Average Error: 16.1 → 15.6
Time: 32.0s
Precision: 64
\[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{\frac{\frac{1}{16} - \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)}{\frac{1}{4} + \left(\frac{1}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{1}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(\frac{\frac{1}{2}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\frac{1}{2}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)}}{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]
1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{\frac{\frac{1}{16} - \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)}{\frac{1}{4} + \left(\frac{1}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{1}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(\frac{\frac{1}{2}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\frac{1}{2}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)}}{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}
double f(double x) {
        double r4813824 = 1.0;
        double r4813825 = 0.5;
        double r4813826 = x;
        double r4813827 = hypot(r4813824, r4813826);
        double r4813828 = r4813824 / r4813827;
        double r4813829 = r4813824 + r4813828;
        double r4813830 = r4813825 * r4813829;
        double r4813831 = sqrt(r4813830);
        double r4813832 = r4813824 - r4813831;
        return r4813832;
}
↓
double f(double x) {
        double r4813833 = 0.0625;
        double r4813834 = 0.5;
        double r4813835 = 1.0;
        double r4813836 = x;
        double r4813837 = hypot(r4813835, r4813836);
        double r4813838 = r4813834 / r4813837;
        double r4813839 = r4813838 * r4813838;
        double r4813840 = r4813839 * r4813839;
        double r4813841 = r4813833 - r4813840;
        double r4813842 = 0.25;
        double r4813843 = cbrt(r4813837);
        double r4813844 = r4813843 * r4813843;
        double r4813845 = r4813835 / r4813844;
        double r4813846 = r4813845 * r4813845;
        double r4813847 = r4813834 / r4813843;
        double r4813848 = r4813847 * r4813847;
        double r4813849 = r4813846 * r4813848;
        double r4813850 = r4813842 + r4813849;
        double r4813851 = r4813841 / r4813850;
        double r4813852 = r4813834 + r4813838;
        double r4813853 = r4813851 / r4813852;
        double r4813854 = sqrt(r4813852);
        double r4813855 = r4813835 + r4813854;
        double r4813856 = r4813853 / r4813855;
        return r4813856;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 16.1

    \[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]

  2. Simplified16.1

    \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]
  3. Using strategy rm

  4. Applied flip--16.1

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

  5. Simplified15.6

    \[\leadsto \frac{\color{blue}{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]
  6. Using strategy rm

  7. Applied flip--15.6

    \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot \frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]
  8. Using strategy rm

  9. Applied flip--15.6

    \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{2}\right) - \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}}{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]
  10. Using strategy rm

  11. Applied add-cube-cbrt15.6

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

  12. Applied *-un-lft-identity15.6

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

  13. Applied times-frac15.6

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

  14. Applied add-cube-cbrt15.6

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

  15. Applied *-un-lft-identity15.6

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

  16. Applied times-frac15.6

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

  17. Applied swap-sqr15.6

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

  18. Final simplification15.6

    \[\leadsto \frac{\frac{\frac{\frac{1}{16} - \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)}{\frac{1}{4} + \left(\frac{1}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{1}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(\frac{\frac{1}{2}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\frac{1}{2}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)}}{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

Reproduce

herbie shell --seed 2019142 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1 (sqrt (* 1/2 (+ 1 (/ 1 (hypot 1 x)))))))