Average Error: 5.7 → 0.9
Time: 19.5s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[x + \frac{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y}}{\frac{y}{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y}}}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
x + \frac{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y}}{\frac{y}{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y}}}
double f(double x, double y, double z) {
        double r384727 = x;
        double r384728 = y;
        double r384729 = z;
        double r384730 = r384729 + r384728;
        double r384731 = r384728 / r384730;
        double r384732 = log(r384731);
        double r384733 = r384728 * r384732;
        double r384734 = exp(r384733);
        double r384735 = r384734 / r384728;
        double r384736 = r384727 + r384735;
        return r384736;
}
↓
double f(double x, double y, double z) {
        double r384737 = x;
        double r384738 = y;
        double r384739 = cbrt(r384738);
        double r384740 = z;
        double r384741 = r384738 + r384740;
        double r384742 = cbrt(r384741);
        double r384743 = r384739 / r384742;
        double r384744 = pow(r384743, r384738);
        double r384745 = r384744 * r384744;
        double r384746 = r384738 / r384744;
        double r384747 = r384745 / r384746;
        double r384748 = r384737 + r384747;
        return r384748;
}

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

Derivation

  1. Initial program 5.7

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

  2. Simplified5.7

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

  4. Applied add-cube-cbrt19.5

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

  5. Applied add-cube-cbrt5.8

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

  6. Applied times-frac5.8

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

  7. Applied unpow-prod-down1.8

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

  8. Applied associate-/l*1.8

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

  10. Applied times-frac1.8

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

  11. Applied unpow-prod-down0.9

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

  12. Final simplification0.9

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

Reproduce

herbie shell --seed 2019174 
(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)))