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Derivation

1. Split input into 4 regimes

2. if b_2 < -1.4031947752424545e+120 or -3.8264402218067264e-70 < b_2 < -2.9566045583213624e-91

   1. Initial program 58.3

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

   3. Applied div-sub59.0

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

   5. Applied *-un-lft-identity59.0

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

   6. Applied associate-/l*59.6

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

   8. Applied associate-/r/59.2

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

   9. Taylor expanded around -inf 3.7

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

   if -1.4031947752424545e+120 < b_2 < -3.8264402218067264e-70

   1. Initial program 42.9

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

   3. Applied flip--43.0

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

   4. Applied associate-/l/45.5

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

   5. Simplified16.0

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

   if -2.9566045583213624e-91 < b_2 < 1.2243609558936807e+127

   1. Initial program 11.8

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

   3. Applied div-sub11.8

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

   if 1.2243609558936807e+127 < b_2

   1. Initial program 52.5

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

   3. Applied div-sub52.5

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

   5. Applied *-un-lft-identity52.5

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

   6. Applied associate-/l*52.5

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

   8. Applied associate-/r/52.5

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

   9. Taylor expanded around 0 2.9

      \[\leadsto \frac{-b_2}{a} - \color{blue}{\frac{b_2}{a}}\]
3. Recombined 4 regimes into one program.

4. Final simplification9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.4031947752424545 \cdot 10^{+120}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.8264402218067264 \cdot 10^{-70}:\\ \;\;\;\;\frac{c \cdot a}{a \cdot \left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}\\ \mathbf{elif}\;b_2 \le -2.9566045583213624 \cdot 10^{-91}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.2243609558936807 \cdot 10^{+127}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 
(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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