Average Error: 5.8 → 1.4
Time: 6.2s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} = -\infty \lor \neg \left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le -6.1512673171776875 \cdot 10^{-275} \lor \neg \left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le -0.0\right)\right):\\ \;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}{\sqrt[3]{z + y}}\right)\right) + y \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\\ \end{array}\]
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} = -\infty \lor \neg \left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le -6.1512673171776875 \cdot 10^{-275} \lor \neg \left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le -0.0\right)\right):\\
\;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}{\sqrt[3]{z + y}}\right)\right) + y \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}}{y}\\

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

\end{array}
double f(double x, double y, double z) {
        double r612298 = x;
        double r612299 = y;
        double r612300 = z;
        double r612301 = r612300 + r612299;
        double r612302 = r612299 / r612301;
        double r612303 = log(r612302);
        double r612304 = r612299 * r612303;
        double r612305 = exp(r612304);
        double r612306 = r612305 / r612299;
        double r612307 = r612298 + r612306;
        return r612307;
}
↓
double f(double x, double y, double z) {
        double r612308 = y;
        double r612309 = z;
        double r612310 = r612309 + r612308;
        double r612311 = r612308 / r612310;
        double r612312 = log(r612311);
        double r612313 = r612308 * r612312;
        double r612314 = exp(r612313);
        double r612315 = r612314 / r612308;
        double r612316 = -inf.0;
        bool r612317 = r612315 <= r612316;
        double r612318 = -6.151267317177687e-275;
        bool r612319 = r612315 <= r612318;
        double r612320 = -0.0;
        bool r612321 = r612315 <= r612320;
        double r612322 = !r612321;
        bool r612323 = r612319 || r612322;
        double r612324 = !r612323;
        bool r612325 = r612317 || r612324;
        double r612326 = x;
        double r612327 = 2.0;
        double r612328 = cbrt(r612308);
        double r612329 = r612328 * r612328;
        double r612330 = cbrt(r612329);
        double r612331 = cbrt(r612328);
        double r612332 = r612330 * r612331;
        double r612333 = cbrt(r612310);
        double r612334 = r612332 / r612333;
        double r612335 = log(r612334);
        double r612336 = r612327 * r612335;
        double r612337 = r612308 * r612336;
        double r612338 = r612328 / r612333;
        double r612339 = log(r612338);
        double r612340 = r612308 * r612339;
        double r612341 = r612337 + r612340;
        double r612342 = exp(r612341);
        double r612343 = r612342 / r612308;
        double r612344 = r612326 + r612343;
        double r612345 = r612326 + r612315;
        double r612346 = r612325 ? r612344 : r612345;
        return r612346;
}

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

Original5.8
Target1.0
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (exp (* y (log (/ y (+ z y))))) y) < -inf.0 or -6.151267317177687e-275 < (/ (exp (* y (log (/ y (+ z y))))) y) < -0.0

    1. Initial program 28.1

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

    3. Applied add-cube-cbrt34.7

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

    4. Applied add-cube-cbrt28.2

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

    5. Applied times-frac28.2

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

    6. Applied log-prod6.1

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

    7. Applied distribute-lft-in6.1

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

    8. Simplified0.2

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

    10. Applied add-cube-cbrt0.9

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

    11. Applied cbrt-prod2.5

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

    if -inf.0 < (/ (exp (* y (log (/ y (+ z y))))) y) < -6.151267317177687e-275 or -0.0 < (/ (exp (* y (log (/ y (+ z y))))) y)

    1. Initial program 1.1

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} = -\infty \lor \neg \left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le -6.1512673171776875 \cdot 10^{-275} \lor \neg \left(\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le -0.0\right)\right):\\ \;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}{\sqrt[3]{z + y}}\right)\right) + y \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +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)))