Average Error: 6.2 → 1.5
Time: 5.2s
Precision: binary64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 58134488.1756577119:\\ \;\;\;\;x + \frac{e^{0}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\sqrt[3]{\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}} \cdot \sqrt[3]{\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}} \cdot \sqrt[3]{\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}}} \cdot \sqrt[3]{\sqrt[3]{\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}}}\right)\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le 58134488.1756577119:\\
\;\;\;\;x + \frac{e^{0}}{y}\\

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

\end{array}
double code(double x, double y, double z) {
	return ((double) (x + ((double) (((double) exp(((double) (y * ((double) log(((double) (y / ((double) (z + y)))))))))) / y))));
}
↓
double code(double x, double y, double z) {
	double VAR;
	if ((y <= 58134488.17565771)) {
		VAR = ((double) (x + ((double) (((double) exp(0.0)) / y))));
	} else {
		VAR = ((double) (x + ((double) (((double) (((double) cbrt(((double) (((double) exp(((double) (y * ((double) log(((double) (y / ((double) (z + y)))))))))) / y)))) * ((double) cbrt(((double) (((double) exp(((double) (y * ((double) log(((double) (y / ((double) (z + y)))))))))) / y)))))) * ((double) (((double) cbrt(((double) (((double) cbrt(((double) (((double) exp(((double) (y * ((double) log(((double) (y / ((double) (z + y)))))))))) / y)))) * ((double) cbrt(((double) (((double) exp(((double) (y * ((double) log(((double) (y / ((double) (z + y)))))))))) / y)))))))) * ((double) cbrt(((double) cbrt(((double) (((double) exp(((double) (y * ((double) log(((double) (y / ((double) (z + y)))))))))) / y))))))))))));
	}
	return VAR;
}

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
Target1.1
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 58134488.1756577119

    1. Initial program 7.8

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

    2. Taylor expanded around inf 1.1

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

    3. Simplified1.1

      \[\leadsto x + \frac{e^{\color{blue}{0}}}{y}\]

    if 58134488.1756577119 < y

    1. Initial program 2.2

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

    3. Applied add-cube-cbrt2.5

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

    5. Applied add-cube-cbrt2.5

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

    6. Applied cbrt-prod2.5

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 58134488.1756577119:\\ \;\;\;\;x + \frac{e^{0}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\sqrt[3]{\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}} \cdot \sqrt[3]{\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}} \cdot \sqrt[3]{\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}}} \cdot \sqrt[3]{\sqrt[3]{\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(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 (/ -1.0 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

  (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))