Average Error: 5.9 → 0.6
Time: 6.2s
Precision: binary64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -5.97184375554952 \cdot 10^{+84} \lor \neg \left(y \leq 1.0446930670683376 \cdot 10^{-37}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -5.97184375554952 \cdot 10^{+84} \lor \neg \left(y \leq 1.0446930670683376 \cdot 10^{-37}\right):\\
\;\;\;\;x + \frac{e^{-z}}{y}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{1}{y}\\


\end{array}

(FPCore (x y z)
 :precision binary64
 (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))
↓
(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5.97184375554952e+84) (not (<= y 1.0446930670683376e-37)))
   (+ x (/ (exp (- z)) y))
   (+ x (/ 1.0 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 ((y <= -5.97184375554952e+84) || !(y <= 1.0446930670683376e-37)) {
		tmp = x + (exp(-z) / y);
	} else {
		tmp = x + (1.0 / 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
Target1.0
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} < 7.11541576 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if y < -5.9718437555495203e84 or 1.0446930670683376e-37 < y

    1. Initial program 1.9

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

    2. Simplified1.9

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

    3. Taylor expanded in y around inf 0.8

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

    if -5.9718437555495203e84 < y < 1.0446930670683376e-37

    1. Initial program 10.0

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

    2. Simplified10.0

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

    3. Taylor expanded in y around 0 0.4

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.97184375554952 \cdot 10^{+84} \lor \neg \left(y \leq 1.0446930670683376 \cdot 10^{-37}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"
  :precision binary64

  :herbie-target
  (if (< (/ y (+ z y)) 7.11541576e-315) (+ x (/ (exp (/ -1.0 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

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