Given's Rotation SVD example, simplified

Average Error: 15.6 → 13.3

Time: 54.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\mathsf{hypot}\left(1, x\right)} \le 0.9889134208281412652397079909860622137785:\\ \;\;\;\;\frac{\left(1 \cdot 1 + \left(\sqrt{0.5 \cdot 1 + \frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 \cdot 1 + \frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}} + \sqrt{0.5 \cdot 1 + \frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}} \cdot 1\right)\right) \cdot \left(1 - \sqrt{0.5 \cdot 1 + \frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}\right)}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\sqrt{0.5 \cdot 1 + \frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}, 1, 0.5 \cdot 1 + \frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x \cdot x}{\sqrt{1} \cdot 1}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)} \cdot 0.09375, \sqrt{\frac{1}{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}}, \mathsf{fma}\left(\frac{\frac{x \cdot x}{1 \cdot 1} \cdot \frac{x \cdot x}{1 \cdot 1}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)} \cdot \sqrt{\frac{1}{{\left(0.5 + \frac{0.5}{\sqrt{1}}\right)}^{9}}}, 0.0263671875, \mathsf{fma}\left(0.1875, \frac{x \cdot x}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right) \cdot \left(\sqrt{1} \cdot 1\right)}, \mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot {\left(\sqrt{1}\right)}^{\frac{7}{2}}\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot {\left(\sqrt{1}\right)}^{\frac{7}{2}}\right)} \cdot 0.046875, \sqrt{\frac{1}{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}}, \frac{0.0234375}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)} \cdot \frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{{\left(\sqrt{1}\right)}^{7}}\right) + \mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot {\left(\sqrt{1}\right)}^{10}} \cdot 0.00439453125, \sqrt{\frac{1}{{\left(0.5 + \frac{0.5}{\sqrt{1}}\right)}^{9}}}, \mathsf{fma}\left(\frac{x \cdot x}{{\left(\sqrt{1}\right)}^{5} \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)} \cdot \sqrt{\frac{1}{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}}, 0.09375, \mathsf{fma}\left(0.015625, \frac{\frac{x \cdot x}{\sqrt{1} \cdot 1} \cdot \frac{x \cdot x}{\sqrt{1} \cdot 1}}{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right)\right)}, \mathsf{fma}\left(0.01171875, \frac{\frac{x \cdot x}{\sqrt{1} \cdot 1} \cdot \frac{x \cdot x}{\sqrt{1} \cdot 1}}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}, \mathsf{fma}\left(0.125, \sqrt{\frac{1}{0.5 + \frac{0.5}{\sqrt{1}}}} \cdot \frac{x \cdot x}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right) \cdot \left(\sqrt{1} \cdot 1\right)}, \mathsf{fma}\left(\frac{\frac{x \cdot x}{1 \cdot 1} \cdot \frac{x \cdot x}{1 \cdot 1}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)} \cdot 0.0234375, \sqrt{\frac{1}{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}}, \mathsf{fma}\left(0.01171875, \frac{\frac{x \cdot x}{1 \cdot 1} \cdot \frac{x \cdot x}{1 \cdot 1}}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}, \mathsf{fma}\left(\frac{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{{\left(\sqrt{1}\right)}^{7}}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)} \cdot \sqrt{\frac{1}{{\left(0.5 + \frac{0.5}{\sqrt{1}}\right)}^{9}}}, 0.017578125, \mathsf{fma}\left(0.0546875, \frac{\frac{x \cdot x}{\sqrt{1} \cdot 1} \cdot \frac{x \cdot x}{\sqrt{1} \cdot 1}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right)} \cdot \sqrt{\frac{1}{0.5 + \frac{0.5}{\sqrt{1}}}}, \mathsf{fma}\left(\frac{\frac{x}{1} \cdot \frac{x}{1}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)} \cdot \sqrt{\frac{1}{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}}, 0.1875, \mathsf{fma}\left(\frac{\frac{x \cdot x}{\sqrt{1} \cdot 1} \cdot \frac{x \cdot x}{\sqrt{1} \cdot 1}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)} \cdot 0.00439453125, \sqrt{\frac{1}{{\left(0.5 + \frac{0.5}{\sqrt{1}}\right)}^{9}}}, \mathsf{fma}\left(0.0390625, \frac{\frac{x \cdot x}{\sqrt{1} \cdot 1} \cdot \frac{x \cdot x}{\sqrt{1} \cdot 1}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)}, \mathsf{fma}\left(0.017578125, \frac{\sqrt{\frac{1}{{\left(0.5 + \frac{0.5}{\sqrt{1}}\right)}^{9}}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{{\left(\sqrt{1}\right)}^{9} \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)}, 0.03125 \cdot \left(\frac{\frac{x \cdot x}{\sqrt{1} \cdot 1} \cdot \frac{x \cdot x}{\sqrt{1} \cdot 1}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right)} + \left(\frac{x \cdot x}{\sqrt{1} \cdot 1} \cdot \frac{x \cdot x}{\sqrt{1} \cdot 1}\right) \cdot \frac{\sqrt{\frac{1}{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)}\right)\right)\right)\right)\right)\right) + \mathsf{fma}\left(0.1875, \frac{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{{\left(\sqrt{1}\right)}^{5}}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)} \cdot \sqrt{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}, \frac{1}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.25, \frac{x \cdot x}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right) \cdot \left(\sqrt{1} \cdot 1\right)} \cdot \sqrt{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}, \mathsf{fma}\left(0.140625, \frac{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{{\left(\sqrt{1}\right)}^{5}}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)}, \mathsf{fma}\left(\frac{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{{\left(\sqrt{1}\right)}^{7}}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)} \cdot \sqrt{\frac{1}{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}}, 0.1171875, \mathsf{fma}\left(0.0703125 \cdot \frac{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{{\left(\sqrt{1}\right)}^{5}}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)}, \sqrt{\frac{1}{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}}, \mathsf{fma}\left(0.0625, \frac{x \cdot x}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot 1\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot 1\right)}, \mathsf{fma}\left(0.0078125, \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \sqrt{\frac{1}{0.5 + \frac{0.5}{\sqrt{1}}}}}{{\left(\sqrt{1}\right)}^{7} \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right)\right)}, \frac{\sqrt{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)} + \mathsf{fma}\left(0.0625, \frac{\frac{x \cdot x}{\sqrt{1} \cdot 1} \cdot \frac{x \cdot x}{\sqrt{1} \cdot 1}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right)} \cdot \sqrt{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}, \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.03125}{{\left(\sqrt{1}\right)}^{7} \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right)\right)} + \mathsf{fma}\left(0.09375, \sqrt{\frac{1}{0.5 + \frac{0.5}{\sqrt{1}}}} \cdot \frac{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{{\left(\sqrt{1}\right)}^{5}}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)}, 0.1875 \cdot \left(\sqrt{\frac{1}{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}} \cdot \frac{\frac{x \cdot x}{\sqrt{1} \cdot 1} \cdot \frac{x \cdot x}{\sqrt{1} \cdot 1}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\\ \end{array}\]

C

double f(double x) {
        double r11315613 = 1.0;
        double r11315614 = 0.5;
        double r11315615 = x;
        double r11315616 = hypot(r11315613, r11315615);
        double r11315617 = r11315613 / r11315616;
        double r11315618 = r11315613 + r11315617;
        double r11315619 = r11315614 * r11315618;
        double r11315620 = sqrt(r11315619);
        double r11315621 = r11315613 - r11315620;
        return r11315621;
}

↓

double f(double x) {
        double r11315622 = 1.0;
        double r11315623 = x;
        double r11315624 = hypot(r11315622, r11315623);
        double r11315625 = r11315622 / r11315624;
        double r11315626 = 0.9889134208281413;
        bool r11315627 = r11315625 <= r11315626;
        double r11315628 = r11315622 * r11315622;
        double r11315629 = 0.5;
        double r11315630 = r11315629 * r11315622;
        double r11315631 = r11315630 / r11315624;
        double r11315632 = r11315630 + r11315631;
        double r11315633 = sqrt(r11315632);
        double r11315634 = r11315633 * r11315633;
        double r11315635 = r11315633 * r11315622;
        double r11315636 = r11315634 + r11315635;
        double r11315637 = r11315628 + r11315636;
        double r11315638 = r11315622 - r11315633;
        double r11315639 = r11315637 * r11315638;
        double r11315640 = fma(r11315633, r11315622, r11315632);
        double r11315641 = fma(r11315622, r11315622, r11315640);
        double r11315642 = r11315639 / r11315641;
        double r11315643 = r11315623 * r11315623;
        double r11315644 = sqrt(r11315622);
        double r11315645 = r11315644 * r11315622;
        double r11315646 = r11315643 / r11315645;
        double r11315647 = r11315629 / r11315644;
        double r11315648 = r11315629 + r11315647;
        double r11315649 = sqrt(r11315648);
        double r11315650 = 1.5;
        double r11315651 = fma(r11315622, r11315649, r11315650);
        double r11315652 = r11315647 + r11315651;
        double r11315653 = r11315646 / r11315652;
        double r11315654 = 0.09375;
        double r11315655 = r11315653 * r11315654;
        double r11315656 = 1.0;
        double r11315657 = r11315648 * r11315648;
        double r11315658 = r11315648 * r11315657;
        double r11315659 = r11315656 / r11315658;
        double r11315660 = sqrt(r11315659);
        double r11315661 = r11315643 / r11315628;
        double r11315662 = r11315661 * r11315661;
        double r11315663 = r11315662 / r11315652;
        double r11315664 = 9.0;
        double r11315665 = pow(r11315648, r11315664);
        double r11315666 = r11315656 / r11315665;
        double r11315667 = sqrt(r11315666);
        double r11315668 = r11315663 * r11315667;
        double r11315669 = 0.0263671875;
        double r11315670 = 0.1875;
        double r11315671 = r11315652 * r11315652;
        double r11315672 = r11315671 * r11315645;
        double r11315673 = r11315643 / r11315672;
        double r11315674 = r11315643 * r11315643;
        double r11315675 = 3.5;
        double r11315676 = pow(r11315644, r11315675);
        double r11315677 = r11315652 * r11315676;
        double r11315678 = r11315677 * r11315677;
        double r11315679 = r11315674 / r11315678;
        double r11315680 = 0.046875;
        double r11315681 = r11315679 * r11315680;
        double r11315682 = 0.0234375;
        double r11315683 = r11315652 * r11315648;
        double r11315684 = r11315683 * r11315683;
        double r11315685 = r11315682 / r11315684;
        double r11315686 = 7.0;
        double r11315687 = pow(r11315644, r11315686);
        double r11315688 = r11315674 / r11315687;
        double r11315689 = r11315685 * r11315688;
        double r11315690 = fma(r11315681, r11315660, r11315689);
        double r11315691 = 10.0;
        double r11315692 = pow(r11315644, r11315691);
        double r11315693 = r11315652 * r11315692;
        double r11315694 = r11315674 / r11315693;
        double r11315695 = 0.00439453125;
        double r11315696 = r11315694 * r11315695;
        double r11315697 = 5.0;
        double r11315698 = pow(r11315644, r11315697);
        double r11315699 = r11315698 * r11315652;
        double r11315700 = r11315643 / r11315699;
        double r11315701 = r11315700 * r11315660;
        double r11315702 = 0.015625;
        double r11315703 = r11315646 * r11315646;
        double r11315704 = r11315652 * r11315671;
        double r11315705 = r11315648 * r11315704;
        double r11315706 = r11315703 / r11315705;
        double r11315707 = 0.01171875;
        double r11315708 = r11315703 / r11315684;
        double r11315709 = 0.125;
        double r11315710 = r11315656 / r11315648;
        double r11315711 = sqrt(r11315710);
        double r11315712 = r11315711 * r11315673;
        double r11315713 = r11315662 / r11315671;
        double r11315714 = r11315713 * r11315682;
        double r11315715 = r11315662 / r11315684;
        double r11315716 = r11315688 / r11315652;
        double r11315717 = r11315716 * r11315667;
        double r11315718 = 0.017578125;
        double r11315719 = 0.0546875;
        double r11315720 = r11315703 / r11315704;
        double r11315721 = r11315720 * r11315711;
        double r11315722 = r11315623 / r11315622;
        double r11315723 = r11315722 * r11315722;
        double r11315724 = r11315723 / r11315652;
        double r11315725 = r11315724 * r11315660;
        double r11315726 = r11315703 / r11315652;
        double r11315727 = r11315726 * r11315695;
        double r11315728 = 0.0390625;
        double r11315729 = r11315703 / r11315671;
        double r11315730 = r11315667 * r11315674;
        double r11315731 = pow(r11315644, r11315664);
        double r11315732 = r11315731 * r11315652;
        double r11315733 = r11315730 / r11315732;
        double r11315734 = 0.03125;
        double r11315735 = r11315660 / r11315671;
        double r11315736 = r11315703 * r11315735;
        double r11315737 = r11315720 + r11315736;
        double r11315738 = r11315734 * r11315737;
        double r11315739 = fma(r11315718, r11315733, r11315738);
        double r11315740 = fma(r11315728, r11315729, r11315739);
        double r11315741 = fma(r11315727, r11315667, r11315740);
        double r11315742 = fma(r11315725, r11315670, r11315741);
        double r11315743 = fma(r11315719, r11315721, r11315742);
        double r11315744 = r11315674 / r11315698;
        double r11315745 = r11315744 / r11315671;
        double r11315746 = sqrt(r11315658);
        double r11315747 = r11315745 * r11315746;
        double r11315748 = r11315622 / r11315652;
        double r11315749 = fma(r11315670, r11315747, r11315748);
        double r11315750 = r11315743 + r11315749;
        double r11315751 = fma(r11315717, r11315718, r11315750);
        double r11315752 = fma(r11315707, r11315715, r11315751);
        double r11315753 = fma(r11315714, r11315660, r11315752);
        double r11315754 = fma(r11315709, r11315712, r11315753);
        double r11315755 = fma(r11315707, r11315708, r11315754);
        double r11315756 = fma(r11315702, r11315706, r11315755);
        double r11315757 = fma(r11315701, r11315654, r11315756);
        double r11315758 = fma(r11315696, r11315667, r11315757);
        double r11315759 = r11315690 + r11315758;
        double r11315760 = fma(r11315670, r11315673, r11315759);
        double r11315761 = fma(r11315668, r11315669, r11315760);
        double r11315762 = 0.25;
        double r11315763 = r11315673 * r11315746;
        double r11315764 = 0.140625;
        double r11315765 = r11315716 * r11315660;
        double r11315766 = 0.1171875;
        double r11315767 = 0.0703125;
        double r11315768 = r11315744 / r11315652;
        double r11315769 = r11315767 * r11315768;
        double r11315770 = 0.0625;
        double r11315771 = r11315652 * r11315622;
        double r11315772 = r11315771 * r11315771;
        double r11315773 = r11315643 / r11315772;
        double r11315774 = 0.0078125;
        double r11315775 = r11315674 * r11315711;
        double r11315776 = r11315687 * r11315704;
        double r11315777 = r11315775 / r11315776;
        double r11315778 = r11315746 / r11315652;
        double r11315779 = r11315720 * r11315746;
        double r11315780 = r11315674 * r11315734;
        double r11315781 = r11315780 / r11315776;
        double r11315782 = r11315711 * r11315745;
        double r11315783 = r11315660 * r11315726;
        double r11315784 = r11315670 * r11315783;
        double r11315785 = fma(r11315654, r11315782, r11315784);
        double r11315786 = r11315781 + r11315785;
        double r11315787 = fma(r11315770, r11315779, r11315786);
        double r11315788 = r11315778 + r11315787;
        double r11315789 = fma(r11315774, r11315777, r11315788);
        double r11315790 = fma(r11315770, r11315773, r11315789);
        double r11315791 = fma(r11315769, r11315660, r11315790);
        double r11315792 = fma(r11315765, r11315766, r11315791);
        double r11315793 = fma(r11315764, r11315745, r11315792);
        double r11315794 = fma(r11315762, r11315763, r11315793);
        double r11315795 = r11315761 - r11315794;
        double r11315796 = fma(r11315655, r11315660, r11315795);
        double r11315797 = r11315627 ? r11315642 : r11315796;
        return r11315797;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if (/ 1.0 (hypot 1.0 x)) < 0.9889134208281413

   1. Initial program 1.0

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

   3. Applied flip3-- 1.5

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

   4. Simplified 1.0

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

   5. Simplified 0.0

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

   7. Applied add-sqr-sqrt 1.0

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

   8. Applied pow3 1.0

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

   9. Applied cube-unmult 1.0

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

   10. Applied difference-cubes 0.0

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

   if 0.9889134208281413 < (/ 1.0 (hypot 1.0 x))

   1. Initial program 30.5

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

   3. Applied flip3-- 30.5

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

   4. Simplified 30.5

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

   5. Simplified 30.5

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

   6. Taylor expanded around 0 30.6

      \[\leadsto \color{blue}{\left(0.09375 \cdot \left(\frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3} \cdot \left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)} \cdot \sqrt{\frac{1}{{\left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)}^{3}}}\right) + \left(0.0263671875 \cdot \left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{8} \cdot \left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)} \cdot \sqrt{\frac{1}{{\left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)}^{9}}}\right) + \left(0.1875 \cdot \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3} \cdot {\left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)}^{2}} + \left(0.046875 \cdot \left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{7} \cdot {\left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)}^{2}} \cdot \sqrt{\frac{1}{{\left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)}^{3}}}\right) + \left(0.0234375 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{7} \cdot \left({\left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)}^{2} \cdot {\left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)}^{2}\right)} + \left(0.00439453125 \cdot \left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{10} \cdot \left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)} \cdot \sqrt{\frac{1}{{\left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)}^{9}}}\right) + \left(0.09375 \cdot \left(\frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{5} \cdot \left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)} \cdot \sqrt{\frac{1}{{\left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)}^{3}}}\right) + \left(0.015625 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{6} \cdot \left({\left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)}^{3} \cdot \left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)\right)} + \left(0.01171875 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{6} \cdot \left({\left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)}^{2} \cdot {\left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)}^{2}\right)} + \left(0.125 \cdot \left(\frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3} \cdot {\left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)}^{2}} \cdot \sqrt{\frac{1}{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5}}\right) + \left(0.0234375 \cdot \left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{8} \cdot {\left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)}^{2}} \cdot \sqrt{\frac{1}{{\left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)}^{3}}}\right) + \left(0.01171875 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{8} \cdot \left({\left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)}^{2} \cdot {\left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)}^{2}\right)} + \left(0.017578125 \cdot \left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{7} \cdot \left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)} \cdot \sqrt{\frac{1}{{\left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)}^{9}}}\right) + \left(0.1875 \cdot \left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5} \cdot {\left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)}^{2}} \cdot \sqrt{{\left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)}^{3}}\right) + \left(1 \cdot \frac{1}{0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)} + \left(0.0546875 \cdot \left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{6} \cdot {\left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)}^{3}} \cdot \sqrt{\frac{1}{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5}}\right) + \left(0.1875 \cdot \left(\frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{4} \cdot \left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)} \cdot \sqrt{\frac{1}{{\left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)}^{3}}}\right) + \left(0.00439453125 \cdot \left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{6} \cdot \left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)} \cdot \sqrt{\frac{1}{{\left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)}^{9}}}\right) + \left(0.0390625 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{6} \cdot {\left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)}^{2}} + \left(0.017578125 \cdot \left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{9} \cdot \left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)} \cdot \sqrt{\frac{1}{{\left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)}^{9}}}\right) + \left(0.03125 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{6} \cdot {\left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)}^{3}} + 0.03125 \cdot \left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{6} \cdot {\left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)}^{2}} \cdot \sqrt{\frac{1}{{\left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)}^{3}}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(0.25 \cdot \left(\frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3} \cdot {\left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)}^{2}} \cdot \sqrt{{\left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)}^{3}}\right) + \left(0.140625 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5} \cdot {\left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)}^{2}} + \left(0.1171875 \cdot \left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{7} \cdot \left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)} \cdot \sqrt{\frac{1}{{\left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)}^{3}}}\right) + \left(0.0703125 \cdot \left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5} \cdot \left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)} \cdot \sqrt{\frac{1}{{\left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)}^{3}}}\right) + \left(0.0625 \cdot \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{4} \cdot {\left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)}^{2}} + \left(0.0078125 \cdot \left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{7} \cdot {\left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)}^{3}} \cdot \sqrt{\frac{1}{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5}}\right) + \left(\frac{1}{0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)} \cdot \sqrt{{\left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)}^{3}} + \left(0.0625 \cdot \left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{6} \cdot {\left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)}^{3}} \cdot \sqrt{{\left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)}^{3}}\right) + \left(0.03125 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{7} \cdot {\left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)}^{3}} + \left(0.1875 \cdot \left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{6} \cdot \left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)} \cdot \sqrt{\frac{1}{{\left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.5\right)}^{3}}}\right) + 0.09375 \cdot \left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5} \cdot {\left(0.5 \cdot \frac{1}{\sqrt{1}} + \left(1 \cdot \sqrt{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5} + 1.5\right)\right)}^{2}} \cdot \sqrt{\frac{1}{0.5 \cdot \frac{1}{\sqrt{1}} + 0.5}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)}\]

   7. Simplified 26.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.09375 \cdot \frac{\frac{x \cdot x}{1 \cdot \sqrt{1}}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)}, \sqrt{\frac{1}{\left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)}}, \mathsf{fma}\left(\frac{\frac{x \cdot x}{1 \cdot 1} \cdot \frac{x \cdot x}{1 \cdot 1}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)} \cdot \sqrt{\frac{1}{{\left(0.5 + \frac{0.5}{\sqrt{1}}\right)}^{9}}}, 0.0263671875, \mathsf{fma}\left(0.1875, \frac{x \cdot x}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right) \cdot \left(1 \cdot \sqrt{1}\right)}, \mathsf{fma}\left(0.046875 \cdot \frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot {\left(\sqrt{1}\right)}^{\frac{7}{2}}\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot {\left(\sqrt{1}\right)}^{\frac{7}{2}}\right)}, \sqrt{\frac{1}{\left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)}}, \frac{0.0234375}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)} \cdot \frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{{\left(\sqrt{1}\right)}^{7}}\right) + \mathsf{fma}\left(0.00439453125 \cdot \frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot {\left(\sqrt{1}\right)}^{10}}, \sqrt{\frac{1}{{\left(0.5 + \frac{0.5}{\sqrt{1}}\right)}^{9}}}, \mathsf{fma}\left(\frac{x \cdot x}{{\left(\sqrt{1}\right)}^{5} \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)} \cdot \sqrt{\frac{1}{\left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)}}, 0.09375, \mathsf{fma}\left(0.015625, \frac{\frac{x \cdot x}{1 \cdot \sqrt{1}} \cdot \frac{x \cdot x}{1 \cdot \sqrt{1}}}{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right)}, \mathsf{fma}\left(0.01171875, \frac{\frac{x \cdot x}{1 \cdot \sqrt{1}} \cdot \frac{x \cdot x}{1 \cdot \sqrt{1}}}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}, \mathsf{fma}\left(0.125, \frac{x \cdot x}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right) \cdot \left(1 \cdot \sqrt{1}\right)} \cdot \sqrt{\frac{1}{0.5 + \frac{0.5}{\sqrt{1}}}}, \mathsf{fma}\left(0.0234375 \cdot \frac{\frac{x \cdot x}{1 \cdot 1} \cdot \frac{x \cdot x}{1 \cdot 1}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)}, \sqrt{\frac{1}{\left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)}}, \mathsf{fma}\left(0.01171875, \frac{\frac{x \cdot x}{1 \cdot 1} \cdot \frac{x \cdot x}{1 \cdot 1}}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}, \mathsf{fma}\left(\frac{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{{\left(\sqrt{1}\right)}^{7}}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)} \cdot \sqrt{\frac{1}{{\left(0.5 + \frac{0.5}{\sqrt{1}}\right)}^{9}}}, 0.017578125, \mathsf{fma}\left(0.1875, \frac{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{{\left(\sqrt{1}\right)}^{5}}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)} \cdot \sqrt{\left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)}, \frac{1}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)}\right) + \mathsf{fma}\left(0.0546875, \frac{\frac{x \cdot x}{1 \cdot \sqrt{1}} \cdot \frac{x \cdot x}{1 \cdot \sqrt{1}}}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)} \cdot \sqrt{\frac{1}{0.5 + \frac{0.5}{\sqrt{1}}}}, \mathsf{fma}\left(\sqrt{\frac{1}{\left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)}} \cdot \frac{\frac{x}{1} \cdot \frac{x}{1}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)}, 0.1875, \mathsf{fma}\left(0.00439453125 \cdot \frac{\frac{x \cdot x}{1 \cdot \sqrt{1}} \cdot \frac{x \cdot x}{1 \cdot \sqrt{1}}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)}, \sqrt{\frac{1}{{\left(0.5 + \frac{0.5}{\sqrt{1}}\right)}^{9}}}, \mathsf{fma}\left(0.0390625, \frac{\frac{x \cdot x}{1 \cdot \sqrt{1}} \cdot \frac{x \cdot x}{1 \cdot \sqrt{1}}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)}, \mathsf{fma}\left(0.017578125, \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \sqrt{\frac{1}{{\left(0.5 + \frac{0.5}{\sqrt{1}}\right)}^{9}}}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot {\left(\sqrt{1}\right)}^{9}}, 0.03125 \cdot \left(\left(\frac{x \cdot x}{1 \cdot \sqrt{1}} \cdot \frac{x \cdot x}{1 \cdot \sqrt{1}}\right) \cdot \frac{\sqrt{\frac{1}{\left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)}}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)} + \frac{\frac{x \cdot x}{1 \cdot \sqrt{1}} \cdot \frac{x \cdot x}{1 \cdot \sqrt{1}}}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.25, \sqrt{\left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)} \cdot \frac{x \cdot x}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right) \cdot \left(1 \cdot \sqrt{1}\right)}, \mathsf{fma}\left(0.140625, \frac{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{{\left(\sqrt{1}\right)}^{5}}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)}, \mathsf{fma}\left(\sqrt{\frac{1}{\left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)}} \cdot \frac{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{{\left(\sqrt{1}\right)}^{7}}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)}, 0.1171875, \mathsf{fma}\left(0.0703125 \cdot \frac{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{{\left(\sqrt{1}\right)}^{5}}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)}, \sqrt{\frac{1}{\left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)}}, \mathsf{fma}\left(0.0625, \frac{x \cdot x}{\left(1 \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right) \cdot \left(1 \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right)}, \mathsf{fma}\left(0.0078125, \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \sqrt{\frac{1}{0.5 + \frac{0.5}{\sqrt{1}}}}}{{\left(\sqrt{1}\right)}^{7} \cdot \left(\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right)}, \mathsf{fma}\left(0.0625, \sqrt{\left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)} \cdot \frac{\frac{x \cdot x}{1 \cdot \sqrt{1}} \cdot \frac{x \cdot x}{1 \cdot \sqrt{1}}}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)}, \mathsf{fma}\left(0.09375, \sqrt{\frac{1}{0.5 + \frac{0.5}{\sqrt{1}}}} \cdot \frac{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{{\left(\sqrt{1}\right)}^{5}}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)}, 0.1875 \cdot \left(\frac{\frac{x \cdot x}{1 \cdot \sqrt{1}} \cdot \frac{x \cdot x}{1 \cdot \sqrt{1}}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)} \cdot \sqrt{\frac{1}{\left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)}}\right)\right) + \frac{0.03125 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{{\left(\sqrt{1}\right)}^{7} \cdot \left(\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right)}\right) + \frac{\sqrt{\left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)}\right)\right)\right)\right)\right)\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 13.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\mathsf{hypot}\left(1, x\right)} \le 0.9889134208281412652397079909860622137785:\\ \;\;\;\;\frac{\left(1 \cdot 1 + \left(\sqrt{0.5 \cdot 1 + \frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 \cdot 1 + \frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}} + \sqrt{0.5 \cdot 1 + \frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}} \cdot 1\right)\right) \cdot \left(1 - \sqrt{0.5 \cdot 1 + \frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}\right)}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\sqrt{0.5 \cdot 1 + \frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}, 1, 0.5 \cdot 1 + \frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x \cdot x}{\sqrt{1} \cdot 1}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)} \cdot 0.09375, \sqrt{\frac{1}{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}}, \mathsf{fma}\left(\frac{\frac{x \cdot x}{1 \cdot 1} \cdot \frac{x \cdot x}{1 \cdot 1}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)} \cdot \sqrt{\frac{1}{{\left(0.5 + \frac{0.5}{\sqrt{1}}\right)}^{9}}}, 0.0263671875, \mathsf{fma}\left(0.1875, \frac{x \cdot x}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right) \cdot \left(\sqrt{1} \cdot 1\right)}, \mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot {\left(\sqrt{1}\right)}^{\frac{7}{2}}\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot {\left(\sqrt{1}\right)}^{\frac{7}{2}}\right)} \cdot 0.046875, \sqrt{\frac{1}{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}}, \frac{0.0234375}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)} \cdot \frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{{\left(\sqrt{1}\right)}^{7}}\right) + \mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot {\left(\sqrt{1}\right)}^{10}} \cdot 0.00439453125, \sqrt{\frac{1}{{\left(0.5 + \frac{0.5}{\sqrt{1}}\right)}^{9}}}, \mathsf{fma}\left(\frac{x \cdot x}{{\left(\sqrt{1}\right)}^{5} \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)} \cdot \sqrt{\frac{1}{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}}, 0.09375, \mathsf{fma}\left(0.015625, \frac{\frac{x \cdot x}{\sqrt{1} \cdot 1} \cdot \frac{x \cdot x}{\sqrt{1} \cdot 1}}{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right)\right)}, \mathsf{fma}\left(0.01171875, \frac{\frac{x \cdot x}{\sqrt{1} \cdot 1} \cdot \frac{x \cdot x}{\sqrt{1} \cdot 1}}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}, \mathsf{fma}\left(0.125, \sqrt{\frac{1}{0.5 + \frac{0.5}{\sqrt{1}}}} \cdot \frac{x \cdot x}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right) \cdot \left(\sqrt{1} \cdot 1\right)}, \mathsf{fma}\left(\frac{\frac{x \cdot x}{1 \cdot 1} \cdot \frac{x \cdot x}{1 \cdot 1}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)} \cdot 0.0234375, \sqrt{\frac{1}{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}}, \mathsf{fma}\left(0.01171875, \frac{\frac{x \cdot x}{1 \cdot 1} \cdot \frac{x \cdot x}{1 \cdot 1}}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}, \mathsf{fma}\left(\frac{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{{\left(\sqrt{1}\right)}^{7}}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)} \cdot \sqrt{\frac{1}{{\left(0.5 + \frac{0.5}{\sqrt{1}}\right)}^{9}}}, 0.017578125, \mathsf{fma}\left(0.0546875, \frac{\frac{x \cdot x}{\sqrt{1} \cdot 1} \cdot \frac{x \cdot x}{\sqrt{1} \cdot 1}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right)} \cdot \sqrt{\frac{1}{0.5 + \frac{0.5}{\sqrt{1}}}}, \mathsf{fma}\left(\frac{\frac{x}{1} \cdot \frac{x}{1}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)} \cdot \sqrt{\frac{1}{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}}, 0.1875, \mathsf{fma}\left(\frac{\frac{x \cdot x}{\sqrt{1} \cdot 1} \cdot \frac{x \cdot x}{\sqrt{1} \cdot 1}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)} \cdot 0.00439453125, \sqrt{\frac{1}{{\left(0.5 + \frac{0.5}{\sqrt{1}}\right)}^{9}}}, \mathsf{fma}\left(0.0390625, \frac{\frac{x \cdot x}{\sqrt{1} \cdot 1} \cdot \frac{x \cdot x}{\sqrt{1} \cdot 1}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)}, \mathsf{fma}\left(0.017578125, \frac{\sqrt{\frac{1}{{\left(0.5 + \frac{0.5}{\sqrt{1}}\right)}^{9}}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{{\left(\sqrt{1}\right)}^{9} \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)}, 0.03125 \cdot \left(\frac{\frac{x \cdot x}{\sqrt{1} \cdot 1} \cdot \frac{x \cdot x}{\sqrt{1} \cdot 1}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right)} + \left(\frac{x \cdot x}{\sqrt{1} \cdot 1} \cdot \frac{x \cdot x}{\sqrt{1} \cdot 1}\right) \cdot \frac{\sqrt{\frac{1}{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)}\right)\right)\right)\right)\right)\right) + \mathsf{fma}\left(0.1875, \frac{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{{\left(\sqrt{1}\right)}^{5}}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)} \cdot \sqrt{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}, \frac{1}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) - \mathsf{fma}\left(0.25, \frac{x \cdot x}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right) \cdot \left(\sqrt{1} \cdot 1\right)} \cdot \sqrt{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}, \mathsf{fma}\left(0.140625, \frac{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{{\left(\sqrt{1}\right)}^{5}}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)}, \mathsf{fma}\left(\frac{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{{\left(\sqrt{1}\right)}^{7}}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)} \cdot \sqrt{\frac{1}{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}}, 0.1171875, \mathsf{fma}\left(0.0703125 \cdot \frac{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{{\left(\sqrt{1}\right)}^{5}}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)}, \sqrt{\frac{1}{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}}, \mathsf{fma}\left(0.0625, \frac{x \cdot x}{\left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot 1\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot 1\right)}, \mathsf{fma}\left(0.0078125, \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \sqrt{\frac{1}{0.5 + \frac{0.5}{\sqrt{1}}}}}{{\left(\sqrt{1}\right)}^{7} \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right)\right)}, \frac{\sqrt{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)} + \mathsf{fma}\left(0.0625, \frac{\frac{x \cdot x}{\sqrt{1} \cdot 1} \cdot \frac{x \cdot x}{\sqrt{1} \cdot 1}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right)} \cdot \sqrt{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}, \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.03125}{{\left(\sqrt{1}\right)}^{7} \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)\right)\right)} + \mathsf{fma}\left(0.09375, \sqrt{\frac{1}{0.5 + \frac{0.5}{\sqrt{1}}}} \cdot \frac{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{{\left(\sqrt{1}\right)}^{5}}}{\left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right) \cdot \left(\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)\right)}, 0.1875 \cdot \left(\sqrt{\frac{1}{\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(\left(0.5 + \frac{0.5}{\sqrt{1}}\right) \cdot \left(0.5 + \frac{0.5}{\sqrt{1}}\right)\right)}} \cdot \frac{\frac{x \cdot x}{\sqrt{1} \cdot 1} \cdot \frac{x \cdot x}{\sqrt{1} \cdot 1}}{\frac{0.5}{\sqrt{1}} + \mathsf{fma}\left(1, \sqrt{0.5 + \frac{0.5}{\sqrt{1}}}, 1.5\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))