Error

Derivation

1. Split input into 2 regimes

2. if b < 2909.9208110583963

   1. Initial program 18.1

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

   2. Simplified18.1

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

   4. Applied flip3--18.2

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

   5. Applied associate-/l/18.2

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

   6. Simplified17.5

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

   if 2909.9208110583963 < b

   1. Initial program 36.8

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

   2. Simplified36.8

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

   3. Taylor expanded around inf 15.9

      \[\leadsto \frac{\color{blue}{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
   4. Using strategy rm

   5. Applied add-sqr-sqrt15.9

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

   6. Applied add-sqr-sqrt16.0

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

   7. Applied unswap-sqr16.0

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

   8. Applied *-un-lft-identity16.0

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

   9. Applied *-un-lft-identity16.0

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

   10. Applied associate-*l*16.0

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

   11. Applied times-frac16.0

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

   12. Applied associate-*r*16.0

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

   13. Applied times-frac16.0

       \[\leadsto \color{blue}{\frac{\frac{-3}{2} \cdot \frac{1}{1}}{\sqrt{3} \cdot \sqrt{a}} \cdot \frac{\frac{a \cdot c}{b}}{\sqrt{3} \cdot \sqrt{a}}}\]

   14. Simplified16.0

       \[\leadsto \color{blue}{\frac{\frac{\frac{-3}{2}}{\sqrt{3}}}{\sqrt{a}}} \cdot \frac{\frac{a \cdot c}{b}}{\sqrt{3} \cdot \sqrt{a}}\]
   15. Using strategy rm

   16. Applied add-cube-cbrt15.9

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

   17. Applied associate-*l*16.0

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

   18. Applied *-un-lft-identity16.0

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

   19. Applied add-sqr-sqrt15.9

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

   20. Applied associate-*l*15.9

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

   21. Applied times-frac15.9

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

   22. Applied times-frac15.9

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

   23. Simplified15.9

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

   24. Simplified15.9

       \[\leadsto \frac{\frac{\frac{-3}{2}}{\sqrt{3}}}{\sqrt{a}} \cdot \left(\frac{\sqrt{a}}{\sqrt[3]{\sqrt{3}} \cdot \sqrt[3]{\sqrt{3}}} \cdot \color{blue}{\frac{\frac{c}{b}}{\sqrt[3]{\sqrt{3}}}}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification16.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2909.9208110583963:\\ \;\;\;\;\frac{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(3 \cdot a\right) \cdot \left(\left(b \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + b \cdot b\right) + \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{c}{b}}{\sqrt[3]{\sqrt{3}}} \cdot \frac{\sqrt{a}}{\sqrt[3]{\sqrt{3}} \cdot \sqrt[3]{\sqrt{3}}}\right) \cdot \frac{\frac{\frac{-3}{2}}{\sqrt{3}}}{\sqrt{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019072 
(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)))