Average Error: 6.1 → 0.9
Time: 5.2s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 6.7189951102485793 \cdot 10^{-26}:\\ \;\;\;\;x + \frac{e^{0}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le 6.7189951102485793 \cdot 10^{-26}:\\
\;\;\;\;x + \frac{e^{0}}{y}\\

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

\end{array}
double f(double x, double y, double z) {
        double r401915 = x;
        double r401916 = y;
        double r401917 = z;
        double r401918 = r401917 + r401916;
        double r401919 = r401916 / r401918;
        double r401920 = log(r401919);
        double r401921 = r401916 * r401920;
        double r401922 = exp(r401921);
        double r401923 = r401922 / r401916;
        double r401924 = r401915 + r401923;
        return r401924;
}
↓
double f(double x, double y, double z) {
        double r401925 = y;
        double r401926 = 6.718995110248579e-26;
        bool r401927 = r401925 <= r401926;
        double r401928 = x;
        double r401929 = 0.0;
        double r401930 = exp(r401929);
        double r401931 = r401930 / r401925;
        double r401932 = r401928 + r401931;
        double r401933 = -1.0;
        double r401934 = z;
        double r401935 = r401933 * r401934;
        double r401936 = exp(r401935);
        double r401937 = r401936 / r401925;
        double r401938 = r401928 + r401937;
        double r401939 = r401927 ? r401932 : r401938;
        return r401939;
}

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

Original6.1
Target1.0
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 6.718995110248579e-26

    1. Initial program 8.3

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

    2. Taylor expanded around inf 0.9

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

    if 6.718995110248579e-26 < y

    1. Initial program 1.6

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

    2. Taylor expanded around inf 1.0

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 6.7189951102485793 \cdot 10^{-26}:\\ \;\;\;\;x + \frac{e^{0}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(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)))