Average Error: 10.0 → 0.4
Time: 3.4s
Precision: binary64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -1.1242647605221334 \lor \neg \left(y \leq 2.9538523048018434 \cdot 10^{-29}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -1.1242647605221334 \lor \neg \left(y \leq 2.9538523048018434 \cdot 10^{-29}\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 -1.1242647605221334) (not (<= y 2.9538523048018434e-29)))
   (+ 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 <= -1.1242647605221334) || !(y <= 2.9538523048018434e-29)) {
		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

Original10.0
Target5.7
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} < 7.11541576 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.1242647605221334 or 2.9538523048018434e-29 < y

    1. Initial program 9.4

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

    2. Simplified9.4

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

    3. Taylor expanded in y around inf 0.4

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

    if -1.1242647605221334 < y < 2.9538523048018434e-29

    1. Initial program 10.9

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

    2. Simplified10.9

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

    3. Taylor expanded in y around 0 0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1242647605221334 \lor \neg \left(y \leq 2.9538523048018434 \cdot 10^{-29}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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)))