Average Error: 0.0 → 0.0
Time: 4.2s
Precision: 64
\[x \cdot e^{y \cdot y}\]
↓
\[\left(x \cdot {\left(e^{\frac{2}{3} \cdot y}\right)}^{y}\right) \cdot {\left(\sqrt[3]{e^{y}}\right)}^{y}\]
x \cdot e^{y \cdot y}
↓
\left(x \cdot {\left(e^{\frac{2}{3} \cdot y}\right)}^{y}\right) \cdot {\left(\sqrt[3]{e^{y}}\right)}^{y}
double f(double x, double y) {
        double r553623 = x;
        double r553624 = y;
        double r553625 = r553624 * r553624;
        double r553626 = exp(r553625);
        double r553627 = r553623 * r553626;
        return r553627;
}
↓
double f(double x, double y) {
        double r553628 = x;
        double r553629 = 0.6666666666666666;
        double r553630 = y;
        double r553631 = r553629 * r553630;
        double r553632 = exp(r553631);
        double r553633 = pow(r553632, r553630);
        double r553634 = r553628 * r553633;
        double r553635 = exp(r553630);
        double r553636 = cbrt(r553635);
        double r553637 = pow(r553636, r553630);
        double r553638 = r553634 * r553637;
        return r553638;
}

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 add-cube-cbrt0.0

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

  7. Applied unpow-prod-down0.0

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

  8. Applied associate-*r*0.0

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

  10. Applied add-exp-log0.0

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

  11. Applied add-exp-log0.0

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

  12. Applied prod-exp0.0

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

  13. Simplified0.0

    \[\leadsto \left(x \cdot {\left(e^{\color{blue}{\frac{2}{3} \cdot y}}\right)}^{y}\right) \cdot {\left(\sqrt[3]{e^{y}}\right)}^{y}\]

  14. Final simplification0.0

    \[\leadsto \left(x \cdot {\left(e^{\frac{2}{3} \cdot y}\right)}^{y}\right) \cdot {\left(\sqrt[3]{e^{y}}\right)}^{y}\]

Reproduce

herbie shell --seed 1978988140 
(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))))