Average Error: 52.0 → 52.0
Time: 2.5s
Precision: 64
\[x = 10864 \land y = 18817\]
\[\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right)\]
↓
\[\mathsf{fma}\left(2 \cdot y, y, \sqrt[3]{{\left(9 \cdot {x}^{4} - {y}^{4}\right)}^{3}}\right)\]
\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right)
↓
\mathsf{fma}\left(2 \cdot y, y, \sqrt[3]{{\left(9 \cdot {x}^{4} - {y}^{4}\right)}^{3}}\right)
double f(double x, double y) {
        double r63738 = 9.0;
        double r63739 = x;
        double r63740 = 4.0;
        double r63741 = pow(r63739, r63740);
        double r63742 = r63738 * r63741;
        double r63743 = y;
        double r63744 = pow(r63743, r63740);
        double r63745 = r63742 - r63744;
        double r63746 = 2.0;
        double r63747 = r63743 * r63743;
        double r63748 = r63746 * r63747;
        double r63749 = r63745 + r63748;
        return r63749;
}
↓
double f(double x, double y) {
        double r63750 = 2.0;
        double r63751 = y;
        double r63752 = r63750 * r63751;
        double r63753 = 9.0;
        double r63754 = x;
        double r63755 = 4.0;
        double r63756 = pow(r63754, r63755);
        double r63757 = r63753 * r63756;
        double r63758 = pow(r63751, r63755);
        double r63759 = r63757 - r63758;
        double r63760 = 3.0;
        double r63761 = pow(r63759, r63760);
        double r63762 = cbrt(r63761);
        double r63763 = fma(r63752, r63751, r63762);
        return r63763;
}

Error

Derivation

  1. Initial program 52.0

    \[\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right)\]

  2. Simplified52.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot y, y, 9 \cdot {x}^{4} - {y}^{4}\right)}\]
  3. Using strategy rm

  4. Applied add-cbrt-cube52.0

    \[\leadsto \mathsf{fma}\left(2 \cdot y, y, \color{blue}{\sqrt[3]{\left(\left(9 \cdot {x}^{4} - {y}^{4}\right) \cdot \left(9 \cdot {x}^{4} - {y}^{4}\right)\right) \cdot \left(9 \cdot {x}^{4} - {y}^{4}\right)}}\right)\]

  5. Simplified52.0

    \[\leadsto \mathsf{fma}\left(2 \cdot y, y, \sqrt[3]{\color{blue}{{\left(9 \cdot {x}^{4} - {y}^{4}\right)}^{3}}}\right)\]

  6. Final simplification52.0

    \[\leadsto \mathsf{fma}\left(2 \cdot y, y, \sqrt[3]{{\left(9 \cdot {x}^{4} - {y}^{4}\right)}^{3}}\right)\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(FPCore (x y)
  :name "From Rump in a 1983 paper"
  :precision binary64
  :pre (and (== x 10864) (== y 18817))
  (+ (- (* 9 (pow x 4)) (pow y 4)) (* 2 (* y y))))