| // Copyright 2010 The Go Authors. All rights reserved. |
| // Use of this source code is governed by a BSD-style |
| // license that can be found in the LICENSE file. |
| |
| package math |
| |
| |
| // The original C code, the long comment, and the constants |
| // below are from FreeBSD's /usr/src/lib/msun/src/s_log1p.c |
| // and came with this notice. The go code is a simplified |
| // version of the original C. |
| // |
| // ==================================================== |
| // 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) |
| // a-0.2929nd |
| // | 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. |
| |
| // Log1p returns the natural logarithm of 1 plus its argument x. |
| // It is more accurate than Log(1 + x) when x is near zero. |
| // |
| // Special cases are: |
| // Log1p(+Inf) = +Inf |
| // Log1p(-1) = -Inf |
| // Log1p(x < -1) = NaN |
| // Log1p(NaN) = NaN |
| func Log1p(x float64) float64 { |
| const ( |
| Sqrt2M1 = 4.142135623730950488017e-01 // Sqrt(2)-1 = 0x3fda827999fcef34 |
| Sqrt2HalfM1 = -2.928932188134524755992e-01 // Sqrt(2)/2-1 = 0xbfd2bec333018866 |
| Small = 1.0 / (1 << 29) // 2**-29 = 0x3e20000000000000 |
| Tiny = 1.0 / (1 << 54) // 2**-54 |
| Two53 = 1 << 53 // 2**53 |
| Ln2Hi = 6.93147180369123816490e-01 // 3fe62e42fee00000 |
| Ln2Lo = 1.90821492927058770002e-10 // 3dea39ef35793c76 |
| Lp1 = 6.666666666666735130e-01 // 3FE5555555555593 |
| Lp2 = 3.999999999940941908e-01 // 3FD999999997FA04 |
| Lp3 = 2.857142874366239149e-01 // 3FD2492494229359 |
| Lp4 = 2.222219843214978396e-01 // 3FCC71C51D8E78AF |
| Lp5 = 1.818357216161805012e-01 // 3FC7466496CB03DE |
| Lp6 = 1.531383769920937332e-01 // 3FC39A09D078C69F |
| Lp7 = 1.479819860511658591e-01 // 3FC2F112DF3E5244 |
| ) |
| |
| // special cases |
| // TODO(rsc): Remove manual inlining of IsNaN, IsInf |
| // when compiler does it for us |
| switch { |
| case x < -1 || x != x: // x < -1 || IsNaN(x): // includes -Inf |
| return NaN() |
| case x == -1: |
| return Inf(-1) |
| case x > MaxFloat64: // IsInf(x, 1): |
| return Inf(1) |
| } |
| |
| absx := x |
| if absx < 0 { |
| absx = -absx |
| } |
| |
| var f float64 |
| var iu uint64 |
| k := 1 |
| if absx < Sqrt2M1 { // |x| < Sqrt(2)-1 |
| if absx < Small { // |x| < 2**-29 |
| if absx < Tiny { // |x| < 2**-54 |
| return x |
| } |
| return x - x*x*0.5 |
| } |
| if x > Sqrt2HalfM1 { // Sqrt(2)/2-1 < x |
| // (Sqrt(2)/2-1) < x < (Sqrt(2)-1) |
| k = 0 |
| f = x |
| iu = 1 |
| } |
| } |
| var c float64 |
| if k != 0 { |
| var u float64 |
| if absx < Two53 { // 1<<53 |
| u = 1.0 + x |
| iu = Float64bits(u) |
| k = int((iu >> 52) - 1023) |
| if k > 0 { |
| c = 1.0 - (u - x) |
| } else { |
| c = x - (u - 1.0) // correction term |
| c /= u |
| } |
| } else { |
| u = x |
| iu = Float64bits(u) |
| k = int((iu >> 52) - 1023) |
| c = 0 |
| } |
| iu &= 0x000fffffffffffff |
| if iu < 0x0006a09e667f3bcd { // mantissa of Sqrt(2) |
| u = Float64frombits(iu | 0x3ff0000000000000) // normalize u |
| } else { |
| k += 1 |
| u = Float64frombits(iu | 0x3fe0000000000000) // normalize u/2 |
| iu = (0x0010000000000000 - iu) >> 2 |
| } |
| f = u - 1.0 // Sqrt(2)/2 < u < Sqrt(2) |
| } |
| hfsq := 0.5 * f * f |
| var s, R, z float64 |
| if iu == 0 { // |f| < 2**-20 |
| if f == 0 { |
| if k == 0 { |
| return 0 |
| } else { |
| c += float64(k) * Ln2Lo |
| return float64(k)*Ln2Hi + c |
| } |
| } |
| R = hfsq * (1.0 - 0.66666666666666666*f) // avoid division |
| if k == 0 { |
| return f - R |
| } |
| return float64(k)*Ln2Hi - ((R - (float64(k)*Ln2Lo + 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)) |
| } |
| return float64(k)*Ln2Hi - ((hfsq - (s*(hfsq+R) + (float64(k)*Ln2Lo + c))) - f) |
| } |