From Rump in a 1983 paper, rewritten

Average Error: 62.0 → 0

Time: 1.7s

Precision: binary64

\[x = 10864 \land y = 18817\]

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(y, y, -2\right) \cdot \left(y \cdot y\right)\\ \mathsf{fma}\left(9, {x}^{4}, -t_0\right) + \mathsf{fma}\left(-\mathsf{fma}\left(y, y, -2\right), y \cdot y, t_0\right) \end{array} \]

FPCore

(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 (* (fma y y -2.0) (* y y))))
   (+ (fma 9.0 (pow x 4.0) (- t_0)) (fma (- (fma y y -2.0)) (* y y) t_0))))

C

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 = fma(y, y, -2.0) * (y * y);
	return fma(9.0, pow(x, 4.0), -t_0) + fma(-fma(y, y, -2.0), (y * y), t_0);
}

Derivation

1. Initial program 62.0

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

2. Simplified 62.0

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

3. Applied prod-diff_binary64 0

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

4. Final simplification 0

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

Reproduce

herbie shell --seed 2022067 
(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))))