```   1 /*
3  *
4  * The contents of this file are subject to the terms of the
6  * You may not use this file except in compliance with the License.
7  *
8  * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9  * or http://www.opensolaris.org/os/licensing.
10  * See the License for the specific language governing permissions
11  * and limitations under the License.
12  *
13  * When distributing Covered Code, include this CDDL HEADER in each
15  * If applicable, add the following below this CDDL HEADER, with the
16  * fields enclosed by brackets "[]" replaced with your own identifying
18  *
20  */
21 /*
23  */
24 /*
26  * Use is subject to license terms.
27  */
28
29 #pragma weak __log1p = log1p
30
31 /* INDENT OFF */
32 /*
33  * Method :
34  *   1. Argument Reduction: find k and f such that
35  *                      1+x = 2^k * (1+f),
36  *         where  sqrt(2)/2 < 1+f < sqrt(2) .
37  *
38  *      Note. If k=0, then f=x is exact. However, if k != 0, then f
39  *      may not be representable exactly. In that case, a correction
40  *      term is need. Let u=1+x rounded. Let c = (1+x)-u, then
41  *      log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
42  *      and add back the correction term c/u.
43  *      (Note: when x > 2**53, one can simply return log(x))
44  *
45  *   2. Approximation of log1p(f).
46  *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
47  *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
48  *               = 2s + s*R
49  *      We use a special Reme algorithm on [0,0.1716] to generate
50  *      a polynomial of degree 14 to approximate R The maximum error
51  *      of this polynomial approximation is bounded by 2**-58.45. In
52  *      other words,
53  *                      2      4      6      8      10      12      14
54  *          R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
55  *      (the values of Lp1 to Lp7 are listed in the program)
56  *      and
57  *          |      2          14          |     -58.45
58  *          | Lp1*s +...+Lp7*s    -  R(z) | <= 2
59  *          |                             |
60  *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
61  *      In order to guarantee error in log below 1ulp, we compute log
62  *      by
63  *              log1p(f) = f - (hfsq - s*(hfsq+R)).
64  *
65  *      3. Finally, log1p(x) = k*ln2 + log1p(f).
66  *                           = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
67  *         Here ln2 is splitted into two floating point number:
68  *                      ln2_hi + ln2_lo,
69  *         where n*ln2_hi is always exact for |n| < 2000.
70  *
71  * Special cases:
72  *      log1p(x) is NaN with signal if x < -1 (including -INF) ;
73  *      log1p(+INF) is +INF; log1p(-1) is -INF with signal;
74  *      log1p(NaN) is that NaN with no signal.
75  *
76  * Accuracy:
77  *      according to an error analysis, the error is always less than
78  *      1 ulp (unit in the last place).
79  *
80  * Constants:
81  * The hexadecimal values are the intended ones for the following
82  * constants. The decimal values may be used, provided that the
83  * compiler will convert from decimal to binary accurately enough
84  * to produce the hexadecimal values shown.
85  *
86  * Note: Assuming log() return accurate answer, the following
87  *       algorithm can be used to compute log1p(x) to within a few ULP:
88  *
89  *              u = 1+x;
90  *              if (u == 1.0) return x ; else
91  *                         return log(u)*(x/(u-1.0));
92  *
93  *       See HP-15C Advanced Functions Handbook, p.193.
94  */
95 /* INDENT ON */
96
97 #include "libm.h"
98
99 static const double xxx[] = {
100 /* ln2_hi */    6.93147180369123816490e-01,     /* 3fe62e42 fee00000 */
101 /* ln2_lo */    1.90821492927058770002e-10,     /* 3dea39ef 35793c76 */
102 /* two54 */     1.80143985094819840000e+16,     /* 43500000 00000000 */
103 /* Lp1 */       6.666666666666735130e-01,       /* 3FE55555 55555593 */
104 /* Lp2 */       3.999999999940941908e-01,       /* 3FD99999 9997FA04 */
105 /* Lp3 */       2.857142874366239149e-01,       /* 3FD24924 94229359 */
106 /* Lp4 */       2.222219843214978396e-01,       /* 3FCC71C5 1D8E78AF */
107 /* Lp5 */       1.818357216161805012e-01,       /* 3FC74664 96CB03DE */
108 /* Lp6 */       1.531383769920937332e-01,       /* 3FC39A09 D078C69F */
109 /* Lp7 */       1.479819860511658591e-01,       /* 3FC2F112 DF3E5244 */
110 /* zero */      0.0
111 };
112 #define ln2_hi  xxx[0]
113 #define ln2_lo  xxx[1]
114 #define two54   xxx[2]
115 #define Lp1     xxx[3]
116 #define Lp2     xxx[4]
117 #define Lp3     xxx[5]
118 #define Lp4     xxx[6]
119 #define Lp5     xxx[7]
120 #define Lp6     xxx[8]
121 #define Lp7     xxx[9]
122 #define zero    xxx[10]
123
124 double
125 log1p(double x) {
126         double  hfsq, f, c = 0.0, s, z, R, u;
127         int     k, hx, hu, ax;
128
129         hx = ((int *)&x)[HIWORD];           /* high word of x */
130         ax = hx & 0x7fffffff;
131
132         if (ax >= 0x7ff00000) { /* x is inf or nan */
133                 if (((hx - 0xfff00000) | ((int *)&x)[LOWORD]) == 0) /* -inf */
134                         return (_SVID_libm_err(x, x, 44));
135                 return (x * x);
136         }
137
138         k = 1;
139         if (hx < 0x3FDA827A) {       /* x < 0.41422  */
140                 if (ax >= 0x3ff00000)        /* x <= -1.0 */
141                         return (_SVID_libm_err(x, x, x == -1.0 ? 43 : 44));
142                 if (ax < 0x3e200000) {       /* |x| < 2**-29 */
143                         if (two54 + x > zero &&      /* raise inexact */
144                             ax < 0x3c900000) /* |x| < 2**-54 */
145                                 return (x);
146                         else
147                                 return (x - x * x * 0.5);
148                 }
149                 if (hx > 0 || hx <= (int)0xbfd2bec3) {    /* -0.2929<x<0.41422 */
150                         k = 0;
151                         f = x;
152                         hu = 1;
153                 }
154         }
155         /* We will initialize 'c' here. */
156         if (k != 0) {
157                 if (hx < 0x43400000) {
158                         u = 1.0 + x;
159                         hu = ((int *)&u)[HIWORD];   /* high word of u */
160                         k = (hu >> 20) - 1023;
161                         /*
162                          * correction term
163                          */
164                         c = k > 0 ? 1.0 - (u - x) : x - (u - 1.0);
165                         c /= u;
166                 } else {
167                         u = x;
168                         hu = ((int *)&u)[HIWORD];   /* high word of u */
169                         k = (hu >> 20) - 1023;
170                         c = 0;
171                 }
172                 hu &= 0x000fffff;
173                 if (hu < 0x6a09e) {  /* normalize u */
174                         ((int *)&u)[HIWORD] = hu | 0x3ff00000;
175                 } else {                        /* normalize u/2 */
176                         k += 1;
177                         ((int *)&u)[HIWORD] = hu | 0x3fe00000;
178                         hu = (0x00100000 - hu) >> 2;
179                 }
180                 f = u - 1.0;
181         }
182         hfsq = 0.5 * f * f;
183         if (hu == 0) {          /* |f| < 2**-20 */
184                 if (f == zero) {
185                         if (k == 0)
186                                 return (zero);
187                         /* We already initialized 'c' before, when (k != 0) */
188                         c += k * ln2_lo;
189                         return (k * ln2_hi + c);
190                 }
191                 R = hfsq * (1.0 - 0.66666666666666666 * f);
192                 if (k == 0)
193                         return (f - R);
194                 return (k * ln2_hi - ((R - (k * ln2_lo + c)) - f));
195         }
196         s = f / (2.0 + f);
197         z = s * s;
198         R = z * (Lp1 + z * (Lp2 + z * (Lp3 + z * (Lp4 + z * (Lp5 +
199                 z * (Lp6 + z * Lp7))))));
200         if (k == 0)
201                 return (f - (hfsq - s * (hfsq + R)));
202         return (k * ln2_hi - ((hfsq - (s * (hfsq + R) +
203                 (k * ln2_lo + c))) - f));
204 }
```