Average Error: 0.0 → 0.0
Time: 735.0ms
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 r446753 = 2.0;
        double r446754 = x;
        double r446755 = r446754 * r446754;
        double r446756 = y;
        double r446757 = r446754 * r446756;
        double r446758 = r446755 + r446757;
        double r446759 = r446753 * r446758;
        return r446759;
}
↓
double f(double x, double y) {
        double r446760 = x;
        double r446761 = y;
        double r446762 = r446760 * r446761;
        double r446763 = fma(r446760, r446760, r446762);
        double r446764 = 2.0;
        double r446765 = r446763 * r446764;
        return r446765;
}

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 2020001 +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))))