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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, write to the Free Software | |
60 | Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA | |
61 | */ | |
62 | ||
63 | #include "math.h" | |
64 | #include "math_private.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 (x) | |
176 | long double x; | |
177 | { | |
178 | long double z; | |
179 | long double y; | |
180 | int e; | |
181 | int64_t hx, lx; | |
182 | ||
183 | /* Test for domain */ | |
184 | GET_LDOUBLE_WORDS64 (hx, lx, x); | |
185 | if (((hx & 0x7fffffffffffffffLL) | (lx & 0x7fffffffffffffffLL)) == 0) | |
186 | return (-1.0L / (x - x)); | |
187 | if (hx < 0) | |
188 | return (x - x) / (x - x); | |
189 | if (hx >= 0x7ff0000000000000LL) | |
190 | return (x + x); | |
191 | ||
192 | /* separate mantissa from exponent */ | |
193 | ||
194 | /* Note, frexp is used so that denormal numbers | |
195 | * will be handled properly. | |
196 | */ | |
197 | x = __frexpl (x, &e); | |
198 | ||
199 | ||
200 | /* logarithm using log(x) = z + z**3 P(z)/Q(z), | |
201 | * where z = 2(x-1)/x+1) | |
202 | */ | |
203 | if ((e > 2) || (e < -2)) | |
204 | { | |
205 | if (x < SQRTH) | |
206 | { /* 2( 2x-1 )/( 2x+1 ) */ | |
207 | e -= 1; | |
208 | z = x - 0.5L; | |
209 | y = 0.5L * z + 0.5L; | |
210 | } | |
211 | else | |
212 | { /* 2 (x-1)/(x+1) */ | |
213 | z = x - 0.5L; | |
214 | z -= 0.5L; | |
215 | y = 0.5L * x + 0.5L; | |
216 | } | |
217 | x = z / y; | |
218 | z = x * x; | |
219 | y = x * (z * neval (z, R, 5) / deval (z, S, 5)); | |
220 | goto done; | |
221 | } | |
222 | ||
223 | ||
224 | /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */ | |
225 | ||
226 | if (x < SQRTH) | |
227 | { | |
228 | e -= 1; | |
229 | x = 2.0 * x - 1.0L; /* 2x - 1 */ | |
230 | } | |
231 | else | |
232 | { | |
233 | x = x - 1.0L; | |
234 | } | |
235 | z = x * x; | |
236 | y = x * (z * neval (x, P, 12) / deval (x, Q, 11)); | |
237 | y = y - 0.5 * z; | |
238 | ||
239 | done: | |
240 | ||
241 | /* Multiply log of fraction by log2(e) | |
242 | * and base 2 exponent by 1 | |
243 | */ | |
244 | z = y * LOG2EA; | |
245 | z += x * LOG2EA; | |
246 | z += y; | |
247 | z += x; | |
248 | z += e; | |
249 | return (z); | |
250 | } | |
0ac5ae23 | 251 | strong_alias (__ieee754_log2l, __log2l_finite) |