Average Error: 6.2 → 1.2
Time: 22.2s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[x + \frac{\left({\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}\right) \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
x + \frac{\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 {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}
double f(double x, double y, double z) {
        double r267664 = x;
        double r267665 = y;
        double r267666 = z;
        double r267667 = r267666 + r267665;
        double r267668 = r267665 / r267667;
        double r267669 = log(r267668);
        double r267670 = r267665 * r267669;
        double r267671 = exp(r267670);
        double r267672 = r267671 / r267665;
        double r267673 = r267664 + r267672;
        return r267673;
}
↓
double f(double x, double y, double z) {
        double r267674 = x;
        double r267675 = y;
        double r267676 = cbrt(r267675);
        double r267677 = z;
        double r267678 = r267677 + r267675;
        double r267679 = cbrt(r267678);
        double r267680 = r267676 / r267679;
        double r267681 = pow(r267680, r267675);
        double r267682 = r267681 * r267681;
        double r267683 = r267682 * r267681;
        double r267684 = r267683 / r267675;
        double r267685 = r267674 + r267684;
        return r267685;
}

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

Derivation

  1. Initial program 6.2

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

  2. Simplified6.2

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

  4. Applied add-cube-cbrt19.6

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

  5. Applied add-cube-cbrt6.2

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

  6. Applied times-frac6.2

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

  7. Applied unpow-prod-down2.3

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

  9. Applied times-frac2.3

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

  10. Applied unpow-prod-down1.2

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

  11. Final simplification1.2

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

Reproduce

herbie shell --seed 2019322 
(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)))