| /* log1pl.c |
| * |
| * Relative error logarithm |
| * Natural logarithm of 1+x for __float128 precision |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * __float128 x, y, log1pl(); |
| * |
| * y = log1pq( 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 & 0x7fffffff) >= 0x7fff0000) |
| return xm1 + fabsq (xm1); |
| |
| /* log1p(+- 0) = +- 0. */ |
| if (((hx & 0x7fffffff) == 0) |
| && (u.words32.w1 | u.words32.w2 | u.words32.w3) == 0) |
| return xm1; |
| |
| if ((hx & 0x7fffffff) < 0x3f8e0000) |
| { |
| math_check_force_underflow (xm1); |
| if ((int) xm1 == 0) |
| return xm1; |
| } |
| |
| if (xm1 >= 0x1p113Q) |
| x = xm1; |
| else |
| x = xm1 + 1.0Q; |
| |
| /* log1p(-1) = -inf */ |
| if (x <= 0.0Q) |
| { |
| if (x == 0.0Q) |
| return (-1.0Q / zero); /* log1p(-1) = -inf */ |
| 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); |
| } |