Average Error: 0.0 → 0.0
Time: 1.8s
Precision: 64
\[x \cdot e^{y \cdot y}\]
↓
\[x \cdot \left({\left(e^{y}\right)}^{\left(\frac{y}{2}\right)} \cdot {\left(e^{y}\right)}^{\left(\frac{y}{2}\right)}\right)\]
x \cdot e^{y \cdot y}
↓
x \cdot \left({\left(e^{y}\right)}^{\left(\frac{y}{2}\right)} \cdot {\left(e^{y}\right)}^{\left(\frac{y}{2}\right)}\right)
double f(double x, double y) {
        double r733702 = x;
        double r733703 = y;
        double r733704 = r733703 * r733703;
        double r733705 = exp(r733704);
        double r733706 = r733702 * r733705;
        return r733706;
}
↓
double f(double x, double y) {
        double r733707 = x;
        double r733708 = y;
        double r733709 = exp(r733708);
        double r733710 = 2.0;
        double r733711 = r733708 / r733710;
        double r733712 = pow(r733709, r733711);
        double r733713 = r733712 * r733712;
        double r733714 = r733707 * r733713;
        return r733714;
}

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. Simplified0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020001 +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))))