Average Error: 6.2 → 1.1
Time: 14.1s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 2.82172074555303536154305440376197327804 \cdot 10^{-37}:\\ \;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right) + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le 2.82172074555303536154305440376197327804 \cdot 10^{-37}:\\
\;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right) + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}}{y}\\

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

\end{array}
double f(double x, double y, double z) {
        double r339733 = x;
        double r339734 = y;
        double r339735 = z;
        double r339736 = r339735 + r339734;
        double r339737 = r339734 / r339736;
        double r339738 = log(r339737);
        double r339739 = r339734 * r339738;
        double r339740 = exp(r339739);
        double r339741 = r339740 / r339734;
        double r339742 = r339733 + r339741;
        return r339742;
}
↓
double f(double x, double y, double z) {
        double r339743 = y;
        double r339744 = 2.8217207455530354e-37;
        bool r339745 = r339743 <= r339744;
        double r339746 = x;
        double r339747 = 2.0;
        double r339748 = cbrt(r339743);
        double r339749 = z;
        double r339750 = r339749 + r339743;
        double r339751 = cbrt(r339750);
        double r339752 = r339748 / r339751;
        double r339753 = log(r339752);
        double r339754 = r339747 * r339753;
        double r339755 = r339754 + r339753;
        double r339756 = r339743 * r339755;
        double r339757 = exp(r339756);
        double r339758 = r339757 / r339743;
        double r339759 = r339746 + r339758;
        double r339760 = -r339749;
        double r339761 = exp(r339760);
        double r339762 = r339761 / r339743;
        double r339763 = r339746 + r339762;
        double r339764 = r339745 ? r339759 : r339763;
        return r339764;
}

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

Original6.2
Target1.1
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.115415759790762719541517221498726780517 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 2.8217207455530354e-37

    1. Initial program 8.4

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt20.0

      \[\leadsto x + \frac{e^{y \cdot \log \left(\frac{y}{\color{blue}{\left(\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}\right) \cdot \sqrt[3]{z + y}}}\right)}}{y}\]

    4. Applied add-cube-cbrt8.4

      \[\leadsto x + \frac{e^{y \cdot \log \left(\frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\left(\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}\right) \cdot \sqrt[3]{z + y}}\right)}}{y}\]

    5. Applied times-frac8.4

      \[\leadsto x + \frac{e^{y \cdot \log \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}}}{y}\]

    6. Applied log-prod2.4

      \[\leadsto x + \frac{e^{y \cdot \color{blue}{\left(\log \left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}\right) + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}}}{y}\]

    7. Simplified0.9

      \[\leadsto x + \frac{e^{y \cdot \left(\color{blue}{2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)} + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}}{y}\]

    if 2.8217207455530354e-37 < y

    1. Initial program 1.7

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

    2. Taylor expanded around inf 1.4

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

    3. Simplified1.4

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 2.82172074555303536154305440376197327804 \cdot 10^{-37}:\\ \;\;\;\;x + \frac{e^{y \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right) + \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"
  :precision binary64

  :herbie-target
  (if (< (/ y (+ z y)) 7.1154157598e-315) (+ x (/ (exp (/ -1 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

  (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))