Average Error: 0.0 → 0.0
Time: 11.0s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{2 + \frac{\left({2}^{3} - {\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}\right) \cdot \left(\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}{\left(\left(2 \cdot 2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + 2\right) + 2 \cdot 2\right)\right) \cdot \left(2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}\]
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}{2 + \frac{\left({2}^{3} - {\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}\right) \cdot \left(\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}{\left(\left(2 \cdot 2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + 2\right) + 2 \cdot 2\right)\right) \cdot \left(2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}
double f(double t) {
        double r49688 = 1.0;
        double r49689 = 2.0;
        double r49690 = t;
        double r49691 = r49689 / r49690;
        double r49692 = r49688 / r49690;
        double r49693 = r49688 + r49692;
        double r49694 = r49691 / r49693;
        double r49695 = r49689 - r49694;
        double r49696 = r49695 * r49695;
        double r49697 = r49689 + r49696;
        double r49698 = r49688 / r49697;
        double r49699 = r49688 - r49698;
        return r49699;
}
↓
double f(double t) {
        double r49700 = 1.0;
        double r49701 = 2.0;
        double r49702 = 3.0;
        double r49703 = pow(r49701, r49702);
        double r49704 = t;
        double r49705 = r49701 / r49704;
        double r49706 = r49700 / r49704;
        double r49707 = r49700 + r49706;
        double r49708 = r49705 / r49707;
        double r49709 = pow(r49708, r49702);
        double r49710 = r49703 - r49709;
        double r49711 = r49701 * r49701;
        double r49712 = r49711 * r49711;
        double r49713 = r49708 * r49708;
        double r49714 = r49713 * r49713;
        double r49715 = r49712 - r49714;
        double r49716 = r49710 * r49715;
        double r49717 = r49711 + r49713;
        double r49718 = r49708 + r49701;
        double r49719 = r49708 * r49718;
        double r49720 = r49719 + r49711;
        double r49721 = r49717 * r49720;
        double r49722 = r49701 + r49708;
        double r49723 = r49721 * r49722;
        double r49724 = r49716 / r49723;
        double r49725 = r49701 + r49724;
        double r49726 = r49700 / r49725;
        double r49727 = r49700 - r49726;
        return r49727;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied flip--0.0

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

  4. Applied associate-*r/0.0

    \[\leadsto 1 - \frac{1}{2 + \color{blue}{\frac{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 \cdot 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}}}}\]
  5. Using strategy rm

  6. Applied flip--0.0

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

  7. Applied flip3--0.0

    \[\leadsto 1 - \frac{1}{2 + \frac{\color{blue}{\frac{{2}^{3} - {\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}}{2 \cdot 2 + \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}} + 2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}} \cdot \frac{\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 \cdot 2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}}}{2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}}}\]

  8. Applied frac-times0.0

    \[\leadsto 1 - \frac{1}{2 + \frac{\color{blue}{\frac{\left({2}^{3} - {\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}\right) \cdot \left(\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}{\left(2 \cdot 2 + \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}} + 2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right) \cdot \left(2 \cdot 2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}}{2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}}}\]

  9. Applied associate-/l/0.0

    \[\leadsto 1 - \frac{1}{2 + \color{blue}{\frac{\left({2}^{3} - {\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}\right) \cdot \left(\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}{\left(2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\left(2 \cdot 2 + \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}} + 2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right) \cdot \left(2 \cdot 2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}}}\]

  10. Simplified0.0

    \[\leadsto 1 - \frac{1}{2 + \frac{\left({2}^{3} - {\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}\right) \cdot \left(\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}{\color{blue}{\left(\left(2 \cdot 2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + 2\right) + 2 \cdot 2\right)\right) \cdot \left(2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}}\]

  11. Final simplification0.0

    \[\leadsto 1 - \frac{1}{2 + \frac{\left({2}^{3} - {\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}\right) \cdot \left(\left(2 \cdot 2\right) \cdot \left(2 \cdot 2\right) - \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}{\left(\left(2 \cdot 2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + 2\right) + 2 \cdot 2\right)\right) \cdot \left(2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}\]

Reproduce

herbie shell --seed 2019308 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))