Average Error: 0.0 → 0.0
Time: 4.2s
Precision: 64
\[x \cdot e^{y \cdot y}\]
↓
\[\left(\left(x \cdot {\left(e^{y \cdot y}\right)}^{\frac{1}{3}}\right) \cdot \sqrt{{\left(e^{y \cdot y}\right)}^{\frac{2}{3}}}\right) \cdot \sqrt[3]{e^{y \cdot y}}\]
x \cdot e^{y \cdot y}
↓
\left(\left(x \cdot {\left(e^{y \cdot y}\right)}^{\frac{1}{3}}\right) \cdot \sqrt{{\left(e^{y \cdot y}\right)}^{\frac{2}{3}}}\right) \cdot \sqrt[3]{e^{y \cdot y}}
double f(double x, double y) {
        double r1356522 = x;
        double r1356523 = y;
        double r1356524 = r1356523 * r1356523;
        double r1356525 = exp(r1356524);
        double r1356526 = r1356522 * r1356525;
        return r1356526;
}
↓
double f(double x, double y) {
        double r1356527 = x;
        double r1356528 = y;
        double r1356529 = r1356528 * r1356528;
        double r1356530 = exp(r1356529);
        double r1356531 = 0.3333333333333333;
        double r1356532 = pow(r1356530, r1356531);
        double r1356533 = r1356527 * r1356532;
        double r1356534 = 0.6666666666666666;
        double r1356535 = pow(r1356530, r1356534);
        double r1356536 = sqrt(r1356535);
        double r1356537 = r1356533 * r1356536;
        double r1356538 = cbrt(r1356530);
        double r1356539 = r1356537 * r1356538;
        return r1356539;
}

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-cube-cbrt0.0

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

  4. Applied associate-*r*0.0

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

  6. Applied pow1/30.0

    \[\leadsto \left(x \cdot \left(\sqrt[3]{e^{y \cdot y}} \cdot \color{blue}{{\left(e^{y \cdot y}\right)}^{\frac{1}{3}}}\right)\right) \cdot \sqrt[3]{e^{y \cdot y}}\]

  7. Applied pow1/30.0

    \[\leadsto \left(x \cdot \left(\color{blue}{{\left(e^{y \cdot y}\right)}^{\frac{1}{3}}} \cdot {\left(e^{y \cdot y}\right)}^{\frac{1}{3}}\right)\right) \cdot \sqrt[3]{e^{y \cdot y}}\]

  8. Applied pow-prod-up0.0

    \[\leadsto \left(x \cdot \color{blue}{{\left(e^{y \cdot y}\right)}^{\left(\frac{1}{3} + \frac{1}{3}\right)}}\right) \cdot \sqrt[3]{e^{y \cdot y}}\]

  9. Simplified0.0

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

  11. Applied add-sqr-sqrt0.0

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

  12. Applied associate-*r*0.0

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

  13. Simplified0.0

    \[\leadsto \left(\color{blue}{\left(x \cdot {\left(e^{y \cdot y}\right)}^{\frac{1}{3}}\right)} \cdot \sqrt{{\left(e^{y \cdot y}\right)}^{\frac{2}{3}}}\right) \cdot \sqrt[3]{e^{y \cdot y}}\]

  14. Final simplification0.0

    \[\leadsto \left(\left(x \cdot {\left(e^{y \cdot y}\right)}^{\frac{1}{3}}\right) \cdot \sqrt{{\left(e^{y \cdot y}\right)}^{\frac{2}{3}}}\right) \cdot \sqrt[3]{e^{y \cdot y}}\]

Reproduce

herbie shell --seed 2020025 +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))))