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12  * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
13  * FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
14  * version 2 for more details (a copy is included in the LICENSE file that
15  * accompanied this code).
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25 
26 /* double log1p(double x)
27  *
28  * Method :
29  *   1. Argument Reduction: find k and f such that
30  *                      1+x = 2^k * (1+f),
31  *         where  sqrt(2)/2 < 1+f < sqrt(2) .
32  *
33  *      Note. If k=0, then f=x is exact. However, if k!=0, then f
34  *      may not be representable exactly. In that case, a correction
35  *      term is need. Let u=1+x rounded. Let c = (1+x)-u, then
36  *      log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
37  *      and add back the correction term c/u.
38  *      (Note: when x > 2**53, one can simply return log(x))
39  *
40  *   2. Approximation of log1p(f).
41  *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
42  *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
43  *               = 2s + s*R
44  *      We use a special Reme algorithm on [0,0.1716] to generate
45  *      a polynomial of degree 14 to approximate R The maximum error
46  *      of this polynomial approximation is bounded by 2**-58.45. In
47  *      other words,
48  *                      2      4      6      8      10      12      14
49  *          R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
50  *      (the values of Lp1 to Lp7 are listed in the program)
51  *      and
52  *          |      2          14          |     -58.45
53  *          | Lp1*s +...+Lp7*s    -  R(z) | <= 2
54  *          |                             |
55  *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
56  *      In order to guarantee error in log below 1ulp, we compute log
57  *      by
58  *              log1p(f) = f - (hfsq - s*(hfsq+R)).
59  *
60  *      3. Finally, log1p(x) = k*ln2 + log1p(f).
61  *                           = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
62  *         Here ln2 is split into two floating point number:
63  *                      ln2_hi + ln2_lo,
64  *         where n*ln2_hi is always exact for |n| < 2000.
65  *
66  * Special cases:
67  *      log1p(x) is NaN with signal if x < -1 (including -INF) ;
68  *      log1p(+INF) is +INF; log1p(-1) is -INF with signal;
69  *      log1p(NaN) is that NaN with no signal.
70  *
71  * Accuracy:
72  *      according to an error analysis, the error is always less than
73  *      1 ulp (unit in the last place).
74  *
75  * Constants:
76  * The hexadecimal values are the intended ones for the following
77  * constants. The decimal values may be used, provided that the
78  * compiler will convert from decimal to binary accurately enough
79  * to produce the hexadecimal values shown.
80  *
81  * Note: Assuming log() return accurate answer, the following
82  *       algorithm can be used to compute log1p(x) to within a few ULP:
83  *
84  *              u = 1+x;
85  *              if(u==1.0) return x ; else
86  *                         return log(u)*(x/(u-1.0));
87  *
88  *       See HP-15C Advanced Functions Handbook, p.193.
89  */
90 
91 #include "fdlibm.h"
92 
93 #ifdef __STDC__
94 static const double
95 #else
96 static double
97 #endif
98 ln2_hi  =  6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
99 ln2_lo  =  1.90821492927058770002e-10,  /* 3dea39ef 35793c76 */
100 two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
101 Lp1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
102 Lp2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
103 Lp3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
104 Lp4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
105 Lp5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
106 Lp6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
107 Lp7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
108 
109 static double zero = 0.0;
110 
111 #ifdef __STDC__
log1p(double x)112         double log1p(double x)
113 #else
114         double log1p(x)
115         double x;
116 #endif
117 {
118         double hfsq,f=0,c=0,s,z,R,u;
119         int k,hx,hu=0,ax;
120 
121         hx = __HI(x);           /* high word of x */
122         ax = hx&0x7fffffff;
123 
124         k = 1;
125         if (hx < 0x3FDA827A) {                  /* x < 0.41422  */
126             if(ax>=0x3ff00000) {                /* x <= -1.0 */
127                 /*
128                  * Added redundant test against hx to work around VC++
129                  * code generation problem.
130                  */
131                 if(x==-1.0 && (hx==0xbff00000)) /* log1p(-1)=-inf */
132                   return -two54/zero;
133                 else
134                   return (x-x)/(x-x);           /* log1p(x<-1)=NaN */
135             }
136             if(ax<0x3e200000) {                 /* |x| < 2**-29 */
137                 if(two54+x>zero                 /* raise inexact */
138                     &&ax<0x3c900000)            /* |x| < 2**-54 */
139                     return x;
140                 else
141                     return x - x*x*0.5;
142             }
143             if(hx>0||hx<=((int)0xbfd2bec3)) {
144                 k=0;f=x;hu=1;}  /* -0.2929<x<0.41422 */
145         }
146         if (hx >= 0x7ff00000) return x+x;
147         if(k!=0) {
148             if(hx<0x43400000) {
149                 u  = 1.0+x;
150                 hu = __HI(u);           /* high word of u */
151                 k  = (hu>>20)-1023;
152                 c  = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
153                 c /= u;
154             } else {
155                 u  = x;
156                 hu = __HI(u);           /* high word of u */
157                 k  = (hu>>20)-1023;
158                 c  = 0;
159             }
160             hu &= 0x000fffff;
161             if(hu<0x6a09e) {
162                 __HI(u) = hu|0x3ff00000;        /* normalize u */
163             } else {
164                 k += 1;
165                 __HI(u) = hu|0x3fe00000;        /* normalize u/2 */
166                 hu = (0x00100000-hu)>>2;
167             }
168             f = u-1.0;
169         }
170         hfsq=0.5*f*f;
171         if(hu==0) {     /* |f| < 2**-20 */
172             if(f==zero) { if(k==0) return zero;
173                           else {c += k*ln2_lo; return k*ln2_hi+c;}}
174             R = hfsq*(1.0-0.66666666666666666*f);
175             if(k==0) return f-R; else
176                      return k*ln2_hi-((R-(k*ln2_lo+c))-f);
177         }
178         s = f/(2.0+f);
179         z = s*s;
180         R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
181         if(k==0) return f-(hfsq-s*(hfsq+R)); else
182                  return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
183 }
184