Error

Derivation

1. Initial program 28.4

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

2. Initial simplification28.4

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

4. Applied flip--28.5

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

5. Applied associate-/l/28.5

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

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 times-frac0.6

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

9. Simplified0.5

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

11. Applied add-sqr-sqrt0.5

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

12. Applied sqrt-prod0.6

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

13. Applied fma-def0.5

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

15. Applied associate-*l*0.5

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

16. Final simplification0.5

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

Runtime

Time bar (total: 20.5s)Debug logProfile

herbie shell --seed 2018295 +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)))