Average Error: 0.0 → 0.1
Time: 3.9s
Precision: 64
\[x \cdot e^{y \cdot y}\]
↓
\[\left(x \cdot \left(\sqrt{e^{y \cdot y}} \cdot \left|\left(\sqrt[3]{\sqrt[3]{e^{y \cdot y}}} \cdot \sqrt[3]{\sqrt[3]{e^{y \cdot y}}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{y \cdot y}}}\right|\right)\right) \cdot \sqrt{\sqrt[3]{e^{y \cdot y}}}\]
x \cdot e^{y \cdot y}
↓
\left(x \cdot \left(\sqrt{e^{y \cdot y}} \cdot \left|\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)\right) \cdot \sqrt{\sqrt[3]{e^{y \cdot y}}}
double f(double x, double y) {
        double r908642 = x;
        double r908643 = y;
        double r908644 = r908643 * r908643;
        double r908645 = exp(r908644);
        double r908646 = r908642 * r908645;
        return r908646;
}
↓
double f(double x, double y) {
        double r908647 = x;
        double r908648 = y;
        double r908649 = r908648 * r908648;
        double r908650 = exp(r908649);
        double r908651 = sqrt(r908650);
        double r908652 = cbrt(r908650);
        double r908653 = cbrt(r908652);
        double r908654 = r908653 * r908653;
        double r908655 = r908654 * r908653;
        double r908656 = fabs(r908655);
        double r908657 = r908651 * r908656;
        double r908658 = r908647 * r908657;
        double r908659 = sqrt(r908652);
        double r908660 = r908658 * r908659;
        return r908660;
}

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-cbrt0.0

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

  7. Applied sqrt-prod0.0

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

  8. Applied associate-*r*0.0

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

  9. Simplified0.0

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

  11. Applied add-cube-cbrt0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019353 +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))))