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Derivation

1. Split input into 4 regimes

2. if b < -1.9917452429919785e+149

   1. Initial program 58.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

   2. Taylor expanded around -inf 2.2

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b} - \frac{2}{3} \cdot \frac{b}{a}}\]

   if -1.9917452429919785e+149 < b < 1.4296093021688485e-116

   1. Initial program 11.3

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

   3. Applied associate-/r*11.3

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}}\]
   4. Using strategy rm

   5. Applied associate-/l/11.3

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{a \cdot 3}}\]

   6. Simplified11.3

      \[\leadsto \frac{\color{blue}{\sqrt{\left(a \cdot -3\right) \cdot c + b \cdot b} - b}}{a \cdot 3}\]

   if 1.4296093021688485e-116 < b < 4.7524414552463597e+86

   1. Initial program 40.4

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

   3. Applied flip-+40.5

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

   4. Applied associate-/l/43.3

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(3 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}\]

   5. Simplified18.6

      \[\leadsto \frac{\color{blue}{3 \cdot \left(c \cdot a\right)}}{\left(3 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}\]

   if 4.7524414552463597e+86 < b

   1. Initial program 58.1

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

   2. Taylor expanded around inf 2.6

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

4. Final simplification9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.9917452429919785 \cdot 10^{+149}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 1.4296093021688485 \cdot 10^{-116}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(-3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{elif}\;b \le 4.7524414552463597 \cdot 10^{+86}:\\ \;\;\;\;\frac{3 \cdot \left(c \cdot a\right)}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(3 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Runtime

Time bar (total: 18.2s)Debug logProfile

herbie shell --seed 2018296 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))