Average Error: 58.4 → 3.6
Time: 40.0s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{a} + \frac{1}{b}\]
\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}
double f(double a, double b, double eps) {
        double r5539617 = eps;
        double r5539618 = a;
        double r5539619 = b;
        double r5539620 = r5539618 + r5539619;
        double r5539621 = r5539620 * r5539617;
        double r5539622 = exp(r5539621);
        double r5539623 = 1.0;
        double r5539624 = r5539622 - r5539623;
        double r5539625 = r5539617 * r5539624;
        double r5539626 = r5539618 * r5539617;
        double r5539627 = exp(r5539626);
        double r5539628 = r5539627 - r5539623;
        double r5539629 = r5539619 * r5539617;
        double r5539630 = exp(r5539629);
        double r5539631 = r5539630 - r5539623;
        double r5539632 = r5539628 * r5539631;
        double r5539633 = r5539625 / r5539632;
        return r5539633;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r5539634 = 1.0;
        double r5539635 = a;
        double r5539636 = r5539634 / r5539635;
        double r5539637 = b;
        double r5539638 = r5539634 / r5539637;
        double r5539639 = r5539636 + r5539638;
        return r5539639;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.4
Target14.5
Herbie3.6
\[\frac{a + b}{a \cdot b}\]

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. Simplified30.8

    \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot b\right)}{\varepsilon}}}\]

  3. Taylor expanded around 0 3.6

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

  4. Final simplification3.6

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(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))))