if c < -6.783646013079365e+175 or 3.1080072161722114e+139 < c

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   43.4
2. Using strategy rm
   43.4
3. Applied div-sub to get
   \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a \cdot d}{{c}^2 + {d}^2}}\]
   43.4
4. Using strategy rm
   43.4
5. Applied associate-/l* to get
   \[\color{red}{\frac{b \cdot c}{{c}^2 + {d}^2}} - \frac{a \cdot d}{{c}^2 + {d}^2} \leadsto \color{blue}{\frac{b}{\frac{{c}^2 + {d}^2}{c}}} - \frac{a \cdot d}{{c}^2 + {d}^2}\]
   42.6
6. Applied taylor to get
   \[\frac{b}{\frac{{c}^2 + {d}^2}{c}} - \frac{a \cdot d}{{c}^2 + {d}^2} \leadsto \frac{b}{c + \frac{{d}^2}{c}} - \frac{a \cdot d}{{c}^2 + {d}^2}\]
   17.7
7. Taylor expanded around 0 to get
   \[\frac{b}{\color{red}{c + \frac{{d}^2}{c}}} - \frac{a \cdot d}{{c}^2 + {d}^2} \leadsto \frac{b}{\color{blue}{c + \frac{{d}^2}{c}}} - \frac{a \cdot d}{{c}^2 + {d}^2}\]
   17.7
8. Applied simplify to get
   \[\frac{b}{c + \frac{{d}^2}{c}} - \frac{a \cdot d}{{c}^2 + {d}^2} \leadsto \frac{b}{\frac{d}{\frac{c}{d}} + c} - \frac{a \cdot d}{(c * c + \left(d \cdot d\right))_*}\]
   13.3

---

9. Applied final simplification
10. Applied simplify to get
    \[\color{red}{\frac{b}{\frac{d}{\frac{c}{d}} + c} - \frac{a \cdot d}{(c * c + \left(d \cdot d\right))_*}} \leadsto \color{blue}{\frac{b}{(\left(\frac{d}{c}\right) * d + c)_*} - \frac{d \cdot a}{(c * c + \left(d \cdot d\right))_*}}\]
    13.4

if -6.783646013079365e+175 < c < 3.1080072161722114e+139

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   19.5
2. Using strategy rm
   19.5
3. Applied div-sub to get
   \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a \cdot d}{{c}^2 + {d}^2}}\]
   19.5
4. Using strategy rm
   19.5
5. Applied associate-/l* to get
   \[\frac{b \cdot c}{{c}^2 + {d}^2} - \color{red}{\frac{a \cdot d}{{c}^2 + {d}^2}} \leadsto \frac{b \cdot c}{{c}^2 + {d}^2} - \color{blue}{\frac{a}{\frac{{c}^2 + {d}^2}{d}}}\]
   18.0
6. Using strategy rm
   18.0
7. Applied add-sqr-sqrt to get
   \[\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a}{\color{red}{\frac{{c}^2 + {d}^2}{d}}} \leadsto \frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a}{\color{blue}{{\left(\sqrt{\frac{{c}^2 + {d}^2}{d}}\right)}^2}}\]
   39.9
8. Applied taylor to get
   \[\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a}{{\left(\sqrt{\frac{{c}^2 + {d}^2}{d}}\right)}^2} \leadsto \frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a}{{\left(\sqrt{d + \frac{{c}^2}{d}}\right)}^2}\]
   35.0
9. Taylor expanded around 0 to get
   \[\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a}{{\left(\sqrt{\color{red}{d + \frac{{c}^2}{d}}}\right)}^2} \leadsto \frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a}{{\left(\sqrt{\color{blue}{d + \frac{{c}^2}{d}}}\right)}^2}\]
   35.0
10. Applied simplify to get
    \[\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a}{{\left(\sqrt{d + \frac{{c}^2}{d}}\right)}^2} \leadsto \frac{c \cdot b}{(d * d + \left(c \cdot c\right))_*} - \frac{a}{d + \frac{c}{d} \cdot c}\]
    8.1

---

11. Applied final simplification
12. Applied simplify to get
    \[\color{red}{\frac{c \cdot b}{(d * d + \left(c \cdot c\right))_*} - \frac{a}{d + \frac{c}{d} \cdot c}} \leadsto \color{blue}{\frac{b \cdot c}{(d * d + \left({c}^2\right))_*} - \frac{a}{(\left(\frac{c}{d}\right) * c + d)_*}}\]
    8.1