| /* 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); |
| } |