Average Error: 0.0 → 0.0
Time: 680.0ms
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 r452948 = 2.0;
        double r452949 = x;
        double r452950 = r452949 * r452949;
        double r452951 = y;
        double r452952 = r452949 * r452951;
        double r452953 = r452950 + r452952;
        double r452954 = r452948 * r452953;
        return r452954;
}
↓
double f(double x, double y) {
        double r452955 = x;
        double r452956 = y;
        double r452957 = r452955 * r452956;
        double r452958 = fma(r452955, r452955, r452957);
        double r452959 = 2.0;
        double r452960 = r452958 * r452959;
        return r452960;
}

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 2020056 +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))))