Average Error: 0.0 → 0.0
Time: 3.1s
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 r463116 = 2.0;
        double r463117 = x;
        double r463118 = r463117 * r463117;
        double r463119 = y;
        double r463120 = r463117 * r463119;
        double r463121 = r463118 + r463120;
        double r463122 = r463116 * r463121;
        return r463122;
}
↓
double f(double x, double y) {
        double r463123 = x;
        double r463124 = y;
        double r463125 = r463123 * r463124;
        double r463126 = fma(r463123, r463123, r463125);
        double r463127 = 2.0;
        double r463128 = r463126 * r463127;
        return r463128;
}

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 2020065 +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))))