Average Error: 5.8 → 0.2
Time: 6.8s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -16812707776681.5078125 \lor \neg \left(y \le 4.508510885357741404461443240209161942289 \cdot 10^{-7}\right):\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{0}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -16812707776681.5078125 \lor \neg \left(y \le 4.508510885357741404461443240209161942289 \cdot 10^{-7}\right):\\
\;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{e^{0}}{y}\\

\end{array}
double f(double x, double y, double z) {
        double r519250 = x;
        double r519251 = y;
        double r519252 = z;
        double r519253 = r519252 + r519251;
        double r519254 = r519251 / r519253;
        double r519255 = log(r519254);
        double r519256 = r519251 * r519255;
        double r519257 = exp(r519256);
        double r519258 = r519257 / r519251;
        double r519259 = r519250 + r519258;
        return r519259;
}
↓
double f(double x, double y, double z) {
        double r519260 = y;
        double r519261 = -16812707776681.508;
        bool r519262 = r519260 <= r519261;
        double r519263 = 4.5085108853577414e-07;
        bool r519264 = r519260 <= r519263;
        double r519265 = !r519264;
        bool r519266 = r519262 || r519265;
        double r519267 = x;
        double r519268 = -1.0;
        double r519269 = z;
        double r519270 = r519268 * r519269;
        double r519271 = exp(r519270);
        double r519272 = r519271 / r519260;
        double r519273 = r519267 + r519272;
        double r519274 = 0.0;
        double r519275 = exp(r519274);
        double r519276 = r519275 / r519260;
        double r519277 = r519267 + r519276;
        double r519278 = r519266 ? r519273 : r519277;
        return r519278;
}

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

Original5.8
Target1.1
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.115415759790762719541517221498726780517 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -16812707776681.508 or 4.5085108853577414e-07 < y

    1. Initial program 2.0

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

    2. Taylor expanded around inf 0.1

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

    if -16812707776681.508 < y < 4.5085108853577414e-07

    1. Initial program 10.1

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

    2. Taylor expanded around inf 0.4

      \[\leadsto x + \frac{e^{\color{blue}{0}}}{y}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -16812707776681.5078125 \lor \neg \left(y \le 4.508510885357741404461443240209161942289 \cdot 10^{-7}\right):\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{0}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(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)))