Average Error: 0.0 → 0.0
Time: 11.9s
Precision: 64
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
↓
\[\mathsf{fma}\left(\frac{1}{x \cdot x - 1 \cdot 1}, x + 1, \sqrt[3]{{\left(\frac{x}{x + 1}\right)}^{3}}\right)\]
\frac{1}{x - 1} + \frac{x}{x + 1}
↓
\mathsf{fma}\left(\frac{1}{x \cdot x - 1 \cdot 1}, x + 1, \sqrt[3]{{\left(\frac{x}{x + 1}\right)}^{3}}\right)
double f(double x) {
        double r122868 = 1.0;
        double r122869 = x;
        double r122870 = r122869 - r122868;
        double r122871 = r122868 / r122870;
        double r122872 = r122869 + r122868;
        double r122873 = r122869 / r122872;
        double r122874 = r122871 + r122873;
        return r122874;
}
↓
double f(double x) {
        double r122875 = 1.0;
        double r122876 = x;
        double r122877 = r122876 * r122876;
        double r122878 = r122875 * r122875;
        double r122879 = r122877 - r122878;
        double r122880 = r122875 / r122879;
        double r122881 = r122876 + r122875;
        double r122882 = r122876 / r122881;
        double r122883 = 3.0;
        double r122884 = pow(r122882, r122883);
        double r122885 = cbrt(r122884);
        double r122886 = fma(r122880, r122881, r122885);
        return r122886;
}

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 add-cbrt-cube20.8

    \[\leadsto \mathsf{fma}\left(\frac{1}{x \cdot x - 1 \cdot 1}, x + 1, \frac{x}{\color{blue}{\sqrt[3]{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(x + 1\right)}}}\right)\]

  8. Applied add-cbrt-cube21.4

    \[\leadsto \mathsf{fma}\left(\frac{1}{x \cdot x - 1 \cdot 1}, x + 1, \frac{\color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot x}}}{\sqrt[3]{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(x + 1\right)}}\right)\]

  9. Applied cbrt-undiv21.4

    \[\leadsto \mathsf{fma}\left(\frac{1}{x \cdot x - 1 \cdot 1}, x + 1, \color{blue}{\sqrt[3]{\frac{\left(x \cdot x\right) \cdot x}{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(x + 1\right)}}}\right)\]

  10. Simplified0.0

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

  11. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(\frac{1}{x \cdot x - 1 \cdot 1}, x + 1, \sqrt[3]{{\left(\frac{x}{x + 1}\right)}^{3}}\right)\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))