Average Error: 5.9 → 0.9
Time: 20.7s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 1.5075292783407231 \cdot 10^{-16}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le 1.5075292783407231 \cdot 10^{-16}:\\
\;\;\;\;x + \frac{1}{y}\\

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

\end{array}
double f(double x, double y, double z) {
        double r18197920 = x;
        double r18197921 = y;
        double r18197922 = z;
        double r18197923 = r18197922 + r18197921;
        double r18197924 = r18197921 / r18197923;
        double r18197925 = log(r18197924);
        double r18197926 = r18197921 * r18197925;
        double r18197927 = exp(r18197926);
        double r18197928 = r18197927 / r18197921;
        double r18197929 = r18197920 + r18197928;
        return r18197929;
}
↓
double f(double x, double y, double z) {
        double r18197930 = y;
        double r18197931 = 1.5075292783407231e-16;
        bool r18197932 = r18197930 <= r18197931;
        double r18197933 = x;
        double r18197934 = 1.0;
        double r18197935 = r18197934 / r18197930;
        double r18197936 = r18197933 + r18197935;
        double r18197937 = z;
        double r18197938 = -r18197937;
        double r18197939 = exp(r18197938);
        double r18197940 = r18197939 / r18197930;
        double r18197941 = r18197933 + r18197940;
        double r18197942 = r18197932 ? r18197936 : r18197941;
        return r18197942;
}

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
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157597908 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 1.5075292783407231e-16

    1. Initial program 7.7

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

    2. Taylor expanded around inf 1.1

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

    if 1.5075292783407231e-16 < y

    1. Initial program 1.7

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

    2. Taylor expanded around inf 0.5

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

    3. Simplified0.5

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 1.5075292783407231 \cdot 10^{-16}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(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)))