if c < -6.643754f+11 or 0.009737393f0 < c

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   18.4
2. Using strategy rm
   18.4
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
   18.4
4. Applied taylor to get
   \[\frac{a \cdot c + b \cdot d}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2}\]
   15.5
5. Taylor expanded around inf to get
   \[\frac{a \cdot c + b \cdot d}{{\color{red}{c}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{c}}^2}\]
   15.5
6. Applied taylor to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2} \leadsto \frac{a}{c} + \frac{b \cdot d}{{c}^2}\]
   5.2
7. Taylor expanded around 0 to get
   \[\color{red}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}} \leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}}\]
   5.2
8. Applied simplify to get
   \[\frac{a}{c} + \frac{b \cdot d}{{c}^2} \leadsto \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\]
   0.4

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9. Applied final simplification

if -6.643754f+11 < c < -6.138259f-23

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   7.6
2. Using strategy rm
   7.6
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
   7.6

if -6.138259f-23 < c < 0.009737393f0

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   11.4
2. Using strategy rm
   11.4
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
   11.3
4. Applied taylor to get
   \[\frac{a \cdot c + b \cdot d}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2} \leadsto \frac{a \cdot c + b \cdot d}{{d}^2}\]
   10.5
5. Taylor expanded around 0 to get
   \[\frac{a \cdot c + b \cdot d}{{\color{red}{d}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{d}}^2}\]
   10.5
6. Applied taylor to get
   \[\frac{a \cdot c + b \cdot d}{{d}^2} \leadsto \frac{c \cdot a}{{d}^2} + \frac{b}{d}\]
   3.2
7. Taylor expanded around 0 to get
   \[\color{red}{\frac{c \cdot a}{{d}^2} + \frac{b}{d}} \leadsto \color{blue}{\frac{c \cdot a}{{d}^2} + \frac{b}{d}}\]
   3.2
8. Applied simplify to get
   \[\frac{c \cdot a}{{d}^2} + \frac{b}{d} \leadsto \frac{c}{d} \cdot \frac{a}{d} + \frac{b}{d}\]
   1.2

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9. Applied final simplification