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Derivation

1. Split input into 5 regimes

2. if b_2 < -1.0373992183972371e+38

   1. Initial program 55.7

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied flip--55.7

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

   4. Applied simplify28.3

      \[\leadsto \frac{\frac{\color{blue}{c \cdot a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

   5. Applied simplify28.3

      \[\leadsto \frac{\frac{c \cdot a}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]

   6. Taylor expanded around -inf 15.3

      \[\leadsto \frac{\frac{c \cdot a}{\color{blue}{\left(\frac{1}{2} \cdot \frac{c \cdot a}{b_2} - b_2\right)} - b_2}}{a}\]

   7. Applied simplify4.0

      \[\leadsto \color{blue}{\frac{c}{\frac{\frac{1}{2} \cdot c}{\frac{b_2}{a}} - \left(b_2 + b_2\right)}}\]

   if -1.0373992183972371e+38 < b_2 < -7.587065739737278e-20

   1. Initial program 43.2

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied flip--43.2

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

   4. Applied simplify12.9

      \[\leadsto \frac{\frac{\color{blue}{c \cdot a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

   5. Applied simplify12.9

      \[\leadsto \frac{\frac{c \cdot a}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]

   if -7.587065739737278e-20 < b_2 < -2.2869707272504606e-173

   1. Initial program 30.8

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied div-sub30.9

      \[\leadsto \color{blue}{\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}\]

   4. Taylor expanded around -inf 60.3

      \[\leadsto \frac{-b_2}{a} - \frac{\color{blue}{\frac{1}{2} \cdot \frac{c \cdot a}{b_2} - b_2}}{a}\]

   5. Applied simplify37.0

      \[\leadsto \color{blue}{0 - \frac{1 \cdot \frac{1}{2}}{\frac{b_2}{c}}}\]

   if -2.2869707272504606e-173 < b_2 < 3.5962548250302795e+78

   1. Initial program 10.1

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied div-sub10.1

      \[\leadsto \color{blue}{\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}\]

   if 3.5962548250302795e+78 < b_2

   1. Initial program 40.5

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Taylor expanded around inf 3.8

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}}\]
3. Recombined 5 regimes into one program.

4. Applied simplify10.3

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b_2 \le -1.0373992183972371 \cdot 10^{+38}:\\ \;\;\;\;\frac{c}{\frac{c \cdot \frac{1}{2}}{\frac{b_2}{a}} - \left(b_2 + b_2\right)}\\ \mathbf{if}\;b_2 \le -7.587065739737278 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{if}\;b_2 \le -2.2869707272504606 \cdot 10^{-173}:\\ \;\;\;\;\frac{-\frac{1}{2}}{\frac{b_2}{c}}\\ \mathbf{if}\;b_2 \le 3.5962548250302795 \cdot 10^{+78}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018199 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))