Average Error: 0.0 → 0.0
Time: 8.8s
Precision: 64
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
↓
\[\frac{\left(\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + 1\right)\right) \cdot \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + 1\right)\right) - \frac{x}{x + 1} \cdot \frac{x}{x + 1}}{\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + 1\right) - \frac{x}{x + 1}}\]
\frac{1}{x - 1} + \frac{x}{x + 1}
↓
\frac{\left(\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + 1\right)\right) \cdot \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + 1\right)\right) - \frac{x}{x + 1} \cdot \frac{x}{x + 1}}{\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + 1\right) - \frac{x}{x + 1}}
double f(double x) {
        double r2189772 = 1.0;
        double r2189773 = x;
        double r2189774 = r2189773 - r2189772;
        double r2189775 = r2189772 / r2189774;
        double r2189776 = r2189773 + r2189772;
        double r2189777 = r2189773 / r2189776;
        double r2189778 = r2189775 + r2189777;
        return r2189778;
}
↓
double f(double x) {
        double r2189779 = 1.0;
        double r2189780 = x;
        double r2189781 = -1.0;
        double r2189782 = fma(r2189780, r2189780, r2189781);
        double r2189783 = r2189779 / r2189782;
        double r2189784 = r2189780 + r2189779;
        double r2189785 = r2189783 * r2189784;
        double r2189786 = r2189785 * r2189785;
        double r2189787 = r2189780 / r2189784;
        double r2189788 = r2189787 * r2189787;
        double r2189789 = r2189786 - r2189788;
        double r2189790 = r2189785 - r2189787;
        double r2189791 = r2189789 / r2189790;
        return r2189791;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[\frac{1}{x - 1} + \frac{x}{x + 1}\]
  2. Using strategy rm

  3. Applied flip--0.0

    \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x + 1}}} + \frac{x}{x + 1}\]

  4. Applied associate-/r/0.0

    \[\leadsto \color{blue}{\frac{1}{x \cdot x - 1 \cdot 1} \cdot \left(x + 1\right)} + \frac{x}{x + 1}\]

  5. Simplified0.0

    \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(x, x, -1\right)}} \cdot \left(x + 1\right) + \frac{x}{x + 1}\]
  6. Using strategy rm

  7. Applied flip-+0.0

    \[\leadsto \color{blue}{\frac{\left(\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + 1\right)\right) \cdot \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + 1\right)\right) - \frac{x}{x + 1} \cdot \frac{x}{x + 1}}{\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + 1\right) - \frac{x}{x + 1}}}\]

  8. Final simplification0.0

    \[\leadsto \frac{\left(\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + 1\right)\right) \cdot \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + 1\right)\right) - \frac{x}{x + 1} \cdot \frac{x}{x + 1}}{\frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + 1\right) - \frac{x}{x + 1}}\]

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))