Average Error: 0.0 → 0.0
Time: 18.5s
Precision: 64
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
↓
\[\frac{\mathsf{fma}\left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}, \frac{x}{1 + x}, \left(\left(\left(1 + x\right) \cdot \frac{1}{x - 1}\right) \cdot \frac{1}{x \cdot x - 1}\right) \cdot \frac{1}{x - 1}\right)}{\left(\frac{x}{1 + x} \cdot \frac{x}{1 + x} - \frac{1}{x - 1} \cdot \frac{x}{1 + x}\right) + \frac{1}{x - 1} \cdot \frac{1}{x - 1}}\]
\frac{1}{x - 1} + \frac{x}{x + 1}
↓
\frac{\mathsf{fma}\left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}, \frac{x}{1 + x}, \left(\left(\left(1 + x\right) \cdot \frac{1}{x - 1}\right) \cdot \frac{1}{x \cdot x - 1}\right) \cdot \frac{1}{x - 1}\right)}{\left(\frac{x}{1 + x} \cdot \frac{x}{1 + x} - \frac{1}{x - 1} \cdot \frac{x}{1 + x}\right) + \frac{1}{x - 1} \cdot \frac{1}{x - 1}}
double f(double x) {
        double r4235838 = 1.0;
        double r4235839 = x;
        double r4235840 = r4235839 - r4235838;
        double r4235841 = r4235838 / r4235840;
        double r4235842 = r4235839 + r4235838;
        double r4235843 = r4235839 / r4235842;
        double r4235844 = r4235841 + r4235843;
        return r4235844;
}
↓
double f(double x) {
        double r4235845 = x;
        double r4235846 = 1.0;
        double r4235847 = r4235846 + r4235845;
        double r4235848 = r4235845 / r4235847;
        double r4235849 = r4235848 * r4235848;
        double r4235850 = r4235845 - r4235846;
        double r4235851 = r4235846 / r4235850;
        double r4235852 = r4235847 * r4235851;
        double r4235853 = r4235845 * r4235845;
        double r4235854 = r4235853 - r4235846;
        double r4235855 = r4235846 / r4235854;
        double r4235856 = r4235852 * r4235855;
        double r4235857 = r4235856 * r4235851;
        double r4235858 = fma(r4235849, r4235848, r4235857);
        double r4235859 = r4235851 * r4235848;
        double r4235860 = r4235849 - r4235859;
        double r4235861 = r4235851 * r4235851;
        double r4235862 = r4235860 + r4235861;
        double r4235863 = r4235858 / r4235862;
        return r4235863;
}

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 flip3-+0.0

    \[\leadsto \color{blue}{\frac{{\left(\frac{1}{x - 1}\right)}^{3} + {\left(\frac{x}{x + 1}\right)}^{3}}{\frac{1}{x - 1} \cdot \frac{1}{x - 1} + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{1}{x - 1} \cdot \frac{x}{x + 1}\right)}}\]

  4. Simplified0.0

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

  6. Applied flip--0.0

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

  7. Applied associate-/r/0.0

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

  8. Applied associate-*l*0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))