if d < -9.1301655f+27 or 3.6376857f+23 < d

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   20.9
2. Using strategy rm
   20.9
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2} + {d}^2} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2}\right)}^2} + {d}^2}\]
   20.9
4. Applied simplify to get
   \[\frac{a \cdot c + b \cdot d}{{\color{red}{\left(\sqrt{{c}^2}\right)}}^2 + {d}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{\left(\left|c\right|\right)}}^2 + {d}^2}\]
   20.9
5. Applied taylor to get
   \[\frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + {d}^2} \leadsto \frac{c \cdot a}{{d}^2} + \frac{b}{d}\]
   5.8
6. Taylor expanded around inf to get
   \[\color{red}{\frac{c \cdot a}{{d}^2} + \frac{b}{d}} \leadsto \color{blue}{\frac{c \cdot a}{{d}^2} + \frac{b}{d}}\]
   5.8
7. Applied taylor to get
   \[\frac{c \cdot a}{{d}^2} + \frac{b}{d} \leadsto \frac{c \cdot a}{{d}^2} + \frac{b}{d}\]
   5.8
8. Taylor expanded around 0 to get
   \[\frac{c \cdot a}{\color{red}{{d}^2}} + \frac{b}{d} \leadsto \frac{c \cdot a}{\color{blue}{{d}^2}} + \frac{b}{d}\]
   5.8
9. Applied simplify to get
   \[\frac{c \cdot a}{{d}^2} + \frac{b}{d} \leadsto \frac{c}{d} \cdot \frac{a}{d} + \frac{b}{d}\]
   0.2

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10. Applied final simplification

if -9.1301655f+27 < d < 3.6376857f+23

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   10.7
2. Using strategy rm
   10.7
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2} + {d}^2} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2}\right)}^2} + {d}^2}\]
   10.7
4. Applied simplify to get
   \[\frac{a \cdot c + b \cdot d}{{\color{red}{\left(\sqrt{{c}^2}\right)}}^2 + {d}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{\left(\left|c\right|\right)}}^2 + {d}^2}\]
   8.2
5. Using strategy rm
   8.2
6. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + \color{red}{{d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
   8.2
7. Applied simplify to get
   \[\frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
   6.5
8. Applied taylor to get
   \[\frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + {\left(\left|d\right|\right)}^2} \leadsto \frac{b \cdot d}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2} + \frac{c \cdot a}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2}\]
   6.5
9. Taylor expanded around 0 to get
   \[\color{red}{\frac{b \cdot d}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2} + \frac{c \cdot a}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2}} \leadsto \color{blue}{\frac{b \cdot d}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2} + \frac{c \cdot a}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2}}\]
   6.5