Error

Derivation

1. Initial program 28.6

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

2. Simplified28.6

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

4. Applied flip--28.6

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

5. Applied associate-/l/28.6

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

6. Simplified0.6

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

8. Applied associate-/r*0.5

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

9. Taylor expanded around -inf 0.3

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

10. Simplified0.3

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

12. Applied add-sqr-sqrt0.3

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

13. Applied sqrt-prod0.4

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

14. Applied fma-def0.3

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

15. Final simplification0.3

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

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical, 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) (* (* 3 a) c)))) (* 3 a)))

Details

Time bar (total: 16.2s)Debug log