- Split input into 3 regimes
if d < -2.04384729301558e+134
Initial program 42.4
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
Initial simplification42.4
\[\leadsto \frac{b \cdot c - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}\]
- Using strategy
rm Applied add-sqr-sqrt42.4
\[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Applied *-un-lft-identity42.4
\[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Applied times-frac42.4
\[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Simplified42.4
\[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Simplified27.6
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}\]
- Using strategy
rm Applied associate-*l/27.6
\[\leadsto \color{blue}{\frac{1 \cdot \frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
Simplified27.6
\[\leadsto \frac{\color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
- Using strategy
rm Applied *-un-lft-identity27.6
\[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(c \cdot b - a \cdot d\right)}}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\]
Applied associate-/l*27.6
\[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{d^2 + c^2}^*}{c \cdot b - a \cdot d}}}}{\sqrt{d^2 + c^2}^*}\]
Taylor expanded around -inf 14.1
\[\leadsto \frac{\color{blue}{a}}{\sqrt{d^2 + c^2}^*}\]
if -2.04384729301558e+134 < d < 7.641604791054253e+211
Initial program 20.3
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
Initial simplification20.3
\[\leadsto \frac{b \cdot c - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}\]
- Using strategy
rm Applied add-sqr-sqrt20.3
\[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Applied *-un-lft-identity20.3
\[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Applied times-frac20.3
\[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Simplified20.3
\[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Simplified12.4
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}\]
- Using strategy
rm Applied associate-*l/12.3
\[\leadsto \color{blue}{\frac{1 \cdot \frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
Simplified12.3
\[\leadsto \frac{\color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
- Using strategy
rm Applied *-un-lft-identity12.3
\[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(c \cdot b - a \cdot d\right)}}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\]
Applied associate-/l*12.4
\[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{d^2 + c^2}^*}{c \cdot b - a \cdot d}}}}{\sqrt{d^2 + c^2}^*}\]
if 7.641604791054253e+211 < d
Initial program 40.9
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
Initial simplification40.9
\[\leadsto \frac{b \cdot c - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}\]
- Using strategy
rm Applied add-sqr-sqrt40.9
\[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Applied *-un-lft-identity40.9
\[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Applied times-frac40.9
\[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Simplified40.9
\[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Simplified31.7
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}\]
- Using strategy
rm Applied associate-*l/31.7
\[\leadsto \color{blue}{\frac{1 \cdot \frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
Simplified31.7
\[\leadsto \frac{\color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
- Using strategy
rm Applied *-un-lft-identity31.7
\[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(c \cdot b - a \cdot d\right)}}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\]
Applied associate-/l*31.7
\[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{d^2 + c^2}^*}{c \cdot b - a \cdot d}}}}{\sqrt{d^2 + c^2}^*}\]
Taylor expanded around inf 9.5
\[\leadsto \frac{\color{blue}{-1 \cdot a}}{\sqrt{d^2 + c^2}^*}\]
Simplified9.5
\[\leadsto \frac{\color{blue}{-a}}{\sqrt{d^2 + c^2}^*}\]
- Recombined 3 regimes into one program.
Final simplification12.4
\[\leadsto \begin{array}{l}
\mathbf{if}\;d \le -2.04384729301558 \cdot 10^{+134}:\\
\;\;\;\;\frac{a}{\sqrt{d^2 + c^2}^*}\\
\mathbf{elif}\;d \le 7.641604791054253 \cdot 10^{+211}:\\
\;\;\;\;\frac{\frac{1}{\frac{\sqrt{d^2 + c^2}^*}{c \cdot b - a \cdot d}}}{\sqrt{d^2 + c^2}^*}\\
\mathbf{else}:\\
\;\;\;\;\frac{-a}{\sqrt{d^2 + c^2}^*}\\
\end{array}\]
