Average Error: 6.0 → 1.5
Time: 21.8s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le 5.320319158246681890891762065017087849432 \cdot 10^{71}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}{y}\\ \mathbf{elif}\;z \le 3.872099534014884485438076645676398898122 \cdot 10^{123}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;z \le 5.320319158246681890891762065017087849432 \cdot 10^{71}:\\
\;\;\;\;x + \frac{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}{y}\\

\mathbf{elif}\;z \le 3.872099534014884485438076645676398898122 \cdot 10^{123}:\\
\;\;\;\;x + \frac{e^{-z}}{y}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}{y}\\

\end{array}
double f(double x, double y, double z) {
        double r49870599 = x;
        double r49870600 = y;
        double r49870601 = z;
        double r49870602 = r49870601 + r49870600;
        double r49870603 = r49870600 / r49870602;
        double r49870604 = log(r49870603);
        double r49870605 = r49870600 * r49870604;
        double r49870606 = exp(r49870605);
        double r49870607 = r49870606 / r49870600;
        double r49870608 = r49870599 + r49870607;
        return r49870608;
}
↓
double f(double x, double y, double z) {
        double r49870609 = z;
        double r49870610 = 5.320319158246682e+71;
        bool r49870611 = r49870609 <= r49870610;
        double r49870612 = x;
        double r49870613 = y;
        double r49870614 = exp(r49870613);
        double r49870615 = r49870609 + r49870613;
        double r49870616 = r49870613 / r49870615;
        double r49870617 = log(r49870616);
        double r49870618 = pow(r49870614, r49870617);
        double r49870619 = r49870618 / r49870613;
        double r49870620 = r49870612 + r49870619;
        double r49870621 = 3.8720995340148845e+123;
        bool r49870622 = r49870609 <= r49870621;
        double r49870623 = -r49870609;
        double r49870624 = exp(r49870623);
        double r49870625 = r49870624 / r49870613;
        double r49870626 = r49870612 + r49870625;
        double r49870627 = r49870622 ? r49870626 : r49870620;
        double r49870628 = r49870611 ? r49870620 : r49870627;
        return r49870628;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.0
Target1.0
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.115415759790762719541517221498726780517 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < 5.320319158246682e+71 or 3.8720995340148845e+123 < z

    1. Initial program 5.8

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
    2. Using strategy rm

    3. Applied add-log-exp34.5

      \[\leadsto x + \frac{e^{\color{blue}{\log \left(e^{y}\right)} \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]

    4. Applied exp-to-pow0.8

      \[\leadsto x + \frac{\color{blue}{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}}{y}\]

    if 5.320319158246682e+71 < z < 3.8720995340148845e+123

    1. Initial program 11.4

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]

    2. Taylor expanded around inf 16.9

      \[\leadsto x + \frac{\color{blue}{e^{-1 \cdot z}}}{y}\]

    3. Simplified16.9

      \[\leadsto x + \frac{\color{blue}{e^{-z}}}{y}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 5.320319158246681890891762065017087849432 \cdot 10^{71}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}{y}\\ \mathbf{elif}\;z \le 3.872099534014884485438076645676398898122 \cdot 10^{123}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"

  :herbie-target
  (if (< (/ y (+ z y)) 7.1154157597908e-315) (+ x (/ (exp (/ -1.0 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

  (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))