Error

Try it out

Derivation

1. Initial program 28.3

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

2. Simplified28.3

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

4. Applied flip--28.3

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

5. Applied associate-/l/28.3

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

6. Simplified0.5

   \[\leadsto \frac{\color{blue}{\left(c \cdot -4\right) \cdot a}}{\left(2 \cdot a\right) \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}\]
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 *-un-lft-identity0.3

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

11. Applied times-frac0.3

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

12. Applied times-frac0.4

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

13. Simplified0.4

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

14. Simplified0.4

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

15. Final simplification0.4

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

Reproduce

herbie shell --seed 2019005 
(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: 30.9s)Debug log