Error

Derivation

1. Initial program 52.8

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

2. Initial simplification52.8

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

4. Applied flip--52.8

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

5. Applied associate-/l/52.8

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

6. Simplified0.5

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

8. Applied associate-/r*0.4

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

10. Applied *-un-lft-identity0.4

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

11. Applied *-un-lft-identity0.4

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

12. Applied times-frac0.4

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

13. Simplified0.4

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

14. Simplified0.1

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

15. Final simplification0.1

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

Runtime

Time bar (total: 31.5s)Debug logProfile

herbie shell --seed 2018297 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical, 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) (* (* 3 a) c)))) (* 3 a)))