Average Error: 6.4 → 2.4
Time: 4.1s
Precision: binary64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 4.45409418936908 \cdot 10^{-307} \lor \neg \left(x \leq 5.450286583532159 \cdot 10^{-245}\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array} \]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 4.45409418936908 \cdot 10^{-307} \lor \neg \left(x \leq 5.450286583532159 \cdot 10^{-245}\right):\\
\;\;\;\;x + \frac{1}{y}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{e^{-z}}{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 (<= x 4.45409418936908e-307) (not (<= x 5.450286583532159e-245)))
   (+ x (/ 1.0 y))
   (+ x (/ (exp (- z)) 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 <= 4.45409418936908e-307) || !(x <= 5.450286583532159e-245)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + (exp(-z) / 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

Original6.4
Target1.2
Herbie2.4
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} < 7.11541576 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if x < 4.45409418936908039e-307 or 5.4502865835321592e-245 < x

    1. Initial program 6.1

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

    2. Simplified6.1

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

    3. Taylor expanded in y around 0 2.0

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

    if 4.45409418936908039e-307 < x < 5.4502865835321592e-245

    1. Initial program 11.9

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

    2. Simplified11.9

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

    3. Taylor expanded in y around inf 11.8

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.45409418936908 \cdot 10^{-307} \lor \neg \left(x \leq 5.450286583532159 \cdot 10^{-245}\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array} \]

Reproduce

herbie shell --seed 2022005 
(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)))