Average Error: 62.0 → 0
Time: 2.0s
Precision: binary64
\[x = 10864 \land y = 18817\]
\[9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt{9 \cdot {x}^{4}}\\ \left(t_0 + y \cdot y\right) \cdot \left(t_0 - y \cdot y\right) - \left(y \cdot y\right) \cdot -2 \end{array} \]
9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)
↓
\begin{array}{l}
t_0 := \sqrt{9 \cdot {x}^{4}}\\
\left(t_0 + y \cdot y\right) \cdot \left(t_0 - y \cdot y\right) - \left(y \cdot y\right) \cdot -2
\end{array}

(FPCore (x y)
 :precision binary64
 (- (* 9.0 (pow x 4.0)) (* (* y y) (- (* y y) 2.0))))
↓
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (* 9.0 (pow x 4.0)))))
   (- (* (+ t_0 (* y y)) (- t_0 (* y y))) (* (* y y) -2.0))))
double code(double x, double y) {
	return (9.0 * pow(x, 4.0)) - ((y * y) * ((y * y) - 2.0));
}
↓
double code(double x, double y) {
	double t_0 = sqrt(9.0 * pow(x, 4.0));
	return ((t_0 + (y * y)) * (t_0 - (y * y))) - ((y * y) * -2.0);
}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  4. Applied fma-udef_binary6462.0

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

  5. Applied distribute-rgt-in_binary6462.0

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

  6. Applied associate--r+_binary6452.0

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

  7. Simplified52.0

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

  9. Applied add-sqr-sqrt_binary6452.0

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

  10. Applied add-sqr-sqrt_binary6452.0

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

  11. Applied difference-of-squares_binary640

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

  12. Simplified0

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

  13. Simplified0

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

  14. Final simplification0

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

Reproduce

herbie shell --seed 2021211 
(FPCore (x y)
  :name "From Rump in a 1983 paper, rewritten"
  :precision binary64
  :pre (and (== x 10864.0) (== y 18817.0))
  (- (* 9.0 (pow x 4.0)) (* (* y y) (- (* y y) 2.0))))