Average Error: 0.0 → 0.0
Time: 50.2s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{(\left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right) + 2)_*}\]
double f(double t) {
        double r4140728 = 1.0;
        double r4140729 = 2.0;
        double r4140730 = t;
        double r4140731 = r4140729 / r4140730;
        double r4140732 = r4140728 / r4140730;
        double r4140733 = r4140728 + r4140732;
        double r4140734 = r4140731 / r4140733;
        double r4140735 = r4140729 - r4140734;
        double r4140736 = r4140735 * r4140735;
        double r4140737 = r4140729 + r4140736;
        double r4140738 = r4140728 / r4140737;
        double r4140739 = r4140728 - r4140738;
        return r4140739;
}
↓
double f(double t) {
        double r4140740 = 1.0;
        double r4140741 = 2.0;
        double r4140742 = t;
        double r4140743 = r4140740 + r4140742;
        double r4140744 = r4140741 / r4140743;
        double r4140745 = r4140741 - r4140744;
        double r4140746 = fma(r4140745, r4140745, r4140741);
        double r4140747 = r4140740 / r4140746;
        double r4140748 = r4140740 - r4140747;
        return r4140748;
}


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}{(\left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right) + 2)_*}

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

    \[\leadsto \color{blue}{1 - \frac{1}{(\left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right) + 2)_*}}\]

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019101 +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)))))))))