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250 lines
6.4 KiB
C
250 lines
6.4 KiB
C
/* log1pl.c
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*
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* Relative error logarithm
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* Natural logarithm of 1+x, 128-bit long double precision
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*
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*
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*
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* SYNOPSIS:
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*
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* long double x, y, log1pl();
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*
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* y = log1pl( x );
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*
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*
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*
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* DESCRIPTION:
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*
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* Returns the base e (2.718...) logarithm of 1+x.
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*
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* The argument 1+x is separated into its exponent and fractional
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* parts. If the exponent is between -1 and +1, the logarithm
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* of the fraction is approximated by
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*
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* log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x).
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*
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* Otherwise, setting z = 2(w-1)/(w+1),
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*
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* log(w) = z + z^3 P(z)/Q(z).
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*
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*
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*
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* ACCURACY:
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*
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* Relative error:
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* arithmetic domain # trials peak rms
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* IEEE -1, 8 100000 1.9e-34 4.3e-35
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*/
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/* Copyright 2001 by Stephen L. Moshier
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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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#include <math.h>
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#include <math_private.h>
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#include <math_ldbl_opt.h>
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/* Coefficients for log(1+x) = x - x^2 / 2 + x^3 P(x)/Q(x)
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* 1/sqrt(2) <= 1+x < sqrt(2)
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* Theoretical peak relative error = 5.3e-37,
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* relative peak error spread = 2.3e-14
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*/
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static const long double
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P12 = 1.538612243596254322971797716843006400388E-6L,
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P11 = 4.998469661968096229986658302195402690910E-1L,
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P10 = 2.321125933898420063925789532045674660756E1L,
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P9 = 4.114517881637811823002128927449878962058E2L,
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P8 = 3.824952356185897735160588078446136783779E3L,
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P7 = 2.128857716871515081352991964243375186031E4L,
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P6 = 7.594356839258970405033155585486712125861E4L,
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P5 = 1.797628303815655343403735250238293741397E5L,
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P4 = 2.854829159639697837788887080758954924001E5L,
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P3 = 3.007007295140399532324943111654767187848E5L,
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P2 = 2.014652742082537582487669938141683759923E5L,
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P1 = 7.771154681358524243729929227226708890930E4L,
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P0 = 1.313572404063446165910279910527789794488E4L,
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/* Q12 = 1.000000000000000000000000000000000000000E0L, */
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Q11 = 4.839208193348159620282142911143429644326E1L,
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Q10 = 9.104928120962988414618126155557301584078E2L,
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Q9 = 9.147150349299596453976674231612674085381E3L,
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Q8 = 5.605842085972455027590989944010492125825E4L,
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Q7 = 2.248234257620569139969141618556349415120E5L,
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Q6 = 6.132189329546557743179177159925690841200E5L,
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Q5 = 1.158019977462989115839826904108208787040E6L,
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Q4 = 1.514882452993549494932585972882995548426E6L,
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Q3 = 1.347518538384329112529391120390701166528E6L,
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Q2 = 7.777690340007566932935753241556479363645E5L,
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Q1 = 2.626900195321832660448791748036714883242E5L,
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Q0 = 3.940717212190338497730839731583397586124E4L;
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/* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
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* where z = 2(x-1)/(x+1)
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* 1/sqrt(2) <= x < sqrt(2)
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* Theoretical peak relative error = 1.1e-35,
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* relative peak error spread 1.1e-9
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*/
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static const long double
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R5 = -8.828896441624934385266096344596648080902E-1L,
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R4 = 8.057002716646055371965756206836056074715E1L,
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R3 = -2.024301798136027039250415126250455056397E3L,
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R2 = 2.048819892795278657810231591630928516206E4L,
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R1 = -8.977257995689735303686582344659576526998E4L,
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R0 = 1.418134209872192732479751274970992665513E5L,
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/* S6 = 1.000000000000000000000000000000000000000E0L, */
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S5 = -1.186359407982897997337150403816839480438E2L,
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S4 = 3.998526750980007367835804959888064681098E3L,
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S3 = -5.748542087379434595104154610899551484314E4L,
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S2 = 4.001557694070773974936904547424676279307E5L,
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S1 = -1.332535117259762928288745111081235577029E6L,
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S0 = 1.701761051846631278975701529965589676574E6L;
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/* C1 + C2 = ln 2 */
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static const long double C1 = 6.93145751953125E-1L;
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static const long double C2 = 1.428606820309417232121458176568075500134E-6L;
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static const long double sqrth = 0.7071067811865475244008443621048490392848L;
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/* ln (2^16384 * (1 - 2^-113)) */
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static const long double zero = 0.0L;
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long double
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__log1pl (long double xm1)
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{
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long double x, y, z, r, s;
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double xhi;
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int32_t hx, lx;
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int e;
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/* Test for NaN or infinity input. */
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xhi = ldbl_high (xm1);
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EXTRACT_WORDS (hx, lx, xhi);
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if ((hx & 0x7fffffff) >= 0x7ff00000)
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return xm1 + xm1 * xm1;
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/* log1p(+- 0) = +- 0. */
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if (((hx & 0x7fffffff) | lx) == 0)
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return xm1;
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if (xm1 >= 0x1p107L)
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x = xm1;
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else
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x = xm1 + 1.0L;
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/* log1p(-1) = -inf */
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if (x <= 0.0L)
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{
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if (x == 0.0L)
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return (-1.0L / 0.0L);
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else
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return (zero / (x - x));
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}
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/* Separate mantissa from exponent. */
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/* Use frexp used so that denormal numbers will be handled properly. */
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x = __frexpl (x, &e);
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/* Logarithm using log(x) = z + z^3 P(z^2)/Q(z^2),
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where z = 2(x-1)/x+1). */
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if ((e > 2) || (e < -2))
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{
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if (x < sqrth)
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{ /* 2( 2x-1 )/( 2x+1 ) */
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e -= 1;
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z = x - 0.5L;
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y = 0.5L * z + 0.5L;
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}
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else
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{ /* 2 (x-1)/(x+1) */
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z = x - 0.5L;
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z -= 0.5L;
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y = 0.5L * x + 0.5L;
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}
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x = z / y;
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z = x * x;
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r = ((((R5 * z
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+ R4) * z
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+ R3) * z
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+ R2) * z
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+ R1) * z
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+ R0;
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s = (((((z
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+ S5) * z
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+ S4) * z
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+ S3) * z
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+ S2) * z
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+ S1) * z
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+ S0;
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z = x * (z * r / s);
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z = z + e * C2;
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z = z + x;
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z = z + e * C1;
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return (z);
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}
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/* Logarithm using log(1+x) = x - .5x^2 + x^3 P(x)/Q(x). */
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if (x < sqrth)
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{
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e -= 1;
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if (e != 0)
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x = 2.0L * x - 1.0L; /* 2x - 1 */
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else
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x = xm1;
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}
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else
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{
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if (e != 0)
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x = x - 1.0L;
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else
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x = xm1;
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}
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z = x * x;
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r = (((((((((((P12 * x
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+ P11) * x
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+ P10) * x
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+ P9) * x
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+ P8) * x
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+ P7) * x
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+ P6) * x
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+ P5) * x
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+ P4) * x
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+ P3) * x
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+ P2) * x
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+ P1) * x
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+ P0;
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s = (((((((((((x
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+ Q11) * x
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+ Q10) * x
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+ Q9) * x
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+ Q8) * x
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+ Q7) * x
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+ Q6) * x
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+ Q5) * x
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+ Q4) * x
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+ Q3) * x
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+ Q2) * x
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+ Q1) * x
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+ Q0;
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y = x * (z * r / s);
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y = y + e * C2;
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z = y - 0.5L * z;
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z = z + x;
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z = z + e * C1;
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return (z);
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}
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