Average Error: 0.0 → 0.0
Time: 10.7s
Precision: 64
\[\left(x + y\right) \cdot \left(x + y\right)\]
↓
\[\mathsf{fma}\left(y + x, x, \left(y + x\right) \cdot y\right)\]
\left(x + y\right) \cdot \left(x + y\right)
↓
\mathsf{fma}\left(y + x, x, \left(y + x\right) \cdot y\right)
double f(double x, double y) {
        double r28664606 = x;
        double r28664607 = y;
        double r28664608 = r28664606 + r28664607;
        double r28664609 = r28664608 * r28664608;
        return r28664609;
}
↓
double f(double x, double y) {
        double r28664610 = y;
        double r28664611 = x;
        double r28664612 = r28664610 + r28664611;
        double r28664613 = r28664612 * r28664610;
        double r28664614 = fma(r28664612, r28664611, r28664613);
        return r28664614;
}

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. Using strategy rm

  5. Applied fma-def0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019179 +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)))