Average Error: 6.3 → 0.9
Time: 4.2s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le -10910987967306.1348 \lor \neg \left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le -7.71869811703901377 \cdot 10^{-252}\right):\\ \;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right) + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + e^{-1 \cdot z} \cdot \frac{1}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le -10910987967306.1348 \lor \neg \left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le -7.71869811703901377 \cdot 10^{-252}\right):\\
\;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right) + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}}{y}\\

\mathbf{else}:\\
\;\;\;\;x + e^{-1 \cdot z} \cdot \frac{1}{y}\\

\end{array}
double f(double x, double y, double z) {
        double r504354 = x;
        double r504355 = y;
        double r504356 = z;
        double r504357 = r504356 + r504355;
        double r504358 = r504355 / r504357;
        double r504359 = log(r504358);
        double r504360 = r504355 * r504359;
        double r504361 = exp(r504360);
        double r504362 = r504361 / r504355;
        double r504363 = r504354 + r504362;
        return r504363;
}
↓
double f(double x, double y, double z) {
        double r504364 = y;
        double r504365 = z;
        double r504366 = r504365 + r504364;
        double r504367 = r504364 / r504366;
        double r504368 = log(r504367);
        double r504369 = r504364 * r504368;
        double r504370 = exp(r504369);
        double r504371 = r504370 / r504364;
        double r504372 = -10910987967306.135;
        bool r504373 = r504371 <= r504372;
        double r504374 = -7.718698117039014e-252;
        bool r504375 = r504371 <= r504374;
        double r504376 = !r504375;
        bool r504377 = r504373 || r504376;
        double r504378 = x;
        double r504379 = 2.0;
        double r504380 = cbrt(r504364);
        double r504381 = cbrt(r504366);
        double r504382 = r504380 / r504381;
        double r504383 = log(r504382);
        double r504384 = r504379 * r504383;
        double r504385 = r504384 + r504383;
        double r504386 = r504364 * r504385;
        double r504387 = exp(r504386);
        double r504388 = r504387 / r504364;
        double r504389 = r504378 + r504388;
        double r504390 = -1.0;
        double r504391 = r504390 * r504365;
        double r504392 = exp(r504391);
        double r504393 = 1.0;
        double r504394 = r504393 / r504364;
        double r504395 = r504392 * r504394;
        double r504396 = r504378 + r504395;
        double r504397 = r504377 ? r504389 : r504396;
        return r504397;
}

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

Original6.3
Target1.3
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (exp (* y (log (/ y (+ z y))))) y) < -10910987967306.135 or -7.718698117039014e-252 < (/ (exp (* y (log (/ y (+ z y))))) y)

    1. Initial program 7.1

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

    3. Applied add-cube-cbrt15.8

      \[\leadsto x + \frac{e^{y \cdot \log \left(\frac{y}{\color{blue}{\left(\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}\right) \cdot \sqrt[3]{z + y}}}\right)}}{y}\]

    4. Applied add-cube-cbrt7.1

      \[\leadsto x + \frac{e^{y \cdot \log \left(\frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\left(\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}\right) \cdot \sqrt[3]{z + y}}\right)}}{y}\]

    5. Applied times-frac7.1

      \[\leadsto x + \frac{e^{y \cdot \log \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}}}{y}\]

    6. Applied log-prod2.2

      \[\leadsto x + \frac{e^{y \cdot \color{blue}{\left(\log \left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}\right) + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}}}{y}\]

    7. Simplified0.8

      \[\leadsto x + \frac{e^{y \cdot \left(\color{blue}{2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)} + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}}{y}\]

    if -10910987967306.135 < (/ (exp (* y (log (/ y (+ z y))))) y) < -7.718698117039014e-252

    1. Initial program 3.0

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

    2. Taylor expanded around inf 1.1

      \[\leadsto \color{blue}{x + \frac{e^{-1 \cdot z}}{y}}\]
    3. Using strategy rm

    4. Applied div-inv1.1

      \[\leadsto x + \color{blue}{e^{-1 \cdot z} \cdot \frac{1}{y}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le -10910987967306.1348 \lor \neg \left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le -7.71869811703901377 \cdot 10^{-252}\right):\\ \;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right) + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + e^{-1 \cdot z} \cdot \frac{1}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"
  :precision binary64

  :herbie-target
  (if (< (/ y (+ z y)) 7.1154157597908e-315) (+ x (/ (exp (/ -1 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

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