Average Error: 0.0 → 0.0
Time: 3.1s
Precision: 64
\[x \cdot e^{y \cdot y}\]
↓
\[\left(x \cdot {\left({\left(\sqrt{\sqrt[3]{e^{y}}}\right)}^{3} \cdot \sqrt{\sqrt[3]{e^{y}}}\right)}^{y}\right) \cdot {\left(\sqrt[3]{e^{y}}\right)}^{y}\]
x \cdot e^{y \cdot y}
↓
\left(x \cdot {\left({\left(\sqrt{\sqrt[3]{e^{y}}}\right)}^{3} \cdot \sqrt{\sqrt[3]{e^{y}}}\right)}^{y}\right) \cdot {\left(\sqrt[3]{e^{y}}\right)}^{y}
double code(double x, double y) {
	return (x * exp((y * y)));
}
↓
double code(double x, double y) {
	return ((x * pow((pow(sqrt(cbrt(exp(y))), 3.0) * sqrt(cbrt(exp(y)))), y)) * pow(cbrt(exp(y)), y));
}

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-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-cube-cbrt0.0

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

  7. Applied unpow-prod-down0.0

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

  8. Applied associate-*r*0.0

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

  10. Applied add-sqr-sqrt0.0

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

  11. Applied associate-*r*0.0

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2020066 +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))))