Average Error: 58.9 → 2.8
Time: 29.3s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.922516355979929 \cdot 10^{-39}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(b \cdot \varepsilon\right)} \cdot \frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]
\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)}
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -7.922516355979929 \cdot 10^{-39}:\\
\;\;\;\;\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(b \cdot \varepsilon\right)} \cdot \frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{b} + \frac{1}{a}\\

\end{array}
double f(double a, double b, double eps) {
        double r1666797 = eps;
        double r1666798 = a;
        double r1666799 = b;
        double r1666800 = r1666798 + r1666799;
        double r1666801 = r1666800 * r1666797;
        double r1666802 = exp(r1666801);
        double r1666803 = 1.0;
        double r1666804 = r1666802 - r1666803;
        double r1666805 = r1666797 * r1666804;
        double r1666806 = r1666798 * r1666797;
        double r1666807 = exp(r1666806);
        double r1666808 = r1666807 - r1666803;
        double r1666809 = r1666799 * r1666797;
        double r1666810 = exp(r1666809);
        double r1666811 = r1666810 - r1666803;
        double r1666812 = r1666808 * r1666811;
        double r1666813 = r1666805 / r1666812;
        return r1666813;
}
↓
double f(double a, double b, double eps) {
        double r1666814 = eps;
        double r1666815 = -7.922516355979929e-39;
        bool r1666816 = r1666814 <= r1666815;
        double r1666817 = a;
        double r1666818 = b;
        double r1666819 = r1666817 + r1666818;
        double r1666820 = r1666819 * r1666814;
        double r1666821 = expm1(r1666820);
        double r1666822 = r1666818 * r1666814;
        double r1666823 = expm1(r1666822);
        double r1666824 = r1666821 / r1666823;
        double r1666825 = r1666817 * r1666814;
        double r1666826 = expm1(r1666825);
        double r1666827 = r1666814 / r1666826;
        double r1666828 = r1666824 * r1666827;
        double r1666829 = 1.0;
        double r1666830 = r1666829 / r1666818;
        double r1666831 = r1666829 / r1666817;
        double r1666832 = r1666830 + r1666831;
        double r1666833 = r1666816 ? r1666828 : r1666832;
        return r1666833;
}

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

Original58.9
Target14.2
Herbie2.8
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -7.922516355979929e-39

    1. Initial program 51.8

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

    2. Simplified15.9

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

    4. Applied times-frac3.1

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

    if -7.922516355979929e-39 < eps

    1. Initial program 59.3

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

    2. Simplified41.1

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

    3. Taylor expanded around 0 2.8

      \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.922516355979929 \cdot 10^{-39}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(b \cdot \varepsilon\right)} \cdot \frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +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))))