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Derivation

1. Initial program 43.9

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

2. Applied simplify43.9

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

4. Applied flip--43.9

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

5. Applied simplify0.5

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

7. Applied distribute-lft-neg-out0.5

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

8. Applied distribute-frac-neg0.5

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

9. Applied distribute-frac-neg0.5

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

10. Applied simplify0.2

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

11. Applied simplify0.2

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

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018206 
(FPCore (a b c d)
  :name "Cubic critical, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))