Average Error: 6.3 → 1.1
Time: 20.2s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[x + \frac{{\left(\left|\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right|\right)}^{y} \cdot {\left(\left|\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right|\right)}^{y}}{\frac{y}{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}}\]
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 r338337 = x;
        double r338338 = y;
        double r338339 = z;
        double r338340 = r338339 + r338338;
        double r338341 = r338338 / r338340;
        double r338342 = log(r338341);
        double r338343 = r338338 * r338342;
        double r338344 = exp(r338343);
        double r338345 = r338344 / r338338;
        double r338346 = r338337 + r338345;
        return r338346;
}
↓
double f(double x, double y, double z) {
        double r338347 = x;
        double r338348 = y;
        double r338349 = cbrt(r338348);
        double r338350 = z;
        double r338351 = r338350 + r338348;
        double r338352 = cbrt(r338351);
        double r338353 = r338349 / r338352;
        double r338354 = fabs(r338353);
        double r338355 = pow(r338354, r338348);
        double r338356 = r338355 * r338355;
        double r338357 = pow(r338353, r338348);
        double r338358 = r338348 / r338357;
        double r338359 = r338356 / r338358;
        double r338360 = r338347 + r338359;
        return r338360;
}

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

Original6.3
Target1.1
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} \lt 7.1154157598 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array}\]

Derivation

  1. Initial program 6.3

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

  2. Simplified6.3

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

  4. Applied add-cube-cbrt19.2

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

  5. Applied add-cube-cbrt6.3

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

  6. Applied times-frac6.3

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

  7. Applied unpow-prod-down2.2

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

  8. Applied associate-/l*2.2

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

  10. Applied add-sqr-sqrt2.2

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

  11. Applied unpow-prod-down2.2

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

  12. Simplified2.2

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

  13. Simplified1.1

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

  14. Final simplification1.1

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

Reproduce

herbie shell --seed 2019198 
(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)))