Average Error: 52.0 → 0
Time: 1.8s
Precision: binary64
\[x = 10864 \land y = 18817\]
\[\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt{{y}^{4}}\\ \left(\left(9 \cdot {x}^{4} - {y}^{4}\right) + \mathsf{fma}\left(-t_0, t_0, t_0 \cdot t_0\right)\right) + 2 \cdot \left(y \cdot y\right) \end{array} \]
\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right)
↓
\begin{array}{l}
t_0 := \sqrt{{y}^{4}}\\
\left(\left(9 \cdot {x}^{4} - {y}^{4}\right) + \mathsf{fma}\left(-t_0, t_0, t_0 \cdot t_0\right)\right) + 2 \cdot \left(y \cdot y\right)
\end{array}

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

Error

Derivation

  1. Initial program 52.0

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

  2. Applied add-sqr-sqrt_binary6452.0

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

  3. Applied prod-diff_binary640

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

  4. Taylor expanded in x around 0 0

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2021307 
(FPCore (x y)
  :name "From Rump in a 1983 paper"
  :precision binary64
  :pre (and (== x 10864.0) (== y 18817.0))
  (+ (- (* 9.0 (pow x 4.0)) (pow y 4.0)) (* 2.0 (* y y))))