Error

Derivation

1. Split input into 3 regimes

2. if b < 5.106436903424975e-284

   1. Initial program 22.6

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

   2. Initial simplification22.7

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

   3. Taylor expanded around 0 22.7

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

   4. Simplified22.6

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

   6. Applied div-inv22.7

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

   if 5.106436903424975e-284 < b < 3.911789772911833e+79

   1. Initial program 33.0

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

   2. Initial simplification33.0

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

   3. Taylor expanded around 0 33.0

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

   4. Simplified33.0

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

   6. Applied associate-/r*33.0

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

   8. Applied flip--33.1

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

   9. Applied associate-/l/33.2

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

   10. Simplified17.0

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

   12. Applied associate-/l*14.9

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

   if 3.911789772911833e+79 < b

   1. Initial program 58.0

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

   2. Initial simplification58.0

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

   3. Taylor expanded around 0 58.0

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

   4. Simplified58.0

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

   6. Applied associate-/r*58.0

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

   8. Applied flip--58.1

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

   9. Applied associate-/l/58.1

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

   10. Simplified30.9

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

   11. Taylor expanded around 0 15.0

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

4. Final simplification18.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 5.106436903424975 \cdot 10^{-284}:\\ \;\;\;\;\frac{1}{3 \cdot a} \cdot \left(\sqrt{(\left(a \cdot -3\right) \cdot c + \left(b \cdot b\right))_*} - b\right)\\ \mathbf{elif}\;b \le 3.911789772911833 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{a \cdot -3}{\frac{\left(b + \sqrt{(\left(a \cdot -3\right) \cdot c + \left(b \cdot b\right))_*}\right) \cdot 3}{c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(a \cdot -3\right) \cdot c}{\left(b + b\right) \cdot 3}}{a}\\ \end{array}\]

Runtime

Time bar (total: 25.5s)Debug logProfile

herbie shell --seed 2018354 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))