Average Error: 16.0 → 16.0
Time: 18.9s
Precision: 64
\[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{\sqrt{\frac{1}{8} - \frac{\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}{\frac{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) + \frac{1}{4}}{\sqrt{\frac{1}{8} - \frac{\frac{\frac{\sqrt[3]{\frac{1}{8}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{\sqrt[3]{\frac{1}{8}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\sqrt[3]{\frac{1}{8}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)}}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]
1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{\sqrt{\frac{1}{8} - \frac{\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}{\frac{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) + \frac{1}{4}}{\sqrt{\frac{1}{8} - \frac{\frac{\frac{\sqrt[3]{\frac{1}{8}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{\sqrt[3]{\frac{1}{8}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\sqrt[3]{\frac{1}{8}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)}}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}
double f(double x) {
        double r5752668 = 1.0;
        double r5752669 = 0.5;
        double r5752670 = x;
        double r5752671 = hypot(r5752668, r5752670);
        double r5752672 = r5752668 / r5752671;
        double r5752673 = r5752668 + r5752672;
        double r5752674 = r5752669 * r5752673;
        double r5752675 = sqrt(r5752674);
        double r5752676 = r5752668 - r5752675;
        return r5752676;
}
↓
double f(double x) {
        double r5752677 = 0.125;
        double r5752678 = 1.0;
        double r5752679 = x;
        double r5752680 = hypot(r5752678, r5752679);
        double r5752681 = r5752677 / r5752680;
        double r5752682 = r5752681 / r5752680;
        double r5752683 = r5752682 / r5752680;
        double r5752684 = r5752677 - r5752683;
        double r5752685 = sqrt(r5752684);
        double r5752686 = 0.5;
        double r5752687 = r5752686 / r5752680;
        double r5752688 = r5752687 + r5752686;
        double r5752689 = r5752687 * r5752688;
        double r5752690 = 0.25;
        double r5752691 = r5752689 + r5752690;
        double r5752692 = cbrt(r5752677);
        double r5752693 = cbrt(r5752680);
        double r5752694 = r5752692 / r5752693;
        double r5752695 = r5752694 / r5752680;
        double r5752696 = r5752695 / r5752680;
        double r5752697 = r5752694 * r5752694;
        double r5752698 = r5752696 * r5752697;
        double r5752699 = r5752677 - r5752698;
        double r5752700 = sqrt(r5752699);
        double r5752701 = r5752691 / r5752700;
        double r5752702 = r5752685 / r5752701;
        double r5752703 = sqrt(r5752688);
        double r5752704 = r5752678 + r5752703;
        double r5752705 = r5752702 / r5752704;
        return r5752705;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 16.0

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

  2. Simplified16.0

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

  4. Applied flip--16.0

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

  5. Simplified15.5

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

  7. Applied flip3--15.5

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

  8. Simplified15.5

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

  9. Simplified15.5

    \[\leadsto \frac{\frac{\frac{1}{8} - \frac{\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{\color{blue}{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) + \frac{1}{4}}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]
  10. Using strategy rm

  11. Applied add-sqr-sqrt16.0

    \[\leadsto \frac{\frac{\color{blue}{\sqrt{\frac{1}{8} - \frac{\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{\frac{1}{8} - \frac{\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}}{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) + \frac{1}{4}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

  12. Applied associate-/l*16.0

    \[\leadsto \frac{\color{blue}{\frac{\sqrt{\frac{1}{8} - \frac{\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}{\frac{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) + \frac{1}{4}}{\sqrt{\frac{1}{8} - \frac{\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]
  13. Using strategy rm

  14. Applied *-un-lft-identity16.0

    \[\leadsto \frac{\frac{\sqrt{\frac{1}{8} - \frac{\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}{\frac{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) + \frac{1}{4}}{\sqrt{\frac{1}{8} - \frac{\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{\color{blue}{1 \cdot \mathsf{hypot}\left(1, x\right)}}}}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

  15. Applied *-un-lft-identity16.0

    \[\leadsto \frac{\frac{\sqrt{\frac{1}{8} - \frac{\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}{\frac{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) + \frac{1}{4}}{\sqrt{\frac{1}{8} - \frac{\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\color{blue}{1 \cdot \mathsf{hypot}\left(1, x\right)}}}{1 \cdot \mathsf{hypot}\left(1, x\right)}}}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

  16. Applied add-cube-cbrt16.0

    \[\leadsto \frac{\frac{\sqrt{\frac{1}{8} - \frac{\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}{\frac{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) + \frac{1}{4}}{\sqrt{\frac{1}{8} - \frac{\frac{\frac{\frac{1}{8}}{\color{blue}{\left(\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}}}{1 \cdot \mathsf{hypot}\left(1, x\right)}}{1 \cdot \mathsf{hypot}\left(1, x\right)}}}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

  17. Applied add-cube-cbrt16.0

    \[\leadsto \frac{\frac{\sqrt{\frac{1}{8} - \frac{\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}{\frac{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) + \frac{1}{4}}{\sqrt{\frac{1}{8} - \frac{\frac{\frac{\color{blue}{\left(\sqrt[3]{\frac{1}{8}} \cdot \sqrt[3]{\frac{1}{8}}\right) \cdot \sqrt[3]{\frac{1}{8}}}}{\left(\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}}{1 \cdot \mathsf{hypot}\left(1, x\right)}}{1 \cdot \mathsf{hypot}\left(1, x\right)}}}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

  18. Applied times-frac16.0

    \[\leadsto \frac{\frac{\sqrt{\frac{1}{8} - \frac{\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}{\frac{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) + \frac{1}{4}}{\sqrt{\frac{1}{8} - \frac{\frac{\color{blue}{\frac{\sqrt[3]{\frac{1}{8}} \cdot \sqrt[3]{\frac{1}{8}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\sqrt[3]{\frac{1}{8}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}}}{1 \cdot \mathsf{hypot}\left(1, x\right)}}{1 \cdot \mathsf{hypot}\left(1, x\right)}}}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

  19. Applied times-frac16.0

    \[\leadsto \frac{\frac{\sqrt{\frac{1}{8} - \frac{\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}{\frac{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) + \frac{1}{4}}{\sqrt{\frac{1}{8} - \frac{\color{blue}{\frac{\frac{\sqrt[3]{\frac{1}{8}} \cdot \sqrt[3]{\frac{1}{8}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}}{1} \cdot \frac{\frac{\sqrt[3]{\frac{1}{8}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}}{\mathsf{hypot}\left(1, x\right)}}}{1 \cdot \mathsf{hypot}\left(1, x\right)}}}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

  20. Applied times-frac16.0

    \[\leadsto \frac{\frac{\sqrt{\frac{1}{8} - \frac{\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}{\frac{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) + \frac{1}{4}}{\sqrt{\frac{1}{8} - \color{blue}{\frac{\frac{\frac{\sqrt[3]{\frac{1}{8}} \cdot \sqrt[3]{\frac{1}{8}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}}{1}}{1} \cdot \frac{\frac{\frac{\sqrt[3]{\frac{1}{8}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

  21. Simplified16.0

    \[\leadsto \frac{\frac{\sqrt{\frac{1}{8} - \frac{\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}{\frac{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) + \frac{1}{4}}{\sqrt{\frac{1}{8} - \color{blue}{\left(\frac{\sqrt[3]{\frac{1}{8}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\sqrt[3]{\frac{1}{8}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)} \cdot \frac{\frac{\frac{\sqrt[3]{\frac{1}{8}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

  22. Final simplification16.0

    \[\leadsto \frac{\frac{\sqrt{\frac{1}{8} - \frac{\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}{\frac{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right) + \frac{1}{4}}{\sqrt{\frac{1}{8} - \frac{\frac{\frac{\sqrt[3]{\frac{1}{8}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{\sqrt[3]{\frac{1}{8}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\sqrt[3]{\frac{1}{8}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)}}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1 (sqrt (* 1/2 (+ 1 (/ 1 (hypot 1 x)))))))