Average Error: 60.2 → 3.5
Time: 9.8s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{b} + \frac{1}{a}\]
\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}{b} + \frac{1}{a}
double f(double a, double b, double eps) {
        double r89837 = eps;
        double r89838 = a;
        double r89839 = b;
        double r89840 = r89838 + r89839;
        double r89841 = r89840 * r89837;
        double r89842 = exp(r89841);
        double r89843 = 1.0;
        double r89844 = r89842 - r89843;
        double r89845 = r89837 * r89844;
        double r89846 = r89838 * r89837;
        double r89847 = exp(r89846);
        double r89848 = r89847 - r89843;
        double r89849 = r89839 * r89837;
        double r89850 = exp(r89849);
        double r89851 = r89850 - r89843;
        double r89852 = r89848 * r89851;
        double r89853 = r89845 / r89852;
        return r89853;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r89854 = 1.0;
        double r89855 = b;
        double r89856 = r89854 / r89855;
        double r89857 = a;
        double r89858 = r89854 / r89857;
        double r89859 = r89856 + r89858;
        return r89859;
}

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

Original60.2
Target15.2
Herbie3.5
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.2

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

  2. Taylor expanded around 0 3.5

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

  3. Final simplification3.5

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

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :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))))