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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 {

        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)

                        if k > 0 {

                                c = 1.0 - (u - x)

                        } else {

                                c = x - (u - 1.0) // correction term

                                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 += 1

                        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

                        } else {

                                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)

}