Average Error: 0.0 → 0.0
Time: 13.2s
Precision: 64
\[x \cdot e^{y \cdot y}\]
↓
\[\sqrt{e^{y \cdot y}} \cdot \left({\left(e^{y}\right)}^{\left(\frac{y}{2}\right)} \cdot x\right)\]
x \cdot e^{y \cdot y}
↓
\sqrt{e^{y \cdot y}} \cdot \left({\left(e^{y}\right)}^{\left(\frac{y}{2}\right)} \cdot x\right)
double f(double x, double y) {
        double r34973515 = x;
        double r34973516 = y;
        double r34973517 = r34973516 * r34973516;
        double r34973518 = exp(r34973517);
        double r34973519 = r34973515 * r34973518;
        return r34973519;
}
↓
double f(double x, double y) {
        double r34973520 = y;
        double r34973521 = r34973520 * r34973520;
        double r34973522 = exp(r34973521);
        double r34973523 = sqrt(r34973522);
        double r34973524 = exp(r34973520);
        double r34973525 = 2.0;
        double r34973526 = r34973520 / r34973525;
        double r34973527 = pow(r34973524, r34973526);
        double r34973528 = x;
        double r34973529 = r34973527 * r34973528;
        double r34973530 = r34973523 * r34973529;
        return r34973530;
}

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

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

  7. Applied exp-to-pow0.0

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

  8. Applied sqrt-pow10.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(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))))