Kahan p13 Example 2

Average Error: 0.0 → 0.0

Time: 16.7s

Precision: 64

Math

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

↓

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

C

double f(double t) {
        double r101760 = 1.0;
        double r101761 = 2.0;
        double r101762 = t;
        double r101763 = r101761 / r101762;
        double r101764 = r101760 / r101762;
        double r101765 = r101760 + r101764;
        double r101766 = r101763 / r101765;
        double r101767 = r101761 - r101766;
        double r101768 = r101767 * r101767;
        double r101769 = r101760 + r101768;
        double r101770 = r101761 + r101768;
        double r101771 = r101769 / r101770;
        return r101771;
}

↓

double f(double t) {
        double r101772 = 1.0;
        double r101773 = 1.0;
        double r101774 = 2.0;
        double r101775 = t;
        double r101776 = r101774 / r101775;
        double r101777 = cbrt(r101776);
        double r101778 = r101772 / r101775;
        double r101779 = r101772 + r101778;
        double r101780 = cbrt(r101779);
        double r101781 = r101777 / r101780;
        double r101782 = 3.0;
        double r101783 = pow(r101781, r101782);
        double r101784 = -r101783;
        double r101785 = fma(r101773, r101774, r101784);
        double r101786 = fma(r101784, r101773, r101783);
        double r101787 = r101785 + r101786;
        double r101788 = r101776 / r101779;
        double r101789 = r101774 - r101788;
        double r101790 = r101787 * r101789;
        double r101791 = r101772 + r101790;
        double r101792 = r101774 - r101783;
        double r101793 = r101777 * r101777;
        double r101794 = r101780 * r101780;
        double r101795 = r101793 / r101794;
        double r101796 = -r101781;
        double r101797 = r101796 + r101781;
        double r101798 = r101795 * r101797;
        double r101799 = r101792 + r101798;
        double r101800 = r101799 * r101789;
        double r101801 = r101774 + r101800;
        double r101802 = r101791 / r101801;
        return r101802;
}

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

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

4. Applied add-cube-cbrt 0.0

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

5. Applied times-frac 0.0

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

6. Applied add-sqr-sqrt 0.5

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

7. Applied prod-diff 0.5

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

8. Simplified 0.0

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

9. Simplified 0.0

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

11. Applied add-cube-cbrt 0.0

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

12. Applied add-cube-cbrt 0.0

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

13. Applied times-frac 0.0

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

14. Applied *-un-lft-identity 0.0

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

15. Applied prod-diff 0.0

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

16. Simplified 0.0

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

17. Simplified 0.0

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

18. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020056 +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))))))))