Average Error: 5.8 → 0.9
Time: 5.0s
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 r353899 = x;
        double r353900 = y;
        double r353901 = z;
        double r353902 = r353901 + r353900;
        double r353903 = r353900 / r353902;
        double r353904 = log(r353903);
        double r353905 = r353900 * r353904;
        double r353906 = exp(r353905);
        double r353907 = r353906 / r353900;
        double r353908 = r353899 + r353907;
        return r353908;
}
↓
double f(double x, double y, double z) {
        double r353909 = x;
        double r353910 = y;
        double r353911 = cbrt(r353910);
        double r353912 = r353911 * r353911;
        double r353913 = exp(r353912);
        double r353914 = z;
        double r353915 = r353914 + r353910;
        double r353916 = r353910 / r353915;
        double r353917 = log(r353916);
        double r353918 = r353911 * r353917;
        double r353919 = pow(r353913, r353918);
        double r353920 = r353919 / r353910;
        double r353921 = r353909 + r353920;
        return r353921;
}

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 +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)))