Average Error: 0.0 → 0.1
Time: 18.6s
Precision: 64
\[x \cdot e^{y \cdot y}\]
↓
\[x \cdot \sqrt[3]{{\left(e^{y}\right)}^{y} \cdot \left(e^{y \cdot y} \cdot {\left(e^{y}\right)}^{y}\right)}\]
x \cdot e^{y \cdot y}
↓
x \cdot \sqrt[3]{{\left(e^{y}\right)}^{y} \cdot \left(e^{y \cdot y} \cdot {\left(e^{y}\right)}^{y}\right)}
double f(double x, double y) {
        double r38310191 = x;
        double r38310192 = y;
        double r38310193 = r38310192 * r38310192;
        double r38310194 = exp(r38310193);
        double r38310195 = r38310191 * r38310194;
        return r38310195;
}
↓
double f(double x, double y) {
        double r38310196 = x;
        double r38310197 = y;
        double r38310198 = exp(r38310197);
        double r38310199 = pow(r38310198, r38310197);
        double r38310200 = r38310197 * r38310197;
        double r38310201 = exp(r38310200);
        double r38310202 = r38310201 * r38310199;
        double r38310203 = r38310199 * r38310202;
        double r38310204 = cbrt(r38310203);
        double r38310205 = r38310196 * r38310204;
        return r38310205;
}

Error

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Bits error versus y

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Results

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Target

Original0.0
Target0.0
Herbie0.1
\[x \cdot {\left(e^{y}\right)}^{y}\]

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-cbrt-cube0.1

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

  8. Applied *-un-lft-identity0.1

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

  9. Applied unpow-prod-down0.1

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

  10. Applied associate-*r*0.1

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019170 
(FPCore (x y)
  :name "Data.Number.Erf:$dmerfcx from erf-2.0.0.0"

  :herbie-target
  (* x (pow (exp y) y))

  (* x (exp (* y y))))