Average Error: 38.9 → 0.0
Time: 16.1s
Precision: 64
\[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]
↓
\[\mathsf{fma}\left(2, x, x \cdot x\right)\]
\left(x + 1\right) \cdot \left(x + 1\right) - 1
↓
\mathsf{fma}\left(2, x, x \cdot x\right)
double f(double x) {
        double r3131967 = x;
        double r3131968 = 1.0;
        double r3131969 = r3131967 + r3131968;
        double r3131970 = r3131969 * r3131969;
        double r3131971 = r3131970 - r3131968;
        return r3131971;
}
↓
double f(double x) {
        double r3131972 = 2.0;
        double r3131973 = x;
        double r3131974 = r3131973 * r3131973;
        double r3131975 = fma(r3131972, r3131973, r3131974);
        return r3131975;
}

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. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))