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Derivation

1. Split input into 4 regimes

2. if b_2 < -7.742439590843584e+150

   1. Initial program 59.9

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

   2. Initial simplification59.9

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

   3. Taylor expanded around -inf 2.2

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

   if -7.742439590843584e+150 < b_2 < -3.6232145061252395e-277

   1. Initial program 7.3

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

   2. Initial simplification7.3

      \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\]
   3. Using strategy rm

   4. Applied sub-neg7.3

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

   if -3.6232145061252395e-277 < b_2 < 1.8008099183440377e+94

   1. Initial program 30.9

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

   2. Initial simplification30.9

      \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\]
   3. Using strategy rm

   4. Applied sub-neg30.9

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

   6. Applied div-inv31.0

      \[\leadsto \color{blue}{\left(\sqrt{b_2 \cdot b_2 + \left(-a \cdot c\right)} - b_2\right) \cdot \frac{1}{a}}\]
   7. Using strategy rm

   8. Applied flip--31.1

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

   9. Applied associate-*l/31.1

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

   10. Simplified15.4

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

   if 1.8008099183440377e+94 < b_2

   1. Initial program 58.2

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

   2. Initial simplification58.2

      \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\]
   3. Using strategy rm

   4. Applied clear-num58.2

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

   5. Taylor expanded around 0 3.5

      \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b_2}{c}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -7.742439590843584 \cdot 10^{+150}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le -3.6232145061252395 \cdot 10^{-277}:\\ \;\;\;\;\frac{\sqrt{\left(-c \cdot a\right) + b_2 \cdot b_2} - b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.8008099183440377 \cdot 10^{+94}:\\ \;\;\;\;\frac{-\frac{c \cdot a}{a}}{\sqrt{\left(-c \cdot a\right) + b_2 \cdot b_2} + b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b_2}{c}}\\ \end{array}\]

Runtime

Time bar (total: 19.0s)Debug logProfile

herbie shell --seed 2018296 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))