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Diffstat (limited to 'libquadmath/math/log1pq.c')
| -rw-r--r-- | libquadmath/math/log1pq.c | 244 |
1 files changed, 244 insertions, 0 deletions
diff --git a/libquadmath/math/log1pq.c b/libquadmath/math/log1pq.c new file mode 100644 index 000000000..a466dc892 --- /dev/null +++ b/libquadmath/math/log1pq.c @@ -0,0 +1,244 @@ +/* log1pl.c + * + * Relative error logarithm + * Natural logarithm of 1+x, 128-bit long double precision + * + * + * + * SYNOPSIS: + * + * long double x, y, log1pl(); + * + * y = log1pl( x ); + * + * + * + * DESCRIPTION: + * + * Returns the base e (2.718...) logarithm of 1+x. + * + * The argument 1+x is separated into its exponent and fractional + * parts. If the exponent is between -1 and +1, the 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(w-1)/(w+1), + * + * log(w) = z + z^3 P(z)/Q(z). + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -1, 8 100000 1.9e-34 4.3e-35 + */ + +/* Copyright 2001 by Stephen L. Moshier + + 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 "quadmath-imp.h" + +/* Coefficients for log(1+x) = x - x^2 / 2 + x^3 P(x)/Q(x) + * 1/sqrt(2) <= 1+x < sqrt(2) + * Theoretical peak relative error = 5.3e-37, + * relative peak error spread = 2.3e-14 + */ +static const __float128 + P12 = 1.538612243596254322971797716843006400388E-6Q, + P11 = 4.998469661968096229986658302195402690910E-1Q, + P10 = 2.321125933898420063925789532045674660756E1Q, + P9 = 4.114517881637811823002128927449878962058E2Q, + P8 = 3.824952356185897735160588078446136783779E3Q, + P7 = 2.128857716871515081352991964243375186031E4Q, + P6 = 7.594356839258970405033155585486712125861E4Q, + P5 = 1.797628303815655343403735250238293741397E5Q, + P4 = 2.854829159639697837788887080758954924001E5Q, + P3 = 3.007007295140399532324943111654767187848E5Q, + P2 = 2.014652742082537582487669938141683759923E5Q, + P1 = 7.771154681358524243729929227226708890930E4Q, + P0 = 1.313572404063446165910279910527789794488E4Q, + /* Q12 = 1.000000000000000000000000000000000000000E0Q, */ + Q11 = 4.839208193348159620282142911143429644326E1Q, + Q10 = 9.104928120962988414618126155557301584078E2Q, + Q9 = 9.147150349299596453976674231612674085381E3Q, + Q8 = 5.605842085972455027590989944010492125825E4Q, + Q7 = 2.248234257620569139969141618556349415120E5Q, + Q6 = 6.132189329546557743179177159925690841200E5Q, + Q5 = 1.158019977462989115839826904108208787040E6Q, + Q4 = 1.514882452993549494932585972882995548426E6Q, + Q3 = 1.347518538384329112529391120390701166528E6Q, + Q2 = 7.777690340007566932935753241556479363645E5Q, + Q1 = 2.626900195321832660448791748036714883242E5Q, + Q0 = 3.940717212190338497730839731583397586124E4Q; + +/* 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 + R5 = -8.828896441624934385266096344596648080902E-1Q, + R4 = 8.057002716646055371965756206836056074715E1Q, + R3 = -2.024301798136027039250415126250455056397E3Q, + R2 = 2.048819892795278657810231591630928516206E4Q, + R1 = -8.977257995689735303686582344659576526998E4Q, + R0 = 1.418134209872192732479751274970992665513E5Q, + /* S6 = 1.000000000000000000000000000000000000000E0Q, */ + S5 = -1.186359407982897997337150403816839480438E2Q, + S4 = 3.998526750980007367835804959888064681098E3Q, + S3 = -5.748542087379434595104154610899551484314E4Q, + S2 = 4.001557694070773974936904547424676279307E5Q, + S1 = -1.332535117259762928288745111081235577029E6Q, + S0 = 1.701761051846631278975701529965589676574E6Q; + +/* C1 + C2 = ln 2 */ +static const __float128 C1 = 6.93145751953125E-1Q; +static const __float128 C2 = 1.428606820309417232121458176568075500134E-6Q; + +static const __float128 sqrth = 0.7071067811865475244008443621048490392848Q; +static const __float128 zero = 0.0Q; + + +__float128 +log1pq (__float128 xm1) +{ + __float128 x, y, z, r, s; + ieee854_float128 u; + int32_t hx; + int e; + + /* Test for NaN or infinity input. */ + u.value = xm1; + hx = u.words32.w0; + if (hx >= 0x7fff0000) + return xm1; + + /* log1p(+- 0) = +- 0. */ + if (((hx & 0x7fffffff) == 0) + && (u.words32.w1 | u.words32.w2 | u.words32.w3) == 0) + return xm1; + + x = xm1 + 1.0Q; + + /* log1p(-1) = -inf */ + if (x <= 0.0Q) + { + if (x == 0.0Q) + return (-1.0Q / (x - x)); + else + return (zero / (x - x)); + } + + /* Separate mantissa from exponent. */ + + /* Use frexp used so that denormal numbers will be handled properly. */ + x = frexpq (x, &e); + + /* Logarithm using log(x) = z + z^3 P(z^2)/Q(z^2), + 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; + r = ((((R5 * z + + R4) * z + + R3) * z + + R2) * z + + R1) * z + + R0; + s = (((((z + + S5) * z + + S4) * z + + S3) * z + + S2) * z + + S1) * z + + S0; + z = x * (z * r / s); + z = z + e * C2; + z = z + x; + z = z + e * C1; + return (z); + } + + + /* Logarithm using log(1+x) = x - .5x^2 + x^3 P(x)/Q(x). */ + + if (x < sqrth) + { + e -= 1; + if (e != 0) + x = 2.0Q * x - 1.0Q; /* 2x - 1 */ + else + x = xm1; + } + else + { + if (e != 0) + x = x - 1.0Q; + else + x = xm1; + } + z = x * x; + r = (((((((((((P12 * x + + P11) * x + + P10) * x + + P9) * x + + P8) * x + + P7) * x + + P6) * x + + P5) * x + + P4) * x + + P3) * x + + P2) * x + + P1) * x + + P0; + s = (((((((((((x + + Q11) * x + + Q10) * x + + Q9) * x + + Q8) * x + + Q7) * x + + Q6) * x + + Q5) * x + + Q4) * x + + Q3) * x + + Q2) * x + + Q1) * x + + Q0; + y = x * (z * r / s); + y = y + e * C2; + z = y - 0.5Q * z; + z = z + x; + z = z + e * C1; + return (z); +} |