Average Error: 0.0 → 0.0
Time: 21.9s
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 r30820 = 1.0;
        double r30821 = 2.0;
        double r30822 = t;
        double r30823 = r30821 / r30822;
        double r30824 = r30820 / r30822;
        double r30825 = r30820 + r30824;
        double r30826 = r30823 / r30825;
        double r30827 = r30821 - r30826;
        double r30828 = r30827 * r30827;
        double r30829 = r30821 + r30828;
        double r30830 = r30820 / r30829;
        double r30831 = r30820 - r30830;
        return r30831;
}
↓
double f(double t) {
        double r30832 = 1.0;
        double r30833 = 2.0;
        double r30834 = t;
        double r30835 = r30834 * r30832;
        double r30836 = r30835 + r30832;
        double r30837 = r30833 / r30836;
        double r30838 = r30833 - r30837;
        double r30839 = 6.0;
        double r30840 = pow(r30838, r30839);
        double r30841 = 3.0;
        double r30842 = pow(r30833, r30841);
        double r30843 = r30840 + r30842;
        double r30844 = r30832 / r30843;
        double r30845 = r30838 * r30838;
        double r30846 = r30845 * r30845;
        double r30847 = r30833 * r30833;
        double r30848 = r30845 * r30833;
        double r30849 = r30847 - r30848;
        double r30850 = r30846 + r30849;
        double r30851 = r30844 * r30850;
        double r30852 = r30832 - r30851;
        return r30852;
}

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 2019303 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))