Error

Derivation

1. Initial program 44.0

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

3. Applied flip-+44.0

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

4. Applied associate-/l/44.0

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

5. Simplified0.5

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

7. Applied times-frac0.5

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

8. Simplified0.5

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

9. Simplified0.5

   \[\leadsto \frac{1}{a} \cdot \color{blue}{\frac{c \cdot a}{\left(-b\right) - \sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}}\]
10. Using strategy rm

11. Applied pow10.5

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

12. Applied pow10.5

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

13. Applied pow-prod-down0.5

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

14. Simplified0.2

    \[\leadsto {\color{blue}{\left(\frac{c}{\left(-b\right) - \sqrt{(\left(a \cdot -3\right) \cdot c + \left(b \cdot b\right))_*}}\right)}}^{1}\]
15. Using strategy rm

16. Applied add-sqr-sqrt0.3

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

17. Applied neg-mul-10.3

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

18. Applied prod-diff0.3

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

19. Simplified0.2

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

20. Simplified0.2

    \[\leadsto {\left(\frac{c}{\left(\left(-b\right) - \sqrt{(a \cdot \left(-3 \cdot c\right) + \left(b \cdot b\right))_*}\right) + \color{blue}{0}}\right)}^{1}\]

21. Final simplification0.2

    \[\leadsto \frac{c}{\left(-b\right) - \sqrt{(a \cdot \left(-3 \cdot c\right) + \left(b \cdot b\right))_*}}\]

Runtime

Time bar (total: 39.3s)Debug logProfile

herbie shell --seed 2018286 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical, 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) (* (* 3 a) c)))) (* 3 a)))