1 | /* log1pl.c |
2 | * |
3 | * Relative error logarithm |
4 | * Natural logarithm of 1+x, 128-bit long double precision |
5 | * |
6 | * |
7 | * |
8 | * SYNOPSIS: |
9 | * |
10 | * long double x, y, log1pl(); |
11 | * |
12 | * y = log1pl( x ); |
13 | * |
14 | * |
15 | * |
16 | * DESCRIPTION: |
17 | * |
18 | * Returns the base e (2.718...) logarithm of 1+x. |
19 | * |
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 |
23 | * |
24 | * log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x). |
25 | * |
26 | * Otherwise, setting z = 2(w-1)/(w+1), |
27 | * |
28 | * log(w) = z + z^3 P(z)/Q(z). |
29 | * |
30 | * |
31 | * |
32 | * ACCURACY: |
33 | * |
34 | * Relative error: |
35 | * arithmetic domain # trials peak rms |
36 | * IEEE -1, 8 100000 1.9e-34 4.3e-35 |
37 | */ |
38 | |
39 | /* Copyright 2001 by Stephen L. Moshier |
40 | |
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. |
45 | |
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. |
50 | |
51 | You should have received a copy of the GNU Lesser General Public |
52 | License along with this library; if not, see |
53 | <https://www.gnu.org/licenses/>. */ |
54 | |
55 | |
56 | #include <math.h> |
57 | #include <math_private.h> |
58 | #include <math_ldbl_opt.h> |
59 | |
60 | /* Coefficients for log(1+x) = x - x^2 / 2 + x^3 P(x)/Q(x) |
61 | * 1/sqrt(2) <= 1+x < sqrt(2) |
62 | * Theoretical peak relative error = 5.3e-37, |
63 | * relative peak error spread = 2.3e-14 |
64 | */ |
65 | static const long double |
66 | P12 = 1.538612243596254322971797716843006400388E-6L, |
67 | P11 = 4.998469661968096229986658302195402690910E-1L, |
68 | P10 = 2.321125933898420063925789532045674660756E1L, |
69 | P9 = 4.114517881637811823002128927449878962058E2L, |
70 | P8 = 3.824952356185897735160588078446136783779E3L, |
71 | P7 = 2.128857716871515081352991964243375186031E4L, |
72 | P6 = 7.594356839258970405033155585486712125861E4L, |
73 | P5 = 1.797628303815655343403735250238293741397E5L, |
74 | P4 = 2.854829159639697837788887080758954924001E5L, |
75 | P3 = 3.007007295140399532324943111654767187848E5L, |
76 | P2 = 2.014652742082537582487669938141683759923E5L, |
77 | P1 = 7.771154681358524243729929227226708890930E4L, |
78 | P0 = 1.313572404063446165910279910527789794488E4L, |
79 | /* Q12 = 1.000000000000000000000000000000000000000E0L, */ |
80 | Q11 = 4.839208193348159620282142911143429644326E1L, |
81 | Q10 = 9.104928120962988414618126155557301584078E2L, |
82 | Q9 = 9.147150349299596453976674231612674085381E3L, |
83 | Q8 = 5.605842085972455027590989944010492125825E4L, |
84 | Q7 = 2.248234257620569139969141618556349415120E5L, |
85 | Q6 = 6.132189329546557743179177159925690841200E5L, |
86 | Q5 = 1.158019977462989115839826904108208787040E6L, |
87 | Q4 = 1.514882452993549494932585972882995548426E6L, |
88 | Q3 = 1.347518538384329112529391120390701166528E6L, |
89 | Q2 = 7.777690340007566932935753241556479363645E5L, |
90 | Q1 = 2.626900195321832660448791748036714883242E5L, |
91 | Q0 = 3.940717212190338497730839731583397586124E4L; |
92 | |
93 | /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2), |
94 | * where z = 2(x-1)/(x+1) |
95 | * 1/sqrt(2) <= x < sqrt(2) |
96 | * Theoretical peak relative error = 1.1e-35, |
97 | * relative peak error spread 1.1e-9 |
98 | */ |
99 | static const long double |
100 | R5 = -8.828896441624934385266096344596648080902E-1L, |
101 | R4 = 8.057002716646055371965756206836056074715E1L, |
102 | R3 = -2.024301798136027039250415126250455056397E3L, |
103 | R2 = 2.048819892795278657810231591630928516206E4L, |
104 | R1 = -8.977257995689735303686582344659576526998E4L, |
105 | R0 = 1.418134209872192732479751274970992665513E5L, |
106 | /* S6 = 1.000000000000000000000000000000000000000E0L, */ |
107 | S5 = -1.186359407982897997337150403816839480438E2L, |
108 | S4 = 3.998526750980007367835804959888064681098E3L, |
109 | S3 = -5.748542087379434595104154610899551484314E4L, |
110 | S2 = 4.001557694070773974936904547424676279307E5L, |
111 | S1 = -1.332535117259762928288745111081235577029E6L, |
112 | S0 = 1.701761051846631278975701529965589676574E6L; |
113 | |
114 | /* C1 + C2 = ln 2 */ |
115 | static const long double C1 = 6.93145751953125E-1L; |
116 | static const long double C2 = 1.428606820309417232121458176568075500134E-6L; |
117 | |
118 | static const long double sqrth = 0.7071067811865475244008443621048490392848L; |
119 | /* ln (2^16384 * (1 - 2^-113)) */ |
120 | static const long double zero = 0.0L; |
121 | |
122 | |
123 | long double |
124 | __log1pl (long double xm1) |
125 | { |
126 | long double x, y, z, r, s; |
127 | double xhi; |
128 | int32_t hx, lx; |
129 | int e; |
130 | |
131 | /* Test for NaN or infinity input. */ |
132 | xhi = ldbl_high (xm1); |
133 | EXTRACT_WORDS (hx, lx, xhi); |
134 | if ((hx & 0x7fffffff) >= 0x7ff00000) |
135 | return xm1 + xm1 * xm1; |
136 | |
137 | /* log1p(+- 0) = +- 0. */ |
138 | if (((hx & 0x7fffffff) | lx) == 0) |
139 | return xm1; |
140 | |
141 | if (xm1 >= 0x1p107L) |
142 | x = xm1; |
143 | else |
144 | x = xm1 + 1.0L; |
145 | |
146 | /* log1p(-1) = -inf */ |
147 | if (x <= 0.0L) |
148 | { |
149 | if (x == 0.0L) |
150 | return (-1.0L / 0.0L); |
151 | else |
152 | return (zero / (x - x)); |
153 | } |
154 | |
155 | /* Separate mantissa from exponent. */ |
156 | |
157 | /* Use frexp used so that denormal numbers will be handled properly. */ |
158 | x = __frexpl (x: x, exponent: &e); |
159 | |
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)) |
163 | { |
164 | if (x < sqrth) |
165 | { /* 2( 2x-1 )/( 2x+1 ) */ |
166 | e -= 1; |
167 | z = x - 0.5L; |
168 | y = 0.5L * z + 0.5L; |
169 | } |
170 | else |
171 | { /* 2 (x-1)/(x+1) */ |
172 | z = x - 0.5L; |
173 | z -= 0.5L; |
174 | y = 0.5L * x + 0.5L; |
175 | } |
176 | x = z / y; |
177 | z = x * x; |
178 | r = ((((R5 * z |
179 | + R4) * z |
180 | + R3) * z |
181 | + R2) * z |
182 | + R1) * z |
183 | + R0; |
184 | s = (((((z |
185 | + S5) * z |
186 | + S4) * z |
187 | + S3) * z |
188 | + S2) * z |
189 | + S1) * z |
190 | + S0; |
191 | z = x * (z * r / s); |
192 | z = z + e * C2; |
193 | z = z + x; |
194 | z = z + e * C1; |
195 | return (z); |
196 | } |
197 | |
198 | |
199 | /* Logarithm using log(1+x) = x - .5x^2 + x^3 P(x)/Q(x). */ |
200 | |
201 | if (x < sqrth) |
202 | { |
203 | e -= 1; |
204 | if (e != 0) |
205 | x = 2.0L * x - 1.0L; /* 2x - 1 */ |
206 | else |
207 | x = xm1; |
208 | } |
209 | else |
210 | { |
211 | if (e != 0) |
212 | x = x - 1.0L; |
213 | else |
214 | x = xm1; |
215 | } |
216 | z = x * x; |
217 | r = (((((((((((P12 * x |
218 | + P11) * x |
219 | + P10) * x |
220 | + P9) * x |
221 | + P8) * x |
222 | + P7) * x |
223 | + P6) * x |
224 | + P5) * x |
225 | + P4) * x |
226 | + P3) * x |
227 | + P2) * x |
228 | + P1) * x |
229 | + P0; |
230 | s = (((((((((((x |
231 | + Q11) * x |
232 | + Q10) * x |
233 | + Q9) * x |
234 | + Q8) * x |
235 | + Q7) * x |
236 | + Q6) * x |
237 | + Q5) * x |
238 | + Q4) * x |
239 | + Q3) * x |
240 | + Q2) * x |
241 | + Q1) * x |
242 | + Q0; |
243 | y = x * (z * r / s); |
244 | y = y + e * C2; |
245 | z = y - 0.5L * z; |
246 | z = z + x; |
247 | z = z + e * C1; |
248 | return (z); |
249 | } |
250 | |