if c < -6.0490543f+21 or 3.800664f+23 < c

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   22.2
2. Using strategy rm
   22.2
3. Applied add-cube-cbrt to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt[3]{{d}^2}\right)}^3}}\]
   22.2
4. Applied taylor to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\sqrt[3]{{d}^2}\right)}^3} \leadsto \frac{a}{c} + \frac{b \cdot d}{{c}^2}\]
   6.0
5. Taylor expanded around inf to get
   \[\color{red}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}} \leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}}\]
   6.0
6. Applied simplify to get
   \[\frac{a}{c} + \frac{b \cdot d}{{c}^2} \leadsto (\left(\frac{b}{c}\right) * \left(\frac{d}{c}\right) + \left(\frac{a}{c}\right))_*\]
   0.2

---

7. Applied final simplification

if -6.0490543f+21 < c < -3.4569357f-25

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   9.2
2. Using strategy rm
   9.2
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
   9.2
4. Applied add-sqr-sqrt to get
   \[\frac{\color{red}{a \cdot c + b \cdot d}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2} \leadsto \frac{\color{blue}{{\left(\sqrt{a \cdot c + b \cdot d}\right)}^2}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}\]
   19.9
5. Applied square-undiv to get
   \[\color{red}{\frac{{\left(\sqrt{a \cdot c + b \cdot d}\right)}^2}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}} \leadsto \color{blue}{{\left(\frac{\sqrt{a \cdot c + b \cdot d}}{\sqrt{{c}^2 + {d}^2}}\right)}^2}\]
   19.9
6. Applied simplify to get
   \[{\color{red}{\left(\frac{\sqrt{a \cdot c + b \cdot d}}{\sqrt{{c}^2 + {d}^2}}\right)}}^2 \leadsto {\color{blue}{\left(\frac{\sqrt{(c * a + \left(d \cdot b\right))_*}}{\sqrt{c^2 + d^2}^*}\right)}}^2\]
   18.5
7. Applied taylor to get
   \[{\left(\frac{\sqrt{(c * a + \left(d \cdot b\right))_*}}{\sqrt{c^2 + d^2}^*}\right)}^2 \leadsto {\left(\frac{\sqrt{(c * a + \left(b \cdot d\right))_*}}{\sqrt{c^2 + d^2}^*}\right)}^2\]
   18.5
8. Taylor expanded around 0 to get
   \[{\left(\frac{\sqrt{(c * a + \color{red}{\left(b \cdot d\right)})_*}}{\sqrt{c^2 + d^2}^*}\right)}^2 \leadsto {\left(\frac{\sqrt{(c * a + \color{blue}{\left(b \cdot d\right)})_*}}{\sqrt{c^2 + d^2}^*}\right)}^2\]
   18.5
9. Applied simplify to get
   \[{\left(\frac{\sqrt{(c * a + \left(b \cdot d\right))_*}}{\sqrt{c^2 + d^2}^*}\right)}^2 \leadsto \frac{(c * a + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^* \cdot \sqrt{c^2 + d^2}^*}\]
   6.2

---

10. Applied final simplification
11. Applied simplify to get
    \[\color{red}{\frac{(c * a + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^* \cdot \sqrt{c^2 + d^2}^*}} \leadsto \color{blue}{\frac{(c * a + \left(b \cdot d\right))_*}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}\]
    6.2

if -3.4569357f-25 < c < 1.4664635f-16

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
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
   10.2
4. Applied simplify to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
   6.1
5. Using strategy rm
   6.1
6. Applied add-cbrt-cube to get
   \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}} \leadsto \color{blue}{\sqrt[3]{{\left(\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}\right)}^3}}\]
   18.4
7. Applied taylor to get
   \[\sqrt[3]{{\left(\frac{a \cdot c + b \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}\right)}^3} \leadsto \sqrt[3]{{\left(\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}\right)}^3}\]
   18.4
8. Taylor expanded around 0 to get
   \[\sqrt[3]{{\color{red}{\left(\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}\right)}}^3} \leadsto \sqrt[3]{{\color{blue}{\left(\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}\right)}}^3}\]
   18.4
9. Applied simplify to get
   \[\sqrt[3]{{\left(\frac{c \cdot a}{{\left(\left|d\right|\right)}^2} + \frac{b \cdot d}{{\left(\left|d\right|\right)}^2}\right)}^3} \leadsto (\left(\frac{d}{\left|d\right|}\right) * \left(\frac{b}{\left|d\right|}\right) + \left(\frac{c}{\left|d\right|} \cdot \frac{a}{\left|d\right|}\right))_*\]
   0.0

---

10. Applied final simplification

if 1.4664635f-16 < c < 3.800664f+23

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   10.1
2. Using strategy rm
   10.1
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
   10.1
4. Applied simplify to get
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
   7.7
5. Using strategy rm
   7.7
6. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2} + {\left(\left|d\right|\right)}^2} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2}\right)}^2} + {\left(\left|d\right|\right)}^2}\]
   7.7
7. Applied simplify to get
   \[\frac{a \cdot c + b \cdot d}{{\color{red}{\left(\sqrt{{c}^2}\right)}}^2 + {\left(\left|d\right|\right)}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{\left(\left|c\right|\right)}}^2 + {\left(\left|d\right|\right)}^2}\]
   6.3