Average Error: 0.0 → 0.0
Time: 11.3s
Precision: 64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(x, x, \left(x \cdot 2\right) \cdot y\right) + y \cdot y\]
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(x, x, \left(x \cdot 2\right) \cdot y\right) + y \cdot y
double f(double x, double y) {
        double r349515 = x;
        double r349516 = r349515 * r349515;
        double r349517 = 2.0;
        double r349518 = r349515 * r349517;
        double r349519 = y;
        double r349520 = r349518 * r349519;
        double r349521 = r349516 + r349520;
        double r349522 = r349519 * r349519;
        double r349523 = r349521 + r349522;
        return r349523;
}
↓
double f(double x, double y) {
        double r349524 = x;
        double r349525 = 2.0;
        double r349526 = r349524 * r349525;
        double r349527 = y;
        double r349528 = r349526 * r349527;
        double r349529 = fma(r349524, r349524, r349528);
        double r349530 = r349527 * r349527;
        double r349531 = r349529 + r349530;
        return r349531;
}

Error

Bits error versus x

Bits error versus y

Target

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

Derivation

  1. Initial program 0.0

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

  3. Applied fma-def0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64

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

  (+ (+ (* x x) (* (* x 2) y)) (* y y)))