Average Error: 6.0 → 1.0
Time: 12.7s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 2.135076845346736920066581671089004732642 \cdot 10^{-34}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le 2.135076845346736920066581671089004732642 \cdot 10^{-34}:\\
\;\;\;\;x + \frac{1}{y}\\

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

\end{array}
double f(double x, double y, double z) {
        double r17146232 = x;
        double r17146233 = y;
        double r17146234 = z;
        double r17146235 = r17146234 + r17146233;
        double r17146236 = r17146233 / r17146235;
        double r17146237 = log(r17146236);
        double r17146238 = r17146233 * r17146237;
        double r17146239 = exp(r17146238);
        double r17146240 = r17146239 / r17146233;
        double r17146241 = r17146232 + r17146240;
        return r17146241;
}
↓
double f(double x, double y, double z) {
        double r17146242 = y;
        double r17146243 = 2.135076845346737e-34;
        bool r17146244 = r17146242 <= r17146243;
        double r17146245 = x;
        double r17146246 = 1.0;
        double r17146247 = r17146246 / r17146242;
        double r17146248 = r17146245 + r17146247;
        double r17146249 = z;
        double r17146250 = -r17146249;
        double r17146251 = exp(r17146250);
        double r17146252 = r17146251 / r17146242;
        double r17146253 = r17146245 + r17146252;
        double r17146254 = r17146244 ? r17146248 : r17146253;
        return r17146254;
}

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
Target1.0
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.115415759790762719541517221498726780517 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 2.135076845346737e-34

    1. Initial program 8.1

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

    2. Taylor expanded around inf 1.0

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

    if 2.135076845346737e-34 < y

    1. Initial program 1.8

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

    2. Taylor expanded around inf 0.9

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

    3. Simplified0.9

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 2.135076845346736920066581671089004732642 \cdot 10^{-34}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(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 (/ -1.0 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

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