1/* Copyright JS Foundation and other contributors, http://js.foundation
2 *
3 * Licensed under the Apache License, Version 2.0 (the "License");
4 * you may not use this file except in compliance with the License.
5 * You may obtain a copy of the License at
6 *
7 *     http://www.apache.org/licenses/LICENSE-2.0
8 *
9 * Unless required by applicable law or agreed to in writing, software
10 * distributed under the License is distributed on an "AS IS" BASIS
11 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
12 * See the License for the specific language governing permissions and
13 * limitations under the License.
14 *
15 * This file is based on work under the following copyright and permission
16 * notice:
17 *
18 *     Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
19 *
20 *     Permission to use, copy, modify, and distribute this
21 *     software is freely granted, provided that this notice
22 *     is preserved.
23 *
24 *     @(#)s_log1p.c 5.1 93/09/24
25 */
26
27#include "jerry-libm-internal.h"
28
29/* log1p(x)
30 * Method :
31 *   1. Argument Reduction: find k and f such that
32 *      1+x = 2^k * (1+f),
33 *     where  sqrt(2)/2 < 1+f < sqrt(2) .
34 *
35 *      Note. If k=0, then f=x is exact. However, if k!=0, then f
36 *  may not be representable exactly. In that case, a correction
37 *  term is need. Let u=1+x rounded. Let c = (1+x)-u, then
38 *  log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
39 *  and add back the correction term c/u.
40 *  (Note: when x > 2**53, one can simply return log(x))
41 *
42 *   2. Approximation of log1p(f).
43 *  Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
44 *     = 2s + 2/3 s**3 + 2/5 s**5 + .....,
45 *          = 2s + s*R
46 *      We use a special Reme algorithm on [0,0.1716] to generate
47 *   a polynomial of degree 14 to approximate R The maximum error
48 *  of this polynomial approximation is bounded by 2**-58.45. In
49 *  other words,
50 *            2      4      6      8      10      12      14
51 *      R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
52 *    (the values of Lp1 to Lp7 are listed in the program)
53 *  and
54 *      |      2          14          |     -58.45
55 *      | Lp1*s +...+Lp7*s    -  R(z) | <= 2
56 *      |                             |
57 *  Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
58 *  In order to guarantee error in log below 1ulp, we compute log
59 *  by
60 *    log1p(f) = f - (hfsq - s*(hfsq+R)).
61 *
62 *  3. Finally, log1p(x) = k*ln2 + log1p(f).
63 *            = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
64 *     Here ln2 is split into two floating point number:
65 *      ln2_hi + ln2_lo,
66 *     where n*ln2_hi is always exact for |n| < 2000.
67 *
68 * Special cases:
69 *  log1p(x) is NaN with signal if x < -1 (including -INF) ;
70 *  log1p(+INF) is +INF; log1p(-1) is -INF with signal;
71 *  log1p(NaN) is that NaN with no signal.
72 *
73 * Accuracy:
74 *  according to an error analysis, the error is always less than
75 *  1 ulp (unit in the last place).
76 *
77 * Constants:
78 * The hexadecimal values are the intended ones for the following
79 * constants. The decimal values may be used, provided that the
80 * compiler will convert from decimal to binary accurately enough
81 * to produce the hexadecimal values shown.
82 *
83 * Note: Assuming log() return accurate answer, the following
84 *    algorithm can be used to compute log1p(x) to within a few ULP:
85 *
86 *    u = 1+x;
87 *    if(u==1.0) return x ; else
88 *         return log(u)*(x/(u-1.0));
89 *
90 *   See HP-15C Advanced Functions Handbook, p.193.
91 */
92
93#define zero 0.0
94#define ln2_hi 6.93147180369123816490e-01 /* 3fe62e42 fee00000 */
95#define ln2_lo 1.90821492927058770002e-10 /* 3dea39ef 35793c76 */
96#define two54 1.80143985094819840000e+16  /* 43500000 00000000 */
97#define Lp1 6.666666666666735130e-01      /* 3FE55555 55555593 */
98#define Lp2 3.999999999940941908e-01      /* 3FD99999 9997FA04 */
99#define Lp3 2.857142874366239149e-01      /* 3FD24924 94229359 */
100#define Lp4 2.222219843214978396e-01      /* 3FCC71C5 1D8E78AF */
101#define Lp5 1.818357216161805012e-01      /* 3FC74664 96CB03DE */
102#define Lp6 1.531383769920937332e-01      /* 3FC39A09 D078C69F */
103#define Lp7 1.479819860511658591e-01      /* 3FC2F112 DF3E5244 */
104
105double
106log1p (double x)
107{
108  double hfsq, f, c, s, z, R;
109  double_accessor u;
110  int k, hx, hu, ax;
111
112  hx = __HI (x);
113  ax = hx & 0x7fffffff;
114  c = 0;
115  k = 1;
116  if (hx < 0x3FDA827A)
117  {
118    /* 1+x < sqrt(2)+ */
119    if (ax >= 0x3ff00000)
120    {
121      /* x <= -1.0 */
122      if (x == -1.0)
123      {
124        /* log1p(-1) = +inf */
125        return -two54 / zero;
126      }
127      else
128      {
129        /* log1p(x<-1) = NaN */
130        return NAN;
131      }
132    }
133    if (ax < 0x3e200000)
134    {                         /* |x| < 2**-29 */
135      if ((two54 + x > zero)    /* raise inexact */
136          && (ax < 0x3c900000)) /* |x| < 2**-54 */
137      {
138        return x;
139      }
140      else
141      {
142        return x - x * x * 0.5;
143      }
144    }
145    if ((hx > 0) || hx <= ((int) 0xbfd2bec4))
146    {
147      /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
148      k = 0;
149      f = x;
150      hu = 1;
151    }
152  }
153  if (hx >= 0x7ff00000)
154  {
155    return x + x;
156  }
157  if (k != 0)
158  {
159    if (hx < 0x43400000)
160    {
161      u.dbl = 1.0 + x;
162      hu = u.as_int.hi;
163      k = (hu >> 20) - 1023;
164      c = (k > 0) ? 1.0 - (u.dbl - x) : x - (u.dbl - 1.0); /* correction term */
165      c /= u.dbl;
166    }
167    else
168    {
169      u.dbl = x;
170      hu = u.as_int.hi;
171      k = (hu >> 20) - 1023;
172      c = 0;
173    }
174    hu &= 0x000fffff;
175    /*
176     * The approximation to sqrt(2) used in thresholds is not
177     * critical.  However, the ones used above must give less
178     * strict bounds than the one here so that the k==0 case is
179     * never reached from here, since here we have committed to
180     * using the correction term but don't use it if k==0.
181     */
182    if (hu < 0x6a09e)
183    {
184      /* u ~< sqrt(2) */
185      u.as_int.hi = hu | 0x3ff00000; /* normalize u */
186    }
187    else
188    {
189      k += 1;
190      u.as_int.hi = hu | 0x3fe00000; /* normalize u/2 */
191      hu = (0x00100000 - hu) >> 2;
192    }
193    f = u.dbl - 1.0;
194  }
195  hfsq = 0.5 * f * f;
196  if (hu == 0)
197  {
198    /* |f| < 2**-20 */
199    if (f == zero)
200    {
201      if (k == 0)
202      {
203        return zero;
204      }
205      else
206      {
207        c += k * ln2_lo;
208        return k * ln2_hi + c;
209      }
210    }
211    R = hfsq * (1.0 - 0.66666666666666666 * f);
212    if (k == 0)
213    {
214      return f - R;
215    }
216    else
217    {
218      return k * ln2_hi - ((R - (k * ln2_lo + c)) - f);
219    }
220  }
221  s = f / (2.0 + f);
222  z = s * s;
223  R = z * (Lp1 +
224           z * (Lp2 + z * (Lp3 + z * (Lp4 + z * (Lp5 + z * (Lp6 + z * Lp7))))));
225  if (k == 0)
226  {
227    return f - (hfsq - s * (hfsq + R));
228  }
229  else
230  {
231    return k * ln2_hi - ((hfsq - (s * (hfsq + R) + (k * ln2_lo + c))) - f);
232  }
233} /* log1p */
234
235#undef zero
236#undef ln2_hi
237#undef ln2_lo
238#undef two54
239#undef Lp1
240#undef Lp2
241#undef Lp3
242#undef Lp4
243#undef Lp5
244#undef Lp6
245#undef Lp7
246