Average Error: 62.0 → 0
Time: 1.3s
Precision: 64
\[x = 10864 \land y = 18817\]
\[9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)\]
↓
\[\mathsf{fma}\left(\sqrt{\mathsf{fma}\left({x}^{4}, 9, 2 \cdot \left(y \cdot y\right)\right)}, \sqrt{\mathsf{fma}\left({x}^{4}, 9, 2 \cdot \left(y \cdot y\right)\right)}, -{y}^{4}\right)\]
9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)
↓
\mathsf{fma}\left(\sqrt{\mathsf{fma}\left({x}^{4}, 9, 2 \cdot \left(y \cdot y\right)\right)}, \sqrt{\mathsf{fma}\left({x}^{4}, 9, 2 \cdot \left(y \cdot y\right)\right)}, -{y}^{4}\right)
double f(double x, double y) {
        double r57257 = 9.0;
        double r57258 = x;
        double r57259 = 4.0;
        double r57260 = pow(r57258, r57259);
        double r57261 = r57257 * r57260;
        double r57262 = y;
        double r57263 = r57262 * r57262;
        double r57264 = 2.0;
        double r57265 = r57263 - r57264;
        double r57266 = r57263 * r57265;
        double r57267 = r57261 - r57266;
        return r57267;
}
↓
double f(double x, double y) {
        double r57268 = x;
        double r57269 = 4.0;
        double r57270 = pow(r57268, r57269);
        double r57271 = 9.0;
        double r57272 = 2.0;
        double r57273 = y;
        double r57274 = r57273 * r57273;
        double r57275 = r57272 * r57274;
        double r57276 = fma(r57270, r57271, r57275);
        double r57277 = sqrt(r57276);
        double r57278 = 4.0;
        double r57279 = pow(r57273, r57278);
        double r57280 = -r57279;
        double r57281 = fma(r57277, r57277, r57280);
        return r57281;
}

Error

Derivation

  1. Initial program 62.0

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

  2. Simplified62.0

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

  4. Applied add-sqr-sqrt62.0

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

  5. Applied fma-neg0

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

  6. Final simplification0

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

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x y)
  :name "From Rump in a 1983 paper, rewritten"
  :precision binary64
  :pre (and (== x 10864) (== y 18817))
  (- (* 9 (pow x 4)) (* (* y y) (- (* y y) 2))))