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 // 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 absx >= Othreshold { // if |x| >= 709.78... return Inf(1) // overflow } if sign { return -1 // x < -56*ln2, return -1.0 } } // 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 }