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Source file src/math/exp.go

Documentation: math

  // Copyright 2009 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
  
  // Exp returns e**x, the base-e exponential of x.
  //
  // Special cases are:
  //	Exp(+Inf) = +Inf
  //	Exp(NaN) = NaN
  // Very large values overflow to 0 or +Inf.
  // Very small values underflow to 1.
  func Exp(x float64) float64
  
  // The original C code, the long comment, and the constants
  // below are from FreeBSD's /usr/src/lib/msun/src/e_exp.c
  // and came with this notice. The go code is a simplified
  // version of the original C.
  //
  // ====================================================
  // Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
  //
  // Permission to use, copy, modify, and distribute this
  // software is freely granted, provided that this notice
  // is preserved.
  // ====================================================
  //
  //
  // exp(x)
  // Returns the exponential of x.
  //
  // Method
  //   1. Argument reduction:
  //      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
  //      Given x, find r and integer k such that
  //
  //               x = k*ln2 + r,  |r| <= 0.5*ln2.
  //
  //      Here r will be represented as r = hi-lo for better
  //      accuracy.
  //
  //   2. Approximation of exp(r) by a special rational function on
  //      the interval [0,0.34658]:
  //      Write
  //          R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
  //      We use a special Remes algorithm on [0,0.34658] to generate
  //      a polynomial of degree 5 to approximate R. The maximum error
  //      of this polynomial approximation is bounded by 2**-59. In
  //      other words,
  //          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
  //      (where z=r*r, and the values of P1 to P5 are listed below)
  //      and
  //          |                  5          |     -59
  //          | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
  //          |                             |
  //      The computation of exp(r) thus becomes
  //                             2*r
  //              exp(r) = 1 + -------
  //                            R - r
  //                                 r*R1(r)
  //                     = 1 + r + ----------- (for better accuracy)
  //                                2 - R1(r)
  //      where
  //                               2       4             10
  //              R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
  //
  //   3. Scale back to obtain exp(x):
  //      From step 1, we have
  //         exp(x) = 2**k * exp(r)
  //
  // Special cases:
  //      exp(INF) is INF, exp(NaN) is NaN;
  //      exp(-INF) is 0, and
  //      for finite argument, only exp(0)=1 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 exp(x) overflow
  //          if x < -7.45133219101941108420e+02 then exp(x) underflow
  //
  // 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.
  
  func exp(x float64) float64 {
  	const (
  		Ln2Hi = 6.93147180369123816490e-01
  		Ln2Lo = 1.90821492927058770002e-10
  		Log2e = 1.44269504088896338700e+00
  
  		Overflow  = 7.09782712893383973096e+02
  		Underflow = -7.45133219101941108420e+02
  		NearZero  = 1.0 / (1 << 28) // 2**-28
  	)
  
  	// special cases
  	switch {
  	case IsNaN(x) || IsInf(x, 1):
  		return x
  	case IsInf(x, -1):
  		return 0
  	case x > Overflow:
  		return Inf(1)
  	case x < Underflow:
  		return 0
  	case -NearZero < x && x < NearZero:
  		return 1 + x
  	}
  
  	// reduce; computed as r = hi - lo for extra precision.
  	var k int
  	switch {
  	case x < 0:
  		k = int(Log2e*x - 0.5)
  	case x > 0:
  		k = int(Log2e*x + 0.5)
  	}
  	hi := x - float64(k)*Ln2Hi
  	lo := float64(k) * Ln2Lo
  
  	// compute
  	return expmulti(hi, lo, k)
  }
  
  // Exp2 returns 2**x, the base-2 exponential of x.
  //
  // Special cases are the same as Exp.
  func Exp2(x float64) float64
  
  func exp2(x float64) float64 {
  	const (
  		Ln2Hi = 6.93147180369123816490e-01
  		Ln2Lo = 1.90821492927058770002e-10
  
  		Overflow  = 1.0239999999999999e+03
  		Underflow = -1.0740e+03
  	)
  
  	// special cases
  	switch {
  	case IsNaN(x) || IsInf(x, 1):
  		return x
  	case IsInf(x, -1):
  		return 0
  	case x > Overflow:
  		return Inf(1)
  	case x < Underflow:
  		return 0
  	}
  
  	// argument reduction; x = r×lg(e) + k with |r| ≤ ln(2)/2.
  	// computed as r = hi - lo for extra precision.
  	var k int
  	switch {
  	case x > 0:
  		k = int(x + 0.5)
  	case x < 0:
  		k = int(x - 0.5)
  	}
  	t := x - float64(k)
  	hi := t * Ln2Hi
  	lo := -t * Ln2Lo
  
  	// compute
  	return expmulti(hi, lo, k)
  }
  
  // exp1 returns e**r × 2**k where r = hi - lo and |r| ≤ ln(2)/2.
  func expmulti(hi, lo float64, k int) float64 {
  	const (
  		P1 = 1.66666666666666019037e-01  /* 0x3FC55555; 0x5555553E */
  		P2 = -2.77777777770155933842e-03 /* 0xBF66C16C; 0x16BEBD93 */
  		P3 = 6.61375632143793436117e-05  /* 0x3F11566A; 0xAF25DE2C */
  		P4 = -1.65339022054652515390e-06 /* 0xBEBBBD41; 0xC5D26BF1 */
  		P5 = 4.13813679705723846039e-08  /* 0x3E663769; 0x72BEA4D0 */
  	)
  
  	r := hi - lo
  	t := r * r
  	c := r - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))))
  	y := 1 - ((lo - (r*c)/(2-c)) - hi)
  	// TODO(rsc): make sure Ldexp can handle boundary k
  	return Ldexp(y, k)
  }
  

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