Average Error: 0.0 → 0.0
Time: 1.6m
Precision: 64
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 1\right)}{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 2\right)}\]
\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
↓
\frac{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 1\right)}{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 2\right)}
double f(double t) {
        double r3111843 = 1.0;
        double r3111844 = 2.0;
        double r3111845 = t;
        double r3111846 = r3111844 / r3111845;
        double r3111847 = r3111843 / r3111845;
        double r3111848 = r3111843 + r3111847;
        double r3111849 = r3111846 / r3111848;
        double r3111850 = r3111844 - r3111849;
        double r3111851 = r3111850 * r3111850;
        double r3111852 = r3111843 + r3111851;
        double r3111853 = r3111844 + r3111851;
        double r3111854 = r3111852 / r3111853;
        return r3111854;
}
↓
double f(double t) {
        double r3111855 = 2.0;
        double r3111856 = 1.0;
        double r3111857 = t;
        double r3111858 = r3111856 + r3111857;
        double r3111859 = r3111855 / r3111858;
        double r3111860 = r3111855 - r3111859;
        double r3111861 = fma(r3111860, r3111860, r3111856);
        double r3111862 = fma(r3111860, r3111860, r3111855);
        double r3111863 = r3111861 / r3111862;
        return r3111863;
}

Error

Bits error versus t

Derivation

  1. Initial program 0.0

    \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 1\right)}{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 2\right)}}\]

  3. Final simplification0.0

    \[\leadsto \frac{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 1\right)}{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 2\right)}\]

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 2"
  (/ (+ 1 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))) (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))))))