Average Error: 0.0 → 0.0
Time: 10.7s
Precision: 64
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
↓
\[\mathsf{fma}\left(\frac{1}{x \cdot x - 1 \cdot 1}, x + 1, \frac{x}{x + 1}\right)\]
\frac{1}{x - 1} + \frac{x}{x + 1}
↓
\mathsf{fma}\left(\frac{1}{x \cdot x - 1 \cdot 1}, x + 1, \frac{x}{x + 1}\right)
double f(double x) {
        double r53882 = 1.0;
        double r53883 = x;
        double r53884 = r53883 - r53882;
        double r53885 = r53882 / r53884;
        double r53886 = r53883 + r53882;
        double r53887 = r53883 / r53886;
        double r53888 = r53885 + r53887;
        return r53888;
}
↓
double f(double x) {
        double r53889 = 1.0;
        double r53890 = x;
        double r53891 = r53890 * r53890;
        double r53892 = r53889 * r53889;
        double r53893 = r53891 - r53892;
        double r53894 = r53889 / r53893;
        double r53895 = r53890 + r53889;
        double r53896 = r53890 / r53895;
        double r53897 = fma(r53894, r53895, r53896);
        return r53897;
}

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. Applied fma-def0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))