Given's Rotation SVD example, simplified

Average Error: 15.2 → 13.0

Time: 1.7m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \le 1.000000109998239405939557400415651500225:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{5}}, 0.1875, \mathsf{fma}\left(\frac{{x}^{2}}{1 \cdot 1}, 0.375, \mathsf{fma}\left(0.1875, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.875\right)\right) - \mathsf{fma}\left(0.375, \frac{1}{\sqrt{1}}, \mathsf{fma}\left(\frac{1}{1}, 0.375, \mathsf{fma}\left(0.140625, \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}, \mathsf{fma}\left(0.234375, \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{7}}, \mathsf{fma}\left(0.375, \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{6}}, 0.125 \cdot \frac{1}{{\left(\sqrt{1}\right)}^{3}}\right)\right)\right)\right)\right)\right)}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot \mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right), {1}^{4}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left({\left(1 \cdot 1\right)}^{3} - \sqrt[3]{{\left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}\right)}^{3}}\right)}^{3}}}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot \mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right), {1}^{4}\right)}\\ \end{array}\]

C

double f(double x) {
        double r384155 = 1.0;
        double r384156 = 0.5;
        double r384157 = x;
        double r384158 = hypot(r384155, r384157);
        double r384159 = r384155 / r384158;
        double r384160 = r384155 + r384159;
        double r384161 = r384156 * r384160;
        double r384162 = sqrt(r384161);
        double r384163 = r384155 - r384162;
        return r384163;
}

↓

double f(double x) {
        double r384164 = 1.0;
        double r384165 = x;
        double r384166 = hypot(r384164, r384165);
        double r384167 = 1.0000001099982394;
        bool r384168 = r384166 <= r384167;
        double r384169 = 2.0;
        double r384170 = pow(r384165, r384169);
        double r384171 = sqrt(r384164);
        double r384172 = 5.0;
        double r384173 = pow(r384171, r384172);
        double r384174 = r384170 / r384173;
        double r384175 = 0.1875;
        double r384176 = r384164 * r384164;
        double r384177 = r384170 / r384176;
        double r384178 = 0.375;
        double r384179 = 3.0;
        double r384180 = pow(r384171, r384179);
        double r384181 = r384170 / r384180;
        double r384182 = 0.875;
        double r384183 = fma(r384175, r384181, r384182);
        double r384184 = fma(r384177, r384178, r384183);
        double r384185 = 1.0;
        double r384186 = r384185 / r384171;
        double r384187 = r384185 / r384164;
        double r384188 = 0.140625;
        double r384189 = 4.0;
        double r384190 = pow(r384165, r384189);
        double r384191 = r384190 / r384173;
        double r384192 = 0.234375;
        double r384193 = 7.0;
        double r384194 = pow(r384171, r384193);
        double r384195 = r384190 / r384194;
        double r384196 = 6.0;
        double r384197 = pow(r384171, r384196);
        double r384198 = r384190 / r384197;
        double r384199 = 0.125;
        double r384200 = r384185 / r384180;
        double r384201 = r384199 * r384200;
        double r384202 = fma(r384178, r384198, r384201);
        double r384203 = fma(r384192, r384195, r384202);
        double r384204 = fma(r384188, r384191, r384203);
        double r384205 = fma(r384187, r384178, r384204);
        double r384206 = fma(r384178, r384186, r384205);
        double r384207 = r384184 - r384206;
        double r384208 = fma(r384174, r384175, r384207);
        double r384209 = 0.5;
        double r384210 = r384164 / r384166;
        double r384211 = r384164 + r384210;
        double r384212 = r384209 * r384211;
        double r384213 = sqrt(r384212);
        double r384214 = r384164 + r384213;
        double r384215 = r384211 * r384209;
        double r384216 = fma(r384164, r384164, r384212);
        double r384217 = pow(r384164, r384189);
        double r384218 = fma(r384215, r384216, r384217);
        double r384219 = r384214 * r384218;
        double r384220 = r384208 / r384219;
        double r384221 = pow(r384176, r384179);
        double r384222 = pow(r384212, r384179);
        double r384223 = pow(r384222, r384179);
        double r384224 = cbrt(r384223);
        double r384225 = r384221 - r384224;
        double r384226 = pow(r384225, r384179);
        double r384227 = cbrt(r384226);
        double r384228 = r384227 / r384219;
        double r384229 = r384168 ? r384220 : r384228;
        return r384229;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if (hypot 1.0 x) < 1.0000001099982394

   1. Initial program 29.6

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

   3. Applied flip-- 29.6

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

   4. Simplified 29.6

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

   6. Applied flip3-- 29.6

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

   7. Applied associate-/l/ 29.6

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

   8. Simplified 29.6

      \[\leadsto \frac{{\left(1 \cdot 1\right)}^{3} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}}{\color{blue}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot \mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right), {1}^{4}\right)}}\]
   9. Using strategy rm

   10. Applied add-cbrt-cube 29.6

       \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left({\left(1 \cdot 1\right)}^{3} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right) \cdot \left({\left(1 \cdot 1\right)}^{3} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)\right) \cdot \left({\left(1 \cdot 1\right)}^{3} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)}}}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot \mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right), {1}^{4}\right)}\]

   11. Simplified 29.6

       \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left({\left(1 \cdot 1\right)}^{3} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)}^{3}}}}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot \mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right), {1}^{4}\right)}\]

   12. Taylor expanded around 0 29.6

       \[\leadsto \frac{\color{blue}{\left(0.1875 \cdot \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{5}} + \left(0.375 \cdot \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{4}} + \left(0.1875 \cdot \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}} + 0.875\right)\right)\right) - \left(0.375 \cdot \frac{1}{\sqrt{1}} + \left(0.375 \cdot \frac{1}{{\left(\sqrt{1}\right)}^{2}} + \left(0.140625 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}} + \left(0.234375 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{7}} + \left(0.375 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{6}} + 0.125 \cdot \frac{1}{{\left(\sqrt{1}\right)}^{3}}\right)\right)\right)\right)\right)}}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot \mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right), {1}^{4}\right)}\]

   13. Simplified 26.0

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{5}}, 0.1875, \mathsf{fma}\left(\frac{{x}^{2}}{1 \cdot 1}, 0.375, \mathsf{fma}\left(0.1875, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.875\right)\right) - \mathsf{fma}\left(0.375, \frac{1}{\sqrt{1}}, \mathsf{fma}\left(\frac{1}{1}, 0.375, \mathsf{fma}\left(0.140625, \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}, \mathsf{fma}\left(0.234375, \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{7}}, \mathsf{fma}\left(0.375, \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{6}}, 0.125 \cdot \frac{1}{{\left(\sqrt{1}\right)}^{3}}\right)\right)\right)\right)\right)\right)}}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot \mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right), {1}^{4}\right)}\]

   if 1.0000001099982394 < (hypot 1.0 x)

   1. Initial program 1.1

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

   3. Applied flip-- 1.1

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

   4. Simplified 0.1

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

   6. Applied flip3-- 0.1

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

   7. Applied associate-/l/ 0.1

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

   8. Simplified 0.1

      \[\leadsto \frac{{\left(1 \cdot 1\right)}^{3} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}}{\color{blue}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot \mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right), {1}^{4}\right)}}\]
   9. Using strategy rm

   10. Applied add-cbrt-cube 0.1

       \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left({\left(1 \cdot 1\right)}^{3} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right) \cdot \left({\left(1 \cdot 1\right)}^{3} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)\right) \cdot \left({\left(1 \cdot 1\right)}^{3} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)}}}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot \mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right), {1}^{4}\right)}\]

   11. Simplified 0.1

       \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left({\left(1 \cdot 1\right)}^{3} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right)}^{3}}}}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot \mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right), {1}^{4}\right)}\]
   12. Using strategy rm

   13. Applied add-cbrt-cube 0.1

       \[\leadsto \frac{\sqrt[3]{{\left({\left(1 \cdot 1\right)}^{3} - \color{blue}{\sqrt[3]{\left({\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3} \cdot {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}\right) \cdot {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}}}\right)}^{3}}}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot \mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right), {1}^{4}\right)}\]

   14. Simplified 0.1

       \[\leadsto \frac{\sqrt[3]{{\left({\left(1 \cdot 1\right)}^{3} - \sqrt[3]{\color{blue}{{\left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}\right)}^{3}}}\right)}^{3}}}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot \mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right), {1}^{4}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 13.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \le 1.000000109998239405939557400415651500225:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{5}}, 0.1875, \mathsf{fma}\left(\frac{{x}^{2}}{1 \cdot 1}, 0.375, \mathsf{fma}\left(0.1875, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.875\right)\right) - \mathsf{fma}\left(0.375, \frac{1}{\sqrt{1}}, \mathsf{fma}\left(\frac{1}{1}, 0.375, \mathsf{fma}\left(0.140625, \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}, \mathsf{fma}\left(0.234375, \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{7}}, \mathsf{fma}\left(0.375, \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{6}}, 0.125 \cdot \frac{1}{{\left(\sqrt{1}\right)}^{3}}\right)\right)\right)\right)\right)\right)}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot \mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right), {1}^{4}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left({\left(1 \cdot 1\right)}^{3} - \sqrt[3]{{\left({\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{3}\right)}^{3}}\right)}^{3}}}{\left(1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot \mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right), {1}^{4}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))