1/*	$OpenBSD: s_log1pl.c,v 1.2 2016/09/12 19:47:02 guenther 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/*							log1pl.c
20 *
21 *      Relative error logarithm
22 *	Natural logarithm of 1+x, 128-bit long double precision
23 *
24 *
25 *
26 * SYNOPSIS:
27 *
28 * long double x, y, log1pl();
29 *
30 * y = log1pl( x );
31 *
32 *
33 *
34 * DESCRIPTION:
35 *
36 * Returns the base e (2.718...) logarithm of 1+x.
37 *
38 * The argument 1+x is separated into its exponent and fractional
39 * parts.  If the exponent is between -1 and +1, the logarithm
40 * of the fraction is approximated by
41 *
42 *     log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x).
43 *
44 * Otherwise, setting  z = 2(w-1)/(w+1),
45 *
46 *     log(w) = z + z^3 P(z)/Q(z).
47 *
48 *
49 *
50 * ACCURACY:
51 *
52 *                      Relative error:
53 * arithmetic   domain     # trials      peak         rms
54 *    IEEE      -1, 8       100000      1.9e-34     4.3e-35
55 */
56
57#include <math.h>
58
59#include "math_private.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 */
66static const long double
67  P12 = 1.538612243596254322971797716843006400388E-6L,
68  P11 = 4.998469661968096229986658302195402690910E-1L,
69  P10 = 2.321125933898420063925789532045674660756E1L,
70  P9 = 4.114517881637811823002128927449878962058E2L,
71  P8 = 3.824952356185897735160588078446136783779E3L,
72  P7 = 2.128857716871515081352991964243375186031E4L,
73  P6 = 7.594356839258970405033155585486712125861E4L,
74  P5 = 1.797628303815655343403735250238293741397E5L,
75  P4 = 2.854829159639697837788887080758954924001E5L,
76  P3 = 3.007007295140399532324943111654767187848E5L,
77  P2 = 2.014652742082537582487669938141683759923E5L,
78  P1 = 7.771154681358524243729929227226708890930E4L,
79  P0 = 1.313572404063446165910279910527789794488E4L,
80  /* Q12 = 1.000000000000000000000000000000000000000E0L, */
81  Q11 = 4.839208193348159620282142911143429644326E1L,
82  Q10 = 9.104928120962988414618126155557301584078E2L,
83  Q9 = 9.147150349299596453976674231612674085381E3L,
84  Q8 = 5.605842085972455027590989944010492125825E4L,
85  Q7 = 2.248234257620569139969141618556349415120E5L,
86  Q6 = 6.132189329546557743179177159925690841200E5L,
87  Q5 = 1.158019977462989115839826904108208787040E6L,
88  Q4 = 1.514882452993549494932585972882995548426E6L,
89  Q3 = 1.347518538384329112529391120390701166528E6L,
90  Q2 = 7.777690340007566932935753241556479363645E5L,
91  Q1 = 2.626900195321832660448791748036714883242E5L,
92  Q0 = 3.940717212190338497730839731583397586124E4L;
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 */
100static const long double
101  R5 = -8.828896441624934385266096344596648080902E-1L,
102  R4 = 8.057002716646055371965756206836056074715E1L,
103  R3 = -2.024301798136027039250415126250455056397E3L,
104  R2 = 2.048819892795278657810231591630928516206E4L,
105  R1 = -8.977257995689735303686582344659576526998E4L,
106  R0 = 1.418134209872192732479751274970992665513E5L,
107  /* S6 = 1.000000000000000000000000000000000000000E0L, */
108  S5 = -1.186359407982897997337150403816839480438E2L,
109  S4 = 3.998526750980007367835804959888064681098E3L,
110  S3 = -5.748542087379434595104154610899551484314E4L,
111  S2 = 4.001557694070773974936904547424676279307E5L,
112  S1 = -1.332535117259762928288745111081235577029E6L,
113  S0 = 1.701761051846631278975701529965589676574E6L;
114
115/* C1 + C2 = ln 2 */
116static const long double C1 = 6.93145751953125E-1L;
117static const long double C2 = 1.428606820309417232121458176568075500134E-6L;
118
119static const long double sqrth = 0.7071067811865475244008443621048490392848L;
120/* ln (2^16384 * (1 - 2^-113)) */
121static const long double zero = 0.0L;
122
123long double
124log1pl(long double xm1)
125{
126  long double x, y, z, r, s;
127  ieee_quad_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.mswhi;
134  if (hx >= 0x7fff0000)
135    return xm1;
136
137  /* log1p(+- 0) = +- 0.  */
138  if (((hx & 0x7fffffff) == 0)
139      && (u.parts32.mswlo | u.parts32.lswhi | u.parts32.lswlo) == 0)
140    return xm1;
141
142  x = xm1 + 1.0L;
143
144  /* log1p(-1) = -inf */
145  if (x <= 0.0L)
146    {
147      if (x == 0.0L)
148	return (-1.0L / (x - x));
149      else
150	return (zero / (x - x));
151    }
152
153  /* Separate mantissa from exponent.  */
154
155  /* Use frexp used so that denormal numbers will be handled properly.  */
156  x = frexpl (x, &e);
157
158  /* Logarithm using log(x) = z + z^3 P(z^2)/Q(z^2),
159     where z = 2(x-1)/x+1).  */
160  if ((e > 2) || (e < -2))
161    {
162      if (x < sqrth)
163	{			/* 2( 2x-1 )/( 2x+1 ) */
164	  e -= 1;
165	  z = x - 0.5L;
166	  y = 0.5L * z + 0.5L;
167	}
168      else
169	{			/*  2 (x-1)/(x+1)   */
170	  z = x - 0.5L;
171	  z -= 0.5L;
172	  y = 0.5L * x + 0.5L;
173	}
174      x = z / y;
175      z = x * x;
176      r = ((((R5 * z
177	      + R4) * z
178	     + R3) * z
179	    + R2) * z
180	   + R1) * z
181	+ R0;
182      s = (((((z
183	       + S5) * z
184	      + S4) * z
185	     + S3) * z
186	    + S2) * z
187	   + S1) * z
188	+ S0;
189      z = x * (z * r / s);
190      z = z + e * C2;
191      z = z + x;
192      z = z + e * C1;
193      return (z);
194    }
195
196
197  /* Logarithm using log(1+x) = x - .5x^2 + x^3 P(x)/Q(x). */
198
199  if (x < sqrth)
200    {
201      e -= 1;
202      if (e != 0)
203	x = 2.0L * x - 1.0L;	/*  2x - 1  */
204      else
205	x = xm1;
206    }
207  else
208    {
209      if (e != 0)
210	x = x - 1.0L;
211      else
212	x = xm1;
213    }
214  z = x * x;
215  r = (((((((((((P12 * x
216		 + P11) * x
217		+ P10) * x
218	       + P9) * x
219	      + P8) * x
220	     + P7) * x
221	    + P6) * x
222	   + P5) * x
223	  + P4) * x
224	 + P3) * x
225	+ P2) * x
226       + P1) * x
227    + P0;
228  s = (((((((((((x
229		 + Q11) * x
230		+ Q10) * x
231	       + Q9) * x
232	      + Q8) * x
233	     + Q7) * x
234	    + Q6) * x
235	   + Q5) * x
236	  + Q4) * x
237	 + Q3) * x
238	+ Q2) * x
239       + Q1) * x
240    + Q0;
241  y = x * (z * r / s);
242  y = y + e * C2;
243  z = y - 0.5L * z;
244  z = z + x;
245  z = z + e * C1;
246  return (z);
247}
248DEF_STD(log1pl);
249