1 /* $OpenBSD: e_log10l.c,v 1.1 2011/07/06 00:02:42 martynas Exp $ */
2
3 /*
4 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
5 *
6 * Permission to use, copy, modify, and distribute this software for any
7 * purpose with or without fee is hereby granted, provided that the above
8 * copyright notice and this permission notice appear in all copies.
9 *
10 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
11 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
12 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
13 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
14 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
15 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
16 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
17 */
18
19 /* log10l.c
20 *
21 * Common logarithm, 128-bit long double precision
22 *
23 *
24 *
25 * SYNOPSIS:
26 *
27 * long double x, y, log10l();
28 *
29 * y = log10l( x );
30 *
31 *
32 *
33 * DESCRIPTION:
34 *
35 * Returns the base 10 logarithm of x.
36 *
37 * The argument is separated into its exponent and fractional
38 * parts. If the exponent is between -1 and +1, the logarithm
39 * of the fraction is approximated by
40 *
41 * log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x).
42 *
43 * Otherwise, setting z = 2(x-1)/x+1),
44 *
45 * log(x) = z + z^3 P(z)/Q(z).
46 *
47 *
48 *
49 * ACCURACY:
50 *
51 * Relative error:
52 * arithmetic domain # trials peak rms
53 * IEEE 0.5, 2.0 30000 2.3e-34 4.9e-35
54 * IEEE exp(+-10000) 30000 1.0e-34 4.1e-35
55 *
56 * In the tests over the interval exp(+-10000), the logarithms
57 * of the random arguments were uniformly distributed over
58 * [-10000, +10000].
59 *
60 */
61
62
63
64 /* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
65 * 1/sqrt(2) <= x < sqrt(2)
66 * Theoretical peak relative error = 5.3e-37,
67 * relative peak error spread = 2.3e-14
68 */
69 static const long double P[13] =
70 {
71 1.313572404063446165910279910527789794488E4L,
72 7.771154681358524243729929227226708890930E4L,
73 2.014652742082537582487669938141683759923E5L,
74 3.007007295140399532324943111654767187848E5L,
75 2.854829159639697837788887080758954924001E5L,
76 1.797628303815655343403735250238293741397E5L,
77 7.594356839258970405033155585486712125861E4L,
78 2.128857716871515081352991964243375186031E4L,
79 3.824952356185897735160588078446136783779E3L,
80 4.114517881637811823002128927449878962058E2L,
81 2.321125933898420063925789532045674660756E1L,
82 4.998469661968096229986658302195402690910E-1L,
83 1.538612243596254322971797716843006400388E-6L
84 };
85 static const long double Q[12] =
86 {
87 3.940717212190338497730839731583397586124E4L,
88 2.626900195321832660448791748036714883242E5L,
89 7.777690340007566932935753241556479363645E5L,
90 1.347518538384329112529391120390701166528E6L,
91 1.514882452993549494932585972882995548426E6L,
92 1.158019977462989115839826904108208787040E6L,
93 6.132189329546557743179177159925690841200E5L,
94 2.248234257620569139969141618556349415120E5L,
95 5.605842085972455027590989944010492125825E4L,
96 9.147150349299596453976674231612674085381E3L,
97 9.104928120962988414618126155557301584078E2L,
98 4.839208193348159620282142911143429644326E1L
99 /* 1.000000000000000000000000000000000000000E0L, */
100 };
101
102 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
103 * where z = 2(x-1)/(x+1)
104 * 1/sqrt(2) <= x < sqrt(2)
105 * Theoretical peak relative error = 1.1e-35,
106 * relative peak error spread 1.1e-9
107 */
108 static const long double R[6] =
109 {
110 1.418134209872192732479751274970992665513E5L,
111 -8.977257995689735303686582344659576526998E4L,
112 2.048819892795278657810231591630928516206E4L,
113 -2.024301798136027039250415126250455056397E3L,
114 8.057002716646055371965756206836056074715E1L,
115 -8.828896441624934385266096344596648080902E-1L
116 };
117 static const long double S[6] =
118 {
119 1.701761051846631278975701529965589676574E6L,
120 -1.332535117259762928288745111081235577029E6L,
121 4.001557694070773974936904547424676279307E5L,
122 -5.748542087379434595104154610899551484314E4L,
123 3.998526750980007367835804959888064681098E3L,
124 -1.186359407982897997337150403816839480438E2L
125 /* 1.000000000000000000000000000000000000000E0L, */
126 };
127
128 static const long double
129 /* log10(2) */
130 L102A = 0.3125L,
131 L102B = -1.14700043360188047862611052755069732318101185E-2L,
132 /* log10(e) */
133 L10EA = 0.5L,
134 L10EB = -6.570551809674817234887108108339491770560299E-2L,
135 /* sqrt(2)/2 */
136 SQRTH = 7.071067811865475244008443621048490392848359E-1L;
137
138
139
140 /* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */
141
142 static long double
neval(long double x,const long double * p,int n)143 neval (long double x, const long double *p, int n)
144 {
145 long double y;
146
147 p += n;
148 y = *p--;
149 do
150 {
151 y = y * x + *p--;
152 }
153 while (--n > 0);
154 return y;
155 }
156
157
158 /* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */
159
160 static long double
deval(long double x,const long double * p,int n)161 deval (long double x, const long double *p, int n)
162 {
163 long double y;
164
165 p += n;
166 y = x + *p--;
167 do
168 {
169 y = y * x + *p--;
170 }
171 while (--n > 0);
172 return y;
173 }
174
175
176
177 long double
log10l(long double x)178 log10l(long double x)
179 {
180 long double z;
181 long double y;
182 int e;
183 int64_t hx, lx;
184
185 /* Test for domain */
186 GET_LDOUBLE_WORDS64 (hx, lx, x);
187 if (((hx & 0x7fffffffffffffffLL) | lx) == 0)
188 return __math_divzerol(1);
189 if (hx < 0)
190 return __math_invalidl(x);
191 if (hx >= 0x7fff000000000000LL)
192 return (x + x);
193
194 /* separate mantissa from exponent */
195
196 /* Note, frexp is used so that denormal numbers
197 * will be handled properly.
198 */
199 x = frexpl (x, &e);
200
201
202 /* logarithm using log(x) = z + z**3 P(z)/Q(z),
203 * where z = 2(x-1)/x+1)
204 */
205 if ((e > 2) || (e < -2))
206 {
207 if (x < SQRTH)
208 { /* 2( 2x-1 )/( 2x+1 ) */
209 e -= 1;
210 z = x - 0.5L;
211 y = 0.5L * z + 0.5L;
212 }
213 else
214 { /* 2 (x-1)/(x+1) */
215 z = x - 0.5L;
216 z -= 0.5L;
217 y = 0.5L * x + 0.5L;
218 }
219 x = z / y;
220 z = x * x;
221 y = x * (z * neval (z, R, 5) / deval (z, S, 5));
222 goto done;
223 }
224
225
226 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
227
228 if (x < SQRTH)
229 {
230 e -= 1;
231 x = 2.0L * x - 1.0L; /* 2x - 1 */
232 }
233 else
234 {
235 x = x - 1.0L;
236 }
237 z = x * x;
238 y = x * (z * neval (x, P, 12) / deval (x, Q, 11));
239 y = y - 0.5L * z;
240
241 done:
242
243 /* Multiply log of fraction by log10(e)
244 * and base 2 exponent by log10(2).
245 */
246 z = y * L10EB;
247 z += x * L10EB;
248 z += e * L102B;
249 z += y * L10EA;
250 z += x * L10EA;
251 z += e * L102A;
252 return (z);
253 }
254
