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Derivation

1. Split input into 3 regimes

2. if b < -1.2165750385518375e+144

   1. Initial program 57.3

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

   2. Taylor expanded around -inf 11.5

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

   if -1.2165750385518375e+144 < b < 3.0061788106952535e-55

   1. Initial program 13.1

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

   2. Simplified13.1

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

   4. Applied add-cube-cbrt13.4

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

   6. Applied add-cube-cbrt13.5

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

   8. Applied add-sqr-sqrt13.5

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

   9. Applied rem-sqrt-square13.5

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

   10. Simplified13.1

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

   if 3.0061788106952535e-55 < 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 18.3

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

4. Final simplification14.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.2165750385518375 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{3}{2} \cdot \frac{a \cdot c}{b} - 2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \le 3.0061788106952535 \cdot 10^{-55}:\\ \;\;\;\;\frac{\left|\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right| - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{a \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019022 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))

Details

Time bar (total: 15.0s)Debug log