Average Error: 0.0 → 0.0
Time: 7.0s
Precision: 64
\[\left(x + y\right) \cdot \left(x + y\right)\]
↓
\[\mathsf{fma}\left(x, x, 2 \cdot \left(x \cdot y\right) + y \cdot y\right)\]
\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(x, x, 2 \cdot \left(x \cdot y\right) + y \cdot y\right)
double f(double x, double y) {
        double r645668 = x;
        double r645669 = y;
        double r645670 = r645668 + r645669;
        double r645671 = r645670 * r645670;
        return r645671;
}
↓
double f(double x, double y) {
        double r645672 = x;
        double r645673 = 2.0;
        double r645674 = y;
        double r645675 = r645672 * r645674;
        double r645676 = r645673 * r645675;
        double r645677 = r645674 * r645674;
        double r645678 = r645676 + r645677;
        double r645679 = fma(r645672, r645672, r645678);
        return r645679;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.0
Target0.0
Herbie0.0
\[x \cdot x + \left(y \cdot y + 2 \cdot \left(y \cdot x\right)\right)\]

Derivation

  1. Initial program 0.0

    \[\left(x + y\right) \cdot \left(x + y\right)\]
  2. Using strategy rm

  3. Applied distribute-lft-in0.0

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

  4. Simplified0.0

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

  5. Taylor expanded around 0 0.0

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

  6. Simplified0.0

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

  8. Applied distribute-lft-in0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f3 from sbv-4.4"
  :precision binary64

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

  (* (+ x y) (+ x y)))