Average Error: 58.1 → 58.6
Time: 1.1s
Precision: binary64
\[x = 77617 \land y = 33096\]
\[\left(\left(333.75 \cdot {y}^{6} + \left(x \cdot x\right) \cdot \left(\left(\left(\left(\left(\left(11 \cdot x\right) \cdot x\right) \cdot y\right) \cdot y - {y}^{6}\right) - 121 \cdot {y}^{4}\right) - 2\right)\right) + 5.5 \cdot {y}^{8}\right) + \frac{x}{2 \cdot y} \]
↓
\[\mathsf{fma}\left(x \cdot x, -\mathsf{fma}\left(121, {y}^{4}, {y}^{6}\right), \mathsf{fma}\left(333.75, {y}^{6}, \mathsf{fma}\left(5.5, {y}^{8}, \frac{x}{y \cdot 2}\right)\right)\right) \]
\left(\left(333.75 \cdot {y}^{6} + \left(x \cdot x\right) \cdot \left(\left(\left(\left(\left(\left(11 \cdot x\right) \cdot x\right) \cdot y\right) \cdot y - {y}^{6}\right) - 121 \cdot {y}^{4}\right) - 2\right)\right) + 5.5 \cdot {y}^{8}\right) + \frac{x}{2 \cdot y}
↓
\mathsf{fma}\left(x \cdot x, -\mathsf{fma}\left(121, {y}^{4}, {y}^{6}\right), \mathsf{fma}\left(333.75, {y}^{6}, \mathsf{fma}\left(5.5, {y}^{8}, \frac{x}{y \cdot 2}\right)\right)\right)

(FPCore (x y)
 :precision binary64
 (+
  (+
   (+
    (* 333.75 (pow y 6.0))
    (*
     (* x x)
     (-
      (- (- (* (* (* (* 11.0 x) x) y) y) (pow y 6.0)) (* 121.0 (pow y 4.0)))
      2.0)))
   (* 5.5 (pow y 8.0)))
  (/ x (* 2.0 y))))
↓
(FPCore (x y)
 :precision binary64
 (fma
  (* x x)
  (- (fma 121.0 (pow y 4.0) (pow y 6.0)))
  (fma 333.75 (pow y 6.0) (fma 5.5 (pow y 8.0) (/ x (* y 2.0))))))
double code(double x, double y) {
	return (((333.75 * pow(y, 6.0)) + ((x * x) * (((((((11.0 * x) * x) * y) * y) - pow(y, 6.0)) - (121.0 * pow(y, 4.0))) - 2.0))) + (5.5 * pow(y, 8.0))) + (x / (2.0 * y));
}
↓
double code(double x, double y) {
	return fma((x * x), -fma(121.0, pow(y, 4.0), pow(y, 6.0)), fma(333.75, pow(y, 6.0), fma(5.5, pow(y, 8.0), (x / (y * 2.0)))));
}

Error

Derivation

  1. Initial program 58.1

    \[\left(\left(333.75 \cdot {y}^{6} + \left(x \cdot x\right) \cdot \left(\left(\left(\left(\left(\left(11 \cdot x\right) \cdot x\right) \cdot y\right) \cdot y - {y}^{6}\right) - 121 \cdot {y}^{4}\right) - 2\right)\right) + 5.5 \cdot {y}^{8}\right) + \frac{x}{2 \cdot y} \]

  2. Simplified63.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(y, y \cdot \left(x \cdot \left(x \cdot 11\right)\right), \mathsf{fma}\left({y}^{4}, -121, -2\right)\right) - {y}^{6}, \mathsf{fma}\left(333.75, {y}^{6}, \mathsf{fma}\left(5.5, {y}^{8}, \frac{x}{y \cdot 2}\right)\right)\right)} \]

  3. Taylor expanded in y around inf 58.6

    \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{-\left(121 \cdot {y}^{4} + {y}^{6}\right)}, \mathsf{fma}\left(333.75, {y}^{6}, \mathsf{fma}\left(5.5, {y}^{8}, \frac{x}{y \cdot 2}\right)\right)\right) \]

  4. Simplified58.6

    \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{-\mathsf{fma}\left(121, {y}^{4}, {y}^{6}\right)}, \mathsf{fma}\left(333.75, {y}^{6}, \mathsf{fma}\left(5.5, {y}^{8}, \frac{x}{y \cdot 2}\right)\right)\right) \]

  5. Final simplification58.6

    \[\leadsto \mathsf{fma}\left(x \cdot x, -\mathsf{fma}\left(121, {y}^{4}, {y}^{6}\right), \mathsf{fma}\left(333.75, {y}^{6}, \mathsf{fma}\left(5.5, {y}^{8}, \frac{x}{y \cdot 2}\right)\right)\right) \]

Reproduce

herbie shell --seed 2022087 
(FPCore (x y)
  :name "Rump's expression from Stadtherr's award speech"
  :precision binary64
  :pre (and (== x 77617.0) (== y 33096.0))
  (+ (+ (+ (* 333.75 (pow y 6.0)) (* (* x x) (- (- (- (* (* (* (* 11.0 x) x) y) y) (pow y 6.0)) (* 121.0 (pow y 4.0))) 2.0))) (* 5.5 (pow y 8.0))) (/ x (* 2.0 y))))