Average Error: 0.0 → 0.0
Time: 2.7s
Precision: binary64
\[x \cdot e^{y \cdot y}\]
↓
\[\left(x \cdot {\left(e^{y \cdot \frac{1}{2}}\right)}^{y}\right) \cdot {\left(\sqrt{e^{y}}\right)}^{y}\]
x \cdot e^{y \cdot y}
↓
\left(x \cdot {\left(e^{y \cdot \frac{1}{2}}\right)}^{y}\right) \cdot {\left(\sqrt{e^{y}}\right)}^{y}
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) pow(((double) exp(((double) (y * 0.5)))), y)))) * ((double) pow(((double) sqrt(((double) exp(y)))), y))));
}

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. Simplified0.0

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

  4. Applied add-sqr-sqrt0.0

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

  5. Applied unpow-prod-down0.0

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

  6. Applied associate-*r*0.0

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

  8. Applied add-exp-log0.0

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

  9. Simplified0.0

    \[\leadsto \left(x \cdot {\left(e^{\color{blue}{y \cdot \frac{1}{2}}}\right)}^{y}\right) \cdot {\left(\sqrt{e^{y}}\right)}^{y}\]

  10. Final simplification0.0

    \[\leadsto \left(x \cdot {\left(e^{y \cdot \frac{1}{2}}\right)}^{y}\right) \cdot {\left(\sqrt{e^{y}}\right)}^{y}\]

Reproduce

herbie shell --seed 2020182 
(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))))