Given's Rotation SVD example, simplified

Average Error: 15.5 → 0.0

Time: 2.1m

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 -0.00990965624909386:\\ \;\;\;\;\frac{\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)}}}\\ \mathbf{elif}\;x \le 0.011507891242233352:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{1}{8} + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{69}{1024} - \frac{11}{128}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{512} - \left(\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)\right) \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\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)\right)\right)}{\mathsf{fma}\left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\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)\right), \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\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)\right), \left(\mathsf{fma}\left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\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)\right), \frac{1}{8}, \frac{1}{64}\right)\right)\right)}}{\mathsf{fma}\left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right), \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right), \frac{1}{4}\right)}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array}\]

C

double f(double x) {
        double r17209127 = 1.0;
        double r17209128 = 0.5;
        double r17209129 = x;
        double r17209130 = hypot(r17209127, r17209129);
        double r17209131 = r17209127 / r17209130;
        double r17209132 = r17209127 + r17209131;
        double r17209133 = r17209128 * r17209132;
        double r17209134 = sqrt(r17209133);
        double r17209135 = r17209127 - r17209134;
        return r17209135;
}

↓

double f(double x) {
        double r17209136 = x;
        double r17209137 = -0.00990965624909386;
        bool r17209138 = r17209136 <= r17209137;
        double r17209139 = 0.5;
        double r17209140 = 1.0;
        double r17209141 = hypot(r17209140, r17209136);
        double r17209142 = r17209139 / r17209141;
        double r17209143 = r17209139 - r17209142;
        double r17209144 = r17209139 + r17209142;
        double r17209145 = sqrt(r17209144);
        double r17209146 = r17209140 + r17209145;
        double r17209147 = r17209143 / r17209146;
        double r17209148 = 0.011507891242233352;
        bool r17209149 = r17209136 <= r17209148;
        double r17209150 = r17209136 * r17209136;
        double r17209151 = 0.125;
        double r17209152 = 0.0673828125;
        double r17209153 = r17209150 * r17209152;
        double r17209154 = 0.0859375;
        double r17209155 = r17209153 - r17209154;
        double r17209156 = r17209150 * r17209155;
        double r17209157 = r17209151 + r17209156;
        double r17209158 = r17209150 * r17209157;
        double r17209159 = 0.001953125;
        double r17209160 = r17209142 * r17209142;
        double r17209161 = r17209160 * r17209160;
        double r17209162 = r17209142 * r17209161;
        double r17209163 = r17209161 * r17209162;
        double r17209164 = r17209159 - r17209163;
        double r17209165 = r17209142 * r17209160;
        double r17209166 = 0.015625;
        double r17209167 = fma(r17209165, r17209151, r17209166);
        double r17209168 = fma(r17209165, r17209165, r17209167);
        double r17209169 = r17209164 / r17209168;
        double r17209170 = 0.25;
        double r17209171 = fma(r17209142, r17209144, r17209170);
        double r17209172 = r17209169 / r17209171;
        double r17209173 = r17209172 / r17209146;
        double r17209174 = r17209149 ? r17209158 : r17209173;
        double r17209175 = r17209138 ? r17209147 : r17209174;
        return r17209175;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.00990965624909386

   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

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

   4. Applied flip-- 1.0

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

   5. Simplified 0.1

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

   if -0.00990965624909386 < x < 0.011507891242233352

   1. Initial program 29.9

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

   2. Simplified 29.9

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

   3. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot {x}^{2} + \frac{69}{1024} \cdot {x}^{6}\right) - \frac{11}{128} \cdot {x}^{4}}\]

   4. Simplified 0.0

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\frac{1}{8} + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{69}{1024} - \frac{11}{128}\right)\right)}\]

   if 0.011507891242233352 < 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

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

   4. Applied flip-- 1.0

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

   5. Simplified 0.0

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

   7. Applied flip3-- 0.1

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

   8. Simplified 0.1

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

   9. Simplified 0.1

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

   11. Applied add-exp-log 1.0

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

   13. Applied flip3-- 1.0

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

   14. Applied log-div 1.0

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

   15. Applied exp-diff 1.6

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

   16. Simplified 1.6

       \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{1}{512} - \left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \left(\left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)}}{e^{\log \left(\frac{1}{8} \cdot \frac{1}{8} + \left(\left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) + \frac{1}{8} \cdot \left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right)}}}{\mathsf{fma}\left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right), \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right), \frac{1}{4}\right)}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]

   17. Simplified 0.1

       \[\leadsto \frac{\frac{\frac{\frac{1}{512} - \left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \left(\left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)}{\color{blue}{\mathsf{fma}\left(\left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right), \left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right), \left(\mathsf{fma}\left(\left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right), \frac{1}{8}, \frac{1}{64}\right)\right)\right)}}}{\mathsf{fma}\left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right), \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right), \frac{1}{4}\right)}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00990965624909386:\\ \;\;\;\;\frac{\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)}}}\\ \mathbf{elif}\;x \le 0.011507891242233352:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{1}{8} + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{69}{1024} - \frac{11}{128}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{512} - \left(\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)\right) \cdot \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\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)\right)\right)}{\mathsf{fma}\left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\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)\right), \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\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)\right), \left(\mathsf{fma}\left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\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)\right), \frac{1}{8}, \frac{1}{64}\right)\right)\right)}}{\mathsf{fma}\left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right), \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right), \frac{1}{4}\right)}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1 (sqrt (* 1/2 (+ 1 (/ 1 (hypot 1 x)))))))