Average Error: 38.4 → 0
Time: 19.3s
Precision: 64
\[\left(x + 1.0\right) \cdot \left(x + 1.0\right) - 1.0\]
↓
\[\mathsf{fma}\left(x, x, x \cdot 2.0\right)\]
\left(x + 1.0\right) \cdot \left(x + 1.0\right) - 1.0
↓
\mathsf{fma}\left(x, x, x \cdot 2.0\right)
double f(double x) {
        double r364921 = x;
        double r364922 = 1.0;
        double r364923 = r364921 + r364922;
        double r364924 = r364923 * r364923;
        double r364925 = r364924 - r364922;
        return r364925;
}
↓
double f(double x) {
        double r364926 = x;
        double r364927 = 2.0;
        double r364928 = r364926 * r364927;
        double r364929 = fma(r364926, r364926, r364928);
        return r364929;
}

Error

Bits error versus x

Derivation

  1. Initial program 38.4

    \[\left(x + 1.0\right) \cdot \left(x + 1.0\right) - 1.0\]

  2. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{2.0 \cdot x + {x}^{2}}\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(2.0, x, x \cdot x\right)}\]

  4. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{2.0 \cdot x + {x}^{2}}\]

  5. Simplified0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 2.0 \cdot x\right)}\]

  6. Final simplification0

    \[\leadsto \mathsf{fma}\left(x, x, x \cdot 2.0\right)\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))