- Split input into 2 regimes
if d < 9.190386957669971e+115
Initial program 23.2
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
- Using strategy
rm Applied add-sqr-sqrt23.2
\[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
Applied *-un-lft-identity23.2
\[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
Applied times-frac23.2
\[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
Applied simplify23.2
\[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
Applied simplify14.5
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
- Using strategy
rm Applied pow114.5
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{{\left(\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}\right)}^{1}}\]
Applied pow114.5
\[\leadsto \color{blue}{{\left(\frac{1}{\sqrt{c^2 + d^2}^*}\right)}^{1}} \cdot {\left(\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}\right)}^{1}\]
Applied pow-prod-down14.5
\[\leadsto \color{blue}{{\left(\frac{1}{\sqrt{c^2 + d^2}^*} \cdot \frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}\right)}^{1}}\]
Applied simplify14.4
\[\leadsto {\color{blue}{\left(\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\right)}}^{1}\]
if 9.190386957669971e+115 < d
Initial program 38.7
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
- Using strategy
rm Applied add-sqr-sqrt38.7
\[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
Applied *-un-lft-identity38.7
\[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
Applied times-frac38.7
\[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
Applied simplify38.7
\[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
Applied simplify26.4
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
- Using strategy
rm Applied pow126.4
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{{\left(\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}\right)}^{1}}\]
Applied pow126.4
\[\leadsto \color{blue}{{\left(\frac{1}{\sqrt{c^2 + d^2}^*}\right)}^{1}} \cdot {\left(\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}\right)}^{1}\]
Applied pow-prod-down26.4
\[\leadsto \color{blue}{{\left(\frac{1}{\sqrt{c^2 + d^2}^*} \cdot \frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}\right)}^{1}}\]
Applied simplify26.3
\[\leadsto {\color{blue}{\left(\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\right)}}^{1}\]
Taylor expanded around 0 29.4
\[\leadsto {\left(\frac{\frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^*}}{\color{blue}{d}}\right)}^{1}\]
Applied simplify9.6
\[\leadsto \color{blue}{\frac{(c \cdot \left(\frac{b}{d}\right) + \left(-a\right))_*}{\sqrt{c^2 + d^2}^*}}\]
- Recombined 2 regimes into one program.
Applied simplify13.6
\[\leadsto \color{blue}{\begin{array}{l}
\mathbf{if}\;d \le 9.190386957669971 \cdot 10^{+115}:\\
\;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\\
\mathbf{else}:\\
\;\;\;\;\frac{(c \cdot \left(\frac{b}{d}\right) + \left(-a\right))_*}{\sqrt{c^2 + d^2}^*}\\
\end{array}}\]
