Average Error: 6.3 → 1.9
Time: 8.7s
Precision: binary64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq -5635525.698233714:\\ \;\;\;\;x + \frac{{\left(\frac{1}{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}}\right)}^{y} \cdot {\left(\frac{y}{\sqrt[3]{y + z}}\right)}^{y}}{y}\\ \mathbf{elif}\;\frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq -2.3393436710919225 \cdot 10^{-246}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;\frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq -5635525.698233714:\\
\;\;\;\;x + \frac{{\left(\frac{1}{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}}\right)}^{y} \cdot {\left(\frac{y}{\sqrt[3]{y + z}}\right)}^{y}}{y}\\

\mathbf{elif}\;\frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq -2.3393436710919225 \cdot 10^{-246}:\\
\;\;\;\;x + \frac{e^{-z}}{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 (<= (/ (exp (* y (log (/ y (+ y z))))) y) -5635525.698233714)
   (+
    x
    (/
     (*
      (pow (/ 1.0 (* (cbrt (+ y z)) (cbrt (+ y z)))) y)
      (pow (/ y (cbrt (+ y z))) y))
     y))
   (if (<= (/ (exp (* y (log (/ y (+ y z))))) y) -2.3393436710919225e-246)
     (+ x (/ (exp (- z)) 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 ((exp(y * log(y / (y + z))) / y) <= -5635525.698233714) {
		tmp = x + ((pow((1.0 / (cbrt(y + z) * cbrt(y + z))), y) * pow((y / cbrt(y + z)), y)) / y);
	} else if ((exp(y * log(y / (y + z))) / y) <= -2.3393436710919225e-246) {
		tmp = x + (exp(-z) / 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

Original6.3
Target1.1
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} < 7.1154157597908 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (exp.f64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y))))) y) < -5635525.6982337143

    1. Initial program 11.8

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

    2. Simplified11.8

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

    4. Applied add-cube-cbrt_binary64_1123011.8

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

    5. Applied *-un-lft-identity_binary64_1119811.8

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

    6. Applied times-frac_binary64_1120411.8

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

    7. Applied unpow-prod-down_binary64_112743.2

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

    if -5635525.6982337143 < (/.f64 (exp.f64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y))))) y) < -2.3393436710919225e-246

    1. Initial program 3.1

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

    2. Simplified3.1

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

    3. Taylor expanded around inf 0.5

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

    if -2.3393436710919225e-246 < (/.f64 (exp.f64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y))))) y)

    1. Initial program 5.1

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

    2. Simplified5.1

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

    4. Applied add-cube-cbrt_binary64_1123017.7

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

    5. Applied add-cube-cbrt_binary64_112305.1

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

    6. Applied times-frac_binary64_112045.1

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

    7. Applied unpow-prod-down_binary64_112741.8

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

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq -5635525.698233714:\\ \;\;\;\;x + \frac{{\left(\frac{1}{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}}\right)}^{y} \cdot {\left(\frac{y}{\sqrt[3]{y + z}}\right)}^{y}}{y}\\ \mathbf{elif}\;\frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq -2.3393436710919225 \cdot 10^{-246}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020292 
(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)))