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Derivation

1. Initial program 52.2

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

2. Initial simplification52.2

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

4. Applied flip--52.2

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

5. Applied associate-/l/52.2

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

6. Simplified0.5

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

8. Applied associate-*l*0.5

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

9. Taylor expanded around 0 0.5

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

10. Final simplification0.5

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

Runtime

Time bar (total: 20.2s)Debug logProfile

herbie shell --seed 2018295 
(FPCore (a b c d)
  :name "Cubic critical, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))