Adding upstream version 1.16.10.upstream/1.16.10upstream

Signed-off-by: Daniel Baumann <daniel.baumann@progress-linux.org>

1 files changed, 237 insertions, 0 deletions

diff --git a/src/math/expm1.go b/src/math/expm1.go
new file mode 100644
index 0000000..8e77398
--- /dev/null
+++ b/src/math/expm1.go
@@ -0,0 +1,237 @@
+// 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 {
+		return x - (x*e - hxs) // c is 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
+}