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Derivation

1. Initial program 52.3

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

2. Simplified52.3

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

4. Applied flip--52.3

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

5. Applied associate-/l/52.3

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

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 flip3--0.2

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

11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019002 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))

Details

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