Average Error: 39.3 → 0.0
Time: 2.1s
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 r3509 = x;
        double r3510 = 1.0;
        double r3511 = r3509 + r3510;
        double r3512 = r3511 * r3511;
        double r3513 = r3512 - r3510;
        return r3513;
}
↓
double f(double x) {
        double r3514 = x;
        double r3515 = 2.0;
        double r3516 = 2.0;
        double r3517 = pow(r3514, r3516);
        double r3518 = fma(r3514, r3515, r3517);
        return r3518;
}

Error

Bits error versus x

Derivation

  1. Initial program 39.3

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

  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 2020083 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1) (+ x 1)) 1))