Average Error: 0.0 → 0.0
Time: 2.3s
Precision: binary64
\[x \cdot e^{y \cdot y}\]
↓
\[\left(x \cdot \left({\left(\sqrt{\sqrt{e^{y}}}\right)}^{y} \cdot {\left(\sqrt{\sqrt{e^{y}}}\right)}^{y}\right)\right) \cdot {\left(e^{y}\right)}^{\left(\frac{y}{2}\right)}\]
x \cdot e^{y \cdot y}
↓
\left(x \cdot \left({\left(\sqrt{\sqrt{e^{y}}}\right)}^{y} \cdot {\left(\sqrt{\sqrt{e^{y}}}\right)}^{y}\right)\right) \cdot {\left(e^{y}\right)}^{\left(\frac{y}{2}\right)}
double code(double x, double y) {
	return ((double) (x * ((double) exp(((double) (y * y))))));
}
↓
double code(double x, double y) {
	return ((double) (((double) (x * ((double) (((double) pow(((double) sqrt(((double) sqrt(((double) exp(y)))))), y)) * ((double) pow(((double) sqrt(((double) sqrt(((double) exp(y)))))), y)))))) * ((double) pow(((double) exp(y)), ((double) (y / 2.0))))));
}

Error

Bits error versus x

Bits error versus y

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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 sqr-pow0.0

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

  7. Applied associate-*r*0.0

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

  8. Simplified0.0

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

  10. Applied add-sqr-sqrt0.0

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

  11. Applied sqrt-prod0.0

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

  12. Applied unpow-prod-down0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2020175 
(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))))