- Split input into 2 regimes
if a < -3.033774745784645e+77 or 2.753179196369121e+48 < a
Initial program 33.3
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
- Using strategy
rm Applied add-sqr-sqrt33.3
\[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
Applied *-un-lft-identity33.3
\[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
Applied times-frac33.3
\[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
Simplified33.3
\[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
Simplified26.4
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
- Using strategy
rm Applied associate-*l/26.3
\[\leadsto \color{blue}{\frac{1 \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}}\]
Simplified26.3
\[\leadsto \frac{\color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
- Using strategy
rm Applied div-sub26.3
\[\leadsto \frac{\color{blue}{\frac{c \cdot b}{\sqrt{c^2 + d^2}^*} - \frac{a \cdot d}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
- Using strategy
rm Applied *-un-lft-identity26.3
\[\leadsto \frac{\frac{c \cdot b}{\sqrt{c^2 + d^2}^*} - \frac{a \cdot d}{\color{blue}{1 \cdot \sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
Applied times-frac9.0
\[\leadsto \frac{\frac{c \cdot b}{\sqrt{c^2 + d^2}^*} - \color{blue}{\frac{a}{1} \cdot \frac{d}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
Simplified9.0
\[\leadsto \frac{\frac{c \cdot b}{\sqrt{c^2 + d^2}^*} - \color{blue}{a} \cdot \frac{d}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\]
if -3.033774745784645e+77 < a < 2.753179196369121e+48
Initial program 21.4
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
- Using strategy
rm Applied add-sqr-sqrt21.4
\[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
Applied *-un-lft-identity21.4
\[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
Applied times-frac21.4
\[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
Simplified21.4
\[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
Simplified10.5
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
- Using strategy
rm Applied associate-*l/10.3
\[\leadsto \color{blue}{\frac{1 \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}}\]
Simplified10.3
\[\leadsto \frac{\color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
- Using strategy
rm Applied div-sub10.3
\[\leadsto \frac{\color{blue}{\frac{c \cdot b}{\sqrt{c^2 + d^2}^*} - \frac{a \cdot d}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
- Using strategy
rm Applied associate-/l*2.4
\[\leadsto \frac{\color{blue}{\frac{c}{\frac{\sqrt{c^2 + d^2}^*}{b}}} - \frac{a \cdot d}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\]
- Recombined 2 regimes into one program.
Final simplification4.8
\[\leadsto \begin{array}{l}
\mathbf{if}\;a \le -3.033774745784645 \cdot 10^{+77} \lor \neg \left(a \le 2.753179196369121 \cdot 10^{+48}\right):\\
\;\;\;\;\frac{\frac{c \cdot b}{\sqrt{c^2 + d^2}^*} - \frac{d}{\sqrt{c^2 + d^2}^*} \cdot a}{\sqrt{c^2 + d^2}^*}\\
\mathbf{else}:\\
\;\;\;\;\frac{\frac{c}{\frac{\sqrt{c^2 + d^2}^*}{b}} - \frac{a \cdot d}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\\
\end{array}\]
