Average Error: 0.0 → 0.0
Time: 4.8s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{2 + \frac{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left({2}^{3} - {\left(\frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}} \cdot \sqrt[3]{1 + \frac{1}{t}}}\right)}^{3} \cdot {\left(\frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}}}\right)}^{3}\right)}{2 \cdot 2 + \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}} + 2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}\]
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 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left({2}^{3} - {\left(\frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}} \cdot \sqrt[3]{1 + \frac{1}{t}}}\right)}^{3} \cdot {\left(\frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}}}\right)}^{3}\right)}{2 \cdot 2 + \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}} + 2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}
double f(double t) {
        double r54135 = 1.0;
        double r54136 = 2.0;
        double r54137 = t;
        double r54138 = r54136 / r54137;
        double r54139 = r54135 / r54137;
        double r54140 = r54135 + r54139;
        double r54141 = r54138 / r54140;
        double r54142 = r54136 - r54141;
        double r54143 = r54142 * r54142;
        double r54144 = r54136 + r54143;
        double r54145 = r54135 / r54144;
        double r54146 = r54135 - r54145;
        return r54146;
}
↓
double f(double t) {
        double r54147 = 1.0;
        double r54148 = 2.0;
        double r54149 = t;
        double r54150 = r54148 / r54149;
        double r54151 = r54147 / r54149;
        double r54152 = r54147 + r54151;
        double r54153 = r54150 / r54152;
        double r54154 = r54148 - r54153;
        double r54155 = 3.0;
        double r54156 = pow(r54148, r54155);
        double r54157 = cbrt(r54148);
        double r54158 = r54157 * r54157;
        double r54159 = cbrt(r54149);
        double r54160 = r54159 * r54159;
        double r54161 = r54158 / r54160;
        double r54162 = cbrt(r54152);
        double r54163 = r54162 * r54162;
        double r54164 = r54161 / r54163;
        double r54165 = pow(r54164, r54155);
        double r54166 = r54157 / r54159;
        double r54167 = r54166 / r54162;
        double r54168 = pow(r54167, r54155);
        double r54169 = r54165 * r54168;
        double r54170 = r54156 - r54169;
        double r54171 = r54154 * r54170;
        double r54172 = r54148 * r54148;
        double r54173 = r54153 * r54153;
        double r54174 = r54148 * r54153;
        double r54175 = r54173 + r54174;
        double r54176 = r54172 + r54175;
        double r54177 = r54171 / r54176;
        double r54178 = r54148 + r54177;
        double r54179 = r54147 / r54178;
        double r54180 = r54147 - r54179;
        return r54180;
}

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. Using strategy rm

  3. Applied flip3--0.0

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

  4. Applied associate-*r/0.0

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

  6. Applied add-cube-cbrt0.0

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

  7. Applied add-cube-cbrt0.0

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

  8. Applied add-cube-cbrt0.0

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

  9. Applied times-frac0.0

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

  10. Applied times-frac0.0

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

  11. Applied unpow-prod-down0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2020001 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))