if d < -1461388.5f0 or 4.759657f+20 < d

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   19.6
2. Using strategy rm
   19.6
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
   19.6
4. Applied taylor to get
   \[\frac{a \cdot c + b \cdot d}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2} \leadsto \frac{a \cdot c + b \cdot d}{{d}^2}\]
   18.2
5. Taylor expanded around 0 to get
   \[\frac{a \cdot c + b \cdot d}{{\color{red}{d}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{d}}^2}\]
   18.2
6. Applied taylor to get
   \[\frac{a \cdot c + b \cdot d}{{d}^2} \leadsto \frac{c \cdot a}{{d}^2} + \frac{b}{d}\]
   5.7
7. Taylor expanded around 0 to get
   \[\color{red}{\frac{c \cdot a}{{d}^2} + \frac{b}{d}} \leadsto \color{blue}{\frac{c \cdot a}{{d}^2} + \frac{b}{d}}\]
   5.7
8. Applied simplify to get
   \[\frac{c \cdot a}{{d}^2} + \frac{b}{d} \leadsto \frac{c}{d} \cdot \frac{a}{d} + \frac{b}{d}\]
   0.3

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9. Applied final simplification

if -1461388.5f0 < d < 3.1068393f-18

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   12.6
2. Using strategy rm
   12.6
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
   12.6
4. Using strategy rm
   12.6
5. Applied add-exp-log to get
   \[\frac{a \cdot c + b \cdot d}{{\color{red}{\left(\sqrt{{c}^2 + {d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{\left(e^{\log \left(\sqrt{{c}^2 + {d}^2}\right)}\right)}}^2}\]
   13.7
6. Applied taylor to get
   \[\frac{a \cdot c + b \cdot d}{{\left(e^{\log \left(\sqrt{{c}^2 + {d}^2}\right)}\right)}^2} \leadsto \frac{a}{c} + \frac{b \cdot d}{{c}^2}\]
   3.0
7. Taylor expanded around inf to get
   \[\color{red}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}} \leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}}\]
   3.0
8. Applied simplify to get
   \[\frac{a}{c} + \frac{b \cdot d}{{c}^2} \leadsto \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\]
   0.8

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9. Applied final simplification

if 3.1068393f-18 < d < 4.759657f+20

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   8.9
2. Using strategy rm
   8.9
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
   8.9