Average Error: 0.0 → 0.0
Time: 3.7s
Precision: 64
\[x \cdot e^{y \cdot y}\]
↓
\[\left(\left(x \cdot {\left({\left(e^{y \cdot y}\right)}^{\frac{2}{3}}\right)}^{\frac{2}{3}}\right) \cdot {\left(\sqrt[3]{e^{y \cdot y}}\right)}^{\frac{2}{3}}\right) \cdot \sqrt[3]{e^{y \cdot y}}\]
x \cdot e^{y \cdot y}
↓
\left(\left(x \cdot {\left({\left(e^{y \cdot y}\right)}^{\frac{2}{3}}\right)}^{\frac{2}{3}}\right) \cdot {\left(\sqrt[3]{e^{y \cdot y}}\right)}^{\frac{2}{3}}\right) \cdot \sqrt[3]{e^{y \cdot y}}
double f(double x, double y) {
        double r728784 = x;
        double r728785 = y;
        double r728786 = r728785 * r728785;
        double r728787 = exp(r728786);
        double r728788 = r728784 * r728787;
        return r728788;
}
↓
double f(double x, double y) {
        double r728789 = x;
        double r728790 = y;
        double r728791 = r728790 * r728790;
        double r728792 = exp(r728791);
        double r728793 = 0.6666666666666666;
        double r728794 = pow(r728792, r728793);
        double r728795 = pow(r728794, r728793);
        double r728796 = r728789 * r728795;
        double r728797 = cbrt(r728792);
        double r728798 = pow(r728797, r728793);
        double r728799 = r728796 * r728798;
        double r728800 = r728799 * r728797;
        return r728800;
}

Error

Bits error versus x

Bits error versus y

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Results

Enter valid numbers for all inputs

Target

Original0.0
Target0.0
Herbie0.0
\[x \cdot {\left(e^{y}\right)}^{y}\]

Derivation

  1. Initial program 0.0

    \[x \cdot e^{y \cdot y}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.1

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

  4. Applied associate-*r*0.1

    \[\leadsto \color{blue}{\left(x \cdot \left(\sqrt[3]{e^{y \cdot y}} \cdot \sqrt[3]{e^{y \cdot y}}\right)\right) \cdot \sqrt[3]{e^{y \cdot y}}}\]
  5. Using strategy rm

  6. Applied pow1/30.0

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

  7. Applied pow1/30.0

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

  8. Applied pow-prod-up0.0

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

  9. Simplified0.0

    \[\leadsto \left(x \cdot {\left(e^{y \cdot y}\right)}^{\color{blue}{\frac{2}{3}}}\right) \cdot \sqrt[3]{e^{y \cdot y}}\]
  10. Using strategy rm

  11. Applied add-cube-cbrt0.1

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

  12. Applied unpow-prod-down0.1

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

  13. Applied associate-*r*0.1

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

  14. Simplified0.0

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

  15. Final simplification0.0

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

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (x y)
  :name "Data.Number.Erf:$dmerfcx from erf-2.0.0.0"
  :precision binary64

  :herbie-target
  (* x (pow (exp y) y))

  (* x (exp (* y y))))