Average Error: 5.9 → 1.1
Time: 17.2s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[x + \frac{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y} \cdot \left({\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y}\right)}{y}\]
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({\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{y + z}}\right)}^{y}\right)}{y}
double f(double x, double y, double z) {
        double r371658 = x;
        double r371659 = y;
        double r371660 = z;
        double r371661 = r371660 + r371659;
        double r371662 = r371659 / r371661;
        double r371663 = log(r371662);
        double r371664 = r371659 * r371663;
        double r371665 = exp(r371664);
        double r371666 = r371665 / r371659;
        double r371667 = r371658 + r371666;
        return r371667;
}
↓
double f(double x, double y, double z) {
        double r371668 = x;
        double r371669 = y;
        double r371670 = cbrt(r371669);
        double r371671 = z;
        double r371672 = r371669 + r371671;
        double r371673 = cbrt(r371672);
        double r371674 = r371670 / r371673;
        double r371675 = pow(r371674, r371669);
        double r371676 = r371675 * r371675;
        double r371677 = r371675 * r371676;
        double r371678 = r371677 / r371669;
        double r371679 = r371668 + r371678;
        return r371679;
}

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

Original5.9
Target1.1
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.115415759790762719541517221498726780517 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Initial program 5.9

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

  2. Simplified5.9

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

  4. Applied add-cube-cbrt19.1

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

  5. Applied add-cube-cbrt5.9

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

  6. Applied times-frac5.9

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

  7. Applied unpow-prod-down2.3

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

  8. Simplified2.3

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

  10. Applied unpow-prod-down1.1

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

  11. Final simplification1.1

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

Reproduce

herbie shell --seed 2019194 
(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)))