Average Error: 0.0 → 0.0
Time: 2.0s
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 r471131 = 2.0;
        double r471132 = x;
        double r471133 = r471132 * r471132;
        double r471134 = y;
        double r471135 = r471132 * r471134;
        double r471136 = r471133 + r471135;
        double r471137 = r471131 * r471136;
        return r471137;
}
↓
double f(double x, double y) {
        double r471138 = x;
        double r471139 = y;
        double r471140 = r471138 * r471139;
        double r471141 = fma(r471138, r471138, r471140);
        double r471142 = 2.0;
        double r471143 = r471141 * r471142;
        return r471143;
}

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 2019353 +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))))