Average Error: 5.9 → 1.4
Time: 7.3s
Precision: binary64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq -4.24261594712533 \cdot 10^{+31}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;x + \frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq -4.24261594712533 \cdot 10^{+31}:\\
\;\;\;\;x + \frac{1}{y}\\

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

\end{array}
(FPCore (x y z)
 :precision binary64
 (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= (+ x (/ (exp (* y (log (/ y (+ y z))))) y)) -4.24261594712533e+31)
   (+ x (/ 1.0 y))
   (+
    x
    (/
     (*
      (pow (/ (* (cbrt y) (cbrt y)) (* (cbrt (+ y z)) (cbrt (+ y z)))) y)
      (pow (/ (cbrt y) (cbrt (+ y z))) y))
     y))))
double code(double x, double y, double z) {
	return x + (exp(y * log(y / (z + y))) / y);
}
↓
double code(double x, double y, double z) {
	double tmp;
	if ((x + (exp(y * log(y / (y + z))) / y)) <= -4.24261594712533e+31) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + ((pow(((cbrt(y) * cbrt(y)) / (cbrt(y + z) * cbrt(y + z))), y) * pow((cbrt(y) / cbrt(y + z)), y)) / y);
	}
	return tmp;
}

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

Original5.9
Target1.1
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} < 7.1154157597908 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (+.f64 x (/.f64 (exp.f64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y))))) y)) < -4.24261594712532992e31

    1. Initial program 8.9

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

    2. Simplified8.9

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

    3. Taylor expanded around 0 0.0

      \[\leadsto x + \frac{\color{blue}{1}}{y}\]

    if -4.24261594712532992e31 < (+.f64 x (/.f64 (exp.f64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y))))) y))

    1. Initial program 4.3

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

    2. Simplified4.3

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

    4. Applied add-cube-cbrt_binary64_1204821.2

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

    5. Applied add-cube-cbrt_binary64_120484.3

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

    6. Applied times-frac_binary64_120194.3

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

    7. Applied unpow-prod-down_binary64_120922.2

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq -4.24261594712533 \cdot 10^{+31}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2021043 
(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)))