Average Error: 3.7 → 1.8
Time: 1.3s
Precision: binary64
\[x \cdot x - y \cdot y \]
↓
\[\mathsf{fma}\left(x, x, -y \cdot y\right) \]
x \cdot x - y \cdot y
↓
\mathsf{fma}\left(x, x, -y \cdot y\right)

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

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 3.7

    \[x \cdot x - y \cdot y \]

  2. Applied fma-neg_binary641.8

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

  3. Final simplification1.8

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

Reproduce

herbie shell --seed 2022088 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))