Error

Try it out

Derivation

1. Initial program 28.8

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

2. Initial simplification28.8

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

4. Applied flip--28.8

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

5. Applied associate-/l/28.8

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

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 *-un-lft-identity0.6

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

11. Applied times-frac0.6

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

12. Simplified0.6

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

13. Simplified0.5

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

15. Applied div-inv0.5

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

16. Applied associate-*r*0.4

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

17. Final simplification0.4

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

Runtime

Time bar (total: 37.5s)Debug logProfile

herbie shell --seed 2018346 
(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)))