Average Error: 0.0 → 0.0
Time: 1.1s
Precision: binary64
\[0.5 \cdot \left(x \cdot x - y\right) \]
↓
\[0.5 \cdot \left(\mathsf{fma}\left(x, x, -y\right) + \mathsf{fma}\left(-y, 1, y\right)\right) \]
0.5 \cdot \left(x \cdot x - y\right)
↓
0.5 \cdot \left(\mathsf{fma}\left(x, x, -y\right) + \mathsf{fma}\left(-y, 1, y\right)\right)

(FPCore (x y) :precision binary64 (* 0.5 (- (* x x) y)))
↓
(FPCore (x y)
 :precision binary64
 (* 0.5 (+ (fma x x (- y)) (fma (- y) 1.0 y))))
double code(double x, double y) {
	return 0.5 * ((x * x) - y);
}
↓
double code(double x, double y) {
	return 0.5 * (fma(x, x, -y) + fma(-y, 1.0, y));
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

    \[0.5 \cdot \left(x \cdot x - y\right) \]

  2. Applied *-un-lft-identity_binary640.0

    \[\leadsto 0.5 \cdot \left(x \cdot x - \color{blue}{1 \cdot y}\right) \]

  3. Applied prod-diff_binary640.0

    \[\leadsto 0.5 \cdot \color{blue}{\left(\mathsf{fma}\left(x, x, -y \cdot 1\right) + \mathsf{fma}\left(-y, 1, y \cdot 1\right)\right)} \]

  4. Final simplification0.0

    \[\leadsto 0.5 \cdot \left(\mathsf{fma}\left(x, x, -y\right) + \mathsf{fma}\left(-y, 1, y\right)\right) \]

Reproduce

herbie shell --seed 2021307 
(FPCore (x y)
  :name "System.Random.MWC.Distributions:standard from mwc-random-0.13.3.2"
  :precision binary64
  (* 0.5 (- (* x x) y)))