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 code(double x, double y) {
	return ((double) (((double) (9.0 * ((double) pow(x, 4.0)))) - ((double) (((double) (y * y)) * ((double) (((double) (y * y)) - 2.0))))));
}
↓
double code(double x, double y) {
	return ((double) fma(((double) sqrt(((double) fma(((double) pow(x, 4.0)), 9.0, ((double) (2.0 * ((double) (y * y)))))))), ((double) sqrt(((double) fma(((double) pow(x, 4.0)), 9.0, ((double) (2.0 * ((double) (y * y)))))))), ((double) -(((double) pow(y, 4.0))))));
}

Error

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Results

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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 2020121 +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))))