if c < -1.2679504393766895e+78 or 1.2361527926053716e+158 < c

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   38.7
2. Using strategy rm
   38.7
3. Applied div-sub to get
   \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a \cdot d}{{c}^2 + {d}^2}}\]
   38.7
4. Using strategy rm
   38.7
5. Applied associate-/l* to get
   \[\frac{b \cdot c}{{c}^2 + {d}^2} - \color{red}{\frac{a \cdot d}{{c}^2 + {d}^2}} \leadsto \frac{b \cdot c}{{c}^2 + {d}^2} - \color{blue}{\frac{a}{\frac{{c}^2 + {d}^2}{d}}}\]
   34.8
6. Using strategy rm
   34.8
7. Applied add-sqr-sqrt to get
   \[\frac{b \cdot c}{\color{red}{{c}^2 + {d}^2}} - \frac{a}{\frac{{c}^2 + {d}^2}{d}} \leadsto \frac{b \cdot c}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}} - \frac{a}{\frac{{c}^2 + {d}^2}{d}}\]
   34.8
8. Applied add-sqr-sqrt to get
   \[\frac{\color{red}{b \cdot c}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2} - \frac{a}{\frac{{c}^2 + {d}^2}{d}} \leadsto \frac{\color{blue}{{\left(\sqrt{b \cdot c}\right)}^2}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2} - \frac{a}{\frac{{c}^2 + {d}^2}{d}}\]
   49.4
9. Applied square-undiv to get
   \[\color{red}{\frac{{\left(\sqrt{b \cdot c}\right)}^2}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}} - \frac{a}{\frac{{c}^2 + {d}^2}{d}} \leadsto \color{blue}{{\left(\frac{\sqrt{b \cdot c}}{\sqrt{{c}^2 + {d}^2}}\right)}^2} - \frac{a}{\frac{{c}^2 + {d}^2}{d}}\]
   49.4
10. Applied simplify to get
    \[{\color{red}{\left(\frac{\sqrt{b \cdot c}}{\sqrt{{c}^2 + {d}^2}}\right)}}^2 - \frac{a}{\frac{{c}^2 + {d}^2}{d}} \leadsto {\color{blue}{\left(\frac{\sqrt{c \cdot b}}{\sqrt{c^2 + d^2}^*}\right)}}^2 - \frac{a}{\frac{{c}^2 + {d}^2}{d}}\]
    43.5
11. Applied taylor to get
    \[{\left(\frac{\sqrt{c \cdot b}}{\sqrt{c^2 + d^2}^*}\right)}^2 - \frac{a}{\frac{{c}^2 + {d}^2}{d}} \leadsto {\left(\frac{\sqrt{c \cdot b}}{\sqrt{c^2 + d^2}^*}\right)}^2 - 0\]
    42.7
12. Taylor expanded around 0 to get
    \[{\left(\frac{\sqrt{c \cdot b}}{\sqrt{c^2 + d^2}^*}\right)}^2 - \color{red}{0} \leadsto {\left(\frac{\sqrt{c \cdot b}}{\sqrt{c^2 + d^2}^*}\right)}^2 - \color{blue}{0}\]
    42.7
13. Applied simplify to get
    \[{\left(\frac{\sqrt{c \cdot b}}{\sqrt{c^2 + d^2}^*}\right)}^2 - 0 \leadsto \frac{c}{\sqrt{c^2 + d^2}^*} \cdot \frac{b}{\sqrt{c^2 + d^2}^*} - 0\]
    0.0

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14. Applied final simplification

if -1.2679504393766895e+78 < c < 1.2361527926053716e+158

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   19.1
2. Using strategy rm
   19.1
3. Applied div-sub to get
   \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a \cdot d}{{c}^2 + {d}^2}}\]
   19.1
4. Using strategy rm
   19.1
5. Applied associate-/l* to get
   \[\frac{b \cdot c}{{c}^2 + {d}^2} - \color{red}{\frac{a \cdot d}{{c}^2 + {d}^2}} \leadsto \frac{b \cdot c}{{c}^2 + {d}^2} - \color{blue}{\frac{a}{\frac{{c}^2 + {d}^2}{d}}}\]
   17.3
6. Using strategy rm
   17.3
7. Applied *-un-lft-identity to get
   \[\frac{b \cdot c}{\color{red}{{c}^2 + {d}^2}} - \frac{a}{\frac{{c}^2 + {d}^2}{d}} \leadsto \frac{b \cdot c}{\color{blue}{1 \cdot \left({c}^2 + {d}^2\right)}} - \frac{a}{\frac{{c}^2 + {d}^2}{d}}\]
   17.3
8. Applied times-frac to get
   \[\color{red}{\frac{b \cdot c}{1 \cdot \left({c}^2 + {d}^2\right)}} - \frac{a}{\frac{{c}^2 + {d}^2}{d}} \leadsto \color{blue}{\frac{b}{1} \cdot \frac{c}{{c}^2 + {d}^2}} - \frac{a}{\frac{{c}^2 + {d}^2}{d}}\]
   16.2
9. Applied taylor to get
   \[\frac{b}{1} \cdot \frac{c}{{c}^2 + {d}^2} - \frac{a}{\frac{{c}^2 + {d}^2}{d}} \leadsto \frac{b}{1} \cdot \frac{c}{{c}^2 + {d}^2} - \frac{a}{d + \frac{{c}^2}{d}}\]
   5.6
10. Taylor expanded around 0 to get
    \[\frac{b}{1} \cdot \frac{c}{{c}^2 + {d}^2} - \frac{a}{\color{red}{d + \frac{{c}^2}{d}}} \leadsto \frac{b}{1} \cdot \frac{c}{{c}^2 + {d}^2} - \frac{a}{\color{blue}{d + \frac{{c}^2}{d}}}\]
    5.6
11. Applied simplify to get
    \[\frac{b}{1} \cdot \frac{c}{{c}^2 + {d}^2} - \frac{a}{d + \frac{{c}^2}{d}} \leadsto \frac{\frac{b \cdot c}{1}}{(d * d + \left(c \cdot c\right))_*} - \frac{a}{\frac{c}{\frac{d}{c}} + d}\]
    6.8

---

12. Applied final simplification
13. Applied simplify to get
    \[\color{red}{\frac{\frac{b \cdot c}{1}}{(d * d + \left(c \cdot c\right))_*} - \frac{a}{\frac{c}{\frac{d}{c}} + d}} \leadsto \color{blue}{\frac{c \cdot b}{(d * d + \left(c \cdot c\right))_*} - \frac{a}{(\left(\frac{c}{d}\right) * c + d)_*}}\]
    6.8