- Split input into 3 regimes
if b < -3.724606600942701e-46
Initial program 53.7
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
Simplified53.7
\[\leadsto \color{blue}{\frac{\frac{\left(-b\right) - \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2}}{a}}\]
- Using strategy
rm Applied *-un-lft-identity53.7
\[\leadsto \frac{\frac{\left(-b\right) - \color{blue}{1 \cdot \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}}{2}}{a}\]
Applied add-cube-cbrt57.0
\[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \sqrt[3]{-b}} - 1 \cdot \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2}}{a}\]
Applied prod-diff57.6
\[\leadsto \frac{\frac{\color{blue}{(\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \left(\sqrt[3]{-b}\right) + \left(-\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} \cdot 1\right))_* + (\left(-\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}\right) \cdot 1 + \left(\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} \cdot 1\right))_*}}{2}}{a}\]
Simplified53.8
\[\leadsto \frac{\frac{\color{blue}{\left(\left(-b\right) - \sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}\right)} + (\left(-\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}\right) \cdot 1 + \left(\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} \cdot 1\right))_*}{2}}{a}\]
Simplified53.8
\[\leadsto \frac{\frac{\left(\left(-b\right) - \sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}\right) + \color{blue}{0}}{2}}{a}\]
Taylor expanded around -inf 7.7
\[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
Simplified7.7
\[\leadsto \color{blue}{-\frac{c}{b}}\]
if -3.724606600942701e-46 < b < 1.4374211915152848e+93
Initial program 13.7
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
Simplified13.7
\[\leadsto \color{blue}{\frac{\frac{\left(-b\right) - \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2}}{a}}\]
- Using strategy
rm Applied *-un-lft-identity13.7
\[\leadsto \frac{\frac{\left(-b\right) - \color{blue}{1 \cdot \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}}{2}}{a}\]
Applied add-cube-cbrt13.9
\[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \sqrt[3]{-b}} - 1 \cdot \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2}}{a}\]
Applied prod-diff14.0
\[\leadsto \frac{\frac{\color{blue}{(\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \left(\sqrt[3]{-b}\right) + \left(-\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} \cdot 1\right))_* + (\left(-\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}\right) \cdot 1 + \left(\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} \cdot 1\right))_*}}{2}}{a}\]
Simplified13.8
\[\leadsto \frac{\frac{\color{blue}{\left(\left(-b\right) - \sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}\right)} + (\left(-\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}\right) \cdot 1 + \left(\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} \cdot 1\right))_*}{2}}{a}\]
Simplified13.7
\[\leadsto \frac{\frac{\left(\left(-b\right) - \sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}\right) + \color{blue}{0}}{2}}{a}\]
if 1.4374211915152848e+93 < b
Initial program 44.2
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
Simplified44.2
\[\leadsto \color{blue}{\frac{\frac{\left(-b\right) - \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2}}{a}}\]
- Using strategy
rm Applied *-un-lft-identity44.2
\[\leadsto \frac{\frac{\left(-b\right) - \color{blue}{1 \cdot \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}}{2}}{a}\]
Applied add-cube-cbrt44.4
\[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \sqrt[3]{-b}} - 1 \cdot \sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}}{2}}{a}\]
Applied prod-diff45.4
\[\leadsto \frac{\frac{\color{blue}{(\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \left(\sqrt[3]{-b}\right) + \left(-\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} \cdot 1\right))_* + (\left(-\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}\right) \cdot 1 + \left(\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} \cdot 1\right))_*}}{2}}{a}\]
Simplified45.2
\[\leadsto \frac{\frac{\color{blue}{\left(\left(-b\right) - \sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}\right)} + (\left(-\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*}\right) \cdot 1 + \left(\sqrt{(\left(a \cdot c\right) \cdot -4 + \left(b \cdot b\right))_*} \cdot 1\right))_*}{2}}{a}\]
Simplified44.2
\[\leadsto \frac{\frac{\left(\left(-b\right) - \sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}\right) + \color{blue}{0}}{2}}{a}\]
- Using strategy
rm Applied clear-num44.2
\[\leadsto \color{blue}{\frac{1}{\frac{a}{\frac{\left(\left(-b\right) - \sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}\right) + 0}{2}}}}\]
Taylor expanded around 0 4.2
\[\leadsto \color{blue}{-1 \cdot \frac{b}{a}}\]
Simplified4.2
\[\leadsto \color{blue}{-\frac{b}{a}}\]
- Recombined 3 regimes into one program.
Final simplification10.0
\[\leadsto \begin{array}{l}
\mathbf{if}\;b \le -3.724606600942701 \cdot 10^{-46}:\\
\;\;\;\;-\frac{c}{b}\\
\mathbf{elif}\;b \le 1.4374211915152848 \cdot 10^{+93}:\\
\;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}{2}}{a}\\
\mathbf{else}:\\
\;\;\;\;\frac{-b}{a}\\
\end{array}\]
