Average Error: 5.8 → 0.2
Time: 6.7s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -16812707776681.5078125 \lor \neg \left(y \le 4.508510885357741404461443240209161942289 \cdot 10^{-7}\right):\\ \;\;\;\;x + \frac{e^{-1 \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{y \cdot 0}}{y}\\ \end{array}\]
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^{y \cdot 0}}{y}\\

\end{array}
double f(double x, double y, double z) {
        double r412060 = x;
        double r412061 = y;
        double r412062 = z;
        double r412063 = r412062 + r412061;
        double r412064 = r412061 / r412063;
        double r412065 = log(r412064);
        double r412066 = r412061 * r412065;
        double r412067 = exp(r412066);
        double r412068 = r412067 / r412061;
        double r412069 = r412060 + r412068;
        return r412069;
}
↓
double f(double x, double y, double z) {
        double r412070 = y;
        double r412071 = -16812707776681.508;
        bool r412072 = r412070 <= r412071;
        double r412073 = 4.5085108853577414e-07;
        bool r412074 = r412070 <= r412073;
        double r412075 = !r412074;
        bool r412076 = r412072 || r412075;
        double r412077 = x;
        double r412078 = -1.0;
        double r412079 = z;
        double r412080 = r412078 * r412079;
        double r412081 = exp(r412080);
        double r412082 = r412081 / r412070;
        double r412083 = r412077 + r412082;
        double r412084 = 0.0;
        double r412085 = r412070 * r412084;
        double r412086 = exp(r412085);
        double r412087 = r412086 / r412070;
        double r412088 = r412077 + r412087;
        double r412089 = r412076 ? r412083 : r412088;
        return r412089;
}

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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(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)))