Average Error: 0.0 → 0.0
Time: 5.7s
Precision: 64
\[\left(x \cdot 2 + x \cdot x\right) + y \cdot y\]
↓
\[x \cdot 2 + \mathsf{fma}\left(x, x, y \cdot y\right)\]
\left(x \cdot 2 + x \cdot x\right) + y \cdot y
↓
x \cdot 2 + \mathsf{fma}\left(x, x, y \cdot y\right)
double f(double x, double y) {
        double r495036 = x;
        double r495037 = 2.0;
        double r495038 = r495036 * r495037;
        double r495039 = r495036 * r495036;
        double r495040 = r495038 + r495039;
        double r495041 = y;
        double r495042 = r495041 * r495041;
        double r495043 = r495040 + r495042;
        return r495043;
}
↓
double f(double x, double y) {
        double r495044 = x;
        double r495045 = 2.0;
        double r495046 = r495044 * r495045;
        double r495047 = y;
        double r495048 = r495047 * r495047;
        double r495049 = fma(r495044, r495044, r495048);
        double r495050 = r495046 + r495049;
        return r495050;
}

Error

Bits error versus x

Bits error versus y

Target

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

Derivation

  1. Initial program 0.0

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

  3. Applied associate-+l+0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, A"
  :precision binary64

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

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