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Derivation

1. Initial program 43.9

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

3. Applied flip-+43.9

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

4. Applied associate-/l/43.9

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

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 associate-*r*0.5

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

9. Applied times-frac0.5

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

10. Simplified0.4

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

12. Applied pow10.4

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

13. Applied pow10.4

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

14. Applied pow-prod-down0.4

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

15. Simplified0.2

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

16. Final simplification0.2

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

Reproduce

herbie shell --seed 2019002 
(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)))

Details

Time bar (total: 24.8s)Debug log