Average Error: 60.4 → 4.8
Time: 9.2s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le 5.3360736243582115 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\log \left(e^{e^{a \cdot \varepsilon} - 1}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \end{array}\]
\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 5.3360736243582115 \cdot 10^{-31}:\\
\;\;\;\;\frac{1}{b} + \frac{1}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\log \left(e^{e^{a \cdot \varepsilon} - 1}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\

\end{array}
double f(double a, double b, double eps) {
        double r57739 = eps;
        double r57740 = a;
        double r57741 = b;
        double r57742 = r57740 + r57741;
        double r57743 = r57742 * r57739;
        double r57744 = exp(r57743);
        double r57745 = 1.0;
        double r57746 = r57744 - r57745;
        double r57747 = r57739 * r57746;
        double r57748 = r57740 * r57739;
        double r57749 = exp(r57748);
        double r57750 = r57749 - r57745;
        double r57751 = r57741 * r57739;
        double r57752 = exp(r57751);
        double r57753 = r57752 - r57745;
        double r57754 = r57750 * r57753;
        double r57755 = r57747 / r57754;
        return r57755;
}
↓
double f(double a, double b, double eps) {
        double r57756 = eps;
        double r57757 = 5.3360736243582115e-31;
        bool r57758 = r57756 <= r57757;
        double r57759 = 1.0;
        double r57760 = b;
        double r57761 = r57759 / r57760;
        double r57762 = a;
        double r57763 = r57759 / r57762;
        double r57764 = r57761 + r57763;
        double r57765 = r57762 + r57760;
        double r57766 = r57765 * r57756;
        double r57767 = exp(r57766);
        double r57768 = 1.0;
        double r57769 = r57767 - r57768;
        double r57770 = r57756 * r57769;
        double r57771 = r57762 * r57756;
        double r57772 = exp(r57771);
        double r57773 = r57772 - r57768;
        double r57774 = exp(r57773);
        double r57775 = log(r57774);
        double r57776 = r57760 * r57756;
        double r57777 = exp(r57776);
        double r57778 = r57777 - r57768;
        double r57779 = r57775 * r57778;
        double r57780 = r57770 / r57779;
        double r57781 = r57758 ? r57764 : r57780;
        return r57781;
}

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

Original60.4
Target14.6
Herbie4.8
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < 5.3360736243582115e-31

    1. Initial program 60.8

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

    2. Taylor expanded around 0 2.9

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

    if 5.3360736243582115e-31 < eps

    1. Initial program 50.7

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
    2. Using strategy rm

    3. Applied add-log-exp50.7

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

    4. Applied add-log-exp50.7

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

    5. Applied diff-log50.7

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

    6. Simplified50.7

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\log \color{blue}{\left(e^{e^{a \cdot \varepsilon} - 1}\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le 5.3360736243582115 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\log \left(e^{e^{a \cdot \varepsilon} - 1}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))))