Average Error: 0.0 → 0.0
Time: 10.6s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, 1, t \cdot 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, 1, t \cdot 1\right)}, 2\right)}\]
1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
↓
1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, 1, t \cdot 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, 1, t \cdot 1\right)}, 2\right)}
double f(double t) {
        double r1333867 = 1.0;
        double r1333868 = 2.0;
        double r1333869 = t;
        double r1333870 = r1333868 / r1333869;
        double r1333871 = r1333867 / r1333869;
        double r1333872 = r1333867 + r1333871;
        double r1333873 = r1333870 / r1333872;
        double r1333874 = r1333868 - r1333873;
        double r1333875 = r1333874 * r1333874;
        double r1333876 = r1333868 + r1333875;
        double r1333877 = r1333867 / r1333876;
        double r1333878 = r1333867 - r1333877;
        return r1333878;
}
↓
double f(double t) {
        double r1333879 = 1.0;
        double r1333880 = 2.0;
        double r1333881 = 1.0;
        double r1333882 = t;
        double r1333883 = r1333882 * r1333879;
        double r1333884 = fma(r1333881, r1333879, r1333883);
        double r1333885 = r1333880 / r1333884;
        double r1333886 = r1333880 - r1333885;
        double r1333887 = fma(r1333886, r1333886, r1333880);
        double r1333888 = r1333879 / r1333887;
        double r1333889 = r1333879 - r1333888;
        return r1333889;
}

Error

Bits error versus t

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 3"
  (- 1.0 (/ 1.0 (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))))))