if c < 7.311325645509872e+152

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   22.6
2. Using strategy rm
   22.6
3. Applied div-inv to get
   \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\left(a \cdot c + b \cdot d\right) \cdot \frac{1}{{c}^2 + {d}^2}}\]
   22.8

if 7.311325645509872e+152 < c < 6.4762574223026695e+289

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   44.0
2. Using strategy rm
   44.0
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
   44.0
4. Applied add-sqr-sqrt to get
   \[\frac{\color{red}{a \cdot c + b \cdot d}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2} \leadsto \frac{\color{blue}{{\left(\sqrt{a \cdot c + b \cdot d}\right)}^2}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}\]
   44.0
5. Applied square-undiv to get
   \[\color{red}{\frac{{\left(\sqrt{a \cdot c + b \cdot d}\right)}^2}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}} \leadsto \color{blue}{{\left(\frac{\sqrt{a \cdot c + b \cdot d}}{\sqrt{{c}^2 + {d}^2}}\right)}^2}\]
   44.0
6. Applied simplify to get
   \[{\color{red}{\left(\frac{\sqrt{a \cdot c + b \cdot d}}{\sqrt{{c}^2 + {d}^2}}\right)}}^2 \leadsto {\color{blue}{\left(\frac{\sqrt{(c * a + \left(d \cdot b\right))_*}}{\sqrt{c^2 + d^2}^*}\right)}}^2\]
   29.1

if 6.4762574223026695e+289 < c

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   42.8
2. Using strategy rm
   42.8
3. Applied add-cube-cbrt to get
   \[\frac{\color{red}{a \cdot c + b \cdot d}}{{c}^2 + {d}^2} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot c + b \cdot d}\right)}^3}}{{c}^2 + {d}^2}\]
   42.8
4. Applied simplify to get
   \[\frac{{\color{red}{\left(\sqrt[3]{a \cdot c + b \cdot d}\right)}}^3}{{c}^2 + {d}^2} \leadsto \frac{{\color{blue}{\left(\sqrt[3]{(c * a + \left(d \cdot b\right))_*}\right)}}^3}{{c}^2 + {d}^2}\]
   42.8
5. Applied taylor to get
   \[\frac{{\left(\sqrt[3]{(c * a + \left(d \cdot b\right))_*}\right)}^3}{{c}^2 + {d}^2} \leadsto \frac{{\left(\sqrt[3]{(\left(\frac{1}{c}\right) * \left(\frac{1}{a}\right) + \left(\frac{1}{b \cdot d}\right))_*}\right)}^3}{{c}^2 + {d}^2}\]
   7.6
6. Taylor expanded around inf to get
   \[\frac{{\color{red}{\left(\sqrt[3]{(\left(\frac{1}{c}\right) * \left(\frac{1}{a}\right) + \left(\frac{1}{b \cdot d}\right))_*}\right)}}^3}{{c}^2 + {d}^2} \leadsto \frac{{\color{blue}{\left(\sqrt[3]{(\left(\frac{1}{c}\right) * \left(\frac{1}{a}\right) + \left(\frac{1}{b \cdot d}\right))_*}\right)}}^3}{{c}^2 + {d}^2}\]
   7.6
7. Applied simplify to get
   \[\color{red}{\frac{{\left(\sqrt[3]{(\left(\frac{1}{c}\right) * \left(\frac{1}{a}\right) + \left(\frac{1}{b \cdot d}\right))_*}\right)}^3}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{(\left(\frac{1}{c}\right) * \left(\frac{1}{a}\right) + \left(\frac{\frac{1}{b}}{d}\right))_*}{(d * d + \left(c \cdot c\right))_*}}\]
   7.6