Average Error: 0.0 → 0.0
Time: 2.0s
Precision: 64
\[2 \cdot \left(x \cdot x + x \cdot y\right)\]
↓
\[\mathsf{fma}\left(x, x, x \cdot y\right) \cdot 2\]
2 \cdot \left(x \cdot x + x \cdot y\right)
↓
\mathsf{fma}\left(x, x, x \cdot y\right) \cdot 2
double f(double x, double y) {
        double r599251 = 2.0;
        double r599252 = x;
        double r599253 = r599252 * r599252;
        double r599254 = y;
        double r599255 = r599252 * r599254;
        double r599256 = r599253 + r599255;
        double r599257 = r599251 * r599256;
        return r599257;
}
↓
double f(double x, double y) {
        double r599258 = x;
        double r599259 = y;
        double r599260 = r599258 * r599259;
        double r599261 = fma(r599258, r599258, r599260);
        double r599262 = 2.0;
        double r599263 = r599261 * r599262;
        return r599263;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.0
Target0.0
Herbie0.0
\[\left(x \cdot 2\right) \cdot \left(x + y\right)\]

Derivation

  1. Initial program 0.0

    \[2 \cdot \left(x \cdot x + x \cdot y\right)\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, x \cdot y\right) \cdot 2}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, x, x \cdot y\right) \cdot 2\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x y)
  :name "Linear.Matrix:fromQuaternion from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (* (* x 2) (+ x y))

  (* 2 (+ (* x x) (* x y))))