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 r56094 = 9.0;
        double r56095 = x;
        double r56096 = 4.0;
        double r56097 = pow(r56095, r56096);
        double r56098 = r56094 * r56097;
        double r56099 = y;
        double r56100 = pow(r56099, r56096);
        double r56101 = r56098 - r56100;
        double r56102 = 2.0;
        double r56103 = r56099 * r56099;
        double r56104 = r56102 * r56103;
        double r56105 = r56101 + r56104;
        return r56105;
}
↓
double f(double x, double y) {
        double r56106 = 2.0;
        double r56107 = y;
        double r56108 = r56106 * r56107;
        double r56109 = 9.0;
        double r56110 = x;
        double r56111 = 4.0;
        double r56112 = pow(r56110, r56111);
        double r56113 = r56109 * r56112;
        double r56114 = pow(r56107, r56111);
        double r56115 = r56113 - r56114;
        double r56116 = 3.0;
        double r56117 = pow(r56115, r56116);
        double r56118 = cbrt(r56117);
        double r56119 = fma(r56108, r56107, r56118);
        return r56119;
}

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 2019354 +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))))