Error

Derivation

1. Initial program 43.9

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

2. Initial simplification43.9

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

4. Applied flip--43.9

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

5. Applied associate-/l/43.9

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

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 associate-/l*0.5

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

10. Applied div-inv0.6

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

11. Applied times-frac0.6

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

12. Simplified0.4

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

13. Simplified0.4

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

15. Applied associate-*l/0.3

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

16. Simplified0.2

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

17. Final simplification0.2

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

Runtime

Time bar (total: 28.4s)Debug logProfile

herbie shell --seed 2018353 +o rules:numerics
(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)))