Average Error: 39.1 → 0.0
Time: 1.7s
Precision: 64
\[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]
↓
\[\mathsf{fma}\left(x, 2, {x}^{2}\right)\]
\left(x + 1\right) \cdot \left(x + 1\right) - 1
↓
\mathsf{fma}\left(x, 2, {x}^{2}\right)
double f(double x) {
        double r4131 = x;
        double r4132 = 1.0;
        double r4133 = r4131 + r4132;
        double r4134 = r4133 * r4133;
        double r4135 = r4134 - r4132;
        return r4135;
}
↓
double f(double x) {
        double r4136 = x;
        double r4137 = 2.0;
        double r4138 = 2.0;
        double r4139 = pow(r4136, r4138);
        double r4140 = fma(r4136, r4137, r4139);
        return r4140;
}

Error

Bits error versus x

Derivation

  1. Initial program 39.1

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

  2. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1) (+ x 1)) 1))