Average Error: 5.8 → 0.3
Time: 1.2m
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -4130317597818000029242101376462356480:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{elif}\;y \le 6.915604877957515934322576682302050368211 \cdot 10^{-19}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -4130317597818000029242101376462356480:\\
\;\;\;\;x + \frac{e^{-z}}{y}\\

\mathbf{elif}\;y \le 6.915604877957515934322576682302050368211 \cdot 10^{-19}:\\
\;\;\;\;x + \frac{1}{y}\\

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

\end{array}
double f(double x, double y, double z) {
        double r12112771 = x;
        double r12112772 = y;
        double r12112773 = z;
        double r12112774 = r12112773 + r12112772;
        double r12112775 = r12112772 / r12112774;
        double r12112776 = log(r12112775);
        double r12112777 = r12112772 * r12112776;
        double r12112778 = exp(r12112777);
        double r12112779 = r12112778 / r12112772;
        double r12112780 = r12112771 + r12112779;
        return r12112780;
}
↓
double f(double x, double y, double z) {
        double r12112781 = y;
        double r12112782 = -4.130317597818e+36;
        bool r12112783 = r12112781 <= r12112782;
        double r12112784 = x;
        double r12112785 = z;
        double r12112786 = -r12112785;
        double r12112787 = exp(r12112786);
        double r12112788 = r12112787 / r12112781;
        double r12112789 = r12112784 + r12112788;
        double r12112790 = 6.915604877957516e-19;
        bool r12112791 = r12112781 <= r12112790;
        double r12112792 = 1.0;
        double r12112793 = r12112792 / r12112781;
        double r12112794 = r12112784 + r12112793;
        double r12112795 = r12112791 ? r12112794 : r12112789;
        double r12112796 = r12112783 ? r12112789 : r12112795;
        return r12112796;
}

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
Target1.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.115415759790762719541517221498726780517 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -4.130317597818e+36 or 6.915604877957516e-19 < y

    1. Initial program 2.3

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

    if -4.130317597818e+36 < y < 6.915604877957516e-19

    1. Initial program 9.6

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

    2. Taylor expanded around inf 0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4130317597818000029242101376462356480:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{elif}\;y \le 6.915604877957515934322576682302050368211 \cdot 10^{-19}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(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)))