- Split input into 3 regimes
if c < -5.107900187833274e+137
Initial program 42.5
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
Initial simplification42.5
\[\leadsto \frac{b \cdot c - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}\]
- Using strategy
rm Applied add-sqr-sqrt42.5
\[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Applied *-un-lft-identity42.5
\[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Applied times-frac42.5
\[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Simplified42.5
\[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Simplified28.2
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}\]
- Using strategy
rm Applied associate-*l/28.1
\[\leadsto \color{blue}{\frac{1 \cdot \frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
Simplified28.1
\[\leadsto \frac{\color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
Taylor expanded around -inf 13.7
\[\leadsto \frac{\color{blue}{-1 \cdot b}}{\sqrt{d^2 + c^2}^*}\]
Simplified13.7
\[\leadsto \frac{\color{blue}{-b}}{\sqrt{d^2 + c^2}^*}\]
if -5.107900187833274e+137 < c < 2.564857354912414e+161
Initial program 19.5
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
Initial simplification19.5
\[\leadsto \frac{b \cdot c - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}\]
- Using strategy
rm Applied add-sqr-sqrt19.5
\[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Applied *-un-lft-identity19.5
\[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Applied times-frac19.5
\[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Simplified19.5
\[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Simplified12.8
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}\]
- Using strategy
rm Applied associate-*r/12.8
\[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{d^2 + c^2}^*} \cdot \left(b \cdot c - a \cdot d\right)}{\sqrt{d^2 + c^2}^*}}\]
- Using strategy
rm Applied sub-neg12.8
\[\leadsto \frac{\frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\left(b \cdot c + \left(-a \cdot d\right)\right)}}{\sqrt{d^2 + c^2}^*}\]
Applied distribute-rgt-in12.8
\[\leadsto \frac{\color{blue}{\left(b \cdot c\right) \cdot \frac{1}{\sqrt{d^2 + c^2}^*} + \left(-a \cdot d\right) \cdot \frac{1}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
Simplified3.7
\[\leadsto \frac{\left(b \cdot c\right) \cdot \frac{1}{\sqrt{d^2 + c^2}^*} + \color{blue}{a \cdot \frac{-d}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
if 2.564857354912414e+161 < c
Initial program 45.4
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
Initial simplification45.4
\[\leadsto \frac{b \cdot c - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}\]
- Using strategy
rm Applied add-sqr-sqrt45.4
\[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Applied *-un-lft-identity45.4
\[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Applied times-frac45.4
\[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Simplified45.4
\[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Simplified29.2
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}\]
- Using strategy
rm Applied associate-*l/29.1
\[\leadsto \color{blue}{\frac{1 \cdot \frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
Simplified29.1
\[\leadsto \frac{\color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
Taylor expanded around inf 12.6
\[\leadsto \frac{\color{blue}{b}}{\sqrt{d^2 + c^2}^*}\]
- Recombined 3 regimes into one program.
Final simplification6.3
\[\leadsto \begin{array}{l}
\mathbf{if}\;c \le -5.107900187833274 \cdot 10^{+137}:\\
\;\;\;\;\frac{-b}{\sqrt{d^2 + c^2}^*}\\
\mathbf{elif}\;c \le 2.564857354912414 \cdot 10^{+161}:\\
\;\;\;\;\frac{a \cdot \frac{-d}{\sqrt{d^2 + c^2}^*} + \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \left(b \cdot c\right)}{\sqrt{d^2 + c^2}^*}\\
\mathbf{else}:\\
\;\;\;\;\frac{b}{\sqrt{d^2 + c^2}^*}\\
\end{array}\]
