Average Error: 60.5 → 3.2
Time: 29.4s
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 r69822 = eps;
        double r69823 = a;
        double r69824 = b;
        double r69825 = r69823 + r69824;
        double r69826 = r69825 * r69822;
        double r69827 = exp(r69826);
        double r69828 = 1.0;
        double r69829 = r69827 - r69828;
        double r69830 = r69822 * r69829;
        double r69831 = r69823 * r69822;
        double r69832 = exp(r69831);
        double r69833 = r69832 - r69828;
        double r69834 = r69824 * r69822;
        double r69835 = exp(r69834);
        double r69836 = r69835 - r69828;
        double r69837 = r69833 * r69836;
        double r69838 = r69830 / r69837;
        return r69838;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r69839 = 1.0;
        double r69840 = b;
        double r69841 = r69839 / r69840;
        double r69842 = a;
        double r69843 = r69839 / r69842;
        double r69844 = r69841 + r69843;
        return r69844;
}

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

Original60.5
Target14.8
Herbie3.2
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.5

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

  2. Taylor expanded around 0 3.2

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

  3. Final simplification3.2

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

Reproduce

herbie shell --seed 2019294 
(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))))