- Split input into 3 regimes
if c < -1.3983465631671893e+162
Initial program 44.6
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
- Using strategy
rm Applied add-sqr-sqrt44.6
\[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
Applied *-un-lft-identity44.6
\[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
Applied times-frac44.6
\[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
Simplified44.6
\[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
Simplified28.6
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
- Using strategy
rm Applied associate-*l/28.5
\[\leadsto \color{blue}{\frac{1 \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}}\]
Simplified28.5
\[\leadsto \frac{\color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
- Using strategy
rm Applied clear-num28.6
\[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{c^2 + d^2}^*}{c \cdot b - a \cdot d}}}}{\sqrt{c^2 + d^2}^*}\]
Taylor expanded around -inf 13.8
\[\leadsto \frac{\color{blue}{-1 \cdot b}}{\sqrt{c^2 + d^2}^*}\]
Simplified13.8
\[\leadsto \frac{\color{blue}{-b}}{\sqrt{c^2 + d^2}^*}\]
if -1.3983465631671893e+162 < c < 3.818555661346783e+183
Initial program 20.3
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
- Using strategy
rm Applied add-sqr-sqrt20.3
\[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
Applied *-un-lft-identity20.3
\[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
Applied times-frac20.3
\[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
Simplified20.3
\[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
Simplified12.7
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
- Using strategy
rm Applied associate-*l/12.6
\[\leadsto \color{blue}{\frac{1 \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}}\]
Simplified12.6
\[\leadsto \frac{\color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
- Using strategy
rm Applied clear-num12.7
\[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{c^2 + d^2}^*}{c \cdot b - a \cdot d}}}}{\sqrt{c^2 + d^2}^*}\]
- Using strategy
rm Applied associate-/r/12.7
\[\leadsto \frac{\color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*} \cdot \left(c \cdot b - a \cdot d\right)}}{\sqrt{c^2 + d^2}^*}\]
if 3.818555661346783e+183 < c
Initial program 42.4
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
- Using strategy
rm Applied add-sqr-sqrt42.4
\[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
Applied *-un-lft-identity42.4
\[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
Applied times-frac42.4
\[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
Simplified42.4
\[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
Simplified28.7
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
- Using strategy
rm Applied associate-*l/28.6
\[\leadsto \color{blue}{\frac{1 \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}}\]
Simplified28.6
\[\leadsto \frac{\color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
- Using strategy
rm Applied clear-num28.6
\[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{c^2 + d^2}^*}{c \cdot b - a \cdot d}}}}{\sqrt{c^2 + d^2}^*}\]
Taylor expanded around inf 10.3
\[\leadsto \frac{\color{blue}{b}}{\sqrt{c^2 + d^2}^*}\]
- Recombined 3 regimes into one program.
Final simplification12.6
\[\leadsto \begin{array}{l}
\mathbf{if}\;c \le -1.3983465631671893 \cdot 10^{+162}:\\
\;\;\;\;\frac{-b}{\sqrt{c^2 + d^2}^*}\\
\mathbf{elif}\;c \le 3.818555661346783 \cdot 10^{+183}:\\
\;\;\;\;\frac{\frac{1}{\sqrt{c^2 + d^2}^*} \cdot \left(b \cdot c - d \cdot a\right)}{\sqrt{c^2 + d^2}^*}\\
\mathbf{else}:\\
\;\;\;\;\frac{b}{\sqrt{c^2 + d^2}^*}\\
\end{array}\]
