Average Error: 58.5 → 3.0
Time: 33.8s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.609900552189287 \cdot 10^{-48}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\sqrt[3]{\varepsilon} \cdot \left(\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right) \cdot \left(a + b\right)\right)\right)}{\mathsf{expm1}\left(b \cdot \varepsilon\right)} \cdot \frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]
\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 -9.609900552189287 \cdot 10^{-48}:\\
\;\;\;\;\frac{\mathsf{expm1}\left(\sqrt[3]{\varepsilon} \cdot \left(\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right) \cdot \left(a + b\right)\right)\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 r4326448 = eps;
        double r4326449 = a;
        double r4326450 = b;
        double r4326451 = r4326449 + r4326450;
        double r4326452 = r4326451 * r4326448;
        double r4326453 = exp(r4326452);
        double r4326454 = 1.0;
        double r4326455 = r4326453 - r4326454;
        double r4326456 = r4326448 * r4326455;
        double r4326457 = r4326449 * r4326448;
        double r4326458 = exp(r4326457);
        double r4326459 = r4326458 - r4326454;
        double r4326460 = r4326450 * r4326448;
        double r4326461 = exp(r4326460);
        double r4326462 = r4326461 - r4326454;
        double r4326463 = r4326459 * r4326462;
        double r4326464 = r4326456 / r4326463;
        return r4326464;
}
↓
double f(double a, double b, double eps) {
        double r4326465 = eps;
        double r4326466 = -9.609900552189287e-48;
        bool r4326467 = r4326465 <= r4326466;
        double r4326468 = cbrt(r4326465);
        double r4326469 = r4326468 * r4326468;
        double r4326470 = a;
        double r4326471 = b;
        double r4326472 = r4326470 + r4326471;
        double r4326473 = r4326469 * r4326472;
        double r4326474 = r4326468 * r4326473;
        double r4326475 = expm1(r4326474);
        double r4326476 = r4326471 * r4326465;
        double r4326477 = expm1(r4326476);
        double r4326478 = r4326475 / r4326477;
        double r4326479 = r4326470 * r4326465;
        double r4326480 = expm1(r4326479);
        double r4326481 = r4326465 / r4326480;
        double r4326482 = r4326478 * r4326481;
        double r4326483 = 1.0;
        double r4326484 = r4326483 / r4326471;
        double r4326485 = r4326483 / r4326470;
        double r4326486 = r4326484 + r4326485;
        double r4326487 = r4326467 ? r4326482 : r4326486;
        return r4326487;
}

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
Target14.2
Herbie3.0
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -9.609900552189287e-48

    1. Initial program 50.3

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

    2. Simplified3.5

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

    4. Applied add-cube-cbrt3.9

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

    5. Applied associate-*r*3.9

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

    if -9.609900552189287e-48 < eps

    1. Initial program 59.1

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

    2. Simplified29.3

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

    3. Taylor expanded around 0 2.9

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

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.609900552189287 \cdot 10^{-48}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\sqrt[3]{\varepsilon} \cdot \left(\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right) \cdot \left(a + b\right)\right)\right)}{\mathsf{expm1}\left(b \cdot \varepsilon\right)} \cdot \frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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))))