/src/moddable/xs/tools/fdlibm/s_log1p.c

Source (jump to first uncovered line)
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */

/* double log1p(double x)
 *
 * Method :
 *   1. Argument Reduction: find k and f such that
 *      1+x = 2^k * (1+f),
 *     where  sqrt(2)/2 < 1+f < sqrt(2) .
 *
 *      Note. If k=0, then f=x is exact. However, if k!=0, then f
 *  may not be representable exactly. In that case, a correction
 *  term is need. Let u=1+x rounded. Let c = (1+x)-u, then
 *  log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
 *  and add back the correction term c/u.
 *  (Note: when x > 2**53, one can simply return log(x))
 *
 *   2. Approximation of log1p(f).
 *  Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
 *     = 2s + 2/3 s**3 + 2/5 s**5 + .....,
 *         = 2s + s*R
 *      We use a special Reme algorithm on [0,0.1716] to generate
 *  a polynomial of degree 14 to approximate R The maximum error
 *  of this polynomial approximation is bounded by 2**-58.45. In
 *  other words,
 *            2      4      6      8      10      12      14
 *      R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
 *    (the values of Lp1 to Lp7 are listed in the program)
 *  and
 *      |      2          14          |     -58.45
 *      | Lp1*s +...+Lp7*s    -  R(z) | <= 2
 *      |                             |
 *  Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
 *  In order to guarantee error in log below 1ulp, we compute log
 *  by
 *    log1p(f) = f - (hfsq - s*(hfsq+R)).
 *
 *  3. Finally, log1p(x) = k*ln2 + log1p(f).
 *           = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
 *     Here ln2 is split into two floating point number:
 *      ln2_hi + ln2_lo,
 *     where n*ln2_hi is always exact for |n| < 2000.
 *
 * Special cases:
 *  log1p(x) is NaN with signal if x < -1 (including -INF) ;
 *  log1p(+INF) is +INF; log1p(-1) is -INF with signal;
 *  log1p(NaN) is that NaN with no signal.
 *
 * Accuracy:
 *  according to an error analysis, the error is always less than
 *  1 ulp (unit in the last place).
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 *
 * Note: Assuming log() return accurate answer, the following
 *   algorithm can be used to compute log1p(x) to within a few ULP:
 *
 *    u = 1+x;
 *    if(u==1.0) return x ; else
 *         return log(u)*(x/(u-1.0));
 *
 *   See HP-15C Advanced Functions Handbook, p.193.
 */

#include "math_private.h"

static const double
ln2_hi  =  6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
ln2_lo  =  1.90821492927058770002e-10,  /* 3dea39ef 35793c76 */
two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
Lp1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
Lp2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
Lp3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
Lp4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
Lp5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
Lp6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
Lp7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */

static const double zero = 0.0;
static volatile double vzero = 0.0;

double
s_log1p(double x)
{
  double hfsq,f,c,s,z,R,u;
  int32_t k,hx,hu,ax;

  GET_HIGH_WORD(hx,x);
  ax = hx&0x7fffffff;

  k = 1;
  if (hx < 0x3FDA827A) {     /* 1+x < sqrt(2)+ */
      if(ax>=0x3ff00000) {   /* x <= -1.0 */
    if(x==-1.0) return -two54/vzero; /* log1p(-1)=+inf */
    else return (x-x)/(x-x); /* log1p(x<-1)=NaN */
      }
      if(ax<0x3e200000) {     /* |x| < 2**-29 */
    if(two54+x>zero      /* raise inexact */
              &&ax<0x3c900000)     /* |x| < 2**-54 */
        return x;
    else
        return x - x*x*0.5;
      }
      if(hx>0||hx<=((int32_t)0xbfd2bec4)) {
    k=0;f=x;hu=1;}    /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
  }
  if (hx >= 0x7ff00000) return x+x;
  if(k!=0) {
      if(hx<0x43400000) {
    STRICT_ASSIGN(double,u,1.0+x);
    GET_HIGH_WORD(hu,u);
          k  = (hu>>20)-1023;
          c  = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
    c /= u;
      } else {
    u  = x;
    GET_HIGH_WORD(hu,u);
          k  = (hu>>20)-1023;
    c  = 0;
      }
      hu &= 0x000fffff;
      /*
       * The approximation to sqrt(2) used in thresholds is not
       * critical.  However, the ones used above must give less
       * strict bounds than the one here so that the k==0 case is
       * never reached from here, since here we have committed to
       * using the correction term but don't use it if k==0.
       */
      if(hu<0x6a09e) {     /* u ~< sqrt(2) */
          SET_HIGH_WORD(u,hu|0x3ff00000);  /* normalize u */
      } else {
          k += 1;
    SET_HIGH_WORD(u,hu|0x3fe00000);  /* normalize u/2 */
          hu = (0x00100000-hu)>>2;
      }
      f = u-1.0;
  }
  hfsq=0.5*f*f;
  if(hu==0) { /* |f| < 2**-20 */
      if(f==zero) {
    if(k==0) {
        return zero;
    } else {
        c += k*ln2_lo;
        return k*ln2_hi+c;
    }
      }
      R = hfsq*(1.0-0.66666666666666666*f);
      if(k==0) return f-R; else
             return k*ln2_hi-((R-(k*ln2_lo+c))-f);
  }
  s = f/(2.0+f);
  z = s*s;
  R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
  if(k==0) return f-(hfsq-s*(hfsq+R)); else
     return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
}

Coverage Report

Created: 2025-06-13 06:17

Line	Count	Source (jump to first uncovered line)
1		/*
2		* ====================================================
3		* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4		*
5		* Developed at SunPro, a Sun Microsystems, Inc. business.
6		* Permission to use, copy, modify, and distribute this
7		* software is freely granted, provided that this notice
8		* is preserved.
9		* ====================================================
10		*/
11
12		/* double log1p(double x)
13		*
14		* Method :
15		* 1. Argument Reduction: find k and f such that
16		* 1+x = 2^k * (1+f),
17		* where sqrt(2)/2 < 1+f < sqrt(2) .
18		*
19		* Note. If k=0, then f=x is exact. However, if k!=0, then f
20		* may not be representable exactly. In that case, a correction
21		* term is need. Let u=1+x rounded. Let c = (1+x)-u, then
22		* log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
23		* and add back the correction term c/u.
24		* (Note: when x > 2**53, one can simply return log(x))
25		*
26		* 2. Approximation of log1p(f).
27		* Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
28		* = 2s + 2/3 s3 + 2/5 s5 + .....,
29		* = 2s + s*R
30		* We use a special Reme algorithm on [0,0.1716] to generate
31		* a polynomial of degree 14 to approximate R The maximum error
32		* of this polynomial approximation is bounded by 2**-58.45. In
33		* other words,
34		* 2 4 6 8 10 12 14
35		* R(z) ~ Lp1s +Lp2s +Lp3s +Lp4s +Lp5s +Lp6s +Lp7*s
36		* (the values of Lp1 to Lp7 are listed in the program)
37		* and
38		* \| 2 14 \| -58.45
39		* \| Lp1s +...+Lp7s - R(z) \| <= 2
40		* \| \|
41		* Note that 2s = f - sf = f - hfsq + shfsq, where hfsq = f*f/2.
42		* In order to guarantee error in log below 1ulp, we compute log
43		* by
44		* log1p(f) = f - (hfsq - s*(hfsq+R)).
45		*
46		* 3. Finally, log1p(x) = k*ln2 + log1p(f).
47		* = kln2_hi+(f-(hfsq-(s(hfsq+R)+k*ln2_lo)))
48		* Here ln2 is split into two floating point number:
49		* ln2_hi + ln2_lo,
50		* where n*ln2_hi is always exact for \|n\| < 2000.
51		*
52		* Special cases:
53		* log1p(x) is NaN with signal if x < -1 (including -INF) ;
54		* log1p(+INF) is +INF; log1p(-1) is -INF with signal;
55		* log1p(NaN) is that NaN with no signal.
56		*
57		* Accuracy:
58		* according to an error analysis, the error is always less than
59		* 1 ulp (unit in the last place).
60		*
61		* Constants:
62		* The hexadecimal values are the intended ones for the following
63		* constants. The decimal values may be used, provided that the
64		* compiler will convert from decimal to binary accurately enough
65		* to produce the hexadecimal values shown.
66		*
67		* Note: Assuming log() return accurate answer, the following
68		* algorithm can be used to compute log1p(x) to within a few ULP:
69		*
70		* u = 1+x;
71		* if(u==1.0) return x ; else
72		* return log(u)*(x/(u-1.0));
73		*
74		* See HP-15C Advanced Functions Handbook, p.193.
75		*/
76
77		#include "math_private.h"
78
79		static const double
80		ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
81		ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
82		two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
83		Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
84		Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
85		Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
86		Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
87		Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
88		Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
89		Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
90
91		static const double zero = 0.0;
92		static volatile double vzero = 0.0;
93
94		double
95		s_log1p(double x)
96	434k	{
97	434k	double hfsq,f,c,s,z,R,u;
98	434k	int32_t k,hx,hu,ax;
99
100	434k	GET_HIGH_WORD(hx,x);
101	434k	ax = hx&0x7fffffff;
102
103	434k	k = 1;
104	434k	if (hx < 0x3FDA827A) { /* 1+x < sqrt(2)+ */
105	3.02k	if(ax>=0x3ff00000) { /* x <= -1.0 */
106	234	if(x==-1.0) return -two54/vzero; /* log1p(-1)=+inf */
107	85	else return (x-x)/(x-x); /* log1p(x<-1)=NaN */
108	234	}
109	2.78k	if(ax<0x3e200000) { /* \|x\| < 2*-29 /
110	461	if(two54+x>zero /* raise inexact */
111	461	&&ax<0x3c900000) /* \|x\| < 2*-54 /
112	461	return x;
113	0	else
114	0	return x - xx0.5;
115	461	}
116	2.32k	if(hx>0\|\|hx<=((int32_t)0xbfd2bec4)) {
117	2.32k	k=0;f=x;hu=1;} /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
118	2.32k	}
119	434k	if (hx >= 0x7ff00000) return x+x;
120	433k	if(k!=0) {
121	430k	if(hx<0x43400000) {
122	155k	STRICT_ASSIGN(double,u,1.0+x);
123	155k	GET_HIGH_WORD(hu,u);
124	155k	k = (hu>>20)-1023;
125	155k	c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
126	155k	c /= u;
127	275k	} else {
128	275k	u = x;
129	275k	GET_HIGH_WORD(hu,u);
130	275k	k = (hu>>20)-1023;
131	275k	c = 0;
132	275k	}
133	430k	hu &= 0x000fffff;
134		/*
135		* The approximation to sqrt(2) used in thresholds is not
136		* critical. However, the ones used above must give less
137		* strict bounds than the one here so that the k==0 case is
138		* never reached from here, since here we have committed to
139		* using the correction term but don't use it if k==0.
140		*/
141	430k	if(hu<0x6a09e) { /* u ~< sqrt(2) */
142	405k	SET_HIGH_WORD(u,hu\|0x3ff00000); /* normalize u */
143	405k	} else {
144	25.2k	k += 1;
145	25.2k	SET_HIGH_WORD(u,hu\|0x3fe00000); /* normalize u/2 */
146	25.2k	hu = (0x00100000-hu)>>2;
147	25.2k	}
148	430k	f = u-1.0;
149	430k	}
150	433k	hfsq=0.5ff;
151	433k	if(hu==0) { /* \|f\| < 2*-20 /
152	128k	if(f==zero) {
153	127k	if(k==0) {
154	0	return zero;
155	127k	} else {
156	127k	c += k*ln2_lo;
157	127k	return k*ln2_hi+c;
158	127k	}
159	127k	}
160	1.73k	R = hfsq(1.0-0.66666666666666666f);
161	1.73k	if(k==0) return f-R; else
162	1.73k	return kln2_hi-((R-(kln2_lo+c))-f);
163	1.73k	}
164	304k	s = f/(2.0+f);
165	304k	z = s*s;
166	304k	R = z(Lp1+z(Lp2+z(Lp3+z(Lp4+z(Lp5+z(Lp6+z*Lp7))))));
167	304k	if(k==0) return f-(hfsq-s*(hfsq+R)); else
168	301k	return kln2_hi-((hfsq-(s(hfsq+R)+(k*ln2_lo+c)))-f);
169	304k	}