Average Error: 0.0 → 0.0
Time: 8.9s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{2 + \frac{\left({2}^{3} - {\left(\frac{2}{1 \cdot \left(1 + t\right)}\right)}^{3}\right) \cdot \left(\left(2 + \frac{2}{1 \cdot \left(1 + t\right)}\right) \cdot \left(2 - \frac{2}{1 \cdot \left(1 + t\right)}\right)\right)}{\left(2 + \frac{2}{1 \cdot \left(1 + t\right)}\right) \cdot \left(\frac{2}{1 + t} \cdot \frac{2 + \frac{2}{1 \cdot \left(1 + t\right)}}{1} + 2 \cdot 2\right)}}\]
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({2}^{3} - {\left(\frac{2}{1 \cdot \left(1 + t\right)}\right)}^{3}\right) \cdot \left(\left(2 + \frac{2}{1 \cdot \left(1 + t\right)}\right) \cdot \left(2 - \frac{2}{1 \cdot \left(1 + t\right)}\right)\right)}{\left(2 + \frac{2}{1 \cdot \left(1 + t\right)}\right) \cdot \left(\frac{2}{1 + t} \cdot \frac{2 + \frac{2}{1 \cdot \left(1 + t\right)}}{1} + 2 \cdot 2\right)}}
double f(double t) {
        double r30046 = 1.0;
        double r30047 = 2.0;
        double r30048 = t;
        double r30049 = r30047 / r30048;
        double r30050 = r30046 / r30048;
        double r30051 = r30046 + r30050;
        double r30052 = r30049 / r30051;
        double r30053 = r30047 - r30052;
        double r30054 = r30053 * r30053;
        double r30055 = r30047 + r30054;
        double r30056 = r30046 / r30055;
        double r30057 = r30046 - r30056;
        return r30057;
}
↓
double f(double t) {
        double r30058 = 1.0;
        double r30059 = 2.0;
        double r30060 = 3.0;
        double r30061 = pow(r30059, r30060);
        double r30062 = 1.0;
        double r30063 = t;
        double r30064 = r30062 + r30063;
        double r30065 = r30058 * r30064;
        double r30066 = r30059 / r30065;
        double r30067 = pow(r30066, r30060);
        double r30068 = r30061 - r30067;
        double r30069 = r30059 + r30066;
        double r30070 = r30059 - r30066;
        double r30071 = r30069 * r30070;
        double r30072 = r30068 * r30071;
        double r30073 = r30059 / r30064;
        double r30074 = r30069 / r30058;
        double r30075 = r30073 * r30074;
        double r30076 = r30059 * r30059;
        double r30077 = r30075 + r30076;
        double r30078 = r30069 * r30077;
        double r30079 = r30072 / r30078;
        double r30080 = r30059 + r30079;
        double r30081 = r30058 / r30080;
        double r30082 = r30058 - r30081;
        return r30082;
}

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 flip--0.0

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

  5. Applied flip3--0.0

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

  6. Applied frac-times0.0

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

  7. Simplified0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019195 
(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)))))))))