Average Error: 0.0 → 0.0
Time: 10.5s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, 1, t \cdot 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, 1, t \cdot 1\right)}, 2\right)}\]
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 r1419756 = 1.0;
        double r1419757 = 2.0;
        double r1419758 = t;
        double r1419759 = r1419757 / r1419758;
        double r1419760 = r1419756 / r1419758;
        double r1419761 = r1419756 + r1419760;
        double r1419762 = r1419759 / r1419761;
        double r1419763 = r1419757 - r1419762;
        double r1419764 = r1419763 * r1419763;
        double r1419765 = r1419757 + r1419764;
        double r1419766 = r1419756 / r1419765;
        double r1419767 = r1419756 - r1419766;
        return r1419767;
}
↓
double f(double t) {
        double r1419768 = 1.0;
        double r1419769 = 2.0;
        double r1419770 = 1.0;
        double r1419771 = t;
        double r1419772 = r1419771 * r1419768;
        double r1419773 = fma(r1419770, r1419768, r1419772);
        double r1419774 = r1419769 / r1419773;
        double r1419775 = r1419769 - r1419774;
        double r1419776 = fma(r1419775, r1419775, r1419769);
        double r1419777 = r1419768 / r1419776;
        double r1419778 = r1419768 - r1419777;
        return r1419778;
}

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 2019171 +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)))))))))