Average Error: 0.0 → 0.1
Time: 3.3s
Precision: 64
\[x \cdot e^{y \cdot y}\]
↓
\[\left(x \cdot \left(\sqrt{e^{y \cdot y}} \cdot \left|\left(\sqrt[3]{\sqrt[3]{e^{y \cdot y}}} \cdot \sqrt[3]{\sqrt[3]{e^{y \cdot y}}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{y \cdot y}}}\right|\right)\right) \cdot \sqrt{\sqrt[3]{e^{y \cdot y}}}\]
x \cdot e^{y \cdot y}
↓
\left(x \cdot \left(\sqrt{e^{y \cdot y}} \cdot \left|\left(\sqrt[3]{\sqrt[3]{e^{y \cdot y}}} \cdot \sqrt[3]{\sqrt[3]{e^{y \cdot y}}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{y \cdot y}}}\right|\right)\right) \cdot \sqrt{\sqrt[3]{e^{y \cdot y}}}
double f(double x, double y) {
        double r880065 = x;
        double r880066 = y;
        double r880067 = r880066 * r880066;
        double r880068 = exp(r880067);
        double r880069 = r880065 * r880068;
        return r880069;
}
↓
double f(double x, double y) {
        double r880070 = x;
        double r880071 = y;
        double r880072 = r880071 * r880071;
        double r880073 = exp(r880072);
        double r880074 = sqrt(r880073);
        double r880075 = cbrt(r880073);
        double r880076 = cbrt(r880075);
        double r880077 = r880076 * r880076;
        double r880078 = r880077 * r880076;
        double r880079 = fabs(r880078);
        double r880080 = r880074 * r880079;
        double r880081 = r880070 * r880080;
        double r880082 = sqrt(r880075);
        double r880083 = r880081 * r880082;
        return r880083;
}

Error

Bits error versus x

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-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-cube-cbrt0.0

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

  7. Applied sqrt-prod0.0

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

  8. Applied associate-*r*0.0

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

  9. Simplified0.0

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

  11. Applied add-cube-cbrt0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019353 
(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))))