Average Error: 0.0 → 0.0
Time: 3.9s
Precision: 64
\[x \cdot e^{y \cdot y}\]
↓
\[\left(x \cdot {\left(e^{y}\right)}^{\left(\frac{y}{2}\right)}\right) \cdot \left(\sqrt{{\left(\sqrt{e^{y}}\right)}^{\left(\frac{y}{2}\right)} \cdot {\left(\sqrt{e^{y}}\right)}^{\left(\frac{y}{2}\right)}} \cdot \sqrt{{\left(e^{y}\right)}^{\left(\frac{y}{2}\right)}}\right)\]
x \cdot e^{y \cdot y}
↓
\left(x \cdot {\left(e^{y}\right)}^{\left(\frac{y}{2}\right)}\right) \cdot \left(\sqrt{{\left(\sqrt{e^{y}}\right)}^{\left(\frac{y}{2}\right)} \cdot {\left(\sqrt{e^{y}}\right)}^{\left(\frac{y}{2}\right)}} \cdot \sqrt{{\left(e^{y}\right)}^{\left(\frac{y}{2}\right)}}\right)
double f(double x, double y) {
        double r819861 = x;
        double r819862 = y;
        double r819863 = r819862 * r819862;
        double r819864 = exp(r819863);
        double r819865 = r819861 * r819864;
        return r819865;
}
↓
double f(double x, double y) {
        double r819866 = x;
        double r819867 = y;
        double r819868 = exp(r819867);
        double r819869 = 2.0;
        double r819870 = r819867 / r819869;
        double r819871 = pow(r819868, r819870);
        double r819872 = r819866 * r819871;
        double r819873 = sqrt(r819868);
        double r819874 = pow(r819873, r819870);
        double r819875 = r819874 * r819874;
        double r819876 = sqrt(r819875);
        double r819877 = sqrt(r819871);
        double r819878 = r819876 * r819877;
        double r819879 = r819872 * r819878;
        return r819879;
}

Error

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Results

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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-log-exp0.0

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

  4. Applied exp-to-pow0.0

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

  6. Applied add-sqr-sqrt0.0

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

  7. Applied unpow-prod-down0.0

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

  8. Applied associate-*r*0.0

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

  9. Simplified0.0

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

  11. Applied add-sqr-sqrt0.0

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

  12. Simplified0.0

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

  13. Simplified0.0

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

  15. Applied add-sqr-sqrt0.0

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

  16. Applied unpow-prod-down0.0

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

  17. Final simplification0.0

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

Reproduce

herbie shell --seed 2020035 +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))))