if d < -1.8956648f+13 or 0.024397144f0 < d

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   19.4
2. Using strategy rm
   19.4
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
   19.4
4. Applied taylor to get
   \[\frac{a \cdot c + b \cdot d}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2} \leadsto \frac{a \cdot c + b \cdot d}{{d}^2}\]
   16.3
5. Taylor expanded around 0 to get
   \[\frac{a \cdot c + b \cdot d}{{\color{red}{d}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{d}}^2}\]
   16.3
6. Applied taylor to get
   \[\frac{a \cdot c + b \cdot d}{{d}^2} \leadsto \frac{c \cdot a}{{d}^2} + \frac{b}{d}\]
   5.3
7. Taylor expanded around 0 to get
   \[\color{red}{\frac{c \cdot a}{{d}^2} + \frac{b}{d}} \leadsto \color{blue}{\frac{c \cdot a}{{d}^2} + \frac{b}{d}}\]
   5.3
8. Applied simplify to get
   \[\frac{c \cdot a}{{d}^2} + \frac{b}{d} \leadsto \frac{c}{d} \cdot \frac{a}{d} + \frac{b}{d}\]
   0.4

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9. Applied final simplification

if -1.8956648f+13 < d < -1.0142954f-12

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   8.4
2. Using strategy rm
   8.4
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
   8.4

if -1.0142954f-12 < d < 0.024397144f0

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. Using strategy rm
   11.3
5. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{{\color{red}{\left(\sqrt{{c}^2 + {d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{\left({\left(\sqrt{\sqrt{{c}^2 + {d}^2}}\right)}^2\right)}}^2}\]
   11.6
6. Applied taylor to get
   \[\frac{a \cdot c + b \cdot d}{{\left({\left(\sqrt{\sqrt{{c}^2 + {d}^2}}\right)}^2\right)}^2} \leadsto \frac{a}{c} + \frac{b \cdot d}{{c}^2}\]
   3.1
7. Taylor expanded around inf to get
   \[\color{red}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}} \leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}}\]
   3.1
8. Applied simplify to get
   \[\frac{a}{c} + \frac{b \cdot d}{{c}^2} \leadsto \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\]
   1.0

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9. Applied final simplification