Average Error: 0.0 → 0.0
Time: 10.1s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{{\left(2 - \frac{2}{t \cdot 1 + 1}\right)}^{6} + {2}^{3}} \cdot \left(\left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) \cdot \left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) + \left(2 \cdot 2 - \left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) \cdot 2\right)\right)\]
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}{{\left(2 - \frac{2}{t \cdot 1 + 1}\right)}^{6} + {2}^{3}} \cdot \left(\left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) \cdot \left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) + \left(2 \cdot 2 - \left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) \cdot 2\right)\right)
double f(double t) {
        double r42791 = 1.0;
        double r42792 = 2.0;
        double r42793 = t;
        double r42794 = r42792 / r42793;
        double r42795 = r42791 / r42793;
        double r42796 = r42791 + r42795;
        double r42797 = r42794 / r42796;
        double r42798 = r42792 - r42797;
        double r42799 = r42798 * r42798;
        double r42800 = r42792 + r42799;
        double r42801 = r42791 / r42800;
        double r42802 = r42791 - r42801;
        return r42802;
}
↓
double f(double t) {
        double r42803 = 1.0;
        double r42804 = 2.0;
        double r42805 = t;
        double r42806 = r42805 * r42803;
        double r42807 = r42806 + r42803;
        double r42808 = r42804 / r42807;
        double r42809 = r42804 - r42808;
        double r42810 = 6.0;
        double r42811 = pow(r42809, r42810);
        double r42812 = 3.0;
        double r42813 = pow(r42804, r42812);
        double r42814 = r42811 + r42813;
        double r42815 = r42803 / r42814;
        double r42816 = r42809 * r42809;
        double r42817 = r42816 * r42816;
        double r42818 = r42804 * r42804;
        double r42819 = r42816 * r42804;
        double r42820 = r42818 - r42819;
        double r42821 = r42817 + r42820;
        double r42822 = r42815 * r42821;
        double r42823 = r42803 - r42822;
        return r42823;
}

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. Simplified0.0

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

  4. Applied flip3-+0.0

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

  5. Applied associate-/r/0.0

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

  6. Simplified0.0

    \[\leadsto 1 - \color{blue}{\frac{1}{{\left(2 - \frac{2}{t \cdot 1 + 1}\right)}^{6} + {2}^{3}}} \cdot \left(\left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) \cdot \left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) + \left(2 \cdot 2 - \left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) \cdot 2\right)\right)\]

  7. Final simplification0.0

    \[\leadsto 1 - \frac{1}{{\left(2 - \frac{2}{t \cdot 1 + 1}\right)}^{6} + {2}^{3}} \cdot \left(\left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) \cdot \left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) + \left(2 \cdot 2 - \left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) \cdot 2\right)\right)\]

Reproduce

herbie shell --seed 2019350 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))