249 lines
6.1 KiB
C
249 lines
6.1 KiB
C
/* log2l.c
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* Base 2 logarithm, 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, log2l();
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*
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* y = log2l( 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 2 logarithm of x.
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*
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* The argument is separated into its exponent and fractional
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* parts. If the exponent is between -1 and +1, the (natural)
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* logarithm 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(x-1)/x+1),
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*
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* log(x) = 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 0.5, 2.0 100,000 2.6e-34 4.9e-35
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* IEEE exp(+-10000) 100,000 9.6e-35 4.0e-35
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*
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* In the tests over the interval exp(+-10000), the logarithms
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* of the random arguments were uniformly distributed over
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* [-10000, +10000].
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*
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*/
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/*
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Cephes Math Library Release 2.2: January, 1991
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Copyright 1984, 1991 by Stephen L. Moshier
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Adapted for glibc November, 2001
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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, write to the Free Software
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Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
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*/
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#include "quadmath-imp.h"
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/* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
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* 1/sqrt(2) <= 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 __float128 P[13] =
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{
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1.313572404063446165910279910527789794488E4Q,
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7.771154681358524243729929227226708890930E4Q,
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2.014652742082537582487669938141683759923E5Q,
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3.007007295140399532324943111654767187848E5Q,
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2.854829159639697837788887080758954924001E5Q,
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1.797628303815655343403735250238293741397E5Q,
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7.594356839258970405033155585486712125861E4Q,
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2.128857716871515081352991964243375186031E4Q,
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3.824952356185897735160588078446136783779E3Q,
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4.114517881637811823002128927449878962058E2Q,
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2.321125933898420063925789532045674660756E1Q,
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4.998469661968096229986658302195402690910E-1Q,
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1.538612243596254322971797716843006400388E-6Q
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};
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static const __float128 Q[12] =
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{
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3.940717212190338497730839731583397586124E4Q,
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2.626900195321832660448791748036714883242E5Q,
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7.777690340007566932935753241556479363645E5Q,
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1.347518538384329112529391120390701166528E6Q,
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1.514882452993549494932585972882995548426E6Q,
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1.158019977462989115839826904108208787040E6Q,
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6.132189329546557743179177159925690841200E5Q,
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2.248234257620569139969141618556349415120E5Q,
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5.605842085972455027590989944010492125825E4Q,
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9.147150349299596453976674231612674085381E3Q,
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9.104928120962988414618126155557301584078E2Q,
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4.839208193348159620282142911143429644326E1Q
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/* 1.000000000000000000000000000000000000000E0Q, */
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};
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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 __float128 R[6] =
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{
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1.418134209872192732479751274970992665513E5Q,
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-8.977257995689735303686582344659576526998E4Q,
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2.048819892795278657810231591630928516206E4Q,
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-2.024301798136027039250415126250455056397E3Q,
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8.057002716646055371965756206836056074715E1Q,
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-8.828896441624934385266096344596648080902E-1Q
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};
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static const __float128 S[6] =
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{
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1.701761051846631278975701529965589676574E6Q,
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-1.332535117259762928288745111081235577029E6Q,
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4.001557694070773974936904547424676279307E5Q,
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-5.748542087379434595104154610899551484314E4Q,
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3.998526750980007367835804959888064681098E3Q,
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-1.186359407982897997337150403816839480438E2Q
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/* 1.000000000000000000000000000000000000000E0Q, */
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};
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static const __float128
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/* log2(e) - 1 */
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LOG2EA = 4.4269504088896340735992468100189213742664595E-1Q,
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/* sqrt(2)/2 */
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SQRTH = 7.071067811865475244008443621048490392848359E-1Q;
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/* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */
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static __float128
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neval (__float128 x, const __float128 *p, int n)
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{
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__float128 y;
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p += n;
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y = *p--;
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do
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{
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y = y * x + *p--;
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}
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while (--n > 0);
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return y;
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}
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/* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */
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static __float128
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deval (__float128 x, const __float128 *p, int n)
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{
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__float128 y;
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p += n;
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y = x + *p--;
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do
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{
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y = y * x + *p--;
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}
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while (--n > 0);
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return y;
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}
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__float128
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log2q (__float128 x)
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{
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__float128 z;
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__float128 y;
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int e;
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int64_t hx, lx;
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/* Test for domain */
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GET_FLT128_WORDS64 (hx, lx, x);
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if (((hx & 0x7fffffffffffffffLL) | lx) == 0)
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return (-1.0Q / (x - x));
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if (hx < 0)
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return (x - x) / (x - x);
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if (hx >= 0x7fff000000000000LL)
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return (x + x);
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/* separate mantissa from exponent */
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/* Note, frexp is used so that denormal numbers
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* will be handled properly.
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*/
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x = frexpq (x, &e);
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/* logarithm using log(x) = z + z**3 P(z)/Q(z),
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* where z = 2(x-1)/x+1)
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*/
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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 - 0.5Q;
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y = 0.5Q * z + 0.5Q;
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}
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else
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{ /* 2 (x-1)/(x+1) */
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z = x - 0.5Q;
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z -= 0.5Q;
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y = 0.5Q * x + 0.5Q;
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}
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x = z / y;
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z = x * x;
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y = x * (z * neval (z, R, 5) / deval (z, S, 5));
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goto done;
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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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x = 2.0 * x - 1.0Q; /* 2x - 1 */
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}
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else
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{
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x = x - 1.0Q;
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}
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z = x * x;
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y = x * (z * neval (x, P, 12) / deval (x, Q, 11));
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y = y - 0.5 * z;
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done:
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/* Multiply log of fraction by log2(e)
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* and base 2 exponent by 1
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*/
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z = y * LOG2EA;
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z += x * LOG2EA;
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z += y;
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z += x;
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z += e;
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return (z);
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}
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