Average Error: 5.7 → 0.8
Time: 18.6s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 18035828.612908236682415008544921875:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \sqrt{\frac{e^{-z}}{y}} \cdot \sqrt{\frac{e^{-z}}{y}}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le 18035828.612908236682415008544921875:\\
\;\;\;\;x + \frac{1}{y}\\

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

\end{array}
double f(double x, double y, double z) {
        double r24269758 = x;
        double r24269759 = y;
        double r24269760 = z;
        double r24269761 = r24269760 + r24269759;
        double r24269762 = r24269759 / r24269761;
        double r24269763 = log(r24269762);
        double r24269764 = r24269759 * r24269763;
        double r24269765 = exp(r24269764);
        double r24269766 = r24269765 / r24269759;
        double r24269767 = r24269758 + r24269766;
        return r24269767;
}
↓
double f(double x, double y, double z) {
        double r24269768 = y;
        double r24269769 = 18035828.612908237;
        bool r24269770 = r24269768 <= r24269769;
        double r24269771 = x;
        double r24269772 = 1.0;
        double r24269773 = r24269772 / r24269768;
        double r24269774 = r24269771 + r24269773;
        double r24269775 = z;
        double r24269776 = -r24269775;
        double r24269777 = exp(r24269776);
        double r24269778 = r24269777 / r24269768;
        double r24269779 = sqrt(r24269778);
        double r24269780 = r24269779 * r24269779;
        double r24269781 = r24269771 + r24269780;
        double r24269782 = r24269770 ? r24269774 : r24269781;
        return r24269782;
}

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

Original5.7
Target0.9
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.115415759790762719541517221498726780517 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 18035828.612908237

    1. Initial program 7.4

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

    2. Taylor expanded around inf 1.1

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

    if 18035828.612908237 < y

    1. Initial program 1.6

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

      \[\leadsto \color{blue}{x + \frac{e^{-z}}{y}}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt0.1

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 18035828.612908236682415008544921875:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \sqrt{\frac{e^{-z}}{y}} \cdot \sqrt{\frac{e^{-z}}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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)))