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| -rw-r--r-- | src/math/expm1.go | 237 |
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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 +} |