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Derivation

1. Split input into 4 regimes

2. if b_2 < -7.629739319014833e+43 or -1.9633653677274763e-72 < b_2 < -1.8183968941270222e-110

   1. Initial program 54.3

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

   2. Initial simplification54.3

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

   4. Applied div-inv54.3

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

   5. Taylor expanded around -inf 7.0

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

   if -7.629739319014833e+43 < b_2 < -1.9633653677274763e-72 or -1.8183968941270222e-110 < b_2 < -1.0355638610472734e-187

   1. Initial program 34.7

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

   2. Initial simplification34.7

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

   4. Applied div-inv34.7

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

   6. Applied flip--34.8

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

   7. Applied associate-*l/34.8

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

   8. Simplified15.7

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

   if -1.0355638610472734e-187 < b_2 < 1.8344422394438925e+28

   1. Initial program 11.6

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

   2. Initial simplification11.6

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

   4. Applied div-inv11.7

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

   if 1.8344422394438925e+28 < b_2

   1. Initial program 34.1

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

   2. Initial simplification34.1

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

   4. Applied div-inv34.2

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

   5. Taylor expanded around inf 5.9

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

4. Final simplification9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -7.629739319014833 \cdot 10^{+43}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.9633653677274763 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{c \cdot a}{a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} + \left(-b_2\right)}\\ \mathbf{elif}\;b_2 \le -1.8183968941270222 \cdot 10^{-110}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.0355638610472734 \cdot 10^{-187}:\\ \;\;\;\;\frac{\frac{c \cdot a}{a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} + \left(-b_2\right)}\\ \mathbf{elif}\;b_2 \le 1.8344422394438925 \cdot 10^{+28}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Runtime

Time bar (total: 40.9s)Debug logProfile

herbie shell --seed 2018252 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))