Average Error: 15.6 → 15.1
Time: 4.7s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{1 \cdot \left(\left(1 \cdot 1 - 0.5 \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 + 1\right)\right)}{\left(1 + 0.5\right) \cdot \mathsf{hypot}\left(1, x\right)}}{{1}^{3} + {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}} \cdot \left(1 \cdot 1 + \frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)} \cdot \left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} - 1\right)}{\sqrt{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\right)\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{1 \cdot \left(\left(1 \cdot 1 - 0.5 \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 + 1\right)\right)}{\left(1 + 0.5\right) \cdot \mathsf{hypot}\left(1, x\right)}}{{1}^{3} + {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}} \cdot \left(1 \cdot 1 + \frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)} \cdot \left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} - 1\right)}{\sqrt{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\right)
double f(double x) {
        double r299763 = 1.0;
        double r299764 = 0.5;
        double r299765 = x;
        double r299766 = hypot(r299763, r299765);
        double r299767 = r299763 / r299766;
        double r299768 = r299763 + r299767;
        double r299769 = r299764 * r299768;
        double r299770 = sqrt(r299769);
        double r299771 = r299763 - r299770;
        return r299771;
}
↓
double f(double x) {
        double r299772 = 1.0;
        double r299773 = r299772 * r299772;
        double r299774 = 0.5;
        double r299775 = r299774 * r299774;
        double r299776 = r299773 - r299775;
        double r299777 = x;
        double r299778 = hypot(r299772, r299777);
        double r299779 = r299776 * r299778;
        double r299780 = r299774 + r299772;
        double r299781 = r299774 * r299780;
        double r299782 = r299779 - r299781;
        double r299783 = r299772 * r299782;
        double r299784 = r299772 + r299774;
        double r299785 = r299784 * r299778;
        double r299786 = r299783 / r299785;
        double r299787 = 3.0;
        double r299788 = pow(r299772, r299787);
        double r299789 = r299772 / r299778;
        double r299790 = r299772 + r299789;
        double r299791 = r299774 * r299790;
        double r299792 = sqrt(r299791);
        double r299793 = pow(r299792, r299787);
        double r299794 = r299788 + r299793;
        double r299795 = r299786 / r299794;
        double r299796 = pow(r299789, r299787);
        double r299797 = r299788 + r299796;
        double r299798 = r299774 * r299797;
        double r299799 = sqrt(r299798);
        double r299800 = r299792 - r299772;
        double r299801 = r299799 * r299800;
        double r299802 = r299789 * r299789;
        double r299803 = r299772 * r299789;
        double r299804 = r299802 - r299803;
        double r299805 = r299773 + r299804;
        double r299806 = sqrt(r299805);
        double r299807 = r299801 / r299806;
        double r299808 = r299773 + r299807;
        double r299809 = r299795 * r299808;
        return r299809;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.6

    \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
  2. Using strategy rm

  3. Applied flip--15.6

    \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\]

  4. Simplified15.1

    \[\leadsto \frac{\color{blue}{1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
  5. Using strategy rm

  6. Applied associate-*r/15.1

    \[\leadsto \frac{1 \cdot \left(1 - 0.5\right) - \color{blue}{\frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  7. Applied flip--15.1

    \[\leadsto \frac{1 \cdot \color{blue}{\frac{1 \cdot 1 - 0.5 \cdot 0.5}{1 + 0.5}} - \frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  8. Applied associate-*r/15.1

    \[\leadsto \frac{\color{blue}{\frac{1 \cdot \left(1 \cdot 1 - 0.5 \cdot 0.5\right)}{1 + 0.5}} - \frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  9. Applied frac-sub15.1

    \[\leadsto \frac{\color{blue}{\frac{\left(1 \cdot \left(1 \cdot 1 - 0.5 \cdot 0.5\right)\right) \cdot \mathsf{hypot}\left(1, x\right) - \left(1 + 0.5\right) \cdot \left(0.5 \cdot 1\right)}{\left(1 + 0.5\right) \cdot \mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  10. Simplified15.1

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(\left(1 \cdot 1 - 0.5 \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 + 1\right)\right)}}{\left(1 + 0.5\right) \cdot \mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
  11. Using strategy rm

  12. Applied flip3-+15.6

    \[\leadsto \frac{\frac{1 \cdot \left(\left(1 \cdot 1 - 0.5 \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 + 1\right)\right)}{\left(1 + 0.5\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\color{blue}{\frac{{1}^{3} + {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}}{1 \cdot 1 + \left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} - 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}}}\]

  13. Applied associate-/r/15.1

    \[\leadsto \color{blue}{\frac{\frac{1 \cdot \left(\left(1 \cdot 1 - 0.5 \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 + 1\right)\right)}{\left(1 + 0.5\right) \cdot \mathsf{hypot}\left(1, x\right)}}{{1}^{3} + {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} - 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}\]
  14. Using strategy rm

  15. Applied flip3-+15.1

    \[\leadsto \frac{\frac{1 \cdot \left(\left(1 \cdot 1 - 0.5 \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 + 1\right)\right)}{\left(1 + 0.5\right) \cdot \mathsf{hypot}\left(1, x\right)}}{{1}^{3} + {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} - 1 \cdot \sqrt{0.5 \cdot \color{blue}{\frac{{1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\right)\right)\]

  16. Applied associate-*r/15.1

    \[\leadsto \frac{\frac{1 \cdot \left(\left(1 \cdot 1 - 0.5 \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 + 1\right)\right)}{\left(1 + 0.5\right) \cdot \mathsf{hypot}\left(1, x\right)}}{{1}^{3} + {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} - 1 \cdot \sqrt{\color{blue}{\frac{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\right)\right)\]

  17. Applied sqrt-div15.1

    \[\leadsto \frac{\frac{1 \cdot \left(\left(1 \cdot 1 - 0.5 \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 + 1\right)\right)}{\left(1 + 0.5\right) \cdot \mathsf{hypot}\left(1, x\right)}}{{1}^{3} + {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} - 1 \cdot \color{blue}{\frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\right)\right)\]

  18. Applied associate-*r/15.1

    \[\leadsto \frac{\frac{1 \cdot \left(\left(1 \cdot 1 - 0.5 \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 + 1\right)\right)}{\left(1 + 0.5\right) \cdot \mathsf{hypot}\left(1, x\right)}}{{1}^{3} + {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} - \color{blue}{\frac{1 \cdot \sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\right)\right)\]

  19. Applied flip3-+15.1

    \[\leadsto \frac{\frac{1 \cdot \left(\left(1 \cdot 1 - 0.5 \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 + 1\right)\right)}{\left(1 + 0.5\right) \cdot \mathsf{hypot}\left(1, x\right)}}{{1}^{3} + {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \color{blue}{\frac{{1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}} - \frac{1 \cdot \sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\right)\right)\]

  20. Applied associate-*r/15.1

    \[\leadsto \frac{\frac{1 \cdot \left(\left(1 \cdot 1 - 0.5 \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 + 1\right)\right)}{\left(1 + 0.5\right) \cdot \mathsf{hypot}\left(1, x\right)}}{{1}^{3} + {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{\color{blue}{\frac{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}} - \frac{1 \cdot \sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\right)\right)\]

  21. Applied sqrt-div15.1

    \[\leadsto \frac{\frac{1 \cdot \left(\left(1 \cdot 1 - 0.5 \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 + 1\right)\right)}{\left(1 + 0.5\right) \cdot \mathsf{hypot}\left(1, x\right)}}{{1}^{3} + {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \color{blue}{\frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}} - \frac{1 \cdot \sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\right)\right)\]

  22. Applied associate-*r/15.1

    \[\leadsto \frac{\frac{1 \cdot \left(\left(1 \cdot 1 - 0.5 \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 + 1\right)\right)}{\left(1 + 0.5\right) \cdot \mathsf{hypot}\left(1, x\right)}}{{1}^{3} + {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\color{blue}{\frac{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}} - \frac{1 \cdot \sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\right)\right)\]

  23. Applied sub-div15.1

    \[\leadsto \frac{\frac{1 \cdot \left(\left(1 \cdot 1 - 0.5 \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 + 1\right)\right)}{\left(1 + 0.5\right) \cdot \mathsf{hypot}\left(1, x\right)}}{{1}^{3} + {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}} \cdot \left(1 \cdot 1 + \color{blue}{\frac{\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)} - 1 \cdot \sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\right)\]

  24. Simplified15.1

    \[\leadsto \frac{\frac{1 \cdot \left(\left(1 \cdot 1 - 0.5 \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 + 1\right)\right)}{\left(1 + 0.5\right) \cdot \mathsf{hypot}\left(1, x\right)}}{{1}^{3} + {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}} \cdot \left(1 \cdot 1 + \frac{\color{blue}{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)} \cdot \left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} - 1\right)}}{\sqrt{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\right)\]

  25. Final simplification15.1

    \[\leadsto \frac{\frac{1 \cdot \left(\left(1 \cdot 1 - 0.5 \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 + 1\right)\right)}{\left(1 + 0.5\right) \cdot \mathsf{hypot}\left(1, x\right)}}{{1}^{3} + {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}} \cdot \left(1 \cdot 1 + \frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)} \cdot \left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} - 1\right)}{\sqrt{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\right)\]

Reproduce

herbie shell --seed 2020001 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))