Average Error: 6.1 → 1.0
Time: 16.1s
Precision: 64
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}\]
↓
\[x + \frac{e^{\left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right) \cdot y}}{\frac{y}{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}}\]
x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
x + \frac{e^{\left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right) \cdot y}}{\frac{y}{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}}
double f(double x, double y, double z) {
        double r468496 = x;
        double r468497 = y;
        double r468498 = z;
        double r468499 = r468498 + r468497;
        double r468500 = r468497 / r468499;
        double r468501 = log(r468500);
        double r468502 = r468497 * r468501;
        double r468503 = exp(r468502);
        double r468504 = r468503 / r468497;
        double r468505 = r468496 + r468504;
        return r468505;
}
↓
double f(double x, double y, double z) {
        double r468506 = x;
        double r468507 = 2.0;
        double r468508 = y;
        double r468509 = cbrt(r468508);
        double r468510 = z;
        double r468511 = r468510 + r468508;
        double r468512 = cbrt(r468511);
        double r468513 = r468509 / r468512;
        double r468514 = log(r468513);
        double r468515 = r468507 * r468514;
        double r468516 = r468515 * r468508;
        double r468517 = exp(r468516);
        double r468518 = pow(r468513, r468508);
        double r468519 = r468508 / r468518;
        double r468520 = r468517 / r468519;
        double r468521 = r468506 + r468520;
        return r468521;
}

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

Original6.1
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 6.1

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

  2. Simplified6.1

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

  4. Applied add-cube-cbrt19.1

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

  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-down2.1

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

  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-exp-log39.4

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

  11. Applied add-exp-log39.4

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

  12. Applied prod-exp39.5

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

  13. Applied add-exp-log39.4

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

  14. Applied add-exp-log35.3

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

  15. Applied prod-exp31.2

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

  16. Applied div-exp31.2

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

  17. Applied pow-exp30.6

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

  18. Simplified1.0

    \[\leadsto x + \frac{e^{\color{blue}{\left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right) \cdot y}}}{\frac{y}{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}}\]

  19. Final simplification1.0

    \[\leadsto x + \frac{e^{\left(2 \cdot \log \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)\right) \cdot y}}{\frac{y}{{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z + y}}\right)}^{y}}}\]

Reproduce

herbie shell --seed 2020042 +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)))