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Derivation

1. Initial program 44.0

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

2. Initial simplification44.0

   \[\leadsto \frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}\]
3. Using strategy rm

4. Applied flip--44.1

   \[\leadsto \frac{\color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b \cdot b}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}}}{2 \cdot a}\]

5. Applied associate-/l/44.1

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b \cdot b}{\left(2 \cdot a\right) \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}}\]

6. Simplified0.4

   \[\leadsto \frac{\color{blue}{\left(c \cdot -4\right) \cdot a}}{\left(2 \cdot a\right) \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}\]
7. Using strategy rm

8. Applied associate-/r*0.2

   \[\leadsto \color{blue}{\frac{\frac{\left(c \cdot -4\right) \cdot a}{2 \cdot a}}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}}\]

9. Taylor expanded around -inf 0.2

   \[\leadsto \frac{\color{blue}{-2 \cdot c}}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\]
10. Using strategy rm

11. Applied flip3--0.2

    \[\leadsto \frac{-2 \cdot c}{\sqrt{\color{blue}{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(c \cdot a\right) \cdot 4\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(c \cdot a\right) \cdot 4\right) \cdot \left(\left(c \cdot a\right) \cdot 4\right) + \left(b \cdot b\right) \cdot \left(\left(c \cdot a\right) \cdot 4\right)\right)}}} + b}\]

12. Final simplification0.2

    \[\leadsto \frac{-2 \cdot c}{b + \sqrt{\frac{{\left(b \cdot b\right)}^{3} - {\left(4 \cdot \left(c \cdot a\right)\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(4 \cdot \left(c \cdot a\right)\right) \cdot \left(4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b\right) \cdot \left(4 \cdot \left(c \cdot a\right)\right)\right)}}}\]

Reproduce

herbie shell --seed 2018365 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))

Details

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