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Derivation

1. Initial program 28.5

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

2. Initial simplification28.5

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

4. Applied flip--28.5

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

5. Applied associate-/l/28.4

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

6. Simplified0.6

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

8. Applied associate-/r*0.5

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

10. Applied times-frac0.5

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

11. Final simplification0.5

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

Runtime

Time bar (total: 30.5s)Debug logProfile

herbie shell --seed 2018336 
(FPCore (a b c d)
  :name "Cubic critical, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))