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Derivation

1. Initial program 28.6

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

2. Initial simplification28.6

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

4. Applied flip--28.6

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

5. Applied associate-/l/28.6

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

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 times-frac0.6

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

9. Simplified0.5

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

11. Applied pow10.5

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

12. Applied pow10.5

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

13. Applied pow-prod-down0.5

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

14. Simplified0.3

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

16. Applied flip-+0.3

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

17. Final simplification0.3

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

Reproduce

herbie shell --seed 2018365 
(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)))

Details

Time bar (total: 13.2s)Debug log