Average Error: 0.0 → 0.0
Time: 20.3s
Precision: 64
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
↓
\[\mathsf{fma}\left(\frac{\frac{1}{x + 1}}{x - 1}, x + 1, \frac{x}{x + 1}\right)\]
\frac{1}{x - 1} + \frac{x}{x + 1}
↓
\mathsf{fma}\left(\frac{\frac{1}{x + 1}}{x - 1}, x + 1, \frac{x}{x + 1}\right)
double f(double x) {
        double r3433870 = 1.0;
        double r3433871 = x;
        double r3433872 = r3433871 - r3433870;
        double r3433873 = r3433870 / r3433872;
        double r3433874 = r3433871 + r3433870;
        double r3433875 = r3433871 / r3433874;
        double r3433876 = r3433873 + r3433875;
        return r3433876;
}
↓
double f(double x) {
        double r3433877 = 1.0;
        double r3433878 = x;
        double r3433879 = r3433878 + r3433877;
        double r3433880 = r3433877 / r3433879;
        double r3433881 = r3433878 - r3433877;
        double r3433882 = r3433880 / r3433881;
        double r3433883 = r3433878 / r3433879;
        double r3433884 = fma(r3433882, r3433879, r3433883);
        return r3433884;
}

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. Using strategy rm

  7. Applied difference-of-squares0.0

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

  8. Applied associate-/r*0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))