Average Error: 5.8 → 0.2
Time: 6.7s
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 r431029 = x;
        double r431030 = y;
        double r431031 = z;
        double r431032 = r431031 + r431030;
        double r431033 = r431030 / r431032;
        double r431034 = log(r431033);
        double r431035 = r431030 * r431034;
        double r431036 = exp(r431035);
        double r431037 = r431036 / r431030;
        double r431038 = r431029 + r431037;
        return r431038;
}
↓
double f(double x, double y, double z) {
        double r431039 = y;
        double r431040 = -16812707776681.508;
        bool r431041 = r431039 <= r431040;
        double r431042 = 4.5085108853577414e-07;
        bool r431043 = r431039 <= r431042;
        double r431044 = !r431043;
        bool r431045 = r431041 || r431044;
        double r431046 = x;
        double r431047 = -1.0;
        double r431048 = z;
        double r431049 = r431047 * r431048;
        double r431050 = exp(r431049);
        double r431051 = r431050 / r431039;
        double r431052 = r431046 + r431051;
        double r431053 = 0.0;
        double r431054 = exp(r431053);
        double r431055 = r431054 / r431039;
        double r431056 = r431046 + r431055;
        double r431057 = r431045 ? r431052 : r431056;
        return r431057;
}

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)))