Average Error: 0.0 → 0.0
Time: 1.1s
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 r687764 = 2.0;
        double r687765 = x;
        double r687766 = r687765 * r687765;
        double r687767 = y;
        double r687768 = r687765 * r687767;
        double r687769 = r687766 + r687768;
        double r687770 = r687764 * r687769;
        return r687770;
}
↓
double f(double x, double y) {
        double r687771 = x;
        double r687772 = y;
        double r687773 = r687771 * r687772;
        double r687774 = fma(r687771, r687771, r687773);
        double r687775 = 2.0;
        double r687776 = r687774 * r687775;
        return r687776;
}

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 2020100 +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))))