Average Error: 15.6 → 15.2
Time: 30.1s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{{\left(1 \cdot \left(1 - 0.5\right)\right)}^{3} - {\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\left(\left(0.5 \cdot \frac{1}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left({\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3} + {\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}\right)\right) \cdot \left(\left(1 \cdot 1 + \left(0.5 \cdot 0.5 + 1 \cdot 0.5\right)\right) \cdot \left(1 \cdot 1 + \left(0.5 \cdot 0.5 + 1 \cdot 0.5\right)\right)\right) + \left(\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \left(\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) + \left(\left(1 \cdot \left(1 - 0.5\right)\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right) - \left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right)\right)\right)\right) \cdot \left(\left(1 \cdot \left({1}^{3} - {0.5}^{3}\right)\right) \cdot \left(1 \cdot \left({1}^{3} - {0.5}^{3}\right)\right)\right)} \cdot \left(\left(\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \left(\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) + \left(\left(1 \cdot \left(1 - 0.5\right)\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right) - \left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right)\right)\right)\right) \cdot \left(\left(1 \cdot 1 + \left(0.5 \cdot 0.5 + 1 \cdot 0.5\right)\right) \cdot \left(1 \cdot 1 + \left(0.5 \cdot 0.5 + 1 \cdot 0.5\right)\right)\right)\right)}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{{\left(1 \cdot \left(1 - 0.5\right)\right)}^{3} - {\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\left(\left(0.5 \cdot \frac{1}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left({\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3} + {\left(1 \cdot \left(1 - 0.5\right)\right)}^{3}\right)\right) \cdot \left(\left(1 \cdot 1 + \left(0.5 \cdot 0.5 + 1 \cdot 0.5\right)\right) \cdot \left(1 \cdot 1 + \left(0.5 \cdot 0.5 + 1 \cdot 0.5\right)\right)\right) + \left(\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \left(\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) + \left(\left(1 \cdot \left(1 - 0.5\right)\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right) - \left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right)\right)\right)\right) \cdot \left(\left(1 \cdot \left({1}^{3} - {0.5}^{3}\right)\right) \cdot \left(1 \cdot \left({1}^{3} - {0.5}^{3}\right)\right)\right)} \cdot \left(\left(\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \left(\left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) + \left(\left(1 \cdot \left(1 - 0.5\right)\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right) - \left(0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right)\right)\right)\right) \cdot \left(\left(1 \cdot 1 + \left(0.5 \cdot 0.5 + 1 \cdot 0.5\right)\right) \cdot \left(1 \cdot 1 + \left(0.5 \cdot 0.5 + 1 \cdot 0.5\right)\right)\right)\right)}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}
double code(double x) {
	return (1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x))))));
}
↓
double code(double x) {
	return ((((pow((1.0 * (1.0 - 0.5)), 3.0) - pow((0.5 * (1.0 / hypot(1.0, x))), 3.0)) / ((((0.5 * (1.0 / (cbrt(hypot(1.0, x)) * cbrt(hypot(1.0, x))))) * (pow((0.5 * (1.0 / hypot(1.0, x))), 3.0) + pow((1.0 * (1.0 - 0.5)), 3.0))) * (((1.0 * 1.0) + ((0.5 * 0.5) + (1.0 * 0.5))) * ((1.0 * 1.0) + ((0.5 * 0.5) + (1.0 * 0.5))))) + ((cbrt(hypot(1.0, x)) * (((0.5 * (1.0 / hypot(1.0, x))) * (0.5 * (1.0 / hypot(1.0, x)))) + (((1.0 * (1.0 - 0.5)) * (1.0 * (1.0 - 0.5))) - ((0.5 * (1.0 / hypot(1.0, x))) * (1.0 * (1.0 - 0.5)))))) * ((1.0 * (pow(1.0, 3.0) - pow(0.5, 3.0))) * (1.0 * (pow(1.0, 3.0) - pow(0.5, 3.0))))))) * ((cbrt(hypot(1.0, x)) * (((0.5 * (1.0 / hypot(1.0, x))) * (0.5 * (1.0 / hypot(1.0, x)))) + (((1.0 * (1.0 - 0.5)) * (1.0 * (1.0 - 0.5))) - ((0.5 * (1.0 / hypot(1.0, x))) * (1.0 * (1.0 - 0.5)))))) * (((1.0 * 1.0) + ((0.5 * 0.5) + (1.0 * 0.5))) * ((1.0 * 1.0) + ((0.5 * 0.5) + (1.0 * 0.5)))))) / (1.0 + sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.6

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

  3. Applied flip--15.6

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

  4. Simplified15.1

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

  6. Applied flip3--15.1

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

  7. Simplified15.1

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

  9. Applied flip3--15.1

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

  10. Applied associate-*r/15.1

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

  11. Applied flip3--15.1

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

  12. Applied associate-*r/15.1

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

  13. Applied frac-times15.1

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

  14. Applied flip3-+15.1

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

  15. Applied add-cube-cbrt15.1

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

  16. Applied associate-/r*15.1

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

  17. Applied associate-*r/15.1

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

  18. Applied frac-times15.1

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

  19. Applied frac-add15.1

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

  20. Applied associate-/r/15.2

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

  21. Final simplification15.2

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

Reproduce

herbie shell --seed 2020066 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))