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 r386470 = 1.0;
        double r386471 = 0.5;
        double r386472 = x;
        double r386473 = hypot(r386470, r386472);
        double r386474 = r386470 / r386473;
        double r386475 = r386470 + r386474;
        double r386476 = r386471 * r386475;
        double r386477 = sqrt(r386476);
        double r386478 = r386470 - r386477;
        return r386478;
}

↓

double f(double x) {
        double r386479 = 1.0;
        double r386480 = x;
        double r386481 = hypot(r386479, r386480);
        double r386482 = 1.0000001099982394;
        bool r386483 = r386481 <= r386482;
        double r386484 = 2.0;
        double r386485 = pow(r386480, r386484);
        double r386486 = sqrt(r386479);
        double r386487 = 5.0;
        double r386488 = pow(r386486, r386487);
        double r386489 = r386485 / r386488;
        double r386490 = 0.1875;
        double r386491 = r386479 * r386479;
        double r386492 = r386485 / r386491;
        double r386493 = 0.375;
        double r386494 = 3.0;
        double r386495 = pow(r386486, r386494);
        double r386496 = r386485 / r386495;
        double r386497 = 0.875;
        double r386498 = fma(r386490, r386496, r386497);
        double r386499 = fma(r386492, r386493, r386498);
        double r386500 = 1.0;
        double r386501 = r386500 / r386486;
        double r386502 = r386500 / r386479;
        double r386503 = 0.140625;
        double r386504 = 4.0;
        double r386505 = pow(r386480, r386504);
        double r386506 = r386505 / r386488;
        double r386507 = 0.234375;
        double r386508 = 7.0;
        double r386509 = pow(r386486, r386508);
        double r386510 = r386505 / r386509;
        double r386511 = 6.0;
        double r386512 = pow(r386486, r386511);
        double r386513 = r386505 / r386512;
        double r386514 = 0.125;
        double r386515 = r386500 / r386495;
        double r386516 = r386514 * r386515;
        double r386517 = fma(r386493, r386513, r386516);
        double r386518 = fma(r386507, r386510, r386517);
        double r386519 = fma(r386503, r386506, r386518);
        double r386520 = fma(r386502, r386493, r386519);
        double r386521 = fma(r386493, r386501, r386520);
        double r386522 = r386499 - r386521;
        double r386523 = fma(r386489, r386490, r386522);
        double r386524 = 0.5;
        double r386525 = r386479 / r386481;
        double r386526 = r386479 + r386525;
        double r386527 = r386524 * r386526;
        double r386528 = sqrt(r386527);
        double r386529 = r386479 + r386528;
        double r386530 = r386526 * r386524;
        double r386531 = fma(r386479, r386479, r386527);
        double r386532 = pow(r386479, r386504);
        double r386533 = fma(r386530, r386531, r386532);
        double r386534 = r386529 * r386533;
        double r386535 = r386523 / r386534;
        double r386536 = pow(r386491, r386494);
        double r386537 = pow(r386527, r386494);
        double r386538 = pow(r386537, r386494);
        double r386539 = cbrt(r386538);
        double r386540 = r386536 - r386539;
        double r386541 = pow(r386540, r386494);
        double r386542 = cbrt(r386541);
        double r386543 = r386542 / r386534;
        double r386544 = r386483 ? r386535 : r386543;
        return r386544;
}

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)))))))