Average Error: 0.0 → 0.0
Time: 9.5s
Precision: 64
\[x \cdot e^{y \cdot y}\]
↓
\[x \cdot \left(\left({\left(\sqrt{\sqrt{e^{y}}}\right)}^{y} \cdot \sqrt{{\left(\sqrt{e^{y}}\right)}^{y}}\right) \cdot \left({\left(\sqrt{\sqrt{e^{y}}}\right)}^{y} \cdot \sqrt{{\left(\sqrt{e^{y}}\right)}^{y}}\right)\right)\]
x \cdot e^{y \cdot y}
↓
x \cdot \left(\left({\left(\sqrt{\sqrt{e^{y}}}\right)}^{y} \cdot \sqrt{{\left(\sqrt{e^{y}}\right)}^{y}}\right) \cdot \left({\left(\sqrt{\sqrt{e^{y}}}\right)}^{y} \cdot \sqrt{{\left(\sqrt{e^{y}}\right)}^{y}}\right)\right)
double f(double x, double y) {
        double r842525 = x;
        double r842526 = y;
        double r842527 = r842526 * r842526;
        double r842528 = exp(r842527);
        double r842529 = r842525 * r842528;
        return r842529;
}
↓
double f(double x, double y) {
        double r842530 = x;
        double r842531 = y;
        double r842532 = exp(r842531);
        double r842533 = sqrt(r842532);
        double r842534 = sqrt(r842533);
        double r842535 = pow(r842534, r842531);
        double r842536 = pow(r842533, r842531);
        double r842537 = sqrt(r842536);
        double r842538 = r842535 * r842537;
        double r842539 = r842538 * r842538;
        double r842540 = r842530 * r842539;
        return r842540;
}

Error

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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-sqr-sqrt0.0

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

  7. Applied unpow-prod-down0.0

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

  9. Applied add-sqr-sqrt0.0

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

  10. Applied add-sqr-sqrt0.0

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

  11. Applied sqrt-prod0.0

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

  12. Applied unpow-prod-down0.0

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

  13. Applied unswap-sqr0.0

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

  14. Final simplification0.0

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

Reproduce

herbie shell --seed 2020045 +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))))