Average Error: 58.5 → 3.5
Time: 45.6s
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 r3159723 = eps;
        double r3159724 = a;
        double r3159725 = b;
        double r3159726 = r3159724 + r3159725;
        double r3159727 = r3159726 * r3159723;
        double r3159728 = exp(r3159727);
        double r3159729 = 1.0;
        double r3159730 = r3159728 - r3159729;
        double r3159731 = r3159723 * r3159730;
        double r3159732 = r3159724 * r3159723;
        double r3159733 = exp(r3159732);
        double r3159734 = r3159733 - r3159729;
        double r3159735 = r3159725 * r3159723;
        double r3159736 = exp(r3159735);
        double r3159737 = r3159736 - r3159729;
        double r3159738 = r3159734 * r3159737;
        double r3159739 = r3159731 / r3159738;
        return r3159739;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r3159740 = 1.0;
        double r3159741 = b;
        double r3159742 = r3159740 / r3159741;
        double r3159743 = a;
        double r3159744 = r3159740 / r3159743;
        double r3159745 = r3159742 + r3159744;
        return r3159745;
}

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

Original58.5
Target14.4
Herbie3.5
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.5

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

  2. Simplified28.0

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

  3. Taylor expanded around 0 3.5

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

  4. Final simplification3.5

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

Reproduce

herbie shell --seed 2019146 +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))))