expq3 (problem 3.4.2)

Average Error: 60.3 → 3.4

Time: 37.8s

Precision: 64

• could not determine a ground truth for program body

  1. a = 1.4639452118847904e+51
  2. b = -1.805670951386701e+285
  3. eps = -1.4405055651571276e-214

\[-1 \lt \varepsilon \land \varepsilon \lt 1\]

Math

\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

↓

\[\frac{1}{a} + \frac{1}{b}\]

C

double f(double a, double b, double eps) {
        double r5051692 = eps;
        double r5051693 = a;
        double r5051694 = b;
        double r5051695 = r5051693 + r5051694;
        double r5051696 = r5051695 * r5051692;
        double r5051697 = exp(r5051696);
        double r5051698 = 1.0;
        double r5051699 = r5051697 - r5051698;
        double r5051700 = r5051692 * r5051699;
        double r5051701 = r5051693 * r5051692;
        double r5051702 = exp(r5051701);
        double r5051703 = r5051702 - r5051698;
        double r5051704 = r5051694 * r5051692;
        double r5051705 = exp(r5051704);
        double r5051706 = r5051705 - r5051698;
        double r5051707 = r5051703 * r5051706;
        double r5051708 = r5051700 / r5051707;
        return r5051708;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r5051709 = 1.0;
        double r5051710 = a;
        double r5051711 = r5051709 / r5051710;
        double r5051712 = b;
        double r5051713 = r5051709 / r5051712;
        double r5051714 = r5051711 + r5051713;
        return r5051714;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.3
Target	14.8
Herbie	3.4

\[\frac{a + b}{a \cdot b}\]

Derivation

1. Initial program 60.3

   \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

2. Taylor expanded around 0 3.4

   \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]

3. Final simplification 3.4

   \[\leadsto \frac{1}{a} + \frac{1}{b}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :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))))