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Diffstat (limited to 'sysdeps/ieee754/ldbl-128ibm/e_log2l.c')
| -rw-r--r-- | sysdeps/ieee754/ldbl-128ibm/e_log2l.c | 250 |
1 files changed, 250 insertions, 0 deletions
diff --git a/sysdeps/ieee754/ldbl-128ibm/e_log2l.c b/sysdeps/ieee754/ldbl-128ibm/e_log2l.c new file mode 100644 index 0000000..fe8a8e1 --- /dev/null +++ b/sysdeps/ieee754/ldbl-128ibm/e_log2l.c @@ -0,0 +1,250 @@ +/* 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, write to the Free Software + Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA + */ + +#include "math.h" +#include "math_private.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 long double P[13] = +{ + 1.313572404063446165910279910527789794488E4L, + 7.771154681358524243729929227226708890930E4L, + 2.014652742082537582487669938141683759923E5L, + 3.007007295140399532324943111654767187848E5L, + 2.854829159639697837788887080758954924001E5L, + 1.797628303815655343403735250238293741397E5L, + 7.594356839258970405033155585486712125861E4L, + 2.128857716871515081352991964243375186031E4L, + 3.824952356185897735160588078446136783779E3L, + 4.114517881637811823002128927449878962058E2L, + 2.321125933898420063925789532045674660756E1L, + 4.998469661968096229986658302195402690910E-1L, + 1.538612243596254322971797716843006400388E-6L +}; +static const long double Q[12] = +{ + 3.940717212190338497730839731583397586124E4L, + 2.626900195321832660448791748036714883242E5L, + 7.777690340007566932935753241556479363645E5L, + 1.347518538384329112529391120390701166528E6L, + 1.514882452993549494932585972882995548426E6L, + 1.158019977462989115839826904108208787040E6L, + 6.132189329546557743179177159925690841200E5L, + 2.248234257620569139969141618556349415120E5L, + 5.605842085972455027590989944010492125825E4L, + 9.147150349299596453976674231612674085381E3L, + 9.104928120962988414618126155557301584078E2L, + 4.839208193348159620282142911143429644326E1L +/* 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 long double R[6] = +{ + 1.418134209872192732479751274970992665513E5L, + -8.977257995689735303686582344659576526998E4L, + 2.048819892795278657810231591630928516206E4L, + -2.024301798136027039250415126250455056397E3L, + 8.057002716646055371965756206836056074715E1L, + -8.828896441624934385266096344596648080902E-1L +}; +static const long double S[6] = +{ + 1.701761051846631278975701529965589676574E6L, + -1.332535117259762928288745111081235577029E6L, + 4.001557694070773974936904547424676279307E5L, + -5.748542087379434595104154610899551484314E4L, + 3.998526750980007367835804959888064681098E3L, + -1.186359407982897997337150403816839480438E2L +/* 1.000000000000000000000000000000000000000E0L, */ +}; + +static const long double +/* log2(e) - 1 */ +LOG2EA = 4.4269504088896340735992468100189213742664595E-1L, +/* sqrt(2)/2 */ +SQRTH = 7.071067811865475244008443621048490392848359E-1L; + + +/* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */ + +static long double +neval (long double x, const long double *p, int n) +{ + long double 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 long double +deval (long double x, const long double *p, int n) +{ + long double y; + + p += n; + y = x + *p--; + do + { + y = y * x + *p--; + } + while (--n > 0); + return y; +} + + + +long double +__ieee754_log2l (x) + long double x; +{ + long double z; + long double y; + int e; + int64_t hx, lx; + +/* Test for domain */ + GET_LDOUBLE_WORDS64 (hx, lx, x); + if (((hx & 0x7fffffffffffffffLL) | (lx & 0x7fffffffffffffffLL)) == 0) + return (-1.0L / (x - x)); + if (hx < 0) + return (x - x) / (x - x); + if (hx >= 0x7ff0000000000000LL) + return (x + x); + +/* separate mantissa from exponent */ + +/* Note, frexp is used so that denormal numbers + * will be handled properly. + */ + x = __frexpl (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 - 0.5L; + y = 0.5L * z + 0.5L; + } + else + { /* 2 (x-1)/(x+1) */ + z = x - 0.5L; + z -= 0.5L; + y = 0.5L * x + 0.5L; + } + 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 - 1.0L; /* 2x - 1 */ + } + else + { + x = x - 1.0L; + } + 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); +} |