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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. Using strategy rm

3. Applied flip-+28.9

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

4. Applied associate-/l/28.9

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

5. Simplified0.6

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

7. Applied associate-*l*0.6

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

8. Final simplification0.6

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

Runtime

Time bar (total: 9.4m)Debug logProfile

herbie shell --seed 2018214 +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)))