Average Error: 5.9 → 0.9
Time: 17.7s
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 r19925175 = x;
        double r19925176 = y;
        double r19925177 = z;
        double r19925178 = r19925177 + r19925176;
        double r19925179 = r19925176 / r19925178;
        double r19925180 = log(r19925179);
        double r19925181 = r19925176 * r19925180;
        double r19925182 = exp(r19925181);
        double r19925183 = r19925182 / r19925176;
        double r19925184 = r19925175 + r19925183;
        return r19925184;
}
↓
double f(double x, double y, double z) {
        double r19925185 = y;
        double r19925186 = 1.5075292783407231e-16;
        bool r19925187 = r19925185 <= r19925186;
        double r19925188 = x;
        double r19925189 = 1.0;
        double r19925190 = r19925189 / r19925185;
        double r19925191 = r19925188 + r19925190;
        double r19925192 = z;
        double r19925193 = -r19925192;
        double r19925194 = exp(r19925193);
        double r19925195 = r19925194 / r19925185;
        double r19925196 = r19925188 + r19925195;
        double r19925197 = r19925187 ? r19925191 : r19925196;
        return r19925197;
}

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 
(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)))