l, t_h, t_l, ay;
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int32_t i, j, k, yisint, n;
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uint32_t ix, iy;
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int32_t hx, hy, hax;
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double ohi, xhi, xlo, yhi, ylo;
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uint32_t lx, ly, lj;
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ldbl_unpack (x, &xhi, &xlo);
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EXTRACT_WORDS (hx, lx, xhi);
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ix = hx & 0x7fffffff;
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ldbl_unpack (y, &yhi, &ylo);
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EXTRACT_WORDS (hy, ly, yhi);
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iy = hy & 0x7fffffff;
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/* y==zero: x**0 = 1 */
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if ((iy | ly) == 0 && !issignaling (x))
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return one;
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/* 1.0**y = 1; -1.0**+-Inf = 1 */
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if (x == one && !issignaling (y))
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return one;
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if (x == -1.0L && ((iy - 0x7ff00000) | ly) == 0)
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return one;
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/* +-NaN return x+y */
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if ((ix >= 0x7ff00000 && ((ix - 0x7ff00000) | lx) != 0)
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|| (iy >= 0x7ff00000 && ((iy - 0x7ff00000) | ly) != 0))
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return x + y;
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/* determine if y is an odd int when x < 0
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* yisint = 0 ... y is not an integer
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* yisint = 1 ... y is an odd int
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* yisint = 2 ... y is an even int
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*/
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yisint = 0;
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if (hx < 0)
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{
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uint32_t low_ye;
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GET_HIGH_WORD (low_ye, ylo);
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if ((low_ye & 0x7fffffff) >= 0x43400000) /* Low part >= 2^53 */
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yisint = 2; /* even integer y */
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else if (iy >= 0x3ff00000) /* 1.0 */
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{
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if (floorl (y) == y)
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{
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z = 0.5 * y;
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if (floorl (z) == z)
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yisint = 2;
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else
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yisint = 1;
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}
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}
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}
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ax = fabsl (x);
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/* special value of y */
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if (ly == 0)
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{
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if (iy == 0x7ff00000) /* y is +-inf */
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{
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if (ax > one)
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/* (|x|>1)**+-inf = inf,0 */
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return (hy >= 0) ? y : zero;
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else
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/* (|x|<1)**-,+inf = inf,0 */
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return (hy < 0) ? -y : zero;
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}
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if (ylo == 0.0)
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{
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if (iy == 0x3ff00000)
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{ /* y is +-1 */
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if (hy < 0)
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return one / x;
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else
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return x;
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}
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if (hy == 0x40000000)
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return x * x; /* y is 2 */
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if (hy == 0x3fe00000)
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{ /* y is 0.5 */
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if (hx >= 0) /* x >= +0 */
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return sqrtl (x);
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}
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}
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}
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/* special value of x */
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if (lx == 0)
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{
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if (ix == 0x7ff00000 || ix == 0 || (ix == 0x3ff00000 && xlo == 0.0))
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{
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z = ax; /*x is +-0,+-inf,+-1 */
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if (hy < 0)
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z = one / z; /* z = (1/|x|) */
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if (hx < 0)
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{
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if (((ix - 0x3ff00000) | yisint) == 0)
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{
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z = (z - z) / (z - z); /* (-1)**non-int is NaN */
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}
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else if (yisint == 1)
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z = -z; /* (x<0)**odd = -(|x|**odd) */
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}
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return z;
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}
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}
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/* (x<0)**(non-int) is NaN */
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if (((((uint32_t) hx >> 31) - 1) | yisint) == 0)
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return (x - x) / (x - x);
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/* sgn (sign of result -ve**odd) = -1 else = 1 */
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sgn = one;
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if (((((uint32_t) hx >> 31) - 1) | (yisint - 1)) == 0)
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sgn = -one; /* (-ve)**(odd int) */
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/* |y| is huge.
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2^-16495 = 1/2 of smallest representable value.
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If (1 - 1/131072)^y underflows, y > 1.4986e9 */
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if (iy > 0x41d654b0)
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{
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/* if (1 - 2^-113)^y underflows, y > 1.1873e38 */
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if (iy > 0x47d654b0)
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{
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if (ix <= 0x3fefffff)
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return (hy < 0) ? sgn * huge * huge : sgn * tiny * tiny;
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if (ix >= 0x3ff00000)
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return (hy > 0) ? sgn * huge * huge : sgn * tiny * tiny;
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}
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/* over/underflow if x is not close to one */
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if (ix < 0x3fefffff)
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return (hy < 0) ? sgn * huge * huge : sgn * tiny * tiny;
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if (ix > 0x3ff00000)
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return (hy > 0) ? sgn * huge * huge : sgn * tiny * tiny;
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}
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ay = y > 0 ? y : -y;
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if (ay < 0x1p-117)
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y = y < 0 ? -0x1p-117 : 0x1p-117;
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n = 0;
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/* take care subnormal number */
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if (ix < 0x00100000)
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{
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ax *= two113;
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n -= 113;
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ohi = ldbl_high (ax);
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GET_HIGH_WORD (ix, ohi);
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}
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n += ((ix) >> 20) - 0x3ff;
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j = ix & 0x000fffff;
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/* determine interval */
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ix = j | 0x3ff00000; /* normalize ix */
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if (j <= 0x39880)
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k = 0; /* |x|<sqrt(3/2) */
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else if (j < 0xbb670)
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k = 1; /* |x|<sqrt(3) */
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else
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{
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k = 0;
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n += 1;
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ix -= 0x00100000;
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}
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ohi = ldbl_high (ax);
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GET_HIGH_WORD (hax, ohi);
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ax = __scalbnl (ax, ((int) ((ix - hax) * 2)) >> 21);
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/* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
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u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
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v = one / (ax + bp[k]);
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s = u * v;
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s_h = ldbl_high (s);
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/* t_h=ax+bp[k] High */
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t_h = ax + bp[k];
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t_h = ldbl_high (t_h);
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t_l = ax - (t_h - bp[k]);
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s_l = v * ((u - s_h * t_h) - s_h * t_l);
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/* compute log(ax) */
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s2 = s * s;
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u = LN[0] + s2 * (LN[1] + s2 * (LN[2] + s2 * (LN[3] + s2 * LN[4])));
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v = LD[0] + s2 * (LD[1] + s2 * (LD[2] + s2 * (LD[3] + s2 * (LD[4] + s2))));
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r = s2 * s2 * u / v;
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r += s_l * (s_h + s);
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s2 = s_h * s_h;
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t_h = 3.0 + s2 + r;
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t_h = ldbl_high (t_h);
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t_l = r - ((t_h - 3.0) - s2);
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/* u+v = s*(1+...) */
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u = s_h * t_h;
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v = s_l * t_h + t_l * s;
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/* 2/(3log2)*(s+...) */
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p_h = u + v;
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p_h = ldbl_high (p_h);
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p_l = v - (p_h - u);
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z_h = cp_h * p_h; /* cp_h+cp_l = 2/(3*log2) */
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z_l = cp_l * p_h + p_l * cp + dp_l[k];
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/* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
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t = (long double) n;
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t1 = (((z_h + z_l) + dp_h[k]) + t);
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t1 = ldbl_high (t1);
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t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
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/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
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y1 = ldbl_high (y);
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p_l = (y - y1) * t1 + y * t2;
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p_h = y1 * t1;
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z = p_l + p_h;
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ohi = ldbl_high (z);
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EXTRACT_WORDS (j, lj, ohi);
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if (j >= 0x40d00000) /* z >= 16384 */
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{
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/* if z > 16384 */
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if (((j - 0x40d00000) | lj) != 0)
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return sgn * huge * huge; /* overflow */
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else
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{
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if (p_l + ovt > z - p_h)
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return sgn * huge * huge; /* overflow */
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}
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}
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else if ((j & 0x7fffffff) >= 0x40d01b90) /* z <= -16495 */
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{
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/* z < -16495 */
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if (((j - 0xc0d01bc0) | lj) != 0)
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return sgn * tiny * tiny; /* underflow */
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else
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{
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if (p_l <= z - p_h)
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return sgn * tiny * tiny; /* underflow */
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}
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}
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/* compute 2**(p_h+p_l) */
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i = j & 0x7fffffff;
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k = (i >> 20) - 0x3ff;
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n = 0;
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if (i > 0x3fe00000)
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{ /* if |z| > 0.5, set n = [z+0.5] */
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n = floorl (z + 0.5L);
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t = n;
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p_h -= t;
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}
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t = p_l + p_h;
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t = ldbl_high (t);
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u = t * lg2_h;
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v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
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z = u + v;
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w = v - (z - u);
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/* exp(z) */
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t = z * z;
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u = PN[0] + t * (PN[1] + t * (PN[2] + t * (PN[3] + t * PN[4])));
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v = PD[0] + t * (PD[1] + t * (PD[2] + t * (PD[3] + t)));
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t1 = z - t * u / v;
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r = (z * t1) / (t1 - two) - (w + z * w);
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z = one - (r - z);
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z = __scalbnl (sgn * z, n);
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math_check_force_underflow (z);
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return z;
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}
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libm_alias_finite (__ieee754_powl, __powl)
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