Average Error: 0.0 → 0.0
Time: 2.3s
Precision: 64
\[x \cdot e^{y \cdot y}\]
↓
\[\left(x \cdot {\left({\left(e^{y}\right)}^{y}\right)}^{\frac{1}{2}}\right) \cdot \left({\left(e^{y}\right)}^{\left(\frac{y}{4}\right)} \cdot {\left(e^{y}\right)}^{\left(\frac{y}{4}\right)}\right)\]
x \cdot e^{y \cdot y}
↓
\left(x \cdot {\left({\left(e^{y}\right)}^{y}\right)}^{\frac{1}{2}}\right) \cdot \left({\left(e^{y}\right)}^{\left(\frac{y}{4}\right)} \cdot {\left(e^{y}\right)}^{\left(\frac{y}{4}\right)}\right)
double code(double x, double y) {
	return (x * exp((y * y)));
}
↓
double code(double x, double y) {
	return ((x * pow(pow(exp(y), y), 0.5)) * (pow(exp(y), (y / 4.0)) * pow(exp(y), (y / 4.0))));
}

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 exp-prod0.0

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

  5. Applied sqr-pow0.0

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

  6. Applied associate-*r*0.0

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

  8. Applied div-inv0.0

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

  9. Applied pow-unpow0.0

    \[\leadsto \left(x \cdot \color{blue}{{\left({\left(e^{y}\right)}^{y}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot {\left(e^{y}\right)}^{\left(\frac{y}{2}\right)}\]
  10. Using strategy rm

  11. Applied sqr-pow0.0

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

  12. Simplified0.0

    \[\leadsto \left(x \cdot {\left({\left(e^{y}\right)}^{y}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{{\left(e^{y}\right)}^{\left(\frac{y}{4}\right)}} \cdot {\left(e^{y}\right)}^{\left(\frac{\frac{y}{2}}{2}\right)}\right)\]

  13. Simplified0.0

    \[\leadsto \left(x \cdot {\left({\left(e^{y}\right)}^{y}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(e^{y}\right)}^{\left(\frac{y}{4}\right)} \cdot \color{blue}{{\left(e^{y}\right)}^{\left(\frac{y}{4}\right)}}\right)\]

  14. Final simplification0.0

    \[\leadsto \left(x \cdot {\left({\left(e^{y}\right)}^{y}\right)}^{\frac{1}{2}}\right) \cdot \left({\left(e^{y}\right)}^{\left(\frac{y}{4}\right)} \cdot {\left(e^{y}\right)}^{\left(\frac{y}{4}\right)}\right)\]

Reproduce

herbie shell --seed 2020078 +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))))