Average Error: 38.9 → 0
Time: 5.7s
Precision: 64
\[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]
↓
\[\mathsf{fma}\left(x, x, x \cdot 2\right)\]
\left(x + 1\right) \cdot \left(x + 1\right) - 1
↓
\mathsf{fma}\left(x, x, x \cdot 2\right)
double f(double x) {
        double r281536 = x;
        double r281537 = 1.0;
        double r281538 = r281536 + r281537;
        double r281539 = r281538 * r281538;
        double r281540 = r281539 - r281537;
        return r281540;
}
↓
double f(double x) {
        double r281541 = x;
        double r281542 = 2.0;
        double r281543 = r281541 * r281542;
        double r281544 = fma(r281541, r281541, r281543);
        return r281544;
}

Error

Bits error versus x

Derivation

  1. Initial program 38.9

    \[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]

  2. Taylor expanded around 0 0.0

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

  3. Simplified0

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

  4. Final simplification0

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))