Average Error: 0.0 → 0.0
Time: 8.5s
Precision: 64
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
↓
\[\frac{x}{x + 1} + \frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + 1\right)\]
\frac{1}{x - 1} + \frac{x}{x + 1}
↓
\frac{x}{x + 1} + \frac{1}{\mathsf{fma}\left(x, x, -1\right)} \cdot \left(x + 1\right)
double f(double x) {
        double r4100909 = 1.0;
        double r4100910 = x;
        double r4100911 = r4100910 - r4100909;
        double r4100912 = r4100909 / r4100911;
        double r4100913 = r4100910 + r4100909;
        double r4100914 = r4100910 / r4100913;
        double r4100915 = r4100912 + r4100914;
        return r4100915;
}
↓
double f(double x) {
        double r4100916 = x;
        double r4100917 = 1.0;
        double r4100918 = r4100916 + r4100917;
        double r4100919 = r4100916 / r4100918;
        double r4100920 = -1.0;
        double r4100921 = fma(r4100916, r4100916, r4100920);
        double r4100922 = r4100917 / r4100921;
        double r4100923 = r4100922 * r4100918;
        double r4100924 = r4100919 + r4100923;
        return r4100924;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))