Kahan p13 Example 2

Average Error: 0.0 → 0.0

Time: 7.4s

Precision: 64

Math

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

↓

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

C

double f(double t) {
        double r81896 = 1.0;
        double r81897 = 2.0;
        double r81898 = t;
        double r81899 = r81897 / r81898;
        double r81900 = r81896 / r81898;
        double r81901 = r81896 + r81900;
        double r81902 = r81899 / r81901;
        double r81903 = r81897 - r81902;
        double r81904 = r81903 * r81903;
        double r81905 = r81896 + r81904;
        double r81906 = r81897 + r81904;
        double r81907 = r81905 / r81906;
        return r81907;
}

↓

double f(double t) {
        double r81908 = 1.0;
        double r81909 = 2.0;
        double r81910 = cbrt(r81909);
        double r81911 = t;
        double r81912 = cbrt(r81911);
        double r81913 = r81910 / r81912;
        double r81914 = r81908 / r81911;
        double r81915 = r81908 + r81914;
        double r81916 = cbrt(r81915);
        double r81917 = r81913 / r81916;
        double r81918 = r81910 * r81910;
        double r81919 = r81912 * r81912;
        double r81920 = r81918 / r81919;
        double r81921 = r81916 * r81916;
        double r81922 = r81920 / r81921;
        double r81923 = r81917 * r81922;
        double r81924 = r81909 - r81923;
        double r81925 = exp(r81924);
        double r81926 = log(r81925);
        double r81927 = -r81917;
        double r81928 = r81927 + r81917;
        double r81929 = r81922 * r81928;
        double r81930 = r81926 + r81929;
        double r81931 = r81909 / r81911;
        double r81932 = r81931 / r81915;
        double r81933 = r81909 - r81932;
        double r81934 = r81930 * r81933;
        double r81935 = r81908 + r81934;
        double r81936 = r81933 * r81933;
        double r81937 = r81909 + r81936;
        double r81938 = r81935 / r81937;
        return r81938;
}

Error

Bits error versus t

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-cbrt 0.0

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

4. Applied add-cube-cbrt 0.0

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

5. Applied add-cube-cbrt 0.0

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

6. Applied times-frac 0.0

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

7. Applied times-frac 0.0

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

8. Applied add-sqr-sqrt 0.5

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

9. Applied prod-diff 0.5

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

10. Simplified 0.0

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

11. Simplified 0.0

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

13. Applied add-log-exp 0.0

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

14. Applied add-log-exp 0.0

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

15. Applied diff-log 0.0

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

16. Simplified 0.0

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

17. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020024 +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))))))))