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Derivation

1. Initial program 43.9

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

2. Initial simplification43.9

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

4. Applied flip--43.9

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

5. Applied associate-/l/43.9

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

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 flip--0.2

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

12. Applied sqrt-div0.2

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

13. Final simplification0.2

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

Runtime

Time bar (total: 36.5s)Debug logProfile

herbie shell --seed 2018297 
(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)))