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Derivation

1. Initial program 43.7

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

2. Initial simplification43.7

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

4. Applied flip--43.7

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

5. Applied associate-/l/43.7

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

6. Simplified0.5

   \[\leadsto \frac{\color{blue}{\left(a \cdot c\right) \cdot -3}}{\left(3 \cdot a\right) \cdot \left(\sqrt{-3 \cdot \left(c \cdot a\right) + b \cdot b} + b\right)}\]
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 *-un-lft-identity0.5

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

11. Applied div-inv0.5

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

12. Applied times-frac0.6

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

13. Simplified0.6

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

14. Simplified0.4

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

15. Final simplification0.4

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

Runtime

Time bar (total: 49.7s)Debug logProfile

herbie shell --seed 2018285 
(FPCore (a b c d)
  :name "Cubic critical, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))