Average Error: 0.0 → 0.0
Time: 1.6s
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 r697572 = x;
        double r697573 = r697572 * r697572;
        double r697574 = 2.0;
        double r697575 = r697572 * r697574;
        double r697576 = y;
        double r697577 = r697575 * r697576;
        double r697578 = r697573 + r697577;
        double r697579 = r697576 * r697576;
        double r697580 = r697578 + r697579;
        return r697580;
}
↓
double f(double x, double y) {
        double r697581 = x;
        double r697582 = 2.0;
        double r697583 = r697581 * r697582;
        double r697584 = y;
        double r697585 = r697583 * r697584;
        double r697586 = fma(r697581, r697581, r697585);
        double r697587 = r697584 * r697584;
        double r697588 = r697586 + r697587;
        return r697588;
}

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 2019353 +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)))