Average Error: 6.1 → 1.0
Time: 18.0s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[x + \left({\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}\right) \cdot \frac{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
x + \left({\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}\right) \cdot \frac{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}
double f(double x, double y, double z) {
        double r538766 = x;
        double r538767 = y;
        double r538768 = z;
        double r538769 = r538768 + r538767;
        double r538770 = r538767 / r538769;
        double r538771 = log(r538770);
        double r538772 = r538767 * r538771;
        double r538773 = exp(r538772);
        double r538774 = r538773 / r538767;
        double r538775 = r538766 + r538774;
        return r538775;
}
↓
double f(double x, double y, double z) {
        double r538776 = x;
        double r538777 = y;
        double r538778 = cbrt(r538777);
        double r538779 = z;
        double r538780 = r538779 + r538777;
        double r538781 = cbrt(r538780);
        double r538782 = r538778 / r538781;
        double r538783 = pow(r538782, r538777);
        double r538784 = r538783 * r538783;
        double r538785 = r538783 / r538777;
        double r538786 = r538784 * r538785;
        double r538787 = r538776 + r538786;
        return r538787;
}

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

Original6.1
Target1.0
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Initial program 6.1

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

  2. Simplified6.1

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

  4. Applied *-un-lft-identity6.1

    \[\leadsto x + \frac{{\left(\frac{y}{z + y}\right)}^{y}}{\color{blue}{1 \cdot y}}\]

  5. Applied add-cube-cbrt19.1

    \[\leadsto x + \frac{{\left(\frac{y}{\color{blue}{\left(\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}\right) \cdot \sqrt[3]{z + y}}}\right)}^{y}}{1 \cdot y}\]

  6. Applied add-cube-cbrt6.2

    \[\leadsto x + \frac{{\left(\frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\left(\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}\right) \cdot \sqrt[3]{z + y}}\right)}^{y}}{1 \cdot y}\]

  7. Applied times-frac6.2

    \[\leadsto x + \frac{{\color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}}^{y}}{1 \cdot y}\]

  8. Applied unpow-prod-down2.1

    \[\leadsto x + \frac{\color{blue}{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}}{1 \cdot y}\]

  9. Applied times-frac2.1

    \[\leadsto x + \color{blue}{\frac{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}\right)}^{y}}{1} \cdot \frac{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}}\]

  10. Simplified2.1

    \[\leadsto x + \color{blue}{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}\right)}^{y}} \cdot \frac{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\]
  11. Using strategy rm

  12. Applied times-frac2.1

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

  13. Applied unpow-prod-down1.0

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

  14. Final simplification1.0

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

Reproduce

herbie shell --seed 2020042 +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)))