Average Error: 0.0 → 0.0
Time: 15.0s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{2 + \frac{\left(\left(2 \cdot 2\right) \cdot 2 - \frac{2}{t \cdot 1 + 1} \cdot \left(\frac{2}{t \cdot 1 + 1} \cdot \frac{2}{t \cdot 1 + 1}\right)\right) \cdot \left(\left(2 \cdot 2\right) \cdot 2 - \frac{2}{t \cdot 1 + 1} \cdot \left(\frac{2}{t \cdot 1 + 1} \cdot \frac{2}{t \cdot 1 + 1}\right)\right)}{\left(\left(2 \cdot \frac{2}{t \cdot 1 + 1} + \frac{2}{t \cdot 1 + 1} \cdot \frac{2}{t \cdot 1 + 1}\right) + 2 \cdot 2\right) \cdot \left(\left(2 \cdot \frac{2}{t \cdot 1 + 1} + \frac{2}{t \cdot 1 + 1} \cdot \frac{2}{t \cdot 1 + 1}\right) + 2 \cdot 2\right)}}\]
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(\left(2 \cdot 2\right) \cdot 2 - \frac{2}{t \cdot 1 + 1} \cdot \left(\frac{2}{t \cdot 1 + 1} \cdot \frac{2}{t \cdot 1 + 1}\right)\right) \cdot \left(\left(2 \cdot 2\right) \cdot 2 - \frac{2}{t \cdot 1 + 1} \cdot \left(\frac{2}{t \cdot 1 + 1} \cdot \frac{2}{t \cdot 1 + 1}\right)\right)}{\left(\left(2 \cdot \frac{2}{t \cdot 1 + 1} + \frac{2}{t \cdot 1 + 1} \cdot \frac{2}{t \cdot 1 + 1}\right) + 2 \cdot 2\right) \cdot \left(\left(2 \cdot \frac{2}{t \cdot 1 + 1} + \frac{2}{t \cdot 1 + 1} \cdot \frac{2}{t \cdot 1 + 1}\right) + 2 \cdot 2\right)}}
double f(double t) {
        double r2028959 = 1.0;
        double r2028960 = 2.0;
        double r2028961 = t;
        double r2028962 = r2028960 / r2028961;
        double r2028963 = r2028959 / r2028961;
        double r2028964 = r2028959 + r2028963;
        double r2028965 = r2028962 / r2028964;
        double r2028966 = r2028960 - r2028965;
        double r2028967 = r2028966 * r2028966;
        double r2028968 = r2028960 + r2028967;
        double r2028969 = r2028959 / r2028968;
        double r2028970 = r2028959 - r2028969;
        return r2028970;
}
↓
double f(double t) {
        double r2028971 = 1.0;
        double r2028972 = 2.0;
        double r2028973 = r2028972 * r2028972;
        double r2028974 = r2028973 * r2028972;
        double r2028975 = t;
        double r2028976 = r2028975 * r2028971;
        double r2028977 = r2028976 + r2028971;
        double r2028978 = r2028972 / r2028977;
        double r2028979 = r2028978 * r2028978;
        double r2028980 = r2028978 * r2028979;
        double r2028981 = r2028974 - r2028980;
        double r2028982 = r2028981 * r2028981;
        double r2028983 = r2028972 * r2028978;
        double r2028984 = r2028983 + r2028979;
        double r2028985 = r2028984 + r2028973;
        double r2028986 = r2028985 * r2028985;
        double r2028987 = r2028982 / r2028986;
        double r2028988 = r2028972 + r2028987;
        double r2028989 = r2028971 / r2028988;
        double r2028990 = r2028971 - r2028989;
        return r2028990;
}

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

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

  4. Applied flip3--0.0

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

  5. Applied flip3--0.0

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

  6. Applied frac-times0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019192 
(FPCore (t)
  :name "Kahan p13 Example 3"
  (- 1.0 (/ 1.0 (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))))))