Average Error: 0.0 → 0.0
Time: 18.8s
Precision: 64
\[x \cdot x + y \cdot y\]
↓
\[\mathsf{fma}\left(x, x, y \cdot y\right)\]
x \cdot x + y \cdot y
↓
\mathsf{fma}\left(x, x, y \cdot y\right)
double f(double x, double y) {
        double r8145497 = x;
        double r8145498 = r8145497 * r8145497;
        double r8145499 = y;
        double r8145500 = r8145499 * r8145499;
        double r8145501 = r8145498 + r8145500;
        return r8145501;
}
↓
double f(double x, double y) {
        double r8145502 = x;
        double r8145503 = y;
        double r8145504 = r8145503 * r8145503;
        double r8145505 = fma(r8145502, r8145502, r8145504);
        return r8145505;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

    \[x \cdot x + y \cdot y\]

  2. Simplified0.0

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

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{{x}^{2} + {y}^{2}}\]

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x y)
  :name "Graphics.Rasterific.Linear:$cquadrance from Rasterific-0.6.1"
  (+ (* x x) (* y y)))