Average Error: 0.0 → 0.0
Time: 8.7s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, 1, t \cdot 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, 1, t \cdot 1\right)}, 2\right)}\]
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 r21801 = 1.0;
        double r21802 = 2.0;
        double r21803 = t;
        double r21804 = r21802 / r21803;
        double r21805 = r21801 / r21803;
        double r21806 = r21801 + r21805;
        double r21807 = r21804 / r21806;
        double r21808 = r21802 - r21807;
        double r21809 = r21808 * r21808;
        double r21810 = r21802 + r21809;
        double r21811 = r21801 / r21810;
        double r21812 = r21801 - r21811;
        return r21812;
}
↓
double f(double t) {
        double r21813 = 1.0;
        double r21814 = 2.0;
        double r21815 = 1.0;
        double r21816 = t;
        double r21817 = r21816 * r21813;
        double r21818 = fma(r21815, r21813, r21817);
        double r21819 = r21814 / r21818;
        double r21820 = r21814 - r21819;
        double r21821 = fma(r21820, r21820, r21814);
        double r21822 = r21813 / r21821;
        double r21823 = r21813 - r21822;
        return r21823;
}

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 2019195 +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)))))))))