Average Error: 0.0 → 0.1
Time: 24.0s
Precision: 64
\[x \cdot e^{y \cdot y}\]
↓
\[\left(x \cdot \left(\left|\sqrt[3]{e^{y \cdot y}}\right| \cdot \sqrt{\left(\sqrt[3]{\sqrt[3]{e^{y \cdot y}}} \cdot \sqrt[3]{\sqrt[3]{e^{y \cdot y}}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{y \cdot y}}}}\right)\right) \cdot \sqrt{e^{y \cdot y}}\]
x \cdot e^{y \cdot y}
↓
\left(x \cdot \left(\left|\sqrt[3]{e^{y \cdot y}}\right| \cdot \sqrt{\left(\sqrt[3]{\sqrt[3]{e^{y \cdot y}}} \cdot \sqrt[3]{\sqrt[3]{e^{y \cdot y}}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{y \cdot y}}}}\right)\right) \cdot \sqrt{e^{y \cdot y}}
double f(double x, double y) {
        double r600871 = x;
        double r600872 = y;
        double r600873 = r600872 * r600872;
        double r600874 = exp(r600873);
        double r600875 = r600871 * r600874;
        return r600875;
}
↓
double f(double x, double y) {
        double r600876 = x;
        double r600877 = y;
        double r600878 = r600877 * r600877;
        double r600879 = exp(r600878);
        double r600880 = cbrt(r600879);
        double r600881 = fabs(r600880);
        double r600882 = cbrt(r600880);
        double r600883 = r600882 * r600882;
        double r600884 = r600883 * r600882;
        double r600885 = sqrt(r600884);
        double r600886 = r600881 * r600885;
        double r600887 = r600876 * r600886;
        double r600888 = sqrt(r600879);
        double r600889 = r600887 * r600888;
        return r600889;
}

Error

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Bits error versus y

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Results

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Target

Original0.0
Target0.0
Herbie0.1
\[x \cdot {\left(e^{y}\right)}^{y}\]

Derivation

  1. Initial program 0.0

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

  3. Applied add-sqr-sqrt0.0

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

  4. Applied associate-*r*0.0

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

  6. Applied add-cube-cbrt0.1

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

  7. Applied sqrt-prod0.1

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

  8. Simplified0.1

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

  10. Applied add-cube-cbrt0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019326 
(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))))