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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

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

4. Applied flip--28.8

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

5. Applied associate-/l/28.8

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

6. Simplified0.6

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

8. Applied times-frac0.6

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

9. Simplified0.4

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

11. Applied frac-times0.4

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

12. Final simplification0.4

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

Runtime

Time bar (total: 47.3s)Debug logProfile

herbie shell --seed 2018234 
(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)))