Error

Derivation

1. Initial program 29.1

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

3. Applied flip-+29.1

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

4. Applied associate-/l/29.1

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

5. Simplified0.6

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

7. Applied times-frac0.6

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

8. Simplified0.5

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

9. Simplified0.5

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

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

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

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

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

13. Applied distribute-lft-out--0.5

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

14. Applied times-frac0.5

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

15. Simplified0.5

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

17. Applied associate-*l/0.4

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

18. Simplified0.5

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

19. Final simplification0.5

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

Runtime

Time bar (total: 44.4s)Debug logProfile

herbie shell --seed 2018339 +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)))