expq3 (problem 3.4.2)

Average Error: 60.2 → 47.5

Time: 1.0min

Precision: binary64

• could not determine a ground truth for program body

  1. a = -2.2047948243548186e+20
  2. b = -2.1684232456167406e+293
  3. eps = -1.221153745695414e-54

\[\]

Math

\[\]

↓

\[\]

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)))))) <= -inf.0) || !(((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)))))) <= 1.0980179087407446e-16))) {
		VAR = ((double) (((double) (1.0 / ((double) (a + ((double) (((double) log(1.0)) + ((double) (eps * ((double) (0.5 * ((double) pow(((double) log(1.0)), 2.0)))))))))))) * ((double) (((double) (((double) pow(((double) exp(((double) (a + b)))), eps)) - 1.0)) / ((double) (((double) pow(((double) exp(b)), eps)) - 1.0))))));
	} 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	60.2
Target	14.5
Herbie	47.5

\[\]

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))) < -inf.0 or 1.09801790874074464e-16 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0)))

   1. Initial program 63.9

      \[\]

   2. Simplified 63.6

      \[\leadsto \]

   3. Taylor expanded around 0 55.3

      \[\leadsto \]

   4. Simplified 55.3

      \[\leadsto \]
   5. Using strategy rm

   6. Applied *-un-lft-identity 55.3

      \[\leadsto \]

   7. Applied times-frac 55.3

      \[\leadsto \]

   8. Applied associate-*r* 55.3

      \[\leadsto \]

   9. Simplified 50.4

      \[\leadsto \]

   if -inf.0 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))) < 1.09801790874074464e-16

   1. Initial program 3.0

      \[\]
3. Recombined 2 regimes into one program.

4. Final simplification 47.5

   \[\leadsto \]

Reproduce

herbie shell --seed 2020192 
(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))))