// Copyright 2010 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package math
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// The original C code, the long comment, and the constants
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// below are from FreeBSD's /usr/src/lib/msun/src/s_expm1.c
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// and came with this notice. The go code is a simplified
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// version of the original C.
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//
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// ====================================================
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// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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//
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// Developed at SunPro, a Sun Microsystems, Inc. business.
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// Permission to use, copy, modify, and distribute this
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// software is freely granted, provided that this notice
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// is preserved.
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// ====================================================
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//
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// expm1(x)
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// Returns exp(x)-1, the exponential of x minus 1.
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//
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// Method
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// 1. Argument reduction:
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// Given x, find r and integer k such that
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//
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// x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
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//
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// Here a correction term c will be computed to compensate
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// the error in r when rounded to a floating-point number.
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//
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// 2. Approximating expm1(r) by a special rational function on
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// the interval [0,0.34658]:
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// Since
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// r*(exp(r)+1)/(exp(r)-1) = 2+ r**2/6 - r**4/360 + ...
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// we define R1(r*r) by
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// r*(exp(r)+1)/(exp(r)-1) = 2+ r**2/6 * R1(r*r)
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// That is,
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// R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
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// = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
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// = 1 - r**2/60 + r**4/2520 - r**6/100800 + ...
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// We use a special Reme algorithm on [0,0.347] to generate
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// a polynomial of degree 5 in r*r to approximate R1. The
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// maximum error of this polynomial approximation is bounded
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// by 2**-61. In other words,
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// R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
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// where Q1 = -1.6666666666666567384E-2,
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// Q2 = 3.9682539681370365873E-4,
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// Q3 = -9.9206344733435987357E-6,
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// Q4 = 2.5051361420808517002E-7,
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// Q5 = -6.2843505682382617102E-9;
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// (where z=r*r, and the values of Q1 to Q5 are listed below)
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// with error bounded by
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// | 5 | -61
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// | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
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// | |
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//
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// expm1(r) = exp(r)-1 is then computed by the following
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// specific way which minimize the accumulation rounding error:
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// 2 3
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// r r [ 3 - (R1 + R1*r/2) ]
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// expm1(r) = r + --- + --- * [--------------------]
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// 2 2 [ 6 - r*(3 - R1*r/2) ]
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//
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// To compensate the error in the argument reduction, we use
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// expm1(r+c) = expm1(r) + c + expm1(r)*c
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// ~ expm1(r) + c + r*c
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// Thus c+r*c will be added in as the correction terms for
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// expm1(r+c). Now rearrange the term to avoid optimization
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// screw up:
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// ( 2 2 )
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// ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
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// expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
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// ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
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// ( )
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//
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// = r - E
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// 3. Scale back to obtain expm1(x):
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// From step 1, we have
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// expm1(x) = either 2**k*[expm1(r)+1] - 1
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// = or 2**k*[expm1(r) + (1-2**-k)]
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// 4. Implementation notes:
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// (A). To save one multiplication, we scale the coefficient Qi
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// to Qi*2**i, and replace z by (x**2)/2.
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// (B). To achieve maximum accuracy, we compute expm1(x) by
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// (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
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// (ii) if k=0, return r-E
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// (iii) if k=-1, return 0.5*(r-E)-0.5
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// (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
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// else return 1.0+2.0*(r-E);
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// (v) if (k<-2||k>56) return 2**k(1-(E-r)) - 1 (or exp(x)-1)
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// (vi) if k <= 20, return 2**k((1-2**-k)-(E-r)), else
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// (vii) return 2**k(1-((E+2**-k)-r))
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//
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// Special cases:
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// expm1(INF) is INF, expm1(NaN) is NaN;
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// expm1(-INF) is -1, and
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// for finite argument, only expm1(0)=0 is exact.
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//
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// Accuracy:
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// according to an error analysis, the error is always less than
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// 1 ulp (unit in the last place).
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//
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// Misc. info.
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// For IEEE double
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// if x > 7.09782712893383973096e+02 then expm1(x) overflow
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//
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// Constants:
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// The hexadecimal values are the intended ones for the following
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// constants. The decimal values may be used, provided that the
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// compiler will convert from decimal to binary accurately enough
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// to produce the hexadecimal values shown.
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//
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// Expm1 returns e**x - 1, the base-e exponential of x minus 1.
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// It is more accurate than Exp(x) - 1 when x is near zero.
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//
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// Special cases are:
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// Expm1(+Inf) = +Inf
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// Expm1(-Inf) = -1
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// Expm1(NaN) = NaN
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// Very large values overflow to -1 or +Inf.
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func Expm1(x float64) float64
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func expm1(x float64) float64 {
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const (
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Othreshold = 7.09782712893383973096e+02 // 0x40862E42FEFA39EF
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Ln2X56 = 3.88162421113569373274e+01 // 0x4043687a9f1af2b1
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Ln2HalfX3 = 1.03972077083991796413e+00 // 0x3ff0a2b23f3bab73
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Ln2Half = 3.46573590279972654709e-01 // 0x3fd62e42fefa39ef
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Ln2Hi = 6.93147180369123816490e-01 // 0x3fe62e42fee00000
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Ln2Lo = 1.90821492927058770002e-10 // 0x3dea39ef35793c76
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InvLn2 = 1.44269504088896338700e+00 // 0x3ff71547652b82fe
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Tiny = 1.0 / (1 << 54) // 2**-54 = 0x3c90000000000000
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// scaled coefficients related to expm1
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Q1 = -3.33333333333331316428e-02 // 0xBFA11111111110F4
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Q2 = 1.58730158725481460165e-03 // 0x3F5A01A019FE5585
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Q3 = -7.93650757867487942473e-05 // 0xBF14CE199EAADBB7
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Q4 = 4.00821782732936239552e-06 // 0x3ED0CFCA86E65239
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Q5 = -2.01099218183624371326e-07 // 0xBE8AFDB76E09C32D
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)
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// special cases
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switch {
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case IsInf(x, 1) || IsNaN(x):
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return x
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case IsInf(x, -1):
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return -1
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}
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absx := x
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sign := false
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if x < 0 {
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absx = -absx
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sign = true
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}
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// filter out huge argument
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if absx >= Ln2X56 { // if |x| >= 56 * ln2
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if sign {
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return -1 // x < -56*ln2, return -1
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}
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if absx >= Othreshold { // if |x| >= 709.78...
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return Inf(1)
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}
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}
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// argument reduction
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var c float64
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var k int
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if absx > Ln2Half { // if |x| > 0.5 * ln2
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var hi, lo float64
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if absx < Ln2HalfX3 { // and |x| < 1.5 * ln2
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if !sign {
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hi = x - Ln2Hi
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lo = Ln2Lo
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k = 1
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} else {
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hi = x + Ln2Hi
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lo = -Ln2Lo
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k = -1
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}
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} else {
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if !sign {
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k = int(InvLn2*x + 0.5)
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} else {
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k = int(InvLn2*x - 0.5)
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}
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t := float64(k)
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hi = x - t*Ln2Hi // t * Ln2Hi is exact here
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lo = t * Ln2Lo
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}
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x = hi - lo
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c = (hi - x) - lo
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} else if absx < Tiny { // when |x| < 2**-54, return x
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return x
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} else {
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k = 0
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}
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// x is now in primary range
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hfx := 0.5 * x
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hxs := x * hfx
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r1 := 1 + hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))))
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t := 3 - r1*hfx
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e := hxs * ((r1 - t) / (6.0 - x*t))
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if k == 0 {
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return x - (x*e - hxs) // c is 0
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}
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e = (x*(e-c) - c)
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e -= hxs
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switch {
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case k == -1:
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return 0.5*(x-e) - 0.5
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case k == 1:
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if x < -0.25 {
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return -2 * (e - (x + 0.5))
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}
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return 1 + 2*(x-e)
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case k <= -2 || k > 56: // suffice to return exp(x)-1
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y := 1 - (e - x)
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y = Float64frombits(Float64bits(y) + uint64(k)<<52) // add k to y's exponent
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return y - 1
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}
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if k < 20 {
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t := Float64frombits(0x3ff0000000000000 - (0x20000000000000 >> uint(k))) // t=1-2**-k
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y := t - (e - x)
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y = Float64frombits(Float64bits(y) + uint64(k)<<52) // add k to y's exponent
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return y
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}
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t = Float64frombits(uint64(0x3ff-k) << 52) // 2**-k
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y := x - (e + t)
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y++
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y = Float64frombits(Float64bits(y) + uint64(k)<<52) // add k to y's exponent
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return y
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}
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