Average Error: 0.0 → 0.0
Time: 3.4s
Precision: 64
\[\left(x + y\right) \cdot \left(x + y\right)\]
↓
\[\mathsf{fma}\left(x, x, \mathsf{fma}\left(2 \cdot x, y, {y}^{2}\right)\right)\]
\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(x, x, \mathsf{fma}\left(2 \cdot x, y, {y}^{2}\right)\right)
double f(double x, double y) {
        double r659671 = x;
        double r659672 = y;
        double r659673 = r659671 + r659672;
        double r659674 = r659673 * r659673;
        return r659674;
}
↓
double f(double x, double y) {
        double r659675 = x;
        double r659676 = 2.0;
        double r659677 = r659676 * r659675;
        double r659678 = y;
        double r659679 = pow(r659678, r659676);
        double r659680 = fma(r659677, r659678, r659679);
        double r659681 = fma(r659675, r659675, r659680);
        return r659681;
}

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 flip-+0.0

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

  4. Applied flip-+0.1

    \[\leadsto \color{blue}{\frac{x \cdot x - y \cdot y}{x - y}} \cdot \frac{x \cdot x - y \cdot y}{x - y}\]

  5. Applied frac-times34.6

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

  6. Taylor expanded around 0 0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020062 +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)))