Average Error: 0.0 → 0.0
Time: 23.6s
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 r39049 = 1.0;
        double r39050 = 2.0;
        double r39051 = t;
        double r39052 = r39050 / r39051;
        double r39053 = r39049 / r39051;
        double r39054 = r39049 + r39053;
        double r39055 = r39052 / r39054;
        double r39056 = r39050 - r39055;
        double r39057 = r39056 * r39056;
        double r39058 = r39050 + r39057;
        double r39059 = r39049 / r39058;
        double r39060 = r39049 - r39059;
        return r39060;
}
↓
double f(double t) {
        double r39061 = 1.0;
        double r39062 = 2.0;
        double r39063 = t;
        double r39064 = r39063 * r39061;
        double r39065 = r39064 + r39061;
        double r39066 = r39062 / r39065;
        double r39067 = r39062 - r39066;
        double r39068 = 6.0;
        double r39069 = pow(r39067, r39068);
        double r39070 = 3.0;
        double r39071 = pow(r39062, r39070);
        double r39072 = r39069 + r39071;
        double r39073 = r39061 / r39072;
        double r39074 = r39067 * r39067;
        double r39075 = r39074 * r39074;
        double r39076 = r39062 * r39062;
        double r39077 = r39074 * r39062;
        double r39078 = r39076 - r39077;
        double r39079 = r39075 + r39078;
        double r39080 = r39073 * r39079;
        double r39081 = r39061 - r39080;
        return r39081;
}

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 2019325 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))