Average Error: 60.3 → 3.5
Time: 29.8s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{a} + \frac{1}{b}\]
\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 r5517830 = eps;
        double r5517831 = a;
        double r5517832 = b;
        double r5517833 = r5517831 + r5517832;
        double r5517834 = r5517833 * r5517830;
        double r5517835 = exp(r5517834);
        double r5517836 = 1.0;
        double r5517837 = r5517835 - r5517836;
        double r5517838 = r5517830 * r5517837;
        double r5517839 = r5517831 * r5517830;
        double r5517840 = exp(r5517839);
        double r5517841 = r5517840 - r5517836;
        double r5517842 = r5517832 * r5517830;
        double r5517843 = exp(r5517842);
        double r5517844 = r5517843 - r5517836;
        double r5517845 = r5517841 * r5517844;
        double r5517846 = r5517838 / r5517845;
        return r5517846;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r5517847 = 1.0;
        double r5517848 = a;
        double r5517849 = r5517847 / r5517848;
        double r5517850 = b;
        double r5517851 = r5517847 / r5517850;
        double r5517852 = r5517849 + r5517851;
        return r5517852;
}

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

Original60.3
Target15.0
Herbie3.5
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.3

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

  2. Taylor expanded around 0 57.8

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

  3. Simplified57.6

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

  4. Taylor expanded around 0 3.5

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

  5. Final simplification3.5

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

Reproduce

herbie shell --seed 2019174 
(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))))