Average Error: 6.0 → 0.8
Time: 15.2s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 0.06656101021950764:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y \cdot e^{z}}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le 0.06656101021950764:\\
\;\;\;\;x + \frac{1}{y}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{1}{y \cdot e^{z}}\\

\end{array}
double f(double x, double y, double z) {
        double r17262646 = x;
        double r17262647 = y;
        double r17262648 = z;
        double r17262649 = r17262648 + r17262647;
        double r17262650 = r17262647 / r17262649;
        double r17262651 = log(r17262650);
        double r17262652 = r17262647 * r17262651;
        double r17262653 = exp(r17262652);
        double r17262654 = r17262653 / r17262647;
        double r17262655 = r17262646 + r17262654;
        return r17262655;
}
↓
double f(double x, double y, double z) {
        double r17262656 = y;
        double r17262657 = 0.06656101021950764;
        bool r17262658 = r17262656 <= r17262657;
        double r17262659 = x;
        double r17262660 = 1.0;
        double r17262661 = r17262660 / r17262656;
        double r17262662 = r17262659 + r17262661;
        double r17262663 = z;
        double r17262664 = exp(r17262663);
        double r17262665 = r17262656 * r17262664;
        double r17262666 = r17262660 / r17262665;
        double r17262667 = r17262659 + r17262666;
        double r17262668 = r17262658 ? r17262662 : r17262667;
        return r17262668;
}

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

Original6.0
Target1.1
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157597908 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 0.06656101021950764

    1. Initial program 7.7

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

    2. Taylor expanded around inf 1.2

      \[\leadsto x + \frac{e^{\color{blue}{0}}}{y}\]

    if 0.06656101021950764 < y

    1. Initial program 2.0

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

    2. Taylor expanded around inf 0.0

      \[\leadsto x + \color{blue}{\frac{e^{-1 \cdot z}}{y}}\]

    3. Simplified0.0

      \[\leadsto x + \color{blue}{\frac{e^{-z}}{y}}\]
    4. Using strategy rm

    5. Applied clear-num0.0

      \[\leadsto x + \color{blue}{\frac{1}{\frac{y}{e^{-z}}}}\]

    6. Simplified0.0

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 0.06656101021950764:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y \cdot e^{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"

  :herbie-target
  (if (< (/ y (+ z y)) 7.1154157597908e-315) (+ x (/ (exp (/ -1 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

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