Error

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

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

4. Applied flip--28.6

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

5. Applied associate-/l/28.6

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

6. Simplified0.6

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

8. Applied distribute-lft-in0.6

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

10. Applied *-un-lft-identity0.6

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

11. Applied associate-/r*0.6

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

12. Simplified0.5

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

14. Applied *-un-lft-identity0.5

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

15. Applied div-inv0.5

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

16. Applied times-frac0.6

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

17. Simplified0.6

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

18. Simplified0.5

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

19. Final simplification0.5

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

Runtime

Time bar (total: 38.3s)Debug logProfile

herbie shell --seed 2018304 +o rules:numerics
(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)))