Average Error: 6.0 → 1.0
Time: 12.6s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[x + \frac{{\left(\left|\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right|\right)}^{y} \cdot {\left(\left|\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right|\right)}^{y}}{\frac{y}{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}}\]
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|\right)}^{y} \cdot {\left(\left|\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right|\right)}^{y}}{\frac{y}{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}}
double f(double x, double y, double z) {
        double r276509 = x;
        double r276510 = y;
        double r276511 = z;
        double r276512 = r276511 + r276510;
        double r276513 = r276510 / r276512;
        double r276514 = log(r276513);
        double r276515 = r276510 * r276514;
        double r276516 = exp(r276515);
        double r276517 = r276516 / r276510;
        double r276518 = r276509 + r276517;
        return r276518;
}
↓
double f(double x, double y, double z) {
        double r276519 = x;
        double r276520 = y;
        double r276521 = cbrt(r276520);
        double r276522 = z;
        double r276523 = r276522 + r276520;
        double r276524 = cbrt(r276523);
        double r276525 = r276521 / r276524;
        double r276526 = fabs(r276525);
        double r276527 = pow(r276526, r276520);
        double r276528 = r276527 * r276527;
        double r276529 = pow(r276525, r276520);
        double r276530 = r276520 / r276529;
        double r276531 = r276528 / r276530;
        double r276532 = r276519 + r276531;
        return r276532;
}

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

Original6.0
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 6.0

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

  2. Simplified6.0

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

  4. Applied add-cube-cbrt19.7

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

  5. Applied add-cube-cbrt6.0

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

  6. Applied times-frac6.0

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

  7. Applied unpow-prod-down2.1

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

  8. Applied associate-/l*2.1

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

  10. Applied add-sqr-sqrt2.1

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

  11. Applied unpow-prod-down2.1

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

  12. Simplified2.1

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

  13. Simplified1.0

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

  14. Final simplification1.0

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

Reproduce

herbie shell --seed 2020046 
(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)))