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Derivation

1. Split input into 4 regimes

2. if b_2 < -5.248460788560141e+152

   1. Initial program 60.4

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

   2. Simplified60.4

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

   3. Taylor expanded around -inf 2.5

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

   if -5.248460788560141e+152 < b_2 < -2.720599285254795e-295

   1. Initial program 8.1

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

   2. Simplified8.1

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

   4. Applied div-inv8.2

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

   if -2.720599285254795e-295 < b_2 < 5.591336446458295e+126

   1. Initial program 32.4

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

   2. Simplified32.4

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

   4. Applied div-inv32.5

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

   6. Applied flip--32.6

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

   7. Applied associate-*l/32.6

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

   8. Simplified15.0

      \[\leadsto \frac{\color{blue}{\frac{\left(-a \cdot c\right) + 0}{a}}}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}\]
   9. Using strategy rm

   10. Applied add-sqr-sqrt15.2

       \[\leadsto \frac{\frac{\left(-a \cdot c\right) + 0}{a}}{\color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}}}\]

   11. Applied *-un-lft-identity15.2

       \[\leadsto \frac{\color{blue}{1 \cdot \frac{\left(-a \cdot c\right) + 0}{a}}}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}}\]

   12. Applied times-frac15.3

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}} \cdot \frac{\frac{\left(-a \cdot c\right) + 0}{a}}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}}}\]

   13. Simplified9.0

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

   if 5.591336446458295e+126 < b_2

   1. Initial program 60.1

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

   2. Simplified60.1

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

   4. Applied div-inv60.1

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

   5. Taylor expanded around inf 1.9

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

4. Final simplification6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.248460788560141 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le -2.720599285254795 \cdot 10^{-295}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)\\ \mathbf{elif}\;b_2 \le 5.591336446458295 \cdot 10^{+126}:\\ \;\;\;\;\frac{-1}{\sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}} \cdot \frac{c}{\sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019018 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

Details

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