Average Error: 0.0 → 0.0
Time: 2.0s
Precision: 64
\[2 \cdot \left(x \cdot x - x \cdot y\right)\]
↓
\[\mathsf{fma}\left(x, x, x \cdot \left(-y\right)\right) \cdot 2\]
2 \cdot \left(x \cdot x - x \cdot y\right)
↓
\mathsf{fma}\left(x, x, x \cdot \left(-y\right)\right) \cdot 2
double f(double x, double y) {
        double r468387 = 2.0;
        double r468388 = x;
        double r468389 = r468388 * r468388;
        double r468390 = y;
        double r468391 = r468388 * r468390;
        double r468392 = r468389 - r468391;
        double r468393 = r468387 * r468392;
        return r468393;
}
↓
double f(double x, double y) {
        double r468394 = x;
        double r468395 = y;
        double r468396 = -r468395;
        double r468397 = r468394 * r468396;
        double r468398 = fma(r468394, r468394, r468397);
        double r468399 = 2.0;
        double r468400 = r468398 * r468399;
        return r468400;
}

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

    \[\leadsto \color{blue}{\left(x \cdot \left(x - y\right)\right) \cdot 2}\]
  3. Using strategy rm

  4. Applied sub-neg0.0

    \[\leadsto \left(x \cdot \color{blue}{\left(x + \left(-y\right)\right)}\right) \cdot 2\]

  5. Applied distribute-lft-in0.0

    \[\leadsto \color{blue}{\left(x \cdot x + x \cdot \left(-y\right)\right)} \cdot 2\]

  6. Simplified0.0

    \[\leadsto \left(\color{blue}{{x}^{2}} + x \cdot \left(-y\right)\right) \cdot 2\]
  7. Using strategy rm

  8. Applied unpow20.0

    \[\leadsto \left(\color{blue}{x \cdot x} + x \cdot \left(-y\right)\right) \cdot 2\]

  9. Applied fma-def0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, x \cdot \left(-y\right)\right)} \cdot 2\]

  10. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(-y\right)\right) \cdot 2\]

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (x y)
  :name "Linear.Matrix:fromQuaternion from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (* (* x 2) (- x y))

  (* 2 (- (* x x) (* x y))))