if c < -1.3770151188228676e+30 or 4.6559433524913966e+129 < c

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   39.8
2. Using strategy rm
   39.8
3. Applied clear-num to get
   \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{1}{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}}}\]
   39.8
4. Using strategy rm
   39.8
5. Applied div-inv to get
   \[\frac{1}{\color{red}{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}}} \leadsto \frac{1}{\color{blue}{\left({c}^2 + {d}^2\right) \cdot \frac{1}{a \cdot c + b \cdot d}}}\]
   39.9
6. Applied associate-/r* to get
   \[\color{red}{\frac{1}{\left({c}^2 + {d}^2\right) \cdot \frac{1}{a \cdot c + b \cdot d}}} \leadsto \color{blue}{\frac{\frac{1}{{c}^2 + {d}^2}}{\frac{1}{a \cdot c + b \cdot d}}}\]
   39.8
7. Using strategy rm
   39.8
8. Applied add-cbrt-cube to get
   \[\frac{\color{red}{\frac{1}{{c}^2 + {d}^2}}}{\frac{1}{a \cdot c + b \cdot d}} \leadsto \frac{\color{blue}{\sqrt[3]{{\left(\frac{1}{{c}^2 + {d}^2}\right)}^3}}}{\frac{1}{a \cdot c + b \cdot d}}\]
   48.1
9. Applied taylor to get
   \[\frac{\sqrt[3]{{\left(\frac{1}{{c}^2 + {d}^2}\right)}^3}}{\frac{1}{a \cdot c + b \cdot d}} \leadsto \frac{a}{c} + \frac{b \cdot d}{{c}^2}\]
   11.1
10. Taylor expanded around 0 to get
    \[\color{red}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}} \leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}}\]
    11.1
11. Applied simplify to get
    \[\frac{a}{c} + \frac{b \cdot d}{{c}^2} \leadsto \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\]
    0.7

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12. Applied final simplification

if -1.3770151188228676e+30 < c < 1.745403512153814e-141 or 4.1244432764021164e-110 < c < 4.6559433524913966e+129

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   18.4

if 1.745403512153814e-141 < c < 4.1244432764021164e-110

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   43.9
2. Using strategy rm
   43.9
3. Applied clear-num to get
   \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{1}{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}}}\]
   44.3
4. Using strategy rm
   44.3
5. Applied div-inv to get
   \[\frac{1}{\color{red}{\frac{{c}^2 + {d}^2}{a \cdot c + b \cdot d}}} \leadsto \frac{1}{\color{blue}{\left({c}^2 + {d}^2\right) \cdot \frac{1}{a \cdot c + b \cdot d}}}\]
   44.3
6. Applied associate-/r* to get
   \[\color{red}{\frac{1}{\left({c}^2 + {d}^2\right) \cdot \frac{1}{a \cdot c + b \cdot d}}} \leadsto \color{blue}{\frac{\frac{1}{{c}^2 + {d}^2}}{\frac{1}{a \cdot c + b \cdot d}}}\]
   43.9
7. Using strategy rm
   43.9
8. Applied add-cbrt-cube to get
   \[\frac{\color{red}{\frac{1}{{c}^2 + {d}^2}}}{\frac{1}{a \cdot c + b \cdot d}} \leadsto \frac{\color{blue}{\sqrt[3]{{\left(\frac{1}{{c}^2 + {d}^2}\right)}^3}}}{\frac{1}{a \cdot c + b \cdot d}}\]
   48.2
9. Applied taylor to get
   \[\frac{\sqrt[3]{{\left(\frac{1}{{c}^2 + {d}^2}\right)}^3}}{\frac{1}{a \cdot c + b \cdot d}} \leadsto \left(\frac{c \cdot a}{{d}^2} + \frac{b}{d}\right) - \frac{b}{{c}^2 \cdot {d}^{3}}\]
   5.8
10. Taylor expanded around inf to get
    \[\color{red}{\left(\frac{c \cdot a}{{d}^2} + \frac{b}{d}\right) - \frac{b}{{c}^2 \cdot {d}^{3}}} \leadsto \color{blue}{\left(\frac{c \cdot a}{{d}^2} + \frac{b}{d}\right) - \frac{b}{{c}^2 \cdot {d}^{3}}}\]
    5.8
11. Applied simplify to get
    \[\left(\frac{c \cdot a}{{d}^2} + \frac{b}{d}\right) - \frac{b}{{c}^2 \cdot {d}^{3}} \leadsto \left(\frac{a}{d} \cdot \frac{c}{d} + \frac{b}{d}\right) - \frac{b}{{d}^3 \cdot {c}^2}\]
    5.9

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12. Applied final simplification