Average Error: 0.0 → 0.0
Time: 7.5s
Precision: 64
\[\left(x + y\right) \cdot \left(x + y\right)\]
↓
\[\mathsf{fma}\left(x, x, y \cdot \left(2 \cdot x + y\right)\right)\]
\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(x, x, y \cdot \left(2 \cdot x + y\right)\right)
double f(double x, double y) {
        double r597125 = x;
        double r597126 = y;
        double r597127 = r597125 + r597126;
        double r597128 = r597127 * r597127;
        return r597128;
}
↓
double f(double x, double y) {
        double r597129 = x;
        double r597130 = y;
        double r597131 = 2.0;
        double r597132 = r597131 * r597129;
        double r597133 = r597132 + r597130;
        double r597134 = r597130 * r597133;
        double r597135 = fma(r597129, r597129, r597134);
        return r597135;
}

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-times34.1

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

  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

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020045 +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)))