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Derivation

1. Initial program 28.6

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

2. Initial simplification28.6

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

4. Applied flip--28.6

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

5. Applied associate-/l/28.6

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

6. Simplified0.6

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

8. Applied distribute-rgt-neg-out0.6

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

9. Applied distribute-frac-neg0.6

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

10. Simplified0.3

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

12. Applied flip3--0.3

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

14. Applied add-cbrt-cube0.3

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

15. Final simplification0.3

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

Runtime

Time bar (total: 51.0s)Debug logProfile

herbie shell --seed 2018225 
(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)))