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Derivation

1. Split input into 3 regimes

2. if b < -2.5438861252780057e+126

   1. Initial program 52.8

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

   2. Taylor expanded around -inf 3.7

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

   if -2.5438861252780057e+126 < b < 1.3220773657163974e-41

   1. Initial program 14.1

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

   3. Applied *-un-lft-identity14.1

      \[\leadsto \frac{\left(-b\right) + \color{blue}{1 \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]

   4. Applied *-un-lft-identity14.1

      \[\leadsto \frac{\color{blue}{1 \cdot \left(-b\right)} + 1 \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

   5. Applied distribute-lft-out14.1

      \[\leadsto \frac{\color{blue}{1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a}\]

   6. Applied times-frac14.2

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{a}}\]

   7. Simplified14.2

      \[\leadsto \color{blue}{\frac{1}{3}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{a}\]

   8. Simplified14.2

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{\sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b} - b}{a}}\]

   if 1.3220773657163974e-41 < b

   1. Initial program 53.4

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

   2. Taylor expanded around inf 8.3

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

4. Final simplification10.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.5438861252780057 \cdot 10^{+126}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 1.3220773657163974 \cdot 10^{-41}:\\ \;\;\;\;\frac{1}{3} \cdot \frac{\sqrt{\left(-3 \cdot a\right) \cdot c + b \cdot b} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))

Details

Time bar (total: 14.7s)Debug log