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Derivation

1. Split input into 4 regimes

2. if b_2 < -8.178143920194742e+98

   1. Initial program 45.0

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

   2. Initial simplification45.0

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

   3. Taylor expanded around -inf 45.0

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

   5. Applied div-inv45.1

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

   6. Taylor expanded around -inf 3.9

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

   if -8.178143920194742e+98 < b_2 < 1.553529129791208e-307

   1. Initial program 10.0

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

   2. Initial simplification10.0

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

   3. Taylor expanded around -inf 10.0

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

   5. Applied div-inv10.1

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

   if 1.553529129791208e-307 < b_2 < 1.3090904531855408e+124

   1. Initial program 33.7

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

   2. Initial simplification33.7

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

   3. Taylor expanded around -inf 33.7

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

   5. Applied div-inv33.7

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

   7. Applied flip--33.8

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

   8. Applied associate-*l/33.8

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

   9. Simplified15.3

      \[\leadsto \frac{\color{blue}{\frac{0 - a \cdot c}{a}}}{\sqrt{{b_2}^{2} - a \cdot c} + b_2}\]
   10. Using strategy rm

   11. Applied sub0-neg15.3

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

   12. Applied distribute-frac-neg15.3

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

   13. Simplified8.8

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

   if 1.3090904531855408e+124 < b_2

   1. Initial program 59.8

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

   2. Initial simplification59.8

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

   3. Taylor expanded around -inf 59.8

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

   5. Applied div-inv59.8

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

   6. Taylor expanded around inf 2.1

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

4. Final simplification7.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -8.178143920194742 \cdot 10^{+98}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.553529129791208 \cdot 10^{-307}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\sqrt{{b_2}^{2} - c \cdot a} - b_2\right)\\ \mathbf{elif}\;b_2 \le 1.3090904531855408 \cdot 10^{+124}:\\ \;\;\;\;\frac{-c}{\sqrt{{b_2}^{2} - c \cdot a} + b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018336 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))