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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. Simplified44.0

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

4. Applied flip--44.0

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

5. Applied associate-/l/44.0

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

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-/l*0.2

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

10. Applied associate-/l*0.2

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

12. Applied *-un-lft-identity0.2

    \[\leadsto \frac{c \cdot -4}{\frac{2 \cdot a}{\color{blue}{1 \cdot \frac{a}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}}}}\]

13. Applied times-frac0.2

    \[\leadsto \frac{c \cdot -4}{\color{blue}{\frac{2}{1} \cdot \frac{a}{\frac{a}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}}}}\]

14. Simplified0.2

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

15. Simplified0.2

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

16. Final simplification0.2

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

Reproduce

herbie shell --seed 2019005 
(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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