Average Error: 5.9 → 0.6
Time: 7.6s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -23989478599247524:\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -23989478599247524:\\
\;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}{y}\\

\end{array}
double f(double x, double y, double z) {
        double r438712 = x;
        double r438713 = y;
        double r438714 = z;
        double r438715 = r438714 + r438713;
        double r438716 = r438713 / r438715;
        double r438717 = log(r438716);
        double r438718 = r438713 * r438717;
        double r438719 = exp(r438718);
        double r438720 = r438719 / r438713;
        double r438721 = r438712 + r438720;
        return r438721;
}
↓
double f(double x, double y, double z) {
        double r438722 = y;
        double r438723 = -23989478599247524.0;
        bool r438724 = r438722 <= r438723;
        double r438725 = x;
        double r438726 = -1.0;
        double r438727 = z;
        double r438728 = r438726 * r438727;
        double r438729 = exp(r438728);
        double r438730 = r438729 / r438722;
        double r438731 = r438725 + r438730;
        double r438732 = exp(r438722);
        double r438733 = r438727 + r438722;
        double r438734 = r438722 / r438733;
        double r438735 = log(r438734);
        double r438736 = pow(r438732, r438735);
        double r438737 = r438736 / r438722;
        double r438738 = r438725 + r438737;
        double r438739 = r438724 ? r438731 : r438738;
        return r438739;
}

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
Target1.1
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -23989478599247524.0

    1. Initial program 2.2

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

    2. Taylor expanded around inf 0.0

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

    if -23989478599247524.0 < y

    1. Initial program 7.0

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
    2. Using strategy rm

    3. Applied add-log-exp27.4

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

    4. Applied exp-to-pow0.8

      \[\leadsto x + \frac{\color{blue}{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}}{y}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -23989478599247524:\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

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