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- /* log2l.c
- * Base 2 logarithm, 128-bit long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, log2l();
- *
- * y = log2l( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the base 2 logarithm of x.
- *
- * The argument is separated into its exponent and fractional
- * parts. If the exponent is between -1 and +1, the (natural)
- * logarithm of the fraction is approximated by
- *
- * log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x).
- *
- * Otherwise, setting z = 2(x-1)/x+1),
- *
- * log(x) = z + z^3 P(z)/Q(z).
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0.5, 2.0 100,000 2.6e-34 4.9e-35
- * IEEE exp(+-10000) 100,000 9.6e-35 4.0e-35
- *
- * In the tests over the interval exp(+-10000), the logarithms
- * of the random arguments were uniformly distributed over
- * [-10000, +10000].
- *
- */
- /*
- Cephes Math Library Release 2.2: January, 1991
- Copyright 1984, 1991 by Stephen L. Moshier
- Adapted for glibc November, 2001
- This library is free software; you can redistribute it and/or
- modify it under the terms of the GNU Lesser General Public
- License as published by the Free Software Foundation; either
- version 2.1 of the License, or (at your option) any later version.
- This library is distributed in the hope that it will be useful,
- but WITHOUT ANY WARRANTY; without even the implied warranty of
- MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
- Lesser General Public License for more details.
- You should have received a copy of the GNU Lesser General Public
- License along with this library; if not, see <http://www.gnu.org/licenses/>.
- */
- #include "quadmath-imp.h"
- /* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
- * 1/sqrt(2) <= x < sqrt(2)
- * Theoretical peak relative error = 5.3e-37,
- * relative peak error spread = 2.3e-14
- */
- static const __float128 P[13] =
- {
- 1.313572404063446165910279910527789794488E4Q,
- 7.771154681358524243729929227226708890930E4Q,
- 2.014652742082537582487669938141683759923E5Q,
- 3.007007295140399532324943111654767187848E5Q,
- 2.854829159639697837788887080758954924001E5Q,
- 1.797628303815655343403735250238293741397E5Q,
- 7.594356839258970405033155585486712125861E4Q,
- 2.128857716871515081352991964243375186031E4Q,
- 3.824952356185897735160588078446136783779E3Q,
- 4.114517881637811823002128927449878962058E2Q,
- 2.321125933898420063925789532045674660756E1Q,
- 4.998469661968096229986658302195402690910E-1Q,
- 1.538612243596254322971797716843006400388E-6Q
- };
- static const __float128 Q[12] =
- {
- 3.940717212190338497730839731583397586124E4Q,
- 2.626900195321832660448791748036714883242E5Q,
- 7.777690340007566932935753241556479363645E5Q,
- 1.347518538384329112529391120390701166528E6Q,
- 1.514882452993549494932585972882995548426E6Q,
- 1.158019977462989115839826904108208787040E6Q,
- 6.132189329546557743179177159925690841200E5Q,
- 2.248234257620569139969141618556349415120E5Q,
- 5.605842085972455027590989944010492125825E4Q,
- 9.147150349299596453976674231612674085381E3Q,
- 9.104928120962988414618126155557301584078E2Q,
- 4.839208193348159620282142911143429644326E1Q
- /* 1.000000000000000000000000000000000000000E0L, */
- };
- /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
- * where z = 2(x-1)/(x+1)
- * 1/sqrt(2) <= x < sqrt(2)
- * Theoretical peak relative error = 1.1e-35,
- * relative peak error spread 1.1e-9
- */
- static const __float128 R[6] =
- {
- 1.418134209872192732479751274970992665513E5Q,
- -8.977257995689735303686582344659576526998E4Q,
- 2.048819892795278657810231591630928516206E4Q,
- -2.024301798136027039250415126250455056397E3Q,
- 8.057002716646055371965756206836056074715E1Q,
- -8.828896441624934385266096344596648080902E-1Q
- };
- static const __float128 S[6] =
- {
- 1.701761051846631278975701529965589676574E6Q,
- -1.332535117259762928288745111081235577029E6Q,
- 4.001557694070773974936904547424676279307E5Q,
- -5.748542087379434595104154610899551484314E4Q,
- 3.998526750980007367835804959888064681098E3Q,
- -1.186359407982897997337150403816839480438E2Q
- /* 1.000000000000000000000000000000000000000E0L, */
- };
- static const __float128
- /* log2(e) - 1 */
- LOG2EA = 4.4269504088896340735992468100189213742664595E-1Q,
- /* sqrt(2)/2 */
- SQRTH = 7.071067811865475244008443621048490392848359E-1Q;
- /* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */
- static __float128
- neval (__float128 x, const __float128 *p, int n)
- {
- __float128 y;
- p += n;
- y = *p--;
- do
- {
- y = y * x + *p--;
- }
- while (--n > 0);
- return y;
- }
- /* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */
- static __float128
- deval (__float128 x, const __float128 *p, int n)
- {
- __float128 y;
- p += n;
- y = x + *p--;
- do
- {
- y = y * x + *p--;
- }
- while (--n > 0);
- return y;
- }
- __float128
- log2q (__float128 x)
- {
- __float128 z;
- __float128 y;
- int e;
- int64_t hx, lx;
- /* Test for domain */
- GET_FLT128_WORDS64 (hx, lx, x);
- if (((hx & 0x7fffffffffffffffLL) | lx) == 0)
- return (-1 / fabsq (x)); /* log2l(+-0)=-inf */
- if (hx < 0)
- return (x - x) / (x - x);
- if (hx >= 0x7fff000000000000LL)
- return (x + x);
- if (x == 1)
- return 0;
- /* separate mantissa from exponent */
- /* Note, frexp is used so that denormal numbers
- * will be handled properly.
- */
- x = frexpq (x, &e);
- /* logarithm using log(x) = z + z**3 P(z)/Q(z),
- * where z = 2(x-1)/x+1)
- */
- if ((e > 2) || (e < -2))
- {
- if (x < SQRTH)
- { /* 2( 2x-1 )/( 2x+1 ) */
- e -= 1;
- z = x - 0.5Q;
- y = 0.5Q * z + 0.5Q;
- }
- else
- { /* 2 (x-1)/(x+1) */
- z = x - 0.5Q;
- z -= 0.5Q;
- y = 0.5Q * x + 0.5Q;
- }
- x = z / y;
- z = x * x;
- y = x * (z * neval (z, R, 5) / deval (z, S, 5));
- goto done;
- }
- /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
- if (x < SQRTH)
- {
- e -= 1;
- x = 2.0 * x - 1; /* 2x - 1 */
- }
- else
- {
- x = x - 1;
- }
- z = x * x;
- y = x * (z * neval (x, P, 12) / deval (x, Q, 11));
- y = y - 0.5 * z;
- done:
- /* Multiply log of fraction by log2(e)
- * and base 2 exponent by 1
- */
- z = y * LOG2EA;
- z += x * LOG2EA;
- z += y;
- z += x;
- z += e;
- return (z);
- }
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