Average Error: 6.2 → 1.5
Time: 8.3s
Precision: binary64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq -1.29307044824209 \cdot 10^{-90} \lor \neg \left(x + \frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq 6.618792720599542 \cdot 10^{-137}\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;x + \frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq -1.29307044824209 \cdot 10^{-90} \lor \neg \left(x + \frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq 6.618792720599542 \cdot 10^{-137}\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 (/ (exp (* y (log (/ y (+ y z))))) y)) -1.29307044824209e-90)
         (not
          (<=
           (+ x (/ (exp (* y (log (/ y (+ y z))))) y))
           6.618792720599542e-137)))
   (+ 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 + (exp(y * log(y / (y + z))) / y)) <= -1.29307044824209e-90) || !((x + (exp(y * log(y / (y + z))) / y)) <= 6.618792720599542e-137)) {
		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

Original6.2
Target1.1
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} < 7.1154157597908 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (+.f64 x (/.f64 (exp.f64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y))))) y)) < -1.2930704482420901e-90 or 6.6187927205995421e-137 < (+.f64 x (/.f64 (exp.f64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y))))) y))

    1. Initial program 5.2

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

    2. Simplified5.2

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

    3. Taylor expanded around 0 0.7

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

    if -1.2930704482420901e-90 < (+.f64 x (/.f64 (exp.f64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y))))) y)) < 6.6187927205995421e-137

    1. Initial program 12.3

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

    2. Simplified12.3

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

    3. Taylor expanded around inf 6.4

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq -1.29307044824209 \cdot 10^{-90} \lor \neg \left(x + \frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq 6.618792720599542 \cdot 10^{-137}\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2021093 
(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)))