Average Error: 0.0 → 0.0
Time: 2.7s
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 r546304 = 2.0;
        double r546305 = x;
        double r546306 = r546305 * r546305;
        double r546307 = y;
        double r546308 = r546305 * r546307;
        double r546309 = r546306 + r546308;
        double r546310 = r546304 * r546309;
        return r546310;
}
↓
double f(double x, double y) {
        double r546311 = x;
        double r546312 = y;
        double r546313 = r546311 * r546312;
        double r546314 = fma(r546311, r546311, r546313);
        double r546315 = 2.0;
        double r546316 = r546314 * r546315;
        return r546316;
}

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 2020002 +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))))