Error

Derivation

1. Split input into 3 regimes

2. if b < -1.127123378506132e+57

   1. Initial program 38.3

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

   3. Applied add-cube-cbrt38.3

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

   5. Applied add-sqr-sqrt38.3

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

   6. Applied rem-sqrt-square38.3

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

   7. Simplified38.3

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

   8. Taylor expanded around inf 10.7

      \[\leadsto \frac{\left(-b\right) + \left|\color{blue}{b - \frac{3}{2} \cdot \frac{a \cdot c}{b}}\right|}{3 \cdot a}\]

   9. Simplified5.5

      \[\leadsto \frac{\left(-b\right) + \left|\color{blue}{(\left(\frac{c}{b} \cdot a\right) \cdot \frac{-3}{2} + b)_*}\right|}{3 \cdot a}\]

   if -1.127123378506132e+57 < b < 2.0611675014481223

   1. Initial program 17.0

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

   3. Applied add-cube-cbrt17.3

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

   5. Applied add-sqr-sqrt17.3

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

   6. Applied rem-sqrt-square17.3

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

   7. Simplified17.1

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

   if 2.0611675014481223 < b

   1. Initial program 54.8

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

   3. Applied associate-*l*54.8

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

   4. Taylor expanded around inf 16.6

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

4. Final simplification14.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.127123378506132 \cdot 10^{+57}:\\ \;\;\;\;\frac{\left|(\left(\frac{c}{b} \cdot a\right) \cdot \frac{-3}{2} + b)_*\right| + \left(-b\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le 2.0611675014481223:\\ \;\;\;\;\frac{\left(-b\right) + \left|\sqrt{b \cdot b - \left(c \cdot 3\right) \cdot a}\right|}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot c}{b} \cdot \frac{-3}{2}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))

Details

Time bar (total: 16.9s)Debug log