Average Error: 52.0 → 52.0
Time: 2.6s
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 r65236 = 9.0;
        double r65237 = x;
        double r65238 = 4.0;
        double r65239 = pow(r65237, r65238);
        double r65240 = r65236 * r65239;
        double r65241 = y;
        double r65242 = pow(r65241, r65238);
        double r65243 = r65240 - r65242;
        double r65244 = 2.0;
        double r65245 = r65241 * r65241;
        double r65246 = r65244 * r65245;
        double r65247 = r65243 + r65246;
        return r65247;
}
↓
double f(double x, double y) {
        double r65248 = 2.0;
        double r65249 = y;
        double r65250 = r65248 * r65249;
        double r65251 = 9.0;
        double r65252 = x;
        double r65253 = 4.0;
        double r65254 = pow(r65252, r65253);
        double r65255 = r65251 * r65254;
        double r65256 = pow(r65249, r65253);
        double r65257 = r65255 - r65256;
        double r65258 = 3.0;
        double r65259 = pow(r65257, r65258);
        double r65260 = cbrt(r65259);
        double r65261 = fma(r65250, r65249, r65260);
        return r65261;
}

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 2020065 +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))))