if c < 3.1088567f+22

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   11.3
2. Using strategy rm
   11.3
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
   11.2
4. Applied simplify to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
   9.0
5. Using strategy rm
   9.0
6. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2} + {\left(\left|d\right|\right)}^2} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2}\right)}^2} + {\left(\left|d\right|\right)}^2}\]
   9.0
7. Applied simplify to get
   \[\frac{a \cdot c + b \cdot d}{{\color{red}{\left(\sqrt{{c}^2}\right)}}^2 + {\left(\left|d\right|\right)}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{\left(\left|c\right|\right)}}^2 + {\left(\left|d\right|\right)}^2}\]
   7.1
8. Using strategy rm
   7.1
9. Applied add-cube-cbrt to get
   \[\color{red}{\frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + {\left(\left|d\right|\right)}^2}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + {\left(\left|d\right|\right)}^2}}\right)}^3}\]
   7.4

if 3.1088567f+22 < c

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   22.1
2. Using strategy rm
   22.1
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
   22.1
4. Applied simplify to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
   22.1
5. Applied taylor to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2} \leadsto \left(\frac{a}{c} + \frac{b \cdot d}{{c}^2}\right) - \frac{b \cdot \left(d \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}\]
   6.5
6. Taylor expanded around inf to get
   \[\color{red}{\left(\frac{a}{c} + \frac{b \cdot d}{{c}^2}\right) - \frac{b \cdot \left(d \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}} \leadsto \color{blue}{\left(\frac{a}{c} + \frac{b \cdot d}{{c}^2}\right) - \frac{b \cdot \left(d \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}}\]
   6.5