Average Error: 0.0 → 0.0
Time: 8.0s
Precision: 64
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\left(2 - \frac{\frac{2}{\sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}}} \cdot \frac{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}} \cdot \sqrt[3]{1 + \frac{1}{t}}}\right) + \frac{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}} \cdot \sqrt[3]{1 + \frac{1}{t}}} \cdot \left(\left(-\frac{\frac{2}{\sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}}}\right) + \frac{\frac{2}{\sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}}}\right)\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
↓
\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\left(2 - \frac{\frac{2}{\sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}}} \cdot \frac{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}} \cdot \sqrt[3]{1 + \frac{1}{t}}}\right) + \frac{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}} \cdot \sqrt[3]{1 + \frac{1}{t}}} \cdot \left(\left(-\frac{\frac{2}{\sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}}}\right) + \frac{\frac{2}{\sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}}}\right)\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
double f(double t) {
        double r74784 = 1.0;
        double r74785 = 2.0;
        double r74786 = t;
        double r74787 = r74785 / r74786;
        double r74788 = r74784 / r74786;
        double r74789 = r74784 + r74788;
        double r74790 = r74787 / r74789;
        double r74791 = r74785 - r74790;
        double r74792 = r74791 * r74791;
        double r74793 = r74784 + r74792;
        double r74794 = r74785 + r74792;
        double r74795 = r74793 / r74794;
        return r74795;
}
↓
double f(double t) {
        double r74796 = 1.0;
        double r74797 = 2.0;
        double r74798 = t;
        double r74799 = r74797 / r74798;
        double r74800 = r74796 / r74798;
        double r74801 = r74796 + r74800;
        double r74802 = r74799 / r74801;
        double r74803 = r74797 - r74802;
        double r74804 = cbrt(r74798);
        double r74805 = r74797 / r74804;
        double r74806 = cbrt(r74801);
        double r74807 = r74805 / r74806;
        double r74808 = 1.0;
        double r74809 = r74804 * r74804;
        double r74810 = r74808 / r74809;
        double r74811 = r74806 * r74806;
        double r74812 = r74810 / r74811;
        double r74813 = r74807 * r74812;
        double r74814 = r74797 - r74813;
        double r74815 = -r74807;
        double r74816 = r74815 + r74807;
        double r74817 = r74812 * r74816;
        double r74818 = r74814 + r74817;
        double r74819 = r74803 * r74818;
        double r74820 = r74796 + r74819;
        double r74821 = r74803 * r74803;
        double r74822 = r74797 + r74821;
        double r74823 = r74820 / r74822;
        return r74823;
}

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

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

  3. Applied add-cube-cbrt0.0

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied *-un-lft-identity0.0

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

  6. Applied times-frac0.0

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

  7. Applied times-frac0.0

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

  8. Applied add-sqr-sqrt0.5

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

  9. Applied prod-diff0.5

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

  10. Simplified0.0

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

  11. Simplified0.0

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

  12. Final simplification0.0

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

Reproduce

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