Average Error: 6.0 → 2.0
Time: 10.7s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 3.711281227157836 \cdot 10^{-137}:\\ \;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right) + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-z}}{y} + x\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le 3.711281227157836 \cdot 10^{-137}:\\
\;\;\;\;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}:\\
\;\;\;\;\frac{e^{-z}}{y} + x\\

\end{array}
double f(double x, double y, double z) {
        double r534287 = x;
        double r534288 = y;
        double r534289 = z;
        double r534290 = r534289 + r534288;
        double r534291 = r534288 / r534290;
        double r534292 = log(r534291);
        double r534293 = r534288 * r534292;
        double r534294 = exp(r534293);
        double r534295 = r534294 / r534288;
        double r534296 = r534287 + r534295;
        return r534296;
}
↓
double f(double x, double y, double z) {
        double r534297 = y;
        double r534298 = 3.711281227157836e-137;
        bool r534299 = r534297 <= r534298;
        double r534300 = x;
        double r534301 = 2.0;
        double r534302 = cbrt(r534297);
        double r534303 = z;
        double r534304 = r534303 + r534297;
        double r534305 = cbrt(r534304);
        double r534306 = r534302 / r534305;
        double r534307 = log(r534306);
        double r534308 = r534301 * r534307;
        double r534309 = r534308 + r534307;
        double r534310 = r534297 * r534309;
        double r534311 = exp(r534310);
        double r534312 = r534311 / r534297;
        double r534313 = r534300 + r534312;
        double r534314 = -r534303;
        double r534315 = exp(r534314);
        double r534316 = r534315 / r534297;
        double r534317 = r534316 + r534300;
        double r534318 = r534299 ? r534313 : r534317;
        return r534318;
}

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
Target1.1
Herbie2.0
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 3.711281227157836e-137

    1. Initial program 8.5

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

    3. Applied add-cube-cbrt21.6

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

    4. Applied add-cube-cbrt8.5

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

    5. Applied times-frac8.5

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

    6. Applied log-prod2.6

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

    7. Simplified0.9

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

    if 3.711281227157836e-137 < y

    1. Initial program 2.1

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

    2. Taylor expanded around inf 3.6

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

    3. Simplified3.6

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

  4. Final simplification2.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 3.711281227157836 \cdot 10^{-137}:\\ \;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right) + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-z}}{y} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(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)))