Average Error: 0.0 → 0.0
Time: 2.2s
Precision: binary64
\[x \cdot e^{y \cdot y}\]
↓
\[\left({\left(e^{y}\right)}^{\left(y \cdot 0.6666666666666666\right)} \cdot x\right) \cdot \sqrt[3]{{\left(e^{y}\right)}^{y}}\]
x \cdot e^{y \cdot y}
↓
\left({\left(e^{y}\right)}^{\left(y \cdot 0.6666666666666666\right)} \cdot x\right) \cdot \sqrt[3]{{\left(e^{y}\right)}^{y}}
(FPCore (x y) :precision binary64 (* x (exp (* y y))))
↓
(FPCore (x y)
 :precision binary64
 (* (* (pow (exp y) (* y 0.6666666666666666)) x) (cbrt (pow (exp y) y))))
double code(double x, double y) {
	return ((double) (x * ((double) exp(((double) (y * y))))));
}
↓
double code(double x, double y) {
	return ((double) (((double) (((double) pow(((double) exp(y)), ((double) (y * 0.6666666666666666)))) * x)) * ((double) cbrt(((double) pow(((double) exp(y)), y))))));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

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 Error: 0.0 bits

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

  3. Applied add-log-expError: 0.0 bits

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

  4. Applied exp-to-powError: 0.0 bits

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

  6. Applied add-cbrt-cubeError: 0.1 bits

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

  7. SimplifiedError: 0.1 bits

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

  9. Applied cube-multError: 0.1 bits

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

  10. Applied cbrt-prodError: 0.1 bits

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

  11. Applied associate-*r*Error: 0.1 bits

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

  12. SimplifiedError: 0.1 bits

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

  14. Applied cbrt-prodError: 0.0 bits

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

  15. Applied associate-*r*Error: 0.0 bits

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

  16. SimplifiedError: 0.0 bits

    \[\leadsto \color{blue}{\left({\left(e^{y}\right)}^{\left(y \cdot 0.6666666666666666\right)} \cdot x\right)} \cdot \sqrt[3]{{\left(e^{y}\right)}^{y}}\]

  17. Final simplificationError: 0.0 bits

    \[\leadsto \left({\left(e^{y}\right)}^{\left(y \cdot 0.6666666666666666\right)} \cdot x\right) \cdot \sqrt[3]{{\left(e^{y}\right)}^{y}}\]

Reproduce

herbie shell --seed 2020205 
(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))))