1// Copyright 2010 The Go Authors. All rights reserved.
2// Use of this source code is governed by a BSD-style
3// license that can be found in the LICENSE file.
4
5package math
6
7// The original C code, the long comment, and the constants
8// below are from FreeBSD's /usr/src/lib/msun/src/s_log1p.c
9// and came with this notice. The go code is a simplified
10// version of the original C.
11//
12// ====================================================
13// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
14//
15// Developed at SunPro, a Sun Microsystems, Inc. business.
16// Permission to use, copy, modify, and distribute this
17// software is freely granted, provided that this notice
18// is preserved.
19// ====================================================
20//
21//
22// double log1p(double x)
23//
24// Method :
25//   1. Argument Reduction: find k and f such that
26//                      1+x = 2**k * (1+f),
27//         where  sqrt(2)/2 < 1+f < sqrt(2) .
28//
29//      Note. If k=0, then f=x is exact. However, if k!=0, then f
30//      may not be representable exactly. In that case, a correction
31//      term is need. Let u=1+x rounded. Let c = (1+x)-u, then
32//      log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
33//      and add back the correction term c/u.
34//      (Note: when x > 2**53, one can simply return log(x))
35//
36//   2. Approximation of log1p(f).
37//      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
38//               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
39//               = 2s + s*R
40//      We use a special Reme algorithm on [0,0.1716] to generate
41//      a polynomial of degree 14 to approximate R The maximum error
42//      of this polynomial approximation is bounded by 2**-58.45. In
43//      other words,
44//                      2      4      6      8      10      12      14
45//          R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
46//      (the values of Lp1 to Lp7 are listed in the program)
47//      and
48//          |      2          14          |     -58.45
49//          | Lp1*s +...+Lp7*s    -  R(z) | <= 2
50//          |                             |
51//      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
52//      In order to guarantee error in log below 1ulp, we compute log
53//      by
54//              log1p(f) = f - (hfsq - s*(hfsq+R)).
55//
56//   3. Finally, log1p(x) = k*ln2 + log1p(f).
57//                        = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
58//      Here ln2 is split into two floating point number:
59//                   ln2_hi + ln2_lo,
60//      where n*ln2_hi is always exact for |n| < 2000.
61//
62// Special cases:
63//      log1p(x) is NaN with signal if x < -1 (including -INF) ;
64//      log1p(+INF) is +INF; log1p(-1) is -INF with signal;
65//      log1p(NaN) is that NaN with no signal.
66//
67// Accuracy:
68//      according to an error analysis, the error is always less than
69//      1 ulp (unit in the last place).
70//
71// Constants:
72// The hexadecimal values are the intended ones for the following
73// constants. The decimal values may be used, provided that the
74// compiler will convert from decimal to binary accurately enough
75// to produce the hexadecimal values shown.
76//
77// Note: Assuming log() return accurate answer, the following
78//       algorithm can be used to compute log1p(x) to within a few ULP:
79//
80//              u = 1+x;
81//              if(u==1.0) return x ; else
82//                         return log(u)*(x/(u-1.0));
83//
84//       See HP-15C Advanced Functions Handbook, p.193.
85
86// Log1p returns the natural logarithm of 1 plus its argument x.
87// It is more accurate than Log(1 + x) when x is near zero.
88//
89// Special cases are:
90//	Log1p(+Inf) = +Inf
91//	Log1p(±0) = ±0
92//	Log1p(-1) = -Inf
93//	Log1p(x < -1) = NaN
94//	Log1p(NaN) = NaN
95func Log1p(x float64) float64
96
97func log1p(x float64) float64 {
98	const (
99		Sqrt2M1     = 4.142135623730950488017e-01  // Sqrt(2)-1 = 0x3fda827999fcef34
100		Sqrt2HalfM1 = -2.928932188134524755992e-01 // Sqrt(2)/2-1 = 0xbfd2bec333018866
101		Small       = 1.0 / (1 << 29)              // 2**-29 = 0x3e20000000000000
102		Tiny        = 1.0 / (1 << 54)              // 2**-54
103		Two53       = 1 << 53                      // 2**53
104		Ln2Hi       = 6.93147180369123816490e-01   // 3fe62e42fee00000
105		Ln2Lo       = 1.90821492927058770002e-10   // 3dea39ef35793c76
106		Lp1         = 6.666666666666735130e-01     // 3FE5555555555593
107		Lp2         = 3.999999999940941908e-01     // 3FD999999997FA04
108		Lp3         = 2.857142874366239149e-01     // 3FD2492494229359
109		Lp4         = 2.222219843214978396e-01     // 3FCC71C51D8E78AF
110		Lp5         = 1.818357216161805012e-01     // 3FC7466496CB03DE
111		Lp6         = 1.531383769920937332e-01     // 3FC39A09D078C69F
112		Lp7         = 1.479819860511658591e-01     // 3FC2F112DF3E5244
113	)
114
115	// special cases
116	switch {
117	case x < -1 || IsNaN(x): // includes -Inf
118		return NaN()
119	case x == -1:
120		return Inf(-1)
121	case IsInf(x, 1):
122		return Inf(1)
123	}
124
125	absx := x
126	if absx < 0 {
127		absx = -absx
128	}
129
130	var f float64
131	var iu uint64
132	k := 1
133	if absx < Sqrt2M1 { //  |x| < Sqrt(2)-1
134		if absx < Small { // |x| < 2**-29
135			if absx < Tiny { // |x| < 2**-54
136				return x
137			}
138			return x - x*x*0.5
139		}
140		if x > Sqrt2HalfM1 { // Sqrt(2)/2-1 < x
141			// (Sqrt(2)/2-1) < x < (Sqrt(2)-1)
142			k = 0
143			f = x
144			iu = 1
145		}
146	}
147	var c float64
148	if k != 0 {
149		var u float64
150		if absx < Two53 { // 1<<53
151			u = 1.0 + x
152			iu = Float64bits(u)
153			k = int((iu >> 52) - 1023)
154			// correction term
155			if k > 0 {
156				c = 1.0 - (u - x)
157			} else {
158				c = x - (u - 1.0)
159			}
160			c /= u
161		} else {
162			u = x
163			iu = Float64bits(u)
164			k = int((iu >> 52) - 1023)
165			c = 0
166		}
167		iu &= 0x000fffffffffffff
168		if iu < 0x0006a09e667f3bcd { // mantissa of Sqrt(2)
169			u = Float64frombits(iu | 0x3ff0000000000000) // normalize u
170		} else {
171			k++
172			u = Float64frombits(iu | 0x3fe0000000000000) // normalize u/2
173			iu = (0x0010000000000000 - iu) >> 2
174		}
175		f = u - 1.0 // Sqrt(2)/2 < u < Sqrt(2)
176	}
177	hfsq := 0.5 * f * f
178	var s, R, z float64
179	if iu == 0 { // |f| < 2**-20
180		if f == 0 {
181			if k == 0 {
182				return 0
183			}
184			c += float64(k) * Ln2Lo
185			return float64(k)*Ln2Hi + c
186		}
187		R = hfsq * (1.0 - 0.66666666666666666*f) // avoid division
188		if k == 0 {
189			return f - R
190		}
191		return float64(k)*Ln2Hi - ((R - (float64(k)*Ln2Lo + c)) - f)
192	}
193	s = f / (2.0 + f)
194	z = s * s
195	R = z * (Lp1 + z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))))
196	if k == 0 {
197		return f - (hfsq - s*(hfsq+R))
198	}
199	return float64(k)*Ln2Hi - ((hfsq - (s*(hfsq+R) + (float64(k)*Ln2Lo + c))) - f)
200}
201