Average Error: 0.0 → 0.0
Time: 1.5s
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 r578826 = x;
        double r578827 = r578826 * r578826;
        double r578828 = 2.0;
        double r578829 = r578826 * r578828;
        double r578830 = y;
        double r578831 = r578829 * r578830;
        double r578832 = r578827 + r578831;
        double r578833 = r578830 * r578830;
        double r578834 = r578832 + r578833;
        return r578834;
}
↓
double f(double x, double y) {
        double r578835 = x;
        double r578836 = 2.0;
        double r578837 = r578835 * r578836;
        double r578838 = y;
        double r578839 = r578837 * r578838;
        double r578840 = fma(r578835, r578835, r578839);
        double r578841 = r578838 * r578838;
        double r578842 = r578840 + r578841;
        return r578842;
}

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)))