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Derivation

1. Initial program 52.5

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

2. Initial simplification52.5

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

4. Applied flip--52.5

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

5. Applied associate-/l/52.5

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

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 times-frac0.4

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

9. Simplified0.4

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

11. Applied pow10.4

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

12. Applied pow10.4

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

13. Applied pow-prod-down0.4

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

14. Simplified0.1

    \[\leadsto {\color{blue}{\left(\frac{c \cdot -2}{\sqrt{\left(c \cdot -4\right) \cdot a + b \cdot b} + b}\right)}}^{1}\]
15. Using strategy rm

16. Applied flip3-+0.1

    \[\leadsto {\left(\frac{c \cdot -2}{\sqrt{\color{blue}{\frac{{\left(\left(c \cdot -4\right) \cdot a\right)}^{3} + {\left(b \cdot b\right)}^{3}}{\left(\left(c \cdot -4\right) \cdot a\right) \cdot \left(\left(c \cdot -4\right) \cdot a\right) + \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(\left(c \cdot -4\right) \cdot a\right) \cdot \left(b \cdot b\right)\right)}}} + b}\right)}^{1}\]

17. Applied sqrt-div0.2

    \[\leadsto {\left(\frac{c \cdot -2}{\color{blue}{\frac{\sqrt{{\left(\left(c \cdot -4\right) \cdot a\right)}^{3} + {\left(b \cdot b\right)}^{3}}}{\sqrt{\left(\left(c \cdot -4\right) \cdot a\right) \cdot \left(\left(c \cdot -4\right) \cdot a\right) + \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(\left(c \cdot -4\right) \cdot a\right) \cdot \left(b \cdot b\right)\right)}}} + b}\right)}^{1}\]

18. Final simplification0.2

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

Runtime

Time bar (total: 21.6s)Debug logProfile

herbie shell --seed 2018339 
(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)))