Average Error: 0.0 → 0.0
Time: 7.5s
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 r510857 = 1.0;
        double r510858 = 2.0;
        double r510859 = t;
        double r510860 = r510858 / r510859;
        double r510861 = r510857 / r510859;
        double r510862 = r510857 + r510861;
        double r510863 = r510860 / r510862;
        double r510864 = r510858 - r510863;
        double r510865 = r510864 * r510864;
        double r510866 = r510858 + r510865;
        double r510867 = r510857 / r510866;
        double r510868 = r510857 - r510867;
        return r510868;
}
↓
double f(double t) {
        double r510869 = 1.0;
        double r510870 = 2.0;
        double r510871 = t;
        double r510872 = r510869 + r510871;
        double r510873 = r510870 / r510872;
        double r510874 = r510870 - r510873;
        double r510875 = fma(r510874, r510874, r510870);
        double r510876 = r510869 / r510875;
        double r510877 = r510869 - r510876;
        return r510877;
}

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 2019152 +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)))))))))