Source file src/pkg/math/log1p.go

     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	
     5	package 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
    95	func Log1p(x float64) float64
    96	
    97	func 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				if k > 0 {
   155					c = 1.0 - (u - x)
   156				} else {
   157					c = x - (u - 1.0) // correction term
   158					c /= u
   159				}
   160			} else {
   161				u = x
   162				iu = Float64bits(u)
   163				k = int((iu >> 52) - 1023)
   164				c = 0
   165			}
   166			iu &= 0x000fffffffffffff
   167			if iu < 0x0006a09e667f3bcd { // mantissa of Sqrt(2)
   168				u = Float64frombits(iu | 0x3ff0000000000000) // normalize u
   169			} else {
   170				k += 1
   171				u = Float64frombits(iu | 0x3fe0000000000000) // normalize u/2
   172				iu = (0x0010000000000000 - iu) >> 2
   173			}
   174			f = u - 1.0 // Sqrt(2)/2 < u < Sqrt(2)
   175		}
   176		hfsq := 0.5 * f * f
   177		var s, R, z float64
   178		if iu == 0 { // |f| < 2**-20
   179			if f == 0 {
   180				if k == 0 {
   181					return 0
   182				} else {
   183					c += float64(k) * Ln2Lo
   184					return float64(k)*Ln2Hi + c
   185				}
   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	}