if c < -0.0047717416f0 or 183.04706f0 < c

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   18.4
2. Using strategy rm
   18.4
3. Applied add-sqr-sqrt to get
   \[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
   18.4
4. Using strategy rm
   18.4
5. Applied add-exp-log to get
   \[\frac{a \cdot c + b \cdot d}{{\color{red}{\left(\sqrt{{c}^2 + {d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{\left(e^{\log \left(\sqrt{{c}^2 + {d}^2}\right)}\right)}}^2}\]
   19.0
6. Applied taylor to get
   \[\frac{a \cdot c + b \cdot d}{{\left(e^{\log \left(\sqrt{{c}^2 + {d}^2}\right)}\right)}^2} \leadsto \frac{a}{c} + \frac{b \cdot d}{{c}^2}\]
   4.8
7. Taylor expanded around inf to get
   \[\color{red}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}} \leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^2}}\]
   4.8
8. Applied simplify to get
   \[\frac{a}{c} + \frac{b \cdot d}{{c}^2} \leadsto \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\]
   0.4

---

9. Applied final simplification

if -0.0047717416f0 < c < 183.04706f0

1. Started with
   \[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
   12.3
2. Using strategy rm
   12.3
3. Applied add-cube-cbrt to get
   \[\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}}\right)}^3}\]
   12.4
4. Using strategy rm
   12.4
5. Applied div-inv to get
   \[{\left(\sqrt[3]{\color{red}{\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}}}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{\left(a \cdot c + b \cdot d\right) \cdot \frac{1}{{c}^2 + {d}^2}}}\right)}^3\]
   12.7
6. Applied cbrt-prod to get
   \[{\color{red}{\left(\sqrt[3]{\left(a \cdot c + b \cdot d\right) \cdot \frac{1}{{c}^2 + {d}^2}}\right)}}^3 \leadsto {\color{blue}{\left(\sqrt[3]{a \cdot c + b \cdot d} \cdot \sqrt[3]{\frac{1}{{c}^2 + {d}^2}}\right)}}^3\]
   12.8
7. Applied cube-prod to get
   \[\color{red}{{\left(\sqrt[3]{a \cdot c + b \cdot d} \cdot \sqrt[3]{\frac{1}{{c}^2 + {d}^2}}\right)}^3} \leadsto \color{blue}{{\left(\sqrt[3]{a \cdot c + b \cdot d}\right)}^3 \cdot {\left(\sqrt[3]{\frac{1}{{c}^2 + {d}^2}}\right)}^3}\]
   12.8
8. Applied simplify to get
   \[\color{red}{{\left(\sqrt[3]{a \cdot c + b \cdot d}\right)}^3} \cdot {\left(\sqrt[3]{\frac{1}{{c}^2 + {d}^2}}\right)}^3 \leadsto \color{blue}{\left(c \cdot a + d \cdot b\right)} \cdot {\left(\sqrt[3]{\frac{1}{{c}^2 + {d}^2}}\right)}^3\]
   12.7
9. Applied taylor to get
   \[\left(c \cdot a + d \cdot b\right) \cdot {\left(\sqrt[3]{\frac{1}{{c}^2 + {d}^2}}\right)}^3 \leadsto \frac{c \cdot a}{{d}^2} + \frac{b}{d}\]
   2.9
10. Taylor expanded around inf to get
    \[\color{red}{\frac{c \cdot a}{{d}^2} + \frac{b}{d}} \leadsto \color{blue}{\frac{c \cdot a}{{d}^2} + \frac{b}{d}}\]
    2.9
11. Applied simplify to get
    \[\frac{c \cdot a}{{d}^2} + \frac{b}{d} \leadsto \frac{c}{d} \cdot \frac{a}{d} + \frac{b}{d}\]
    0.8

---

12. Applied final simplification