if re < -1.7812046f-12

1. Started with
   \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
   28.1
2. Using strategy rm
   28.1
3. Applied flip-+ to get
   \[0.5 \cdot \sqrt{2.0 \cdot \color{red}{\left(\sqrt{re \cdot re + im \cdot im} + re\right)}} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2}{\sqrt{re \cdot re + im \cdot im} - re}}}\]
   28.8
4. Applied associate-*r/ to get
   \[0.5 \cdot \sqrt{\color{red}{2.0 \cdot \frac{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2}{\sqrt{re \cdot re + im \cdot im} - re}}} \leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2.0 \cdot \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2\right)}{\sqrt{re \cdot re + im \cdot im} - re}}}\]
   28.8
5. Applied sqrt-div to get
   \[0.5 \cdot \color{red}{\sqrt{\frac{2.0 \cdot \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2\right)}{\sqrt{re \cdot re + im \cdot im} - re}}} \leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2.0 \cdot \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]
   28.8
6. Applied simplify to get
   \[0.5 \cdot \frac{\color{red}{\sqrt{2.0 \cdot \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^2 - {re}^2\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{\left(2.0 \cdot im\right) \cdot im}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
   20.6
7. Applied simplify to get
   \[0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\color{red}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}} \leadsto 0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\color{blue}{\sqrt{\sqrt{{re}^2 + im \cdot im} - re}}}\]
   20.6
8. Applied taylor to get
   \[0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\sqrt{\sqrt{{re}^2 + im \cdot im} - re}} \leadsto 0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\sqrt{-1 \cdot re - re}}\]
   11.3
9. Taylor expanded around -inf to get
   \[0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\sqrt{\color{red}{-1 \cdot re} - re}} \leadsto 0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\sqrt{\color{blue}{-1 \cdot re} - re}}\]
   11.3
10. Applied simplify to get
    \[0.5 \cdot \frac{\sqrt{\left(2.0 \cdot im\right) \cdot im}}{\sqrt{-1 \cdot re - re}} \leadsto \frac{0.5 \cdot \sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\left(-re\right) - re}}\]
    11.4

---

11. Applied final simplification

if -1.7812046f-12 < re < 1.1667896f+15

1. Started with
   \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
   10.7
2. Using strategy rm
   10.7
3. Applied add-sqr-sqrt to get
   \[0.5 \cdot \sqrt{2.0 \cdot \left(\color{red}{\sqrt{re \cdot re + im \cdot im}} + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{{\left(\sqrt{\sqrt{re \cdot re + im \cdot im}}\right)}^2} + re\right)}\]
   10.8
4. Applied simplify to get
   \[0.5 \cdot \sqrt{2.0 \cdot \left({\color{red}{\left(\sqrt{\sqrt{re \cdot re + im \cdot im}}\right)}}^2 + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left({\color{blue}{\left(\sqrt{\sqrt{{re}^2 + im \cdot im}}\right)}}^2 + re\right)}\]
   10.8

if 1.1667896f+15 < re

1. Started with
   \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
   25.6
2. Applied taylor to get
   \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(re + re\right)}\]
   0.8
3. Taylor expanded around inf to get
   \[0.5 \cdot \sqrt{2.0 \cdot \left(\color{red}{re} + re\right)} \leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{re} + re\right)}\]
   0.8