Average Error: 0.0 → 0.0
Time: 10.2s
Precision: 64
\[x \cdot e^{y \cdot y}\]
↓
\[\left(\left({\left(\sqrt{e^{y}}\right)}^{\left(\frac{y}{2}\right)} \cdot \sqrt{e^{y \cdot y}}\right) \cdot x\right) \cdot {\left(\sqrt{e^{y}}\right)}^{\left(\frac{y}{2}\right)}\]
x \cdot e^{y \cdot y}
↓
\left(\left({\left(\sqrt{e^{y}}\right)}^{\left(\frac{y}{2}\right)} \cdot \sqrt{e^{y \cdot y}}\right) \cdot x\right) \cdot {\left(\sqrt{e^{y}}\right)}^{\left(\frac{y}{2}\right)}
double f(double x, double y) {
        double r559352 = x;
        double r559353 = y;
        double r559354 = r559353 * r559353;
        double r559355 = exp(r559354);
        double r559356 = r559352 * r559355;
        return r559356;
}
↓
double f(double x, double y) {
        double r559357 = y;
        double r559358 = exp(r559357);
        double r559359 = sqrt(r559358);
        double r559360 = 2.0;
        double r559361 = r559357 / r559360;
        double r559362 = pow(r559359, r559361);
        double r559363 = r559357 * r559357;
        double r559364 = exp(r559363);
        double r559365 = sqrt(r559364);
        double r559366 = r559362 * r559365;
        double r559367 = x;
        double r559368 = r559366 * r559367;
        double r559369 = r559368 * r559362;
        return r559369;
}

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-sqr-sqrt0.0

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

  4. Applied associate-*r*0.0

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

  6. Applied add-log-exp0.0

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

  7. Applied exp-to-pow0.0

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

  8. Applied sqrt-pow10.0

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

  10. Applied add-sqr-sqrt0.0

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

  11. Applied unpow-prod-down0.0

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

  12. Applied associate-*r*0.0

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

  13. Simplified0.0

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

  15. Applied associate-*r*0.0

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

  16. Final simplification0.0

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

Reproduce

herbie shell --seed 2019305 
(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))))