Average Error: 5.8 → 0.9
Time: 5.1s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[x + \frac{{\left(e^{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right)}^{\left(\sqrt[3]{y} \cdot \log \left(\frac{y}{z + y}\right)\right)}}{y}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
x + \frac{{\left(e^{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right)}^{\left(\sqrt[3]{y} \cdot \log \left(\frac{y}{z + y}\right)\right)}}{y}
double f(double x, double y, double z) {
        double r524857 = x;
        double r524858 = y;
        double r524859 = z;
        double r524860 = r524859 + r524858;
        double r524861 = r524858 / r524860;
        double r524862 = log(r524861);
        double r524863 = r524858 * r524862;
        double r524864 = exp(r524863);
        double r524865 = r524864 / r524858;
        double r524866 = r524857 + r524865;
        return r524866;
}
↓
double f(double x, double y, double z) {
        double r524867 = x;
        double r524868 = y;
        double r524869 = cbrt(r524868);
        double r524870 = r524869 * r524869;
        double r524871 = exp(r524870);
        double r524872 = z;
        double r524873 = r524872 + r524868;
        double r524874 = r524868 / r524873;
        double r524875 = log(r524874);
        double r524876 = r524869 * r524875;
        double r524877 = pow(r524871, r524876);
        double r524878 = r524877 / r524868;
        double r524879 = r524867 + r524878;
        return r524879;
}

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
Target0.9
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Initial program 5.8

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

  3. Applied add-log-exp34.8

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

  4. Applied exp-to-pow0.9

    \[\leadsto x + \frac{\color{blue}{{\left(e^{y}\right)}^{\left(\log \left(\frac{y}{z + y}\right)\right)}}}{y}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.9

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

  7. Applied exp-prod0.9

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

  8. Applied pow-pow0.9

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

  9. Final simplification0.9

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

Reproduce

herbie shell --seed 2020047 
(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)))