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 // 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 logarithm 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 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 */ ) // special cases switch { case 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) }