if c < -1.674345f+14 or 6.144087f+15 < c

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   21.8
2. Using strategy rm
   21.8
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
   21.8
4. Applied simplify to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
   21.8
5. Applied taylor to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2} \leadsto \left(\frac{a}{c} + \frac{b \cdot d}{{c}^2}\right) - \frac{b \cdot \left(d \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}\]
   6.5
6. Taylor expanded around inf to get
   \[\color{red}{\left(\frac{a}{c} + \frac{b \cdot d}{{c}^2}\right) - \frac{b \cdot \left(d \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}} \leadsto \color{blue}{\left(\frac{a}{c} + \frac{b \cdot d}{{c}^2}\right) - \frac{b \cdot \left(d \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}}\]
   6.5
7. Applied simplify to get
   \[\left(\frac{a}{c} + \frac{b \cdot d}{{c}^2}\right) - \frac{b \cdot \left(d \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}} \leadsto \left(\frac{a}{c} + \frac{d \cdot b}{c \cdot c}\right) - \frac{\left(d \cdot b\right) \cdot {\left(\left|\frac{1}{d}\right|\right)}^2}{{c}^{4}}\]
   6.5

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8. Applied final simplification

if -1.674345f+14 < c < -1.8106279f-24 or 9.359174f-16 < c < 6.144087f+15

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   8.0
2. Using strategy rm
   8.0
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
   7.9
4. Applied simplify to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
   5.6

if -1.8106279f-24 < c < 9.359174f-16

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   10.8
2. Using strategy rm
   10.8
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2} + {d}^2} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2}\right)}^2} + {d}^2}\]
   10.8
4. Applied simplify to get
   \[\frac{a \cdot c + b \cdot d}{{\color{red}{\left(\sqrt{{c}^2}\right)}}^2 + {d}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{\left(\left|c\right|\right)}}^2 + {d}^2}\]
   10.8
5. Applied taylor to get
   \[\frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + {d}^2} \leadsto \frac{c \cdot a}{{d}^2} + \frac{b}{d}\]
   3.0
6. Taylor expanded around inf to get
   \[\color{red}{\frac{c \cdot a}{{d}^2} + \frac{b}{d}} \leadsto \color{blue}{\frac{c \cdot a}{{d}^2} + \frac{b}{d}}\]
   3.0
7. Applied taylor to get
   \[\frac{c \cdot a}{{d}^2} + \frac{b}{d} \leadsto \frac{c \cdot a}{{d}^2} + \frac{b}{d}\]
   3.0
8. Taylor expanded around 0 to get
   \[\frac{c \cdot a}{{d}^2} + \color{red}{\frac{b}{d}} \leadsto \frac{c \cdot a}{{d}^2} + \color{blue}{\frac{b}{d}}\]
   3.0
9. Applied simplify to get
   \[\frac{c \cdot a}{{d}^2} + \frac{b}{d} \leadsto \frac{c}{d} \cdot \frac{a}{d} + \frac{b}{d}\]
   1.0

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10. Applied final simplification