Average Error: 0.0 → 0.0
Time: 17.8s
Precision: 64
\[x \cdot x + y \cdot y\]
↓
\[\mathsf{fma}\left(y, y, x \cdot x\right)\]
x \cdot x + y \cdot y
↓
\mathsf{fma}\left(y, y, x \cdot x\right)
double f(double x, double y) {
        double r130247 = x;
        double r130248 = r130247 * r130247;
        double r130249 = y;
        double r130250 = r130249 * r130249;
        double r130251 = r130248 + r130250;
        return r130251;
}
↓
double f(double x, double y) {
        double r130252 = y;
        double r130253 = x;
        double r130254 = r130253 * r130253;
        double r130255 = fma(r130252, r130252, r130254);
        return r130255;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, y \cdot y\right)}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.0

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

  5. Simplified0.0

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

  6. Simplified0.0

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

  7. Taylor expanded around 0 0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y)
  :name "Graphics.Rasterific.Linear:$cquadrance from Rasterific-0.6.1"
  :precision binary64
  (+ (* x x) (* y y)))