Average Error: 59.1 → 2.9
Time: 38.2s
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 r5105737 = eps;
        double r5105738 = a;
        double r5105739 = b;
        double r5105740 = r5105738 + r5105739;
        double r5105741 = r5105740 * r5105737;
        double r5105742 = exp(r5105741);
        double r5105743 = 1.0;
        double r5105744 = r5105742 - r5105743;
        double r5105745 = r5105737 * r5105744;
        double r5105746 = r5105738 * r5105737;
        double r5105747 = exp(r5105746);
        double r5105748 = r5105747 - r5105743;
        double r5105749 = r5105739 * r5105737;
        double r5105750 = exp(r5105749);
        double r5105751 = r5105750 - r5105743;
        double r5105752 = r5105748 * r5105751;
        double r5105753 = r5105745 / r5105752;
        return r5105753;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r5105754 = 1.0;
        double r5105755 = a;
        double r5105756 = r5105754 / r5105755;
        double r5105757 = b;
        double r5105758 = r5105754 / r5105757;
        double r5105759 = r5105756 + r5105758;
        return r5105759;
}

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

Original59.1
Target13.8
Herbie2.9
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 59.1

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

  2. Taylor expanded around 0 56.1

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

  3. Simplified55.1

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

  4. Taylor expanded around 0 2.9

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

  5. Final simplification2.9

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

Reproduce

herbie shell --seed 2019163 
(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))))