| // Copyright 2009 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 |
| |
| /* |
| Floating-point logarithm. |
| */ |
| |
| // The original C code, the long comment, and the constants |
| // below are from FreeBSD's /usr/src/lib/msun/src/e_log.c |
| // and came with this notice. The go code is a simpler |
| // 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. |
| // ==================================================== |
| // |
| // __ieee754_log(x) |
| // Return the logrithm of x |
| // |
| // Method : |
| // 1. Argument Reduction: find k and f such that |
| // x = 2**k * (1+f), |
| // where sqrt(2)/2 < 1+f < sqrt(2) . |
| // |
| // 2. Approximation of log(1+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) ~ L1*s +L2*s +L3*s +L4*s +L5*s +L6*s +L7*s |
| // (the values of L1 to L7 are listed in the program) and |
| // | 2 14 | -58.45 |
| // | L1*s +...+L7*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 |
| // log(1+f) = f - s*(f - R) (if f is not too large) |
| // log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) |
| // |
| // 3. Finally, log(x) = k*Ln2 + log(1+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: |
| // log(x) is NaN with signal if x < 0 (including -INF) ; |
| // log(+INF) is +INF; log(0) is -INF with signal; |
| // log(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. |
| |
| // Log returns the natural logarithm of x. |
| // |
| // Special cases are: |
| // Log(+Inf) = +Inf |
| // Log(0) = -Inf |
| // Log(x < 0) = NaN |
| // Log(NaN) = NaN |
| func Log(x float64) float64 { |
| const ( |
| Ln2Hi = 6.93147180369123816490e-01 /* 3fe62e42 fee00000 */ |
| Ln2Lo = 1.90821492927058770002e-10 /* 3dea39ef 35793c76 */ |
| L1 = 6.666666666666735130e-01 /* 3FE55555 55555593 */ |
| L2 = 3.999999999940941908e-01 /* 3FD99999 9997FA04 */ |
| L3 = 2.857142874366239149e-01 /* 3FD24924 94229359 */ |
| L4 = 2.222219843214978396e-01 /* 3FCC71C5 1D8E78AF */ |
| L5 = 1.818357216161805012e-01 /* 3FC74664 96CB03DE */ |
| L6 = 1.531383769920937332e-01 /* 3FC39A09 D078C69F */ |
| L7 = 1.479819860511658591e-01 /* 3FC2F112 DF3E5244 */ |
| ) |
| |
| // TODO(rsc): Remove manual inlining of IsNaN, IsInf |
| // when compiler does it for us |
| // special cases |
| switch { |
| case x != x || x > MaxFloat64: // IsNaN(x) || IsInf(x, 1): |
| return x |
| case x < 0: |
| return NaN() |
| case x == 0: |
| return Inf(-1) |
| } |
| |
| // reduce |
| f1, ki := Frexp(x) |
| if f1 < Sqrt2/2 { |
| f1 *= 2 |
| ki-- |
| } |
| f := f1 - 1 |
| k := float64(ki) |
| |
| // compute |
| s := f / (2 + f) |
| s2 := s * s |
| s4 := s2 * s2 |
| t1 := s2 * (L1 + s4*(L3+s4*(L5+s4*L7))) |
| t2 := s4 * (L2 + s4*(L4+s4*L6)) |
| R := t1 + t2 |
| hfsq := 0.5 * f * f |
| return k*Ln2Hi - ((hfsq - (s*(hfsq+R) + k*Ln2Lo)) - f) |
| } |