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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
import "math"
// 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) ~ lg1*s +lg2*s +lg3*s +lg4*s +lg5*s +lg6*s +lg7*s
// (the values of lg1 to lg7 are listed in the program)
// and
// | 2 14 | -58.45
// | lg1*s +...+lg7*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.
const (
ln2Hi = 6.93147180369123816490e-01; /* 3fe62e42 fee00000 */
ln2Lo = 1.90821492927058770002e-10; /* 3dea39ef 35793c76 */
lg1 = 6.666666666666735130e-01; /* 3FE55555 55555593 */
lg2 = 3.999999999940941908e-01; /* 3FD99999 9997FA04 */
lg3 = 2.857142874366239149e-01; /* 3FD24924 94229359 */
lg4 = 2.222219843214978396e-01; /* 3FCC71C5 1D8E78AF */
lg5 = 1.818357216161805012e-01; /* 3FC74664 96CB03DE */
lg6 = 1.531383769920937332e-01; /* 3FC39A09 D078C69F */
lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
)
export func Log(x float64) float64 {
// special cases
switch {
case sys.isNaN(x) || sys.isInf(x, 1):
return x;
case x < 0:
return sys.NaN();
case x == 0:
return sys.Inf(-1);
}
// reduce
f1, ki := sys.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*(lg1 + s4*(lg3 + s4*(lg5 + s4*lg7)));
t2 := s4*(lg2 + s4*(lg4 + s4*lg6));
R := t1 + t2;
hfsq := 0.5*f*f;
return k*ln2Hi - ((hfsq-(s*(hfsq+R)+k*ln2Lo)) - f);
}
const
(
ln10u1 = .4342944819032518276511;
)
export func Log10(arg float64) float64 {
if arg <= 0 {
return sys.NaN();
}
return Log(arg) * ln10u1;
}
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