Average Error: 0.0 → 0.0
Time: 2.6s
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 r562072 = 2.0;
        double r562073 = x;
        double r562074 = r562073 * r562073;
        double r562075 = y;
        double r562076 = r562073 * r562075;
        double r562077 = r562074 + r562076;
        double r562078 = r562072 * r562077;
        return r562078;
}
↓
double f(double x, double y) {
        double r562079 = x;
        double r562080 = y;
        double r562081 = r562079 * r562080;
        double r562082 = fma(r562079, r562079, r562081);
        double r562083 = 2.0;
        double r562084 = r562082 * r562083;
        return r562084;
}

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 2020057 +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))))