if d < -1.01464965f+11 or 4.628637f+09 < d

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   20.2
2. Using strategy rm
   20.2
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
   20.2
4. Applied simplify to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
   15.6
5. Using strategy rm
   15.6
6. Applied add-cube-cbrt to get
   \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}}\right)}^3}\]
   15.7
7. Applied taylor to get
   \[{\left(\sqrt[3]{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}}\right)}^3 \leadsto {\left(\sqrt[3]{\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}}\right)}^3\]
   12.8
8. Taylor expanded around 0 to get
   \[{\left(\sqrt[3]{\color{red}{\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}}}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}}}\right)}^3\]
   12.8
9. Applied simplify to get
   \[{\left(\sqrt[3]{\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}}\right)}^3 \leadsto (\left(\frac{b}{\left|d\right|}\right) * \left(\frac{d}{\left|d\right|}\right) + \left(\frac{a}{\left|d\right|} \cdot \frac{c}{\left|d\right|}\right))_*\]
   0.0

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10. Applied final simplification

if -1.01464965f+11 < d < -2.1758005f-17 or 1.421191f-23 < d < 4.628637f+09

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   7.7
2. Using strategy rm
   7.7
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2} + {d}^2} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2}\right)}^2} + {d}^2}\]
   7.7
4. Applied simplify to get
   \[\frac{a \cdot c + b \cdot d}{{\color{red}{\left(\sqrt{{c}^2}\right)}}^2 + {d}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{\left(\left|c\right|\right)}}^2 + {d}^2}\]
   4.9

if -2.1758005f-17 < d < 1.421191f-23

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   10.5
2. Using strategy rm
   10.5
3. Applied add-cube-cbrt to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt[3]{{d}^2}\right)}^3}}\]
   10.5
4. Applied taylor to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\sqrt[3]{{d}^2}\right)}^3} \leadsto \frac{a}{c} + \frac{b \cdot d}{{c}^2}\]
   3.2
5. Taylor expanded around inf to get
   \[\color{red}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}} \leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}}\]
   3.2
6. Applied simplify to get
   \[\frac{a}{c} + \frac{b \cdot d}{{c}^2} \leadsto (\left(\frac{b}{c}\right) * \left(\frac{d}{c}\right) + \left(\frac{a}{c}\right))_*\]
   1.3

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7. Applied final simplification