Average Error: 6.2 → 0.8
Time: 4.6s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 4.6364166897926171 \cdot 10^{-20}:\\ \;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right) + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le 4.6364166897926171 \cdot 10^{-20}:\\
\;\;\;\;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 + \frac{e^{-1 \cdot z}}{y}\\

\end{array}
double f(double x, double y, double z) {
        double r492020 = x;
        double r492021 = y;
        double r492022 = z;
        double r492023 = r492022 + r492021;
        double r492024 = r492021 / r492023;
        double r492025 = log(r492024);
        double r492026 = r492021 * r492025;
        double r492027 = exp(r492026);
        double r492028 = r492027 / r492021;
        double r492029 = r492020 + r492028;
        return r492029;
}
↓
double f(double x, double y, double z) {
        double r492030 = y;
        double r492031 = 4.636416689792617e-20;
        bool r492032 = r492030 <= r492031;
        double r492033 = x;
        double r492034 = 2.0;
        double r492035 = cbrt(r492030);
        double r492036 = z;
        double r492037 = r492036 + r492030;
        double r492038 = cbrt(r492037);
        double r492039 = r492035 / r492038;
        double r492040 = log(r492039);
        double r492041 = r492034 * r492040;
        double r492042 = r492041 + r492040;
        double r492043 = r492030 * r492042;
        double r492044 = exp(r492043);
        double r492045 = r492044 / r492030;
        double r492046 = r492033 + r492045;
        double r492047 = -1.0;
        double r492048 = r492047 * r492036;
        double r492049 = exp(r492048);
        double r492050 = r492049 / r492030;
        double r492051 = r492033 + r492050;
        double r492052 = r492032 ? r492046 : r492051;
        return r492052;
}

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

Original6.2
Target1.2
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 4.636416689792617e-20

    1. Initial program 7.9

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

    3. Applied add-cube-cbrt19.0

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

    4. Applied add-cube-cbrt7.9

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

    5. Applied times-frac7.9

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

    6. Applied log-prod2.3

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

    7. Simplified0.7

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

    if 4.636416689792617e-20 < y

    1. Initial program 2.1

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

    2. Taylor expanded around inf 0.8

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 4.6364166897926171 \cdot 10^{-20}:\\ \;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right) + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(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)))