- Split input into 3 regimes
if c < -1.4416508390481902e+156
Initial program 44.0
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
- Using strategy
rm Applied add-sqr-sqrt44.0
\[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
Applied *-un-lft-identity44.0
\[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
Applied times-frac44.0
\[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
Simplified44.0
\[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
Simplified27.9
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
- Using strategy
rm Applied associate-*l/27.8
\[\leadsto \color{blue}{\frac{1 \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}}\]
Simplified27.8
\[\leadsto \frac{\color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
- Using strategy
rm Applied div-inv27.9
\[\leadsto \frac{\color{blue}{\left(c \cdot b - a \cdot d\right) \cdot \frac{1}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
Taylor expanded around -inf 12.9
\[\leadsto \frac{\color{blue}{-1 \cdot b}}{\sqrt{c^2 + d^2}^*}\]
Simplified12.9
\[\leadsto \frac{\color{blue}{-b}}{\sqrt{c^2 + d^2}^*}\]
if -1.4416508390481902e+156 < c < 1.2007049931859238e+159
Initial program 19.8
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
- Using strategy
rm Applied add-sqr-sqrt19.8
\[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
Applied *-un-lft-identity19.8
\[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
Applied times-frac19.8
\[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
Simplified19.8
\[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
Simplified12.8
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
- 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
\[\leadsto \frac{\color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
if 1.2007049931859238e+159 < c
Initial program 44.4
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
- Using strategy
rm Applied add-sqr-sqrt44.4
\[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
Applied *-un-lft-identity44.4
\[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
Applied times-frac44.4
\[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
Simplified44.4
\[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
Simplified27.9
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
- Using strategy
rm Applied associate-*l/27.8
\[\leadsto \color{blue}{\frac{1 \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}}\]
Simplified27.8
\[\leadsto \frac{\color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
- Using strategy
rm Applied div-inv27.8
\[\leadsto \frac{\color{blue}{\left(c \cdot b - a \cdot d\right) \cdot \frac{1}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
Taylor expanded around inf 13.7
\[\leadsto \frac{\color{blue}{b}}{\sqrt{c^2 + d^2}^*}\]
- Recombined 3 regimes into one program.
Final simplification12.8
\[\leadsto \begin{array}{l}
\mathbf{if}\;c \le -1.4416508390481902 \cdot 10^{+156}:\\
\;\;\;\;\frac{-b}{\sqrt{c^2 + d^2}^*}\\
\mathbf{elif}\;c \le 1.2007049931859238 \cdot 10^{+159}:\\
\;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\\
\mathbf{else}:\\
\;\;\;\;\frac{b}{\sqrt{c^2 + d^2}^*}\\
\end{array}\]
