expq3 (problem 3.4.2)

Average Error: 58.4 → 3.5

Time: 35.3s

Precision: 64

• could not determine a ground truth for program body

  1. a = 2.429037555075112e+93
  2. b = 1.6927219136688534e-103
  3. eps = 2.7426708530025896e-55

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

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 r2402621 = eps;
        double r2402622 = a;
        double r2402623 = b;
        double r2402624 = r2402622 + r2402623;
        double r2402625 = r2402624 * r2402621;
        double r2402626 = exp(r2402625);
        double r2402627 = 1.0;
        double r2402628 = r2402626 - r2402627;
        double r2402629 = r2402621 * r2402628;
        double r2402630 = r2402622 * r2402621;
        double r2402631 = exp(r2402630);
        double r2402632 = r2402631 - r2402627;
        double r2402633 = r2402623 * r2402621;
        double r2402634 = exp(r2402633);
        double r2402635 = r2402634 - r2402627;
        double r2402636 = r2402632 * r2402635;
        double r2402637 = r2402629 / r2402636;
        return r2402637;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r2402638 = 1.0;
        double r2402639 = a;
        double r2402640 = r2402638 / r2402639;
        double r2402641 = b;
        double r2402642 = r2402638 / r2402641;
        double r2402643 = r2402640 + r2402642;
        return r2402643;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	58.4
Target	14.4
Herbie	3.5

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

Derivation

1. Initial program 58.4

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

2. Taylor expanded around 0 3.5

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

3. Final simplification 3.5

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

Reproduce

herbie shell --seed 2019142 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :pre (and (< -1 eps) (< eps 1))

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

  (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))))