Average Error: 0.0 → 0.0
Time: 7.4s
Precision: 64
\[\left(x + y\right) \cdot \left(x + y\right)\]
↓
\[\mathsf{fma}\left(y, y, \mathsf{fma}\left(2, y, x\right) \cdot x\right)\]
\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(y, y, \mathsf{fma}\left(2, y, x\right) \cdot x\right)
double f(double x, double y) {
        double r41215859 = x;
        double r41215860 = y;
        double r41215861 = r41215859 + r41215860;
        double r41215862 = r41215861 * r41215861;
        return r41215862;
}
↓
double f(double x, double y) {
        double r41215863 = y;
        double r41215864 = 2.0;
        double r41215865 = x;
        double r41215866 = fma(r41215864, r41215863, r41215865);
        double r41215867 = r41215866 * r41215865;
        double r41215868 = fma(r41215863, r41215863, r41215867);
        return r41215868;
}

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. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Taylor expanded around 0 0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f3 from sbv-4.4"

  :herbie-target
  (+ (* x x) (+ (* y y) (* 2.0 (* y x))))

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