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Derivation

1. Split input into 3 regimes

2. if b_2 < -1.974394570283321e+129

   1. Initial program 53.3

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

   2. Initial simplification53.3

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

   3. Taylor expanded around 0 53.3

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

   4. Taylor expanded around -inf 3.3

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

   if -1.974394570283321e+129 < b_2 < 9.075353573798497e-157

   1. Initial program 10.6

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

   2. Initial simplification10.6

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

   3. Taylor expanded around 0 10.6

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

   if 9.075353573798497e-157 < b_2

   1. Initial program 48.9

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

   2. Initial simplification48.9

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

   3. Taylor expanded around 0 48.9

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

   4. Taylor expanded around inf 13.3

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

4. Final simplification10.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.974394570283321 \cdot 10^{+129}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 9.075353573798497 \cdot 10^{-157}:\\ \;\;\;\;\frac{\sqrt{{b_2}^{2} - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Runtime

Time bar (total: 14.7s)Debug logProfile

herbie shell --seed 2018295 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))