Average Error: 6.0 → 1.5
Time: 21.9s
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 r75296375 = x;
        double r75296376 = y;
        double r75296377 = z;
        double r75296378 = r75296377 + r75296376;
        double r75296379 = r75296376 / r75296378;
        double r75296380 = log(r75296379);
        double r75296381 = r75296376 * r75296380;
        double r75296382 = exp(r75296381);
        double r75296383 = r75296382 / r75296376;
        double r75296384 = r75296375 + r75296383;
        return r75296384;
}
↓
double f(double x, double y, double z) {
        double r75296385 = z;
        double r75296386 = 5.320319158246682e+71;
        bool r75296387 = r75296385 <= r75296386;
        double r75296388 = x;
        double r75296389 = y;
        double r75296390 = exp(r75296389);
        double r75296391 = r75296385 + r75296389;
        double r75296392 = r75296389 / r75296391;
        double r75296393 = log(r75296392);
        double r75296394 = pow(r75296390, r75296393);
        double r75296395 = r75296394 / r75296389;
        double r75296396 = r75296388 + r75296395;
        double r75296397 = 3.8720995340148845e+123;
        bool r75296398 = r75296385 <= r75296397;
        double r75296399 = -r75296385;
        double r75296400 = exp(r75296399);
        double r75296401 = r75296400 / r75296389;
        double r75296402 = r75296388 + r75296401;
        double r75296403 = r75296398 ? r75296402 : r75296396;
        double r75296404 = r75296387 ? r75296396 : r75296403;
        return r75296404;
}

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 
(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)))