Average Error: 60.1 → 3.6
Time: 31.2s
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 r5702683 = eps;
        double r5702684 = a;
        double r5702685 = b;
        double r5702686 = r5702684 + r5702685;
        double r5702687 = r5702686 * r5702683;
        double r5702688 = exp(r5702687);
        double r5702689 = 1.0;
        double r5702690 = r5702688 - r5702689;
        double r5702691 = r5702683 * r5702690;
        double r5702692 = r5702684 * r5702683;
        double r5702693 = exp(r5702692);
        double r5702694 = r5702693 - r5702689;
        double r5702695 = r5702685 * r5702683;
        double r5702696 = exp(r5702695);
        double r5702697 = r5702696 - r5702689;
        double r5702698 = r5702694 * r5702697;
        double r5702699 = r5702691 / r5702698;
        return r5702699;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r5702700 = 1.0;
        double r5702701 = a;
        double r5702702 = r5702700 / r5702701;
        double r5702703 = b;
        double r5702704 = r5702700 / r5702703;
        double r5702705 = r5702702 + r5702704;
        return r5702705;
}

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

Original60.1
Target15.2
Herbie3.6
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.1

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

  2. Taylor expanded around 0 57.9

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + \left(\frac{1}{6} \cdot \left({a}^{3} \cdot {\varepsilon}^{3}\right) + a \cdot \varepsilon\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

  3. Simplified57.8

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right)\right) + \left(\varepsilon \cdot a + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

  4. Taylor expanded around 0 3.6

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

  5. Final simplification3.6

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

Reproduce

herbie shell --seed 2019171 
(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))))