Average Error: 0.0 → 0.0
Time: 749.0ms
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 r494749 = 2.0;
        double r494750 = x;
        double r494751 = r494750 * r494750;
        double r494752 = y;
        double r494753 = r494750 * r494752;
        double r494754 = r494751 + r494753;
        double r494755 = r494749 * r494754;
        return r494755;
}
↓
double f(double x, double y) {
        double r494756 = x;
        double r494757 = y;
        double r494758 = r494756 * r494757;
        double r494759 = fma(r494756, r494756, r494758);
        double r494760 = 2.0;
        double r494761 = r494759 * r494760;
        return r494761;
}

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 2020018 +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))))