Average Error: 0.0 → 0.0
Time: 4.9s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{{\left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right)}^{6} + {2}^{3}} \cdot \left(2 \cdot 2 + \left(\left(\left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right) \cdot \left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right)\right) \cdot \left(\left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right) \cdot \left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right)\right) - 2 \cdot \left(\left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right) \cdot \left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right)\right)\right)\right)\]
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}{1 \cdot \left(t + 1\right)}\right)}^{6} + {2}^{3}} \cdot \left(2 \cdot 2 + \left(\left(\left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right) \cdot \left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right)\right) \cdot \left(\left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right) \cdot \left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right)\right) - 2 \cdot \left(\left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right) \cdot \left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right)\right)\right)\right)
double f(double t) {
        double r55944 = 1.0;
        double r55945 = 2.0;
        double r55946 = t;
        double r55947 = r55945 / r55946;
        double r55948 = r55944 / r55946;
        double r55949 = r55944 + r55948;
        double r55950 = r55947 / r55949;
        double r55951 = r55945 - r55950;
        double r55952 = r55951 * r55951;
        double r55953 = r55945 + r55952;
        double r55954 = r55944 / r55953;
        double r55955 = r55944 - r55954;
        return r55955;
}
↓
double f(double t) {
        double r55956 = 1.0;
        double r55957 = 2.0;
        double r55958 = t;
        double r55959 = 1.0;
        double r55960 = r55958 + r55959;
        double r55961 = r55956 * r55960;
        double r55962 = r55957 / r55961;
        double r55963 = r55957 - r55962;
        double r55964 = 6.0;
        double r55965 = pow(r55963, r55964);
        double r55966 = 3.0;
        double r55967 = pow(r55957, r55966);
        double r55968 = r55965 + r55967;
        double r55969 = r55956 / r55968;
        double r55970 = r55957 * r55957;
        double r55971 = r55963 * r55963;
        double r55972 = r55971 * r55971;
        double r55973 = r55957 * r55971;
        double r55974 = r55972 - r55973;
        double r55975 = r55970 + r55974;
        double r55976 = r55969 * r55975;
        double r55977 = r55956 - r55976;
        return r55977;
}

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

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

  4. Applied flip3-+0.0

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

  5. Applied associate-/r/0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020046 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))