Average Error: 6.3 → 0.9
Time: 4.3s
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 r517239 = x;
        double r517240 = y;
        double r517241 = z;
        double r517242 = r517241 + r517240;
        double r517243 = r517240 / r517242;
        double r517244 = log(r517243);
        double r517245 = r517240 * r517244;
        double r517246 = exp(r517245);
        double r517247 = r517246 / r517240;
        double r517248 = r517239 + r517247;
        return r517248;
}
↓
double f(double x, double y, double z) {
        double r517249 = y;
        double r517250 = z;
        double r517251 = r517250 + r517249;
        double r517252 = r517249 / r517251;
        double r517253 = log(r517252);
        double r517254 = r517249 * r517253;
        double r517255 = exp(r517254);
        double r517256 = r517255 / r517249;
        double r517257 = -10910987967306.135;
        bool r517258 = r517256 <= r517257;
        double r517259 = -7.718698117039014e-252;
        bool r517260 = r517256 <= r517259;
        double r517261 = !r517260;
        bool r517262 = r517258 || r517261;
        double r517263 = x;
        double r517264 = 2.0;
        double r517265 = cbrt(r517249);
        double r517266 = cbrt(r517251);
        double r517267 = r517265 / r517266;
        double r517268 = log(r517267);
        double r517269 = r517264 * r517268;
        double r517270 = r517269 + r517268;
        double r517271 = r517249 * r517270;
        double r517272 = exp(r517271);
        double r517273 = r517272 / r517249;
        double r517274 = r517263 + r517273;
        double r517275 = -1.0;
        double r517276 = r517275 * r517250;
        double r517277 = exp(r517276);
        double r517278 = 1.0;
        double r517279 = r517278 / r517249;
        double r517280 = r517277 * r517279;
        double r517281 = r517263 + r517280;
        double r517282 = r517262 ? r517274 : r517281;
        return r517282;
}

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)))