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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

//extern log
func libc_log(float64) float64

func Log(x float64) float64 {
        return libc_log(x)
}

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)
}