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 // 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) // 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. // 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(±0) = ±0 // Log1p(-1) = -Inf // Log1p(x < -1) = NaN // Log1p(NaN) = NaN //extern log1p func libc_log1p(float64) float64 func Log1p(x float64) float64 { if x == 0 { return x } return libc_log1p(x) } 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 switch { case x < -1 || IsNaN(x): // includes -Inf return NaN() case x == -1: return Inf(-1) case 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) // correction term if k > 0 { c = 1.0 - (u - x) } else { c = x - (u - 1.0) } 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++ 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 } 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) }