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Derivation

1. Split input into 3 regimes

2. if b_2 < -2.8328889513248502e-121

   1. Initial program 50.8

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

   2. Initial simplification50.8

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

   3. Taylor expanded around -inf 11.6

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

   if -2.8328889513248502e-121 < b_2 < 3.0964691378674393e+108

   1. Initial program 11.9

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

   2. Initial simplification11.9

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

   3. Taylor expanded around 0 11.9

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

   5. Applied *-un-lft-identity11.9

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

   6. Applied *-un-lft-identity11.9

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

   7. Applied distribute-lft-out--11.9

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

   8. Applied associate-/l*12.0

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

   if 3.0964691378674393e+108 < b_2

   1. Initial program 46.3

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

   2. Initial simplification46.3

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

   3. Taylor expanded around inf 3.5

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

4. Final simplification10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.8328889513248502 \cdot 10^{-121}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.0964691378674393 \cdot 10^{+108}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{{b_2}^{2} - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2018365 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

Details

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