Average Error: 0.0 → 0.0
Time: 21.3s
Precision: 64
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{1 + \left(\left(2 + \frac{-\frac{{\left(\frac{\sqrt[3]{2}}{\sqrt[3]{t}}\right)}^{3}}{\sqrt[3]{1 + \frac{1}{t}}}}{\sqrt[3]{1 + \frac{1}{t}} \cdot \sqrt[3]{1 + \frac{1}{t}}}\right) + \frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}} \cdot \sqrt[3]{1 + \frac{1}{t}}} \cdot \left(\left(-\frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}}}\right) + \frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}}}\right)\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(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(\left(2 + \frac{-\frac{{\left(\frac{\sqrt[3]{2}}{\sqrt[3]{t}}\right)}^{3}}{\sqrt[3]{1 + \frac{1}{t}}}}{\sqrt[3]{1 + \frac{1}{t}} \cdot \sqrt[3]{1 + \frac{1}{t}}}\right) + \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 \left(\left(-\frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}}}\right) + \frac{\frac{\sqrt[3]{2}}{\sqrt[3]{t}}}{\sqrt[3]{1 + \frac{1}{t}}}\right)\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)}
double f(double t) {
        double r49947 = 1.0;
        double r49948 = 2.0;
        double r49949 = t;
        double r49950 = r49948 / r49949;
        double r49951 = r49947 / r49949;
        double r49952 = r49947 + r49951;
        double r49953 = r49950 / r49952;
        double r49954 = r49948 - r49953;
        double r49955 = r49954 * r49954;
        double r49956 = r49947 + r49955;
        double r49957 = r49948 + r49955;
        double r49958 = r49956 / r49957;
        return r49958;
}
↓
double f(double t) {
        double r49959 = 1.0;
        double r49960 = 2.0;
        double r49961 = cbrt(r49960);
        double r49962 = t;
        double r49963 = cbrt(r49962);
        double r49964 = r49961 / r49963;
        double r49965 = 3.0;
        double r49966 = pow(r49964, r49965);
        double r49967 = r49959 / r49962;
        double r49968 = r49959 + r49967;
        double r49969 = cbrt(r49968);
        double r49970 = r49966 / r49969;
        double r49971 = -r49970;
        double r49972 = r49969 * r49969;
        double r49973 = r49971 / r49972;
        double r49974 = r49960 + r49973;
        double r49975 = r49961 * r49961;
        double r49976 = r49963 * r49963;
        double r49977 = r49975 / r49976;
        double r49978 = r49977 / r49972;
        double r49979 = r49964 / r49969;
        double r49980 = -r49979;
        double r49981 = r49980 + r49979;
        double r49982 = r49978 * r49981;
        double r49983 = r49974 + r49982;
        double r49984 = r49960 / r49962;
        double r49985 = r49984 / r49968;
        double r49986 = r49960 - r49985;
        double r49987 = r49983 * r49986;
        double r49988 = r49959 + r49987;
        double r49989 = r49986 * r49986;
        double r49990 = r49960 + r49989;
        double r49991 = r49988 / r49990;
        return r49991;
}

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

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied add-cube-cbrt0.0

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

  6. Applied times-frac0.0

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

  7. Applied times-frac0.0

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

  8. Applied add-sqr-sqrt0.5

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

  9. Applied prod-diff0.5

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

  10. Simplified0.0

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

  11. Simplified0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 2"
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))