expq3 (problem 3.4.2)

Average Error: 60.0 → 0.6

Time: 9.7s

Precision: binary64

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Math

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↓

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C

double code(double a, double b, double eps) {
	return ((double) (((double) (eps * ((double) (((double) exp(((double) (((double) (a + b)) * eps)))) - 1.0)))) / ((double) (((double) (((double) exp(((double) (a * eps)))) - 1.0)) * ((double) (((double) exp(((double) (b * eps)))) - 1.0))))));
}

↓

double code(double a, double b, double eps) {
	double VAR;
	if (((((double) (((double) (eps * ((double) (((double) exp(((double) (eps * ((double) (a + b)))))) - 1.0)))) / ((double) (((double) (((double) exp(((double) (eps * a)))) - 1.0)) * ((double) (((double) exp(((double) (eps * b)))) - 1.0)))))) <= -2.6990817458068544e-16) || !(((double) (((double) (eps * ((double) (((double) exp(((double) (eps * ((double) (a + b)))))) - 1.0)))) / ((double) (((double) (((double) exp(((double) (eps * a)))) - 1.0)) * ((double) (((double) exp(((double) (eps * b)))) - 1.0)))))) <= 6.872128992539374e-18))) {
		VAR = ((double) (((double) (1.0 / b)) + ((double) (1.0 / a))));
	} else {
		VAR = ((double) (((double) (eps * ((double) (((double) exp(((double) (eps * ((double) (a + b)))))) - 1.0)))) / ((double) (((double) (((double) exp(((double) (eps * a)))) - 1.0)) * ((double) (((double) exp(((double) (eps * b)))) - 1.0))))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.0
Target	15.4
Herbie	0.6

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Derivation

1. Split input into 2 regimes

2. if (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))) < -2.6990817458068544e-16 or 6.87212899253937371e-18 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0)))

   1. Initial program 63.6

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   2. Taylor expanded around 0 0.4

      \[\leadsto \]

   if -2.6990817458068544e-16 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))) < 6.87212899253937371e-18

   1. Initial program 4.4

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3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \]

Reproduce

herbie shell --seed 2020190 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :pre (and (< -1.0 eps) (< eps 1.0))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))