Error

Derivation

1. Split input into 2 regimes

2. if b < 1.3370483339861573e+141

   1. Initial program 27.2

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

   2. Simplified27.3

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

   4. Applied *-un-lft-identity27.3

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

   5. Applied times-frac27.3

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

   6. Simplified27.3

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

   if 1.3370483339861573e+141 < b

   1. Initial program 61.1

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

   2. Simplified61.1

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

   3. Taylor expanded around 0 39.9

      \[\leadsto \color{blue}{0}\]
3. Recombined 2 regimes into one program.

4. Final simplification29.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1.3370483339861573 \cdot 10^{+141}:\\ \;\;\;\;\frac{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{a} \cdot \frac{1}{3}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019100 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))