1 | /* log2l.c |
2 | * Base 2 logarithm, 128-bit long double precision |
3 | * |
4 | * |
5 | * |
6 | * SYNOPSIS: |
7 | * |
8 | * long double x, y, log2l(); |
9 | * |
10 | * y = log2l( x ); |
11 | * |
12 | * |
13 | * |
14 | * DESCRIPTION: |
15 | * |
16 | * Returns the base 2 logarithm of x. |
17 | * |
18 | * The argument is separated into its exponent and fractional |
19 | * parts. If the exponent is between -1 and +1, the (natural) |
20 | * logarithm of the fraction is approximated by |
21 | * |
22 | * log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x). |
23 | * |
24 | * Otherwise, setting z = 2(x-1)/x+1), |
25 | * |
26 | * log(x) = z + z^3 P(z)/Q(z). |
27 | * |
28 | * |
29 | * |
30 | * ACCURACY: |
31 | * |
32 | * Relative error: |
33 | * arithmetic domain # trials peak rms |
34 | * IEEE 0.5, 2.0 100,000 2.6e-34 4.9e-35 |
35 | * IEEE exp(+-10000) 100,000 9.6e-35 4.0e-35 |
36 | * |
37 | * In the tests over the interval exp(+-10000), the logarithms |
38 | * of the random arguments were uniformly distributed over |
39 | * [-10000, +10000]. |
40 | * |
41 | */ |
42 | |
43 | /* |
44 | Cephes Math Library Release 2.2: January, 1991 |
45 | Copyright 1984, 1991 by Stephen L. Moshier |
46 | Adapted for glibc November, 2001 |
47 | |
48 | This library is free software; you can redistribute it and/or |
49 | modify it under the terms of the GNU Lesser General Public |
50 | License as published by the Free Software Foundation; either |
51 | version 2.1 of the License, or (at your option) any later version. |
52 | |
53 | This library is distributed in the hope that it will be useful, |
54 | but WITHOUT ANY WARRANTY; without even the implied warranty of |
55 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
56 | Lesser General Public License for more details. |
57 | |
58 | You should have received a copy of the GNU Lesser General Public |
59 | License along with this library; if not, see <https://www.gnu.org/licenses/>. |
60 | */ |
61 | |
62 | #include <math.h> |
63 | #include <math_private.h> |
64 | #include <libm-alias-finite.h> |
65 | |
66 | /* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x) |
67 | * 1/sqrt(2) <= x < sqrt(2) |
68 | * Theoretical peak relative error = 5.3e-37, |
69 | * relative peak error spread = 2.3e-14 |
70 | */ |
71 | static const long double P[13] = |
72 | { |
73 | 1.313572404063446165910279910527789794488E4L, |
74 | 7.771154681358524243729929227226708890930E4L, |
75 | 2.014652742082537582487669938141683759923E5L, |
76 | 3.007007295140399532324943111654767187848E5L, |
77 | 2.854829159639697837788887080758954924001E5L, |
78 | 1.797628303815655343403735250238293741397E5L, |
79 | 7.594356839258970405033155585486712125861E4L, |
80 | 2.128857716871515081352991964243375186031E4L, |
81 | 3.824952356185897735160588078446136783779E3L, |
82 | 4.114517881637811823002128927449878962058E2L, |
83 | 2.321125933898420063925789532045674660756E1L, |
84 | 4.998469661968096229986658302195402690910E-1L, |
85 | 1.538612243596254322971797716843006400388E-6L |
86 | }; |
87 | static const long double Q[12] = |
88 | { |
89 | 3.940717212190338497730839731583397586124E4L, |
90 | 2.626900195321832660448791748036714883242E5L, |
91 | 7.777690340007566932935753241556479363645E5L, |
92 | 1.347518538384329112529391120390701166528E6L, |
93 | 1.514882452993549494932585972882995548426E6L, |
94 | 1.158019977462989115839826904108208787040E6L, |
95 | 6.132189329546557743179177159925690841200E5L, |
96 | 2.248234257620569139969141618556349415120E5L, |
97 | 5.605842085972455027590989944010492125825E4L, |
98 | 9.147150349299596453976674231612674085381E3L, |
99 | 9.104928120962988414618126155557301584078E2L, |
100 | 4.839208193348159620282142911143429644326E1L |
101 | /* 1.000000000000000000000000000000000000000E0L, */ |
102 | }; |
103 | |
104 | /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2), |
105 | * where z = 2(x-1)/(x+1) |
106 | * 1/sqrt(2) <= x < sqrt(2) |
107 | * Theoretical peak relative error = 1.1e-35, |
108 | * relative peak error spread 1.1e-9 |
109 | */ |
110 | static const long double R[6] = |
111 | { |
112 | 1.418134209872192732479751274970992665513E5L, |
113 | -8.977257995689735303686582344659576526998E4L, |
114 | 2.048819892795278657810231591630928516206E4L, |
115 | -2.024301798136027039250415126250455056397E3L, |
116 | 8.057002716646055371965756206836056074715E1L, |
117 | -8.828896441624934385266096344596648080902E-1L |
118 | }; |
119 | static const long double S[6] = |
120 | { |
121 | 1.701761051846631278975701529965589676574E6L, |
122 | -1.332535117259762928288745111081235577029E6L, |
123 | 4.001557694070773974936904547424676279307E5L, |
124 | -5.748542087379434595104154610899551484314E4L, |
125 | 3.998526750980007367835804959888064681098E3L, |
126 | -1.186359407982897997337150403816839480438E2L |
127 | /* 1.000000000000000000000000000000000000000E0L, */ |
128 | }; |
129 | |
130 | static const long double |
131 | /* log2(e) - 1 */ |
132 | LOG2EA = 4.4269504088896340735992468100189213742664595E-1L, |
133 | /* sqrt(2)/2 */ |
134 | SQRTH = 7.071067811865475244008443621048490392848359E-1L; |
135 | |
136 | |
137 | /* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */ |
138 | |
139 | static long double |
140 | neval (long double x, const long double *p, int n) |
141 | { |
142 | long double y; |
143 | |
144 | p += n; |
145 | y = *p--; |
146 | do |
147 | { |
148 | y = y * x + *p--; |
149 | } |
150 | while (--n > 0); |
151 | return y; |
152 | } |
153 | |
154 | |
155 | /* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */ |
156 | |
157 | static long double |
158 | deval (long double x, const long double *p, int n) |
159 | { |
160 | long double y; |
161 | |
162 | p += n; |
163 | y = x + *p--; |
164 | do |
165 | { |
166 | y = y * x + *p--; |
167 | } |
168 | while (--n > 0); |
169 | return y; |
170 | } |
171 | |
172 | |
173 | |
174 | long double |
175 | __ieee754_log2l (long double x) |
176 | { |
177 | long double z; |
178 | long double y; |
179 | int e; |
180 | int64_t hx; |
181 | double xhi; |
182 | |
183 | /* Test for domain */ |
184 | xhi = ldbl_high (x); |
185 | EXTRACT_WORDS64 (hx, xhi); |
186 | if ((hx & 0x7fffffffffffffffLL) == 0) |
187 | return (-1.0L / fabsl (x: x)); /* log2l(+-0)=-inf */ |
188 | if (hx < 0) |
189 | return (x - x) / (x - x); |
190 | if (hx >= 0x7ff0000000000000LL) |
191 | return (x + x); |
192 | |
193 | if (x == 1.0L) |
194 | return 0.0L; |
195 | |
196 | /* separate mantissa from exponent */ |
197 | |
198 | /* Note, frexp is used so that denormal numbers |
199 | * will be handled properly. |
200 | */ |
201 | x = __frexpl (x: x, exponent: &e); |
202 | |
203 | |
204 | /* logarithm using log(x) = z + z**3 P(z)/Q(z), |
205 | * where z = 2(x-1)/x+1) |
206 | */ |
207 | if ((e > 2) || (e < -2)) |
208 | { |
209 | if (x < SQRTH) |
210 | { /* 2( 2x-1 )/( 2x+1 ) */ |
211 | e -= 1; |
212 | z = x - 0.5L; |
213 | y = 0.5L * z + 0.5L; |
214 | } |
215 | else |
216 | { /* 2 (x-1)/(x+1) */ |
217 | z = x - 0.5L; |
218 | z -= 0.5L; |
219 | y = 0.5L * x + 0.5L; |
220 | } |
221 | x = z / y; |
222 | z = x * x; |
223 | y = x * (z * neval (x: z, p: R, n: 5) / deval (x: z, p: S, n: 5)); |
224 | goto done; |
225 | } |
226 | |
227 | |
228 | /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */ |
229 | |
230 | if (x < SQRTH) |
231 | { |
232 | e -= 1; |
233 | x = 2.0 * x - 1.0L; /* 2x - 1 */ |
234 | } |
235 | else |
236 | { |
237 | x = x - 1.0L; |
238 | } |
239 | z = x * x; |
240 | y = x * (z * neval (x, p: P, n: 12) / deval (x, p: Q, n: 11)); |
241 | y = y - 0.5 * z; |
242 | |
243 | done: |
244 | |
245 | /* Multiply log of fraction by log2(e) |
246 | * and base 2 exponent by 1 |
247 | */ |
248 | z = y * LOG2EA; |
249 | z += x * LOG2EA; |
250 | z += y; |
251 | z += x; |
252 | z += e; |
253 | return (z); |
254 | } |
255 | libm_alias_finite (__ieee754_log2l, __log2l) |
256 | |