Average Error: 58.5 → 4.0
Time: 41.6s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.923527264493171 \cdot 10^{+136}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]
\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}\;b \le -4.923527264493171 \cdot 10^{+136}:\\
\;\;\;\;\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\

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

\end{array}
double f(double a, double b, double eps) {
        double r3548235 = eps;
        double r3548236 = a;
        double r3548237 = b;
        double r3548238 = r3548236 + r3548237;
        double r3548239 = r3548238 * r3548235;
        double r3548240 = exp(r3548239);
        double r3548241 = 1.0;
        double r3548242 = r3548240 - r3548241;
        double r3548243 = r3548235 * r3548242;
        double r3548244 = r3548236 * r3548235;
        double r3548245 = exp(r3548244);
        double r3548246 = r3548245 - r3548241;
        double r3548247 = r3548237 * r3548235;
        double r3548248 = exp(r3548247);
        double r3548249 = r3548248 - r3548241;
        double r3548250 = r3548246 * r3548249;
        double r3548251 = r3548243 / r3548250;
        return r3548251;
}
↓
double f(double a, double b, double eps) {
        double r3548252 = b;
        double r3548253 = -4.923527264493171e+136;
        bool r3548254 = r3548252 <= r3548253;
        double r3548255 = a;
        double r3548256 = r3548255 + r3548252;
        double r3548257 = eps;
        double r3548258 = r3548256 * r3548257;
        double r3548259 = expm1(r3548258);
        double r3548260 = r3548255 * r3548257;
        double r3548261 = expm1(r3548260);
        double r3548262 = r3548257 / r3548261;
        double r3548263 = r3548259 * r3548262;
        double r3548264 = r3548257 * r3548252;
        double r3548265 = expm1(r3548264);
        double r3548266 = r3548263 / r3548265;
        double r3548267 = 1.0;
        double r3548268 = r3548267 / r3548252;
        double r3548269 = r3548267 / r3548255;
        double r3548270 = r3548268 + r3548269;
        double r3548271 = r3548254 ? r3548266 : r3548270;
        return r3548271;
}

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
Target13.9
Herbie4.0
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if b < -4.923527264493171e+136

    1. Initial program 49.6

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

    2. Simplified17.2

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

    4. Applied associate-*l/17.2

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

    if -4.923527264493171e+136 < b

    1. Initial program 59.6

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

    2. Simplified29.0

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

    3. Taylor expanded around 0 2.4

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

  4. Final simplification4.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.923527264493171 \cdot 10^{+136}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +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))))