Average Error: 0.0 → 0.0
Time: 11.7s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{1 + t}, 2 - \frac{2}{1 + t}, 2\right)}\]
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}{1 + t}, 2 - \frac{2}{1 + t}, 2\right)}
double f(double t) {
        double r762854 = 1.0;
        double r762855 = 2.0;
        double r762856 = t;
        double r762857 = r762855 / r762856;
        double r762858 = r762854 / r762856;
        double r762859 = r762854 + r762858;
        double r762860 = r762857 / r762859;
        double r762861 = r762855 - r762860;
        double r762862 = r762861 * r762861;
        double r762863 = r762855 + r762862;
        double r762864 = r762854 / r762863;
        double r762865 = r762854 - r762864;
        return r762865;
}
↓
double f(double t) {
        double r762866 = 1.0;
        double r762867 = 2.0;
        double r762868 = t;
        double r762869 = r762866 + r762868;
        double r762870 = r762867 / r762869;
        double r762871 = r762867 - r762870;
        double r762872 = fma(r762871, r762871, r762867);
        double r762873 = r762866 / r762872;
        double r762874 = r762866 - r762873;
        return r762874;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 3"
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))