Average Error: 58.7 → 3.3
Time: 30.1s
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 r4170663 = eps;
        double r4170664 = a;
        double r4170665 = b;
        double r4170666 = r4170664 + r4170665;
        double r4170667 = r4170666 * r4170663;
        double r4170668 = exp(r4170667);
        double r4170669 = 1.0;
        double r4170670 = r4170668 - r4170669;
        double r4170671 = r4170663 * r4170670;
        double r4170672 = r4170664 * r4170663;
        double r4170673 = exp(r4170672);
        double r4170674 = r4170673 - r4170669;
        double r4170675 = r4170665 * r4170663;
        double r4170676 = exp(r4170675);
        double r4170677 = r4170676 - r4170669;
        double r4170678 = r4170674 * r4170677;
        double r4170679 = r4170671 / r4170678;
        return r4170679;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r4170680 = 1.0;
        double r4170681 = a;
        double r4170682 = r4170680 / r4170681;
        double r4170683 = b;
        double r4170684 = r4170680 / r4170683;
        double r4170685 = r4170682 + r4170684;
        return r4170685;
}

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

Original58.7
Target14.5
Herbie3.3
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.7

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

  2. Simplified28.1

    \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}\]

  3. Taylor expanded around 0 3.3

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

  4. Final simplification3.3

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(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))))