1 /* @(#)s_log1p.c 5.1 93/09/24 */
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
10 * ====================================================
12 /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
13 for performance improvement on pipelined processors.
16 /* double log1p(double x)
19 * 1. Argument Reduction: find k and f such that
21 * where sqrt(2)/2 < 1+f < sqrt(2) .
23 * Note. If k=0, then f=x is exact. However, if k!=0, then f
24 * may not be representable exactly. In that case, a correction
25 * term is need. Let u=1+x rounded. Let c = (1+x)-u, then
26 * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
27 * and add back the correction term c/u.
28 * (Note: when x > 2**53, one can simply return log(x))
30 * 2. Approximation of log1p(f).
31 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
32 * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
34 * We use a special Reme algorithm on [0,0.1716] to generate
35 * a polynomial of degree 14 to approximate R The maximum error
36 * of this polynomial approximation is bounded by 2**-58.45. In
39 * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
40 * (the values of Lp1 to Lp7 are listed in the program)
43 * | Lp1*s +...+Lp7*s - R(z) | <= 2
45 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
46 * In order to guarantee error in log below 1ulp, we compute log
48 * log1p(f) = f - (hfsq - s*(hfsq+R)).
50 * 3. Finally, log1p(x) = k*ln2 + log1p(f).
51 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
52 * Here ln2 is split into two floating point number:
54 * where n*ln2_hi is always exact for |n| < 2000.
57 * log1p(x) is NaN with signal if x < -1 (including -INF) ;
58 * log1p(+INF) is +INF; log1p(-1) is -INF with signal;
59 * log1p(NaN) is that NaN with no signal.
62 * according to an error analysis, the error is always less than
63 * 1 ulp (unit in the last place).
66 * The hexadecimal values are the intended ones for the following
67 * constants. The decimal values may be used, provided that the
68 * compiler will convert from decimal to binary accurately enough
69 * to produce the hexadecimal values shown.
71 * Note: Assuming log() return accurate answer, the following
72 * algorithm can be used to compute log1p(x) to within a few ULP:
75 * if(u==1.0) return x ; else
76 * return log(u)*(x/(u-1.0));
78 * See HP-15C Advanced Functions Handbook, p.193.
83 #include <math_private.h>
86 ln2_hi
= 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
87 ln2_lo
= 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
88 two54
= 1.80143985094819840000e+16, /* 43500000 00000000 */
89 Lp
[] = { 0.0, 6.666666666666735130e-01, /* 3FE55555 55555593 */
90 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
91 2.857142874366239149e-01, /* 3FD24924 94229359 */
92 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
93 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
94 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
95 1.479819860511658591e-01 }; /* 3FC2F112 DF3E5244 */
97 static const double zero
= 0.0;
102 double hfsq
, f
, c
, s
, z
, R
, u
, z2
, z4
, z6
, R1
, R2
, R3
, R4
;
103 int32_t k
, hx
, hu
, ax
;
105 GET_HIGH_WORD (hx
, x
);
106 ax
= hx
& 0x7fffffff;
109 if (hx
< 0x3FDA827A) /* x < 0.41422 */
111 if (__glibc_unlikely (ax
>= 0x3ff00000)) /* x <= -1.0 */
114 return -two54
/ zero
; /* log1p(-1)=-inf */
116 return (x
- x
) / (x
- x
); /* log1p(x<-1)=NaN */
118 if (__glibc_unlikely (ax
< 0x3e200000)) /* |x| < 2**-29 */
120 math_force_eval (two54
+ x
); /* raise inexact */
121 if (ax
< 0x3c900000) /* |x| < 2**-54 */
123 if (fabs (x
) < DBL_MIN
)
125 double force_underflow
= x
* x
;
126 math_force_eval (force_underflow
);
131 return x
- x
* x
* 0.5;
133 if (hx
> 0 || hx
<= ((int32_t) 0xbfd2bec3))
135 k
= 0; f
= x
; hu
= 1;
136 } /* -0.2929<x<0.41422 */
138 else if (__glibc_unlikely (hx
>= 0x7ff00000))
145 GET_HIGH_WORD (hu
, u
);
146 k
= (hu
>> 20) - 1023;
147 c
= (k
> 0) ? 1.0 - (u
- x
) : x
- (u
- 1.0); /* correction term */
153 GET_HIGH_WORD (hu
, u
);
154 k
= (hu
>> 20) - 1023;
160 SET_HIGH_WORD (u
, hu
| 0x3ff00000); /* normalize u */
165 SET_HIGH_WORD (u
, hu
| 0x3fe00000); /* normalize u/2 */
166 hu
= (0x00100000 - hu
) >> 2;
171 if (hu
== 0) /* |f| < 2**-20 */
179 c
+= k
* ln2_lo
; return k
* ln2_hi
+ c
;
182 R
= hfsq
* (1.0 - 0.66666666666666666 * f
);
186 return k
* ln2_hi
- ((R
- (k
* ln2_lo
+ c
)) - f
);
190 R1
= z
* Lp
[1]; z2
= z
* z
;
191 R2
= Lp
[2] + z
* Lp
[3]; z4
= z2
* z2
;
192 R3
= Lp
[4] + z
* Lp
[5]; z6
= z4
* z2
;
193 R4
= Lp
[6] + z
* Lp
[7];
194 R
= R1
+ z2
* R2
+ z4
* R3
+ z6
* R4
;
196 return f
- (hfsq
- s
* (hfsq
+ R
));
198 return k
* ln2_hi
- ((hfsq
- (s
* (hfsq
+ R
) + (k
* ln2_lo
+ c
))) - f
);