Average Error: 0.0 → 0.1
Time: 36.8s
Precision: 64
\[e^{\left(x \cdot y\right) \cdot y}\]
↓
\[{\left(\sqrt[3]{e^{\left(y \cdot y\right) \cdot \mathsf{fma}\left(2, x, x\right)}} \cdot \left(\sqrt[3]{e^{\left(y \cdot y\right) \cdot \mathsf{fma}\left(2, x, x\right)}} \cdot \sqrt[3]{e^{\left(y \cdot y\right) \cdot \mathsf{fma}\left(2, x, x\right)}}\right)\right)}^{\frac{1}{3}}\]
e^{\left(x \cdot y\right) \cdot y}
↓
{\left(\sqrt[3]{e^{\left(y \cdot y\right) \cdot \mathsf{fma}\left(2, x, x\right)}} \cdot \left(\sqrt[3]{e^{\left(y \cdot y\right) \cdot \mathsf{fma}\left(2, x, x\right)}} \cdot \sqrt[3]{e^{\left(y \cdot y\right) \cdot \mathsf{fma}\left(2, x, x\right)}}\right)\right)}^{\frac{1}{3}}
double f(double x, double y) {
        double r8960981 = x;
        double r8960982 = y;
        double r8960983 = r8960981 * r8960982;
        double r8960984 = r8960983 * r8960982;
        double r8960985 = exp(r8960984);
        return r8960985;
}
↓
double f(double x, double y) {
        double r8960986 = y;
        double r8960987 = r8960986 * r8960986;
        double r8960988 = 2.0;
        double r8960989 = x;
        double r8960990 = fma(r8960988, r8960989, r8960989);
        double r8960991 = r8960987 * r8960990;
        double r8960992 = exp(r8960991);
        double r8960993 = cbrt(r8960992);
        double r8960994 = r8960993 * r8960993;
        double r8960995 = r8960993 * r8960994;
        double r8960996 = 0.3333333333333333;
        double r8960997 = pow(r8960995, r8960996);
        return r8960997;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

    \[e^{\left(x \cdot y\right) \cdot y}\]
  2. Using strategy rm

  3. Applied add-cbrt-cube0.0

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

  4. Simplified0.1

    \[\leadsto \sqrt[3]{\color{blue}{e^{\left(y \cdot y\right) \cdot \mathsf{fma}\left(2, x, x\right)}}}\]
  5. Using strategy rm

  6. Applied pow1/30.1

    \[\leadsto \color{blue}{{\left(e^{\left(y \cdot y\right) \cdot \mathsf{fma}\left(2, x, x\right)}\right)}^{\frac{1}{3}}}\]
  7. Using strategy rm

  8. Applied add-cube-cbrt0.1

    \[\leadsto {\color{blue}{\left(\left(\sqrt[3]{e^{\left(y \cdot y\right) \cdot \mathsf{fma}\left(2, x, x\right)}} \cdot \sqrt[3]{e^{\left(y \cdot y\right) \cdot \mathsf{fma}\left(2, x, x\right)}}\right) \cdot \sqrt[3]{e^{\left(y \cdot y\right) \cdot \mathsf{fma}\left(2, x, x\right)}}\right)}}^{\frac{1}{3}}\]

  9. Final simplification0.1

    \[\leadsto {\left(\sqrt[3]{e^{\left(y \cdot y\right) \cdot \mathsf{fma}\left(2, x, x\right)}} \cdot \left(\sqrt[3]{e^{\left(y \cdot y\right) \cdot \mathsf{fma}\left(2, x, x\right)}} \cdot \sqrt[3]{e^{\left(y \cdot y\right) \cdot \mathsf{fma}\left(2, x, x\right)}}\right)\right)}^{\frac{1}{3}}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  (exp (* (* x y) y)))