Average Error: 58.9 → 2.8
Time: 29.4s
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 r1936433 = eps;
        double r1936434 = a;
        double r1936435 = b;
        double r1936436 = r1936434 + r1936435;
        double r1936437 = r1936436 * r1936433;
        double r1936438 = exp(r1936437);
        double r1936439 = 1.0;
        double r1936440 = r1936438 - r1936439;
        double r1936441 = r1936433 * r1936440;
        double r1936442 = r1936434 * r1936433;
        double r1936443 = exp(r1936442);
        double r1936444 = r1936443 - r1936439;
        double r1936445 = r1936435 * r1936433;
        double r1936446 = exp(r1936445);
        double r1936447 = r1936446 - r1936439;
        double r1936448 = r1936444 * r1936447;
        double r1936449 = r1936441 / r1936448;
        return r1936449;
}
↓
double f(double a, double b, double eps) {
        double r1936450 = eps;
        double r1936451 = -7.922516355979929e-39;
        bool r1936452 = r1936450 <= r1936451;
        double r1936453 = a;
        double r1936454 = b;
        double r1936455 = r1936453 + r1936454;
        double r1936456 = r1936455 * r1936450;
        double r1936457 = expm1(r1936456);
        double r1936458 = r1936454 * r1936450;
        double r1936459 = expm1(r1936458);
        double r1936460 = r1936457 / r1936459;
        double r1936461 = r1936453 * r1936450;
        double r1936462 = expm1(r1936461);
        double r1936463 = r1936450 / r1936462;
        double r1936464 = r1936460 * r1936463;
        double r1936465 = 1.0;
        double r1936466 = r1936465 / r1936454;
        double r1936467 = r1936465 / r1936453;
        double r1936468 = r1936466 + r1936467;
        double r1936469 = r1936452 ? r1936464 : r1936468;
        return r1936469;
}

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))))