Given's Rotation SVD example, simplified

Average Error: 14.8 → 0.0

Time: 22.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.012079716481524932:\\ \;\;\;\;\frac{\frac{\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1\right) \cdot \frac{\mathsf{fma}\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, \mathsf{fma}\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, -1\right), \mathsf{fma}\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, -1\right)\right)}{\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} - 1} - \frac{1}{8}}{\mathsf{fma}\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \mathsf{fma}\left(\frac{1}{2}, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \frac{1}{4}\right)\right)}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\\ \mathbf{elif}\;x \le 0.011674747997097862:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{4}, x \cdot x, \frac{5}{32} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) - \frac{3}{16} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \mathsf{fma}\left(\frac{1}{2}, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \frac{1}{4}\right)\right)}{\mathsf{fma}\left(\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1\right) \cdot \left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1\right), \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \frac{-1}{8}\right)}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\\ \end{array}\]

C

double f(double x) {
        double r2186617 = 1.0;
        double r2186618 = 0.5;
        double r2186619 = x;
        double r2186620 = hypot(r2186617, r2186619);
        double r2186621 = r2186617 / r2186620;
        double r2186622 = r2186617 + r2186621;
        double r2186623 = r2186618 * r2186622;
        double r2186624 = sqrt(r2186623);
        double r2186625 = r2186617 - r2186624;
        return r2186625;
}

↓

double f(double x) {
        double r2186626 = x;
        double r2186627 = -0.012079716481524932;
        bool r2186628 = r2186626 <= r2186627;
        double r2186629 = -0.5;
        double r2186630 = 1.0;
        double r2186631 = hypot(r2186630, r2186626);
        double r2186632 = r2186629 / r2186631;
        double r2186633 = r2186632 + r2186630;
        double r2186634 = -1.0;
        double r2186635 = fma(r2186632, r2186632, r2186634);
        double r2186636 = fma(r2186632, r2186635, r2186635);
        double r2186637 = r2186632 - r2186630;
        double r2186638 = r2186636 / r2186637;
        double r2186639 = r2186633 * r2186638;
        double r2186640 = 0.125;
        double r2186641 = r2186639 - r2186640;
        double r2186642 = 0.5;
        double r2186643 = 0.25;
        double r2186644 = fma(r2186642, r2186633, r2186643);
        double r2186645 = fma(r2186633, r2186633, r2186644);
        double r2186646 = r2186641 / r2186645;
        double r2186647 = r2186642 / r2186631;
        double r2186648 = r2186647 + r2186642;
        double r2186649 = sqrt(r2186648);
        double r2186650 = r2186630 + r2186649;
        double r2186651 = r2186646 / r2186650;
        double r2186652 = 0.011674747997097862;
        bool r2186653 = r2186626 <= r2186652;
        double r2186654 = r2186626 * r2186626;
        double r2186655 = 0.15625;
        double r2186656 = r2186654 * r2186654;
        double r2186657 = r2186654 * r2186656;
        double r2186658 = r2186655 * r2186657;
        double r2186659 = fma(r2186643, r2186654, r2186658);
        double r2186660 = 0.1875;
        double r2186661 = r2186660 * r2186656;
        double r2186662 = r2186659 - r2186661;
        double r2186663 = r2186662 / r2186650;
        double r2186664 = r2186633 * r2186633;
        double r2186665 = -0.125;
        double r2186666 = fma(r2186664, r2186633, r2186665);
        double r2186667 = r2186645 / r2186666;
        double r2186668 = r2186630 / r2186667;
        double r2186669 = r2186668 / r2186650;
        double r2186670 = r2186653 ? r2186663 : r2186669;
        double r2186671 = r2186628 ? r2186651 : r2186670;
        return r2186671;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.012079716481524932

   1. Initial program 1.0

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

   2. Simplified 1.0

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

   4. Applied flip-- 1.0

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

   5. Simplified 0.1

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

   7. Applied flip3-- 0.1

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

   8. Simplified 0.1

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

   9. Simplified 0.1

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

   11. Applied flip-+ 0.1

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

   12. Applied associate-*r/ 0.1

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

   13. Simplified 0.1

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

   if -0.012079716481524932 < x < 0.011674747997097862

   1. Initial program 29.1

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

   2. Simplified 29.1

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

   4. Applied flip-- 29.1

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

   5. Simplified 29.1

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

   6. Taylor expanded around 0 0.0

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} \cdot {x}^{2} + \frac{5}{32} \cdot {x}^{6}\right) - \frac{3}{16} \cdot {x}^{4}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

   7. Simplified 0.0

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{4}, x \cdot x, \frac{5}{32} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) - \frac{3}{16} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

   if 0.011674747997097862 < x

   1. Initial program 1.0

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

   2. Simplified 1.0

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

   4. Applied flip-- 1.0

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

   5. Simplified 0.1

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

   7. Applied flip3-- 0.1

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

   8. Simplified 0.1

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

   9. Simplified 0.1

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

   11. Applied *-un-lft-identity 0.1

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

   12. Applied associate-/l* 0.1

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

   13. Simplified 0.1

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\mathsf{fma}\left(1 + \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, 1 + \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, \mathsf{fma}\left(\frac{1}{2}, 1 + \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{1}{4}\right)\right)}{\mathsf{fma}\left(\left(1 + \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 + \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}\right), 1 + \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{-1}{8}\right)}}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.012079716481524932:\\ \;\;\;\;\frac{\frac{\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1\right) \cdot \frac{\mathsf{fma}\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, \mathsf{fma}\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, -1\right), \mathsf{fma}\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}, -1\right)\right)}{\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} - 1} - \frac{1}{8}}{\mathsf{fma}\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \mathsf{fma}\left(\frac{1}{2}, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \frac{1}{4}\right)\right)}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\\ \mathbf{elif}\;x \le 0.011674747997097862:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{4}, x \cdot x, \frac{5}{32} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) - \frac{3}{16} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \mathsf{fma}\left(\frac{1}{2}, \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \frac{1}{4}\right)\right)}{\mathsf{fma}\left(\left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1\right) \cdot \left(\frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1\right), \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)} + 1, \frac{-1}{8}\right)}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1 (sqrt (* 1/2 (+ 1 (/ 1 (hypot 1 x)))))))