Average Error: 58.7 → 2.8
Time: 50.7s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le 1.4538299418999213 \cdot 10^{-51}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{expm1}\left(\varepsilon \cdot a\right)}{\frac{\varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right)} \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}}\\ \end{array}\]
\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 1.4538299418999213 \cdot 10^{-51}:\\
\;\;\;\;\frac{1}{b} + \frac{1}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{\mathsf{expm1}\left(\varepsilon \cdot a\right)}{\frac{\varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right)} \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}}\\

\end{array}
double f(double a, double b, double eps) {
        double r4479047 = eps;
        double r4479048 = a;
        double r4479049 = b;
        double r4479050 = r4479048 + r4479049;
        double r4479051 = r4479050 * r4479047;
        double r4479052 = exp(r4479051);
        double r4479053 = 1.0;
        double r4479054 = r4479052 - r4479053;
        double r4479055 = r4479047 * r4479054;
        double r4479056 = r4479048 * r4479047;
        double r4479057 = exp(r4479056);
        double r4479058 = r4479057 - r4479053;
        double r4479059 = r4479049 * r4479047;
        double r4479060 = exp(r4479059);
        double r4479061 = r4479060 - r4479053;
        double r4479062 = r4479058 * r4479061;
        double r4479063 = r4479055 / r4479062;
        return r4479063;
}
↓
double f(double a, double b, double eps) {
        double r4479064 = eps;
        double r4479065 = 1.4538299418999213e-51;
        bool r4479066 = r4479064 <= r4479065;
        double r4479067 = 1.0;
        double r4479068 = b;
        double r4479069 = r4479067 / r4479068;
        double r4479070 = a;
        double r4479071 = r4479067 / r4479070;
        double r4479072 = r4479069 + r4479071;
        double r4479073 = r4479064 * r4479070;
        double r4479074 = expm1(r4479073);
        double r4479075 = r4479068 * r4479064;
        double r4479076 = expm1(r4479075);
        double r4479077 = r4479064 / r4479076;
        double r4479078 = r4479070 + r4479068;
        double r4479079 = r4479064 * r4479078;
        double r4479080 = expm1(r4479079);
        double r4479081 = r4479077 * r4479080;
        double r4479082 = r4479074 / r4479081;
        double r4479083 = r4479067 / r4479082;
        double r4479084 = r4479066 ? r4479072 : r4479083;
        return r4479084;
}

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
Target14.7
Herbie2.8
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < 1.4538299418999213e-51

    1. Initial program 59.2

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

    2. Simplified29.3

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

    3. Taylor expanded around 0 2.8

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

    if 1.4538299418999213e-51 < eps

    1. Initial program 51.3

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

    2. Simplified3.0

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

    4. Applied clear-num3.0

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(\varepsilon \cdot a\right)}{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \cdot \mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le 1.4538299418999213 \cdot 10^{-51}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{expm1}\left(\varepsilon \cdot a\right)}{\frac{\varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right)} \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 +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))))