3 * Relative error logarithm
4 * Natural logarithm of 1+x, 128-bit long double precision
10 * long double x, y, log1pl();
18 * Returns the base e (2.718...) logarithm of 1+x.
20 * The argument 1+x is separated into its exponent and fractional
21 * parts. If the exponent is between -1 and +1, the logarithm
22 * of the fraction is approximated by
24 * log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x).
26 * Otherwise, setting z = 2(w-1)/(w+1),
28 * log(w) = z + z^3 P(z)/Q(z).
35 * arithmetic domain # trials peak rms
36 * IEEE -1, 8 100000 1.9e-34 4.3e-35
39 /* Copyright 2001 by Stephen L. Moshier
41 This library is free software; you can redistribute it and/or
42 modify it under the terms of the GNU Lesser General Public
43 License as published by the Free Software Foundation; either
44 version 2.1 of the License, or (at your option) any later version.
46 This library is distributed in the hope that it will be useful,
47 but WITHOUT ANY WARRANTY; without even the implied warranty of
48 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
49 Lesser General Public License for more details.
51 You should have received a copy of the GNU Lesser General Public
52 License along with this library; if not, write to the Free Software
53 Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */
57 #include "math_private.h"
58 #include <gnu/option-groups.h>
60 #if __OPTION_EGLIBC_LIBM_BIG
62 /* Coefficients for log(1+x) = x - x^2 / 2 + x^3 P(x)/Q(x)
63 * 1/sqrt(2) <= 1+x < sqrt(2)
64 * Theoretical peak relative error = 5.3e-37,
65 * relative peak error spread = 2.3e-14
67 static const long double
68 P12 = 1.538612243596254322971797716843006400388E-6L,
69 P11 = 4.998469661968096229986658302195402690910E-1L,
70 P10 = 2.321125933898420063925789532045674660756E1L,
71 P9 = 4.114517881637811823002128927449878962058E2L,
72 P8 = 3.824952356185897735160588078446136783779E3L,
73 P7 = 2.128857716871515081352991964243375186031E4L,
74 P6 = 7.594356839258970405033155585486712125861E4L,
75 P5 = 1.797628303815655343403735250238293741397E5L,
76 P4 = 2.854829159639697837788887080758954924001E5L,
77 P3 = 3.007007295140399532324943111654767187848E5L,
78 P2 = 2.014652742082537582487669938141683759923E5L,
79 P1 = 7.771154681358524243729929227226708890930E4L,
80 P0 = 1.313572404063446165910279910527789794488E4L,
81 /* Q12 = 1.000000000000000000000000000000000000000E0L, */
82 Q11 = 4.839208193348159620282142911143429644326E1L,
83 Q10 = 9.104928120962988414618126155557301584078E2L,
84 Q9 = 9.147150349299596453976674231612674085381E3L,
85 Q8 = 5.605842085972455027590989944010492125825E4L,
86 Q7 = 2.248234257620569139969141618556349415120E5L,
87 Q6 = 6.132189329546557743179177159925690841200E5L,
88 Q5 = 1.158019977462989115839826904108208787040E6L,
89 Q4 = 1.514882452993549494932585972882995548426E6L,
90 Q3 = 1.347518538384329112529391120390701166528E6L,
91 Q2 = 7.777690340007566932935753241556479363645E5L,
92 Q1 = 2.626900195321832660448791748036714883242E5L,
93 Q0 = 3.940717212190338497730839731583397586124E4L;
95 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
96 * where z = 2(x-1)/(x+1)
97 * 1/sqrt(2) <= x < sqrt(2)
98 * Theoretical peak relative error = 1.1e-35,
99 * relative peak error spread 1.1e-9
101 static const long double
102 R5 = -8.828896441624934385266096344596648080902E-1L,
103 R4 = 8.057002716646055371965756206836056074715E1L,
104 R3 = -2.024301798136027039250415126250455056397E3L,
105 R2 = 2.048819892795278657810231591630928516206E4L,
106 R1 = -8.977257995689735303686582344659576526998E4L,
107 R0 = 1.418134209872192732479751274970992665513E5L,
108 /* S6 = 1.000000000000000000000000000000000000000E0L, */
109 S5 = -1.186359407982897997337150403816839480438E2L,
110 S4 = 3.998526750980007367835804959888064681098E3L,
111 S3 = -5.748542087379434595104154610899551484314E4L,
112 S2 = 4.001557694070773974936904547424676279307E5L,
113 S1 = -1.332535117259762928288745111081235577029E6L,
114 S0 = 1.701761051846631278975701529965589676574E6L;
117 static const long double C1 = 6.93145751953125E-1L;
118 static const long double C2 = 1.428606820309417232121458176568075500134E-6L;
120 static const long double sqrth = 0.7071067811865475244008443621048490392848L;
121 /* ln (2^16384 * (1 - 2^-113)) */
122 static const long double maxlog = 1.1356523406294143949491931077970764891253E4L;
123 static const long double zero = 0.0L;
126 __log1pl (long double xm1)
128 long double x, y, z, r, s;
129 ieee854_long_double_shape_type u;
133 /* Test for NaN or infinity input. */
136 if (hx >= 0x7fff0000)
139 /* log1p(+- 0) = +- 0. */
140 if (((hx & 0x7fffffff) == 0)
141 && (u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0)
146 /* log1p(-1) = -inf */
150 return (-1.0L / (x - x));
152 return (zero / (x - x));
155 /* Separate mantissa from exponent. */
157 /* Use frexp used so that denormal numbers will be handled properly. */
158 x = __frexpl (x, &e);
160 /* Logarithm using log(x) = z + z^3 P(z^2)/Q(z^2),
161 where z = 2(x-1)/x+1). */
162 if ((e > 2) || (e < -2))
165 { /* 2( 2x-1 )/( 2x+1 ) */
171 { /* 2 (x-1)/(x+1) */
199 /* Logarithm using log(1+x) = x - .5x^2 + x^3 P(x)/Q(x). */
205 x = 2.0L * x - 1.0L; /* 2x - 1 */
217 r = (((((((((((P12 * x
251 #else /* !__OPTION_EGLIBC_LIBM_BIG */
252 # include <sysdeps/ieee754/ldbl-wrap/s_log1pl-wrap.c>
253 #endif /* __OPTION_EGLIBC_LIBM_BIG */
255 weak_alias (__log1pl, log1pl)