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 | <http://www.gnu.org/licenses/>. */ |
54 | |
55 | |
56 | #include <float.h> |
57 | #include <math.h> |
58 | #include <math_private.h> |
59 | #include <math-underflow.h> |
60 | |
61 | /* Coefficients for log(1+x) = x - x^2 / 2 + x^3 P(x)/Q(x) |
62 | * 1/sqrt(2) <= 1+x < sqrt(2) |
63 | * Theoretical peak relative error = 5.3e-37, |
64 | * relative peak error spread = 2.3e-14 |
65 | */ |
66 | static const _Float128 |
67 | P12 = L(1.538612243596254322971797716843006400388E-6), |
68 | P11 = L(4.998469661968096229986658302195402690910E-1), |
69 | P10 = L(2.321125933898420063925789532045674660756E1), |
70 | P9 = L(4.114517881637811823002128927449878962058E2), |
71 | P8 = L(3.824952356185897735160588078446136783779E3), |
72 | P7 = L(2.128857716871515081352991964243375186031E4), |
73 | P6 = L(7.594356839258970405033155585486712125861E4), |
74 | P5 = L(1.797628303815655343403735250238293741397E5), |
75 | P4 = L(2.854829159639697837788887080758954924001E5), |
76 | P3 = L(3.007007295140399532324943111654767187848E5), |
77 | P2 = L(2.014652742082537582487669938141683759923E5), |
78 | P1 = L(7.771154681358524243729929227226708890930E4), |
79 | P0 = L(1.313572404063446165910279910527789794488E4), |
80 | /* Q12 = 1.000000000000000000000000000000000000000E0L, */ |
81 | Q11 = L(4.839208193348159620282142911143429644326E1), |
82 | Q10 = L(9.104928120962988414618126155557301584078E2), |
83 | Q9 = L(9.147150349299596453976674231612674085381E3), |
84 | Q8 = L(5.605842085972455027590989944010492125825E4), |
85 | Q7 = L(2.248234257620569139969141618556349415120E5), |
86 | Q6 = L(6.132189329546557743179177159925690841200E5), |
87 | Q5 = L(1.158019977462989115839826904108208787040E6), |
88 | Q4 = L(1.514882452993549494932585972882995548426E6), |
89 | Q3 = L(1.347518538384329112529391120390701166528E6), |
90 | Q2 = L(7.777690340007566932935753241556479363645E5), |
91 | Q1 = L(2.626900195321832660448791748036714883242E5), |
92 | Q0 = L(3.940717212190338497730839731583397586124E4); |
93 | |
94 | /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2), |
95 | * where z = 2(x-1)/(x+1) |
96 | * 1/sqrt(2) <= x < sqrt(2) |
97 | * Theoretical peak relative error = 1.1e-35, |
98 | * relative peak error spread 1.1e-9 |
99 | */ |
100 | static const _Float128 |
101 | R5 = L(-8.828896441624934385266096344596648080902E-1), |
102 | R4 = L(8.057002716646055371965756206836056074715E1), |
103 | R3 = L(-2.024301798136027039250415126250455056397E3), |
104 | R2 = L(2.048819892795278657810231591630928516206E4), |
105 | R1 = L(-8.977257995689735303686582344659576526998E4), |
106 | R0 = L(1.418134209872192732479751274970992665513E5), |
107 | /* S6 = 1.000000000000000000000000000000000000000E0L, */ |
108 | S5 = L(-1.186359407982897997337150403816839480438E2), |
109 | S4 = L(3.998526750980007367835804959888064681098E3), |
110 | S3 = L(-5.748542087379434595104154610899551484314E4), |
111 | S2 = L(4.001557694070773974936904547424676279307E5), |
112 | S1 = L(-1.332535117259762928288745111081235577029E6), |
113 | S0 = L(1.701761051846631278975701529965589676574E6); |
114 | |
115 | /* C1 + C2 = ln 2 */ |
116 | static const _Float128 C1 = L(6.93145751953125E-1); |
117 | static const _Float128 C2 = L(1.428606820309417232121458176568075500134E-6); |
118 | |
119 | static const _Float128 sqrth = L(0.7071067811865475244008443621048490392848); |
120 | /* ln (2^16384 * (1 - 2^-113)) */ |
121 | static const _Float128 zero = 0; |
122 | |
123 | _Float128 |
124 | __log1pl (_Float128 xm1) |
125 | { |
126 | _Float128 x, y, z, r, s; |
127 | ieee854_long_double_shape_type u; |
128 | int32_t hx; |
129 | int e; |
130 | |
131 | /* Test for NaN or infinity input. */ |
132 | u.value = xm1; |
133 | hx = u.parts32.w0; |
134 | if ((hx & 0x7fffffff) >= 0x7fff0000) |
135 | return xm1 + fabsl (xm1); |
136 | |
137 | /* log1p(+- 0) = +- 0. */ |
138 | if (((hx & 0x7fffffff) == 0) |
139 | && (u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0) |
140 | return xm1; |
141 | |
142 | if ((hx & 0x7fffffff) < 0x3f8e0000) |
143 | { |
144 | math_check_force_underflow (xm1); |
145 | if ((int) xm1 == 0) |
146 | return xm1; |
147 | } |
148 | |
149 | if (xm1 >= L(0x1p113)) |
150 | x = xm1; |
151 | else |
152 | x = xm1 + 1; |
153 | |
154 | /* log1p(-1) = -inf */ |
155 | if (x <= 0) |
156 | { |
157 | if (x == 0) |
158 | return (-1 / zero); /* log1p(-1) = -inf */ |
159 | else |
160 | return (zero / (x - x)); |
161 | } |
162 | |
163 | /* Separate mantissa from exponent. */ |
164 | |
165 | /* Use frexp used so that denormal numbers will be handled properly. */ |
166 | x = __frexpl (x, &e); |
167 | |
168 | /* Logarithm using log(x) = z + z^3 P(z^2)/Q(z^2), |
169 | where z = 2(x-1)/x+1). */ |
170 | if ((e > 2) || (e < -2)) |
171 | { |
172 | if (x < sqrth) |
173 | { /* 2( 2x-1 )/( 2x+1 ) */ |
174 | e -= 1; |
175 | z = x - L(0.5); |
176 | y = L(0.5) * z + L(0.5); |
177 | } |
178 | else |
179 | { /* 2 (x-1)/(x+1) */ |
180 | z = x - L(0.5); |
181 | z -= L(0.5); |
182 | y = L(0.5) * x + L(0.5); |
183 | } |
184 | x = z / y; |
185 | z = x * x; |
186 | r = ((((R5 * z |
187 | + R4) * z |
188 | + R3) * z |
189 | + R2) * z |
190 | + R1) * z |
191 | + R0; |
192 | s = (((((z |
193 | + S5) * z |
194 | + S4) * z |
195 | + S3) * z |
196 | + S2) * z |
197 | + S1) * z |
198 | + S0; |
199 | z = x * (z * r / s); |
200 | z = z + e * C2; |
201 | z = z + x; |
202 | z = z + e * C1; |
203 | return (z); |
204 | } |
205 | |
206 | |
207 | /* Logarithm using log(1+x) = x - .5x^2 + x^3 P(x)/Q(x). */ |
208 | |
209 | if (x < sqrth) |
210 | { |
211 | e -= 1; |
212 | if (e != 0) |
213 | x = 2 * x - 1; /* 2x - 1 */ |
214 | else |
215 | x = xm1; |
216 | } |
217 | else |
218 | { |
219 | if (e != 0) |
220 | x = x - 1; |
221 | else |
222 | x = xm1; |
223 | } |
224 | z = x * x; |
225 | r = (((((((((((P12 * x |
226 | + P11) * x |
227 | + P10) * x |
228 | + P9) * x |
229 | + P8) * x |
230 | + P7) * x |
231 | + P6) * x |
232 | + P5) * x |
233 | + P4) * x |
234 | + P3) * x |
235 | + P2) * x |
236 | + P1) * x |
237 | + P0; |
238 | s = (((((((((((x |
239 | + Q11) * x |
240 | + Q10) * x |
241 | + Q9) * x |
242 | + Q8) * x |
243 | + Q7) * x |
244 | + Q6) * x |
245 | + Q5) * x |
246 | + Q4) * x |
247 | + Q3) * x |
248 | + Q2) * x |
249 | + Q1) * x |
250 | + Q0; |
251 | y = x * (z * r / s); |
252 | y = y + e * C2; |
253 | z = y - L(0.5) * z; |
254 | z = z + x; |
255 | z = z + e * C1; |
256 | return (z); |
257 | } |
258 | |