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Derivation

1. Initial program 28.6

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

2. Initial simplification28.6

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

4. Applied flip--28.6

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

5. Applied associate-/l/28.6

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

6. Simplified0.5

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

8. Applied associate-/r*0.3

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

10. Applied flip--0.4

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

11. Applied sqrt-div0.4

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

13. Applied pow20.4

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

14. Applied pow20.4

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

15. Applied pow-prod-up0.4

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

16. Simplified0.4

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

17. Final simplification0.4

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

Reproduce

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

Details

Time bar (total: 44.9s)Debug log