if c < -8.89023270141141e+107 or 1.0222346456089473e+104 < c

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   40.8
2. Using strategy rm
   40.8
3. Applied add-exp-log to get
   \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{e^{\log \left({c}^2 + {d}^2\right)}}}\]
   41.5
4. Applied taylor to get
   \[\frac{a \cdot c + b \cdot d}{e^{\log \left({c}^2 + {d}^2\right)}} \leadsto \frac{a}{c} + \frac{b \cdot d}{{c}^2}\]
   10.9
5. Taylor expanded around inf to get
   \[\color{red}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}} \leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}}\]
   10.9
6. Applied simplify to get
   \[\frac{a}{c} + \frac{b \cdot d}{{c}^2} \leadsto (\left(\frac{b}{c}\right) * \left(\frac{d}{c}\right) + \left(\frac{a}{c}\right))_*\]
   0.9

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7. Applied final simplification

if -8.89023270141141e+107 < c < 1.0222346456089473e+104

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   18.4
2. Using strategy rm
   18.4
3. Applied clear-num to get
   \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{1}{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}}}\]
   18.6
4. Applied simplify to get
   \[\frac{1}{\color{red}{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}}} \leadsto \frac{1}{\color{blue}{\frac{(d * d + \left({c}^2\right))_*}{(c * a + \left(b \cdot d\right))_*}}}\]
   18.6