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
   \[\frac{a}{c} + \frac{b \cdot d}{{c}^2} \leadsto (\left(\frac{b}{c}\right) * \left(\frac{d}{c}\right) + \left(\frac{a}{c}\right))_*\]
   0.3

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7. Applied final simplification

if -9.398266f+14 < c < -4.298775f-18 or 3.2195558f-26 < c < 6.416234f+14

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   8.1
2. Using strategy rm
   8.1
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
   8.1
4. Applied add-sqr-sqrt to get
   \[\frac{\color{red}{a \cdot c + b \cdot d}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2} \leadsto \frac{\color{blue}{{\left(\sqrt{a \cdot c + b \cdot d}\right)}^2}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}\]
   20.1
5. Applied square-undiv to get
   \[\color{red}{\frac{{\left(\sqrt{a \cdot c + b \cdot d}\right)}^2}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}} \leadsto \color{blue}{{\left(\frac{\sqrt{a \cdot c + b \cdot d}}{\sqrt{{c}^2 + {d}^2}}\right)}^2}\]
   20.1
6. Applied simplify to get
   \[{\color{red}{\left(\frac{\sqrt{a \cdot c + b \cdot d}}{\sqrt{{c}^2 + {d}^2}}\right)}}^2 \leadsto {\color{blue}{\left(\frac{\sqrt{(c * a + \left(d \cdot b\right))_*}}{\sqrt{c^2 + d^2}^*}\right)}}^2\]
   18.9
7. Applied taylor to get
   \[{\left(\frac{\sqrt{(c * a + \left(d \cdot b\right))_*}}{\sqrt{c^2 + d^2}^*}\right)}^2 \leadsto {\left(\frac{\sqrt{(c * a + \left(b \cdot d\right))_*}}{\sqrt{c^2 + d^2}^*}\right)}^2\]
   18.9
8. Taylor expanded around 0 to get
   \[{\left(\frac{\color{red}{\sqrt{(c * a + \left(b \cdot d\right))_*}}}{\sqrt{c^2 + d^2}^*}\right)}^2 \leadsto {\left(\frac{\color{blue}{\sqrt{(c * a + \left(b \cdot d\right))_*}}}{\sqrt{c^2 + d^2}^*}\right)}^2\]
   18.9
9. Applied simplify to get
   \[{\left(\frac{\sqrt{(c * a + \left(b \cdot d\right))_*}}{\sqrt{c^2 + d^2}^*}\right)}^2 \leadsto \frac{(c * a + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^* \cdot \sqrt{c^2 + d^2}^*}\]
   5.4

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10. Applied final simplification
11. Applied simplify to get
    \[\color{red}{\frac{(c * a + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^* \cdot \sqrt{c^2 + d^2}^*}} \leadsto \color{blue}{\frac{(c * a + \left(b \cdot d\right))_*}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}\]
    5.4

if -4.298775f-18 < c < 3.2195558f-26

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
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
   6.3
5. Using strategy rm
   6.3
6. Applied add-cube-cbrt to get
   \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}}\right)}^3}\]
   6.6
7. Applied taylor to get
   \[{\left(\sqrt[3]{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}}\right)}^3 \leadsto {\left(\sqrt[3]{\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}}\right)}^3\]
   6.2
8. Taylor expanded around 0 to get
   \[{\left(\sqrt[3]{\color{red}{\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}}}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}}}\right)}^3\]
   6.2
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