if c < -9.398266f+14 or 6.416234f+14 < c

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-exp-log to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + \color{blue}{e^{\log \left({d}^2\right)}}}\]
   20.9
4. Applied taylor to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + e^{\log \left({d}^2\right)}} \leadsto \frac{a}{c} + \frac{b \cdot d}{{c}^2}\]
   6.3
5. Taylor expanded around inf to get
   \[\color{red}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}} \leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}}\]
   6.3
6. Applied simplify to get
   \[\frac{a}{c} + \frac{b \cdot d}{{c}^2} \leadsto \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\]
   0.3

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7. Applied final simplification

if -9.398266f+14 < c < -4.298775f-18 or 6.361403f-15 < c < 6.416234f+14

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
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
   8.0
4. Applied simplify to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
   5.5
5. Using strategy rm
   5.5
6. Applied add-cube-cbrt to get
   \[\frac{\color{red}{a \cdot c + b \cdot d}}{{c}^2 + {\left(\left|d\right|\right)}^2} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot c + b \cdot d}\right)}^3}}{{c}^2 + {\left(\left|d\right|\right)}^2}\]
   5.9

if -4.298775f-18 < c < 6.361403f-15

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   10.3
2. Using strategy rm
   10.3
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
   10.3
4. Applied simplify to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
   6.4
5. Using strategy rm
   6.4
6. Applied add-cube-cbrt to get
   \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}}\right)}^3}\]
   6.7
7. Using strategy rm
   6.7
8. Applied add-cube-cbrt to get
   \[{\color{red}{\left(\sqrt[3]{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}}\right)}}^3 \leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}}}\right)}^3\right)}}^3\]
   7.2
9. Applied taylor to get
   \[{\left({\left(\sqrt[3]{\sqrt[3]{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\sqrt[3]{\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}}}\right)}^3\right)}^3\]
   6.5
10. Taylor expanded around 0 to get
    \[{\left({\left(\sqrt[3]{\sqrt[3]{\color{red}{\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}}}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\sqrt[3]{\color{blue}{\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}}}}\right)}^3\right)}^3\]
    6.5
11. Applied simplify to get
    \[{\left({\left(\sqrt[3]{\sqrt[3]{\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}}}\right)}^3\right)}^3 \leadsto \frac{b}{\left|d\right|} \cdot \frac{d}{\left|d\right|} + \frac{a}{\left|d\right|} \cdot \frac{c}{\left|d\right|}\]
    0

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12. Applied final simplification