// 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_expm1.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.
// ====================================================
//
// expm1(x)
// Returns exp(x)-1, the exponential of x minus 1.
//
// Method
// 1. Argument reduction:
// Given x, find r and integer k such that
//
// x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
//
// Here a correction term c will be computed to compensate
// the error in r when rounded to a floating-point number.
//
// 2. Approximating expm1(r) by a special rational function on
// the interval [0,0.34658]:
// Since
// r*(exp(r)+1)/(exp(r)-1) = 2+ r**2/6 - r**4/360 + ...
// we define R1(r*r) by
// r*(exp(r)+1)/(exp(r)-1) = 2+ r**2/6 * R1(r*r)
// That is,
// R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
// = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
// = 1 - r**2/60 + r**4/2520 - r**6/100800 + ...
// We use a special Reme algorithm on [0,0.347] to generate
// a polynomial of degree 5 in r*r to approximate R1. The
// maximum error of this polynomial approximation is bounded
// by 2**-61. In other words,
// R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
// where Q1 = -1.6666666666666567384E-2,
// Q2 = 3.9682539681370365873E-4,
// Q3 = -9.9206344733435987357E-6,
// Q4 = 2.5051361420808517002E-7,
// Q5 = -6.2843505682382617102E-9;
// (where z=r*r, and the values of Q1 to Q5 are listed below)
// with error bounded by
// | 5 | -61
// | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
// | |
//
// expm1(r) = exp(r)-1 is then computed by the following
// specific way which minimize the accumulation rounding error:
// 2 3
// r r [ 3 - (R1 + R1*r/2) ]
// expm1(r) = r + --- + --- * [--------------------]
// 2 2 [ 6 - r*(3 - R1*r/2) ]
//
// To compensate the error in the argument reduction, we use
// expm1(r+c) = expm1(r) + c + expm1(r)*c
// ~ expm1(r) + c + r*c
// Thus c+r*c will be added in as the correction terms for
// expm1(r+c). Now rearrange the term to avoid optimization
// screw up:
// ( 2 2 )
// ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
// expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
// ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
// ( )
//
// = r - E
// 3. Scale back to obtain expm1(x):
// From step 1, we have
// expm1(x) = either 2**k*[expm1(r)+1] - 1
// = or 2**k*[expm1(r) + (1-2**-k)]
// 4. Implementation notes:
// (A). To save one multiplication, we scale the coefficient Qi
// to Qi*2**i, and replace z by (x**2)/2.
// (B). To achieve maximum accuracy, we compute expm1(x) by
// (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
// (ii) if k=0, return r-E
// (iii) if k=-1, return 0.5*(r-E)-0.5
// (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
// else return 1.0+2.0*(r-E);
// (v) if (k<-2||k>56) return 2**k(1-(E-r)) - 1 (or exp(x)-1)
// (vi) if k <= 20, return 2**k((1-2**-k)-(E-r)), else
// (vii) return 2**k(1-((E+2**-k)-r))
//
// Special cases:
// expm1(INF) is INF, expm1(NaN) is NaN;
// expm1(-INF) is -1, and
// for finite argument, only expm1(0)=0 is exact.
//
// Accuracy:
// according to an error analysis, the error is always less than
// 1 ulp (unit in the last place).
//
// Misc. info.
// For IEEE double
// if x > 7.09782712893383973096e+02 then expm1(x) overflow
//
// 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.
//
// Expm1 returns e**x - 1, the base-e exponential of x minus 1.
// It is more accurate than Exp(x) - 1 when x is near zero.
//
// Special cases are:
// Expm1(+Inf) = +Inf
// Expm1(-Inf) = -1
// Expm1(NaN) = NaN
// Very large values overflow to -1 or +Inf.
func Expm1(x float64) float64
func expm1(x float64) float64 {
const (
Othreshold = 7.09782712893383973096e+02 // 0x40862E42FEFA39EF
Ln2X56 = 3.88162421113569373274e+01 // 0x4043687a9f1af2b1
Ln2HalfX3 = 1.03972077083991796413e+00 // 0x3ff0a2b23f3bab73
Ln2Half = 3.46573590279972654709e-01 // 0x3fd62e42fefa39ef
Ln2Hi = 6.93147180369123816490e-01 // 0x3fe62e42fee00000
Ln2Lo = 1.90821492927058770002e-10 // 0x3dea39ef35793c76
InvLn2 = 1.44269504088896338700e+00 // 0x3ff71547652b82fe
Tiny = 1.0 / (1 << 54) // 2**-54 = 0x3c90000000000000
// scaled coefficients related to expm1
Q1 = -3.33333333333331316428e-02 // 0xBFA11111111110F4
Q2 = 1.58730158725481460165e-03 // 0x3F5A01A019FE5585
Q3 = -7.93650757867487942473e-05 // 0xBF14CE199EAADBB7
Q4 = 4.00821782732936239552e-06 // 0x3ED0CFCA86E65239
Q5 = -2.01099218183624371326e-07 // 0xBE8AFDB76E09C32D
)
// special cases
switch {
case IsInf(x, 1) || IsNaN(x):
return x
case IsInf(x, -1):
return -1
}
absx := x
sign := false
if x < 0 {
absx = -absx
sign = true
}
// filter out huge argument
if absx >= Ln2X56 { // if |x| >= 56 * ln2
if sign {
return -1 // x < -56*ln2, return -1
}
if absx >= Othreshold { // if |x| >= 709.78...
return Inf(1)
}
}
// argument reduction
var c float64
var k int
if absx > Ln2Half { // if |x| > 0.5 * ln2
var hi, lo float64
if absx < Ln2HalfX3 { // and |x| < 1.5 * ln2
if !sign {
hi = x - Ln2Hi
lo = Ln2Lo
k = 1
} else {
hi = x + Ln2Hi
lo = -Ln2Lo
k = -1
}
} else {
if !sign {
k = int(InvLn2*x + 0.5)
} else {
k = int(InvLn2*x - 0.5)
}
t := float64(k)
hi = x - t*Ln2Hi // t * Ln2Hi is exact here
lo = t * Ln2Lo
}
x = hi - lo
c = (hi - x) - lo
} else if absx < Tiny { // when |x| < 2**-54, return x
return x
} else {
k = 0
}
// x is now in primary range
hfx := 0.5 * x
hxs := x * hfx
r1 := 1 + hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))))
t := 3 - r1*hfx
e := hxs * ((r1 - t) / (6.0 - x*t))
if k != 0 {
e = (x*(e-c) - c)
e -= hxs
switch {
case k == -1:
return 0.5*(x-e) - 0.5
case k == 1:
if x < -0.25 {
return -2 * (e - (x + 0.5))
}
return 1 + 2*(x-e)
case k <= -2 || k > 56: // suffice to return exp(x)-1
y := 1 - (e - x)
y = Float64frombits(Float64bits(y) + uint64(k)<<52) // add k to y's exponent
return y - 1
}
if k < 20 {
t := Float64frombits(0x3ff0000000000000 - (0x20000000000000 >> uint(k))) // t=1-2**-k
y := t - (e - x)
y = Float64frombits(Float64bits(y) + uint64(k)<<52) // add k to y's exponent
return y
}
t := Float64frombits(uint64((0x3ff - k) << 52)) // 2**-k
y := x - (e + t)
y += 1
y = Float64frombits(Float64bits(y) + uint64(k)<<52) // add k to y's exponent
return y
}
return x - (x*e - hxs) // c is 0
}