if d < -6.978547f+10 or 2.1848217f+18 < d

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   21.0
2. Using strategy rm
   21.0
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2} + {d}^2} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2}\right)}^2} + {d}^2}\]
   21.0
4. Applied simplify to get
   \[\frac{a \cdot c + b \cdot d}{{\color{red}{\left(\sqrt{{c}^2}\right)}}^2 + {d}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{\left(\left|c\right|\right)}}^2 + {d}^2}\]
   20.9
5. Applied taylor to get
   \[\frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + {d}^2} \leadsto \frac{c \cdot a}{{d}^2} + \frac{b}{d}\]
   6.0
6. Taylor expanded around inf to get
   \[\color{red}{\frac{c \cdot a}{{d}^2} + \frac{b}{d}} \leadsto \color{blue}{\frac{c \cdot a}{{d}^2} + \frac{b}{d}}\]
   6.0
7. Applied taylor to get
   \[\frac{c \cdot a}{{d}^2} + \frac{b}{d} \leadsto \frac{c \cdot a}{{d}^2} + \frac{b}{d}\]
   6.0
8. Taylor expanded around 0 to get
   \[\frac{c \cdot a}{\color{red}{{d}^2}} + \frac{b}{d} \leadsto \frac{c \cdot a}{\color{blue}{{d}^2}} + \frac{b}{d}\]
   6.0
9. Applied simplify to get
   \[\frac{c \cdot a}{{d}^2} + \frac{b}{d} \leadsto (\left(\frac{a}{d}\right) * \left(\frac{c}{d}\right) + \left(\frac{b}{d}\right))_*\]
   0.3

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10. Applied final simplification

if -6.978547f+10 < d < 2.1848217f+18

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
   \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2} + {d}^2} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2}\right)}^2} + {d}^2}\]
   8.9
4. Applied simplify to get
   \[\frac{a \cdot c + b \cdot d}{{\color{red}{\left(\sqrt{{c}^2}\right)}}^2 + {d}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{\left(\left|c\right|\right)}}^2 + {d}^2}\]
   6.0
5. Using strategy rm
   6.0
6. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + \color{red}{{d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
   6.0
7. Applied simplify to get
   \[\frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
   5.6
8. Applied taylor to get
   \[\frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + {\left(\left|d\right|\right)}^2} \leadsto \frac{b \cdot d}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2} + \frac{c \cdot a}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2}\]
   5.6
9. Taylor expanded around 0 to get
   \[\color{red}{\frac{b \cdot d}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2} + \frac{c \cdot a}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2}} \leadsto \color{blue}{\frac{b \cdot d}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2} + \frac{c \cdot a}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2}}\]
   5.6