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Source file src/math/expm1.go

  // 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++
  		y = Float64frombits(Float64bits(y) + uint64(k)<<52) // add k to y's exponent
  		return y
  	}
  	return x - (x*e - hxs) // c is 0
  }
  

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