if c < -3.3148205f+22 or 2.8363323f+14 < c

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
   \[\frac{a \cdot c + b \cdot d}{{\color{red}{\left(\sqrt{{c}^2}\right)}}^2 + {d}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{\left(\left|c\right|\right)}}^2 + {d}^2}\]
   15.6
5. Using strategy rm
   15.6
6. Applied add-cube-cbrt to get
   \[\color{red}{\frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + {d}^2}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + {d}^2}}\right)}^3}\]
   15.7
7. Applied taylor to get
   \[{\left(\sqrt[3]{\frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + {d}^2}}\right)}^3 \leadsto {\left(\sqrt[3]{\frac{b \cdot d}{{\left(\left|c\right|\right)}^2} + \frac{c \cdot a}{{\left(\left|c\right|\right)}^2}}\right)}^3\]
   13.5
8. Taylor expanded around 0 to get
   \[{\left(\sqrt[3]{\color{red}{\frac{b \cdot d}{{\left(\left|c\right|\right)}^2} + \frac{c \cdot a}{{\left(\left|c\right|\right)}^2}}}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{\frac{b \cdot d}{{\left(\left|c\right|\right)}^2} + \frac{c \cdot a}{{\left(\left|c\right|\right)}^2}}}\right)}^3\]
   13.5
9. Applied simplify to get
   \[{\left(\sqrt[3]{\frac{b \cdot d}{{\left(\left|c\right|\right)}^2} + \frac{c \cdot a}{{\left(\left|c\right|\right)}^2}}\right)}^3 \leadsto (\left(\frac{a}{\left|c\right|}\right) * \left(\frac{c}{\left|c\right|}\right) + \left(\frac{d}{\left|c\right|} \cdot \frac{b}{\left|c\right|}\right))_*\]
   0.0

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10. Applied final simplification

if -3.3148205f+22 < c < 2.8363323f+14

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   9.5
2. Using strategy rm
   9.5
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2} + {d}^2} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2}\right)}^2} + {d}^2}\]
   9.4
4. Applied simplify to get
   \[\frac{a \cdot c + b \cdot d}{{\color{red}{\left(\sqrt{{c}^2}\right)}}^2 + {d}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{\left(\left|c\right|\right)}}^2 + {d}^2}\]
   8.8
5. Using strategy rm
   8.8
6. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + \color{red}{{d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
   8.8
7. Applied simplify to get
   \[\frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\left(\left|c\right|\right)}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
   6.0