Average Error: 6.3 → 2.0
Time: 24.8s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le 2798330066458407314915328 \lor \neg \left(z \le 4.76689686693070055018888846744602636955 \cdot 10^{118}\right):\\ \;\;\;\;x + \frac{\left({\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}\right) \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}\right)}^{y} \cdot e^{\frac{-1}{3} \cdot z}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;z \le 2798330066458407314915328 \lor \neg \left(z \le 4.76689686693070055018888846744602636955 \cdot 10^{118}\right):\\
\;\;\;\;x + \frac{\left({\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}\right) \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}\right)}^{y} \cdot e^{\frac{-1}{3} \cdot z}}{y}\\

\end{array}
double f(double x, double y, double z) {
        double r254289 = x;
        double r254290 = y;
        double r254291 = z;
        double r254292 = r254291 + r254290;
        double r254293 = r254290 / r254292;
        double r254294 = log(r254293);
        double r254295 = r254290 * r254294;
        double r254296 = exp(r254295);
        double r254297 = r254296 / r254290;
        double r254298 = r254289 + r254297;
        return r254298;
}
↓
double f(double x, double y, double z) {
        double r254299 = z;
        double r254300 = 2.7983300664584073e+24;
        bool r254301 = r254299 <= r254300;
        double r254302 = 4.7668968669307006e+118;
        bool r254303 = r254299 <= r254302;
        double r254304 = !r254303;
        bool r254305 = r254301 || r254304;
        double r254306 = x;
        double r254307 = y;
        double r254308 = cbrt(r254307);
        double r254309 = r254299 + r254307;
        double r254310 = cbrt(r254309);
        double r254311 = r254308 / r254310;
        double r254312 = pow(r254311, r254307);
        double r254313 = r254312 * r254312;
        double r254314 = r254313 * r254312;
        double r254315 = r254314 / r254307;
        double r254316 = r254306 + r254315;
        double r254317 = r254308 * r254308;
        double r254318 = r254310 * r254310;
        double r254319 = r254317 / r254318;
        double r254320 = pow(r254319, r254307);
        double r254321 = -0.3333333333333333;
        double r254322 = r254321 * r254299;
        double r254323 = exp(r254322);
        double r254324 = r254320 * r254323;
        double r254325 = r254324 / r254307;
        double r254326 = r254306 + r254325;
        double r254327 = r254305 ? r254316 : r254326;
        return r254327;
}

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
Target1.1
Herbie2.0
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.115415759790762719541517221498726780517 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < 2.7983300664584073e+24 or 4.7668968669307006e+118 < z

    1. Initial program 5.9

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

    2. Simplified5.9

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{z + y}\right)}^{y}}{y}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt19.3

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

    5. Applied add-cube-cbrt5.9

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

    6. Applied times-frac5.9

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

    7. Applied unpow-prod-down1.8

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

    9. Applied times-frac1.8

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

    10. Applied unpow-prod-down0.7

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

    if 2.7983300664584073e+24 < z < 4.7668968669307006e+118

    1. Initial program 10.9

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

    2. Simplified10.9

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{z + y}\right)}^{y}}{y}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt17.9

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

    5. Applied add-cube-cbrt10.9

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

    6. Applied times-frac10.9

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

    7. Applied unpow-prod-down6.3

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

    8. Taylor expanded around inf 18.4

      \[\leadsto x + \frac{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}\right)}^{y} \cdot \color{blue}{e^{-\frac{1}{3} \cdot z}}}{y}\]

    9. Simplified18.4

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

  4. Final simplification2.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 2798330066458407314915328 \lor \neg \left(z \le 4.76689686693070055018888846744602636955 \cdot 10^{118}\right):\\ \;\;\;\;x + \frac{\left({\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}\right) \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}\right)}^{y} \cdot e^{\frac{-1}{3} \cdot z}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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)))