Average Error: 6.0 → 0.7
Time: 6.1s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le 0.0:\\ \;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right) + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}}{y}\\ \mathbf{elif}\;\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le 1.4696829624869402 \cdot 10^{-78}:\\ \;\;\;\;1 \cdot \left(x + \frac{e^{-1 \cdot z}}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\frac{{\left(\frac{y}{z + y}\right)}^{y}}{y}, 1, x\right)\\ \end{array}\]
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} \le 0.0:\\
\;\;\;\;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{elif}\;\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le 1.4696829624869402 \cdot 10^{-78}:\\
\;\;\;\;1 \cdot \left(x + \frac{e^{-1 \cdot z}}{y}\right)\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \mathsf{fma}\left(\frac{{\left(\frac{y}{z + y}\right)}^{y}}{y}, 1, x\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 ((((double) (((double) exp(((double) (y * ((double) log(((double) (y / ((double) (z + y)))))))))) / y)) <= 0.0)) {
		VAR = ((double) (x + ((double) (((double) exp(((double) (y * ((double) (((double) (2.0 * ((double) log(((double) (((double) cbrt(y)) / ((double) cbrt(((double) (z + y)))))))))) + ((double) log(((double) (((double) cbrt(y)) / ((double) cbrt(((double) (z + y)))))))))))))) / y))));
	} else {
		double VAR_1;
		if ((((double) (((double) exp(((double) (y * ((double) log(((double) (y / ((double) (z + y)))))))))) / y)) <= 1.4696829624869402e-78)) {
			VAR_1 = ((double) (1.0 * ((double) (x + ((double) (((double) exp(((double) (-1.0 * z)))) / y))))));
		} else {
			VAR_1 = ((double) (1.0 * ((double) fma(((double) (((double) pow(((double) (y / ((double) (z + y)))), y)) / y)), 1.0, x))));
		}
		VAR = VAR_1;
	}
	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

Original6.0
Target1.1
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (exp (* y (log (/ y (+ z y))))) y) < 0.0

    1. Initial program 9.5

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

    3. Applied add-cube-cbrt23.6

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

    4. Applied add-cube-cbrt9.5

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

    5. Applied times-frac9.5

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

    6. Applied log-prod2.9

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

    7. Simplified0.9

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

    if 0.0 < (/ (exp (* y (log (/ y (+ z y))))) y) < 1.4696829624869402e-78

    1. Initial program 2.5

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

    3. Applied *-un-lft-identity2.5

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

    4. Applied *-un-lft-identity2.5

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

    5. Applied distribute-lft-out2.5

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

    6. Simplified2.5

      \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(\frac{y}{z + y}\right)}^{y}}{y}, 1, x\right)}\]

    7. Taylor expanded around inf 0.0

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

    8. Simplified0.0

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

    if 1.4696829624869402e-78 < (/ (exp (* y (log (/ y (+ z y))))) y)

    1. Initial program 0.5

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

    3. Applied *-un-lft-identity0.5

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

    4. Applied *-un-lft-identity0.5

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

    5. Applied distribute-lft-out0.5

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

    6. Simplified0.5

      \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(\frac{y}{z + y}\right)}^{y}}{y}, 1, x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le 0.0:\\ \;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right) + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}}{y}\\ \mathbf{elif}\;\frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \le 1.4696829624869402 \cdot 10^{-78}:\\ \;\;\;\;1 \cdot \left(x + \frac{e^{-1 \cdot z}}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\frac{{\left(\frac{y}{z + y}\right)}^{y}}{y}, 1, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 +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)))