Average Error: 14.9 → 14.9
Time: 1.3m
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{\left(\sqrt{{\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}} + \sqrt{{\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}\right) \cdot \frac{\sqrt{{\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}} - \sqrt{{\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}}{\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} + 1 \cdot \left(1 - 0.5\right)\right) + \left(1 \cdot \left(1 - 0.5\right)\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right)}}{\sqrt{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}}{\sqrt{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{\left(\sqrt{{\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}} + \sqrt{{\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}\right) \cdot \frac{\sqrt{{\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}} - \sqrt{{\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}}{\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} + 1 \cdot \left(1 - 0.5\right)\right) + \left(1 \cdot \left(1 - 0.5\right)\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right)}}{\sqrt{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}}{\sqrt{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}
double f(double x) {
        double r247562 = 1.0;
        double r247563 = 0.5;
        double r247564 = x;
        double r247565 = hypot(r247562, r247564);
        double r247566 = r247562 / r247565;
        double r247567 = r247562 + r247566;
        double r247568 = r247563 * r247567;
        double r247569 = sqrt(r247568);
        double r247570 = r247562 - r247569;
        return r247570;
}
↓
double f(double x) {
        double r247571 = 1.0;
        double r247572 = 0.5;
        double r247573 = r247571 - r247572;
        double r247574 = r247571 * r247573;
        double r247575 = 3.0;
        double r247576 = pow(r247574, r247575);
        double r247577 = sqrt(r247576);
        double r247578 = x;
        double r247579 = hypot(r247571, r247578);
        double r247580 = r247571 / r247579;
        double r247581 = r247572 * r247580;
        double r247582 = pow(r247581, r247575);
        double r247583 = sqrt(r247582);
        double r247584 = r247577 + r247583;
        double r247585 = r247577 - r247583;
        double r247586 = r247581 + r247574;
        double r247587 = r247581 * r247586;
        double r247588 = r247574 * r247574;
        double r247589 = r247587 + r247588;
        double r247590 = r247585 / r247589;
        double r247591 = r247584 * r247590;
        double r247592 = r247571 + r247580;
        double r247593 = r247572 * r247592;
        double r247594 = sqrt(r247593);
        double r247595 = r247571 + r247594;
        double r247596 = sqrt(r247595);
        double r247597 = r247591 / r247596;
        double r247598 = r247597 / r247596;
        return r247598;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.9

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

  3. Applied flip--14.9

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

  4. Simplified14.4

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

  6. Applied add-sqr-sqrt14.9

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

  7. Applied associate-/r*14.4

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

  9. Applied flip3--14.4

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

  10. Simplified14.4

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

  12. Applied add-sqr-sqrt14.4

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

  13. Applied sqrt-prod14.4

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

  14. Applied add-sqr-sqrt14.4

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

  15. Applied sqrt-prod14.4

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

  16. Applied *-un-lft-identity14.4

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

  17. Applied add-sqr-sqrt30.2

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

  18. Applied add-sqr-sqrt14.9

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

  19. Applied difference-of-squares14.9

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

  20. Applied times-frac14.9

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

  21. Applied times-frac14.9

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

  22. Applied times-frac14.4

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

  23. Final simplification14.9

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

Reproduce

herbie shell --seed 2019298 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))