Average Error: 6.1 → 0.6
Time: 5.6s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.611458593987227695782152019682377982421 \cdot 10^{54}:\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right) + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.611458593987227695782152019682377982421 \cdot 10^{54}:\\
\;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\

\mathbf{else}:\\
\;\;\;\;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}\\

\end{array}
double f(double x, double y, double z) {
        double r641331 = x;
        double r641332 = y;
        double r641333 = z;
        double r641334 = r641333 + r641332;
        double r641335 = r641332 / r641334;
        double r641336 = log(r641335);
        double r641337 = r641332 * r641336;
        double r641338 = exp(r641337);
        double r641339 = r641338 / r641332;
        double r641340 = r641331 + r641339;
        return r641340;
}
↓
double f(double x, double y, double z) {
        double r641341 = y;
        double r641342 = -1.6114585939872277e+54;
        bool r641343 = r641341 <= r641342;
        double r641344 = x;
        double r641345 = -1.0;
        double r641346 = z;
        double r641347 = r641345 * r641346;
        double r641348 = exp(r641347);
        double r641349 = r641348 / r641341;
        double r641350 = r641344 + r641349;
        double r641351 = 2.0;
        double r641352 = cbrt(r641341);
        double r641353 = r641346 + r641341;
        double r641354 = cbrt(r641353);
        double r641355 = r641352 / r641354;
        double r641356 = log(r641355);
        double r641357 = r641351 * r641356;
        double r641358 = r641357 + r641356;
        double r641359 = r641341 * r641358;
        double r641360 = exp(r641359);
        double r641361 = r641360 / r641341;
        double r641362 = r641344 + r641361;
        double r641363 = r641343 ? r641350 : r641362;
        return r641363;
}

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

Original6.1
Target1.2
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.115415759790762719541517221498726780517 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.6114585939872277e+54

    1. Initial program 2.7

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

    2. Taylor expanded around inf 0.0

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

    if -1.6114585939872277e+54 < y

    1. Initial program 6.9

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

    3. Applied add-cube-cbrt15.6

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

    4. Applied add-cube-cbrt6.9

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

    5. Applied times-frac6.9

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

    6. Applied log-prod2.0

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

    7. Simplified0.8

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.611458593987227695782152019682377982421 \cdot 10^{54}:\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right) + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(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)))