Average Error: 6.0 → 1.0
Time: 12.8s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[x + \left({\left(\left|\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right|\right)}^{y} \cdot {\left(\left|\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right|\right)}^{y}\right) \cdot \frac{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
x + \left({\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}\right) \cdot \frac{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}
double f(double x, double y, double z) {
        double r383368 = x;
        double r383369 = y;
        double r383370 = z;
        double r383371 = r383370 + r383369;
        double r383372 = r383369 / r383371;
        double r383373 = log(r383372);
        double r383374 = r383369 * r383373;
        double r383375 = exp(r383374);
        double r383376 = r383375 / r383369;
        double r383377 = r383368 + r383376;
        return r383377;
}
↓
double f(double x, double y, double z) {
        double r383378 = x;
        double r383379 = y;
        double r383380 = cbrt(r383379);
        double r383381 = z;
        double r383382 = r383381 + r383379;
        double r383383 = cbrt(r383382);
        double r383384 = r383380 / r383383;
        double r383385 = fabs(r383384);
        double r383386 = pow(r383385, r383379);
        double r383387 = r383386 * r383386;
        double r383388 = pow(r383384, r383379);
        double r383389 = r383388 / r383379;
        double r383390 = r383387 * r383389;
        double r383391 = r383378 + r383390;
        return r383391;
}

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 *-un-lft-identity6.0

    \[\leadsto x + \frac{{\left(\frac{y}{z + y}\right)}^{y}}{\color{blue}{1 \cdot y}}\]

  5. Applied add-cube-cbrt19.7

    \[\leadsto x + \frac{{\left(\frac{y}{\color{blue}{\left(\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}\right) \cdot \sqrt[3]{z + y}}}\right)}^{y}}{1 \cdot y}\]

  6. Applied add-cube-cbrt6.0

    \[\leadsto x + \frac{{\left(\frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\left(\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}\right) \cdot \sqrt[3]{z + y}}\right)}^{y}}{1 \cdot y}\]

  7. Applied times-frac6.0

    \[\leadsto x + \frac{{\color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}}^{y}}{1 \cdot y}\]

  8. Applied unpow-prod-down2.1

    \[\leadsto x + \frac{\color{blue}{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}\right)}^{y} \cdot {\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}}{1 \cdot y}\]

  9. Applied times-frac2.1

    \[\leadsto x + \color{blue}{\frac{{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}\right)}^{y}}{1} \cdot \frac{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}{y}}\]

  10. Simplified2.1

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

  12. Applied add-sqr-sqrt2.1

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

  13. Applied unpow-prod-down2.1

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

  14. Simplified2.1

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

  15. Simplified1.0

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

  16. Final simplification1.0

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(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)))