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Derivation

1. Initial program 28.9

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

2. Initial simplification28.9

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

4. Applied flip--28.9

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

5. Applied associate-/l/28.9

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

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-/l*0.6

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

10. Applied div-inv0.6

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

11. Applied times-frac0.6

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

12. Simplified0.5

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

13. Simplified0.5

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

15. Applied associate-*l/0.4

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

16. Simplified0.3

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

17. Final simplification0.3

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

Runtime

Time bar (total: 22.4s)Debug logProfile

herbie shell --seed 2018286 
(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)))