Average Error: 0.0 → 0.0
Time: 8.5s
Precision: 64
\[\left(x + y\right) \cdot \left(x + y\right)\]
↓
\[\mathsf{fma}\left(x, x, y \cdot y + 2 \cdot \left(x \cdot y\right)\right)\]
\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(x, x, y \cdot y + 2 \cdot \left(x \cdot y\right)\right)
double f(double x, double y) {
        double r725102 = x;
        double r725103 = y;
        double r725104 = r725102 + r725103;
        double r725105 = r725104 * r725104;
        return r725105;
}
↓
double f(double x, double y) {
        double r725106 = x;
        double r725107 = y;
        double r725108 = r725107 * r725107;
        double r725109 = 2.0;
        double r725110 = r725106 * r725107;
        double r725111 = r725109 * r725110;
        double r725112 = r725108 + r725111;
        double r725113 = fma(r725106, r725106, r725112);
        return r725113;
}

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

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

  8. Applied distribute-lft-in0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

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)))