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)); /* 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, &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 (z, R, 5) / deval (z, S, 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, 12) / deval (x, Q, 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)