Error

Derivation

1. Split input into 3 regimes

2. if b < -1.2165750385518375e+144

   1. Initial program 57.3

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

   2. Taylor expanded around 0 57.3

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

   3. Taylor expanded around -inf 11.5

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

   4. Simplified11.5

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

   if -1.2165750385518375e+144 < b < 3.0061788106952535e-55

   1. Initial program 13.1

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

   2. Taylor expanded around 0 13.1

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

   if 3.0061788106952535e-55 < b

   1. Initial program 53.4

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

   2. Taylor expanded around inf 18.3

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

4. Final simplification14.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.2165750385518375 \cdot 10^{+144}:\\ \;\;\;\;\frac{(-2 \cdot b + \left(\left(a \cdot c\right) \cdot \frac{\frac{3}{2}}{b}\right))_*}{3 \cdot a}\\ \mathbf{elif}\;b \le 3.0061788106952535 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} + \left(-b\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))

Details

Time bar (total: 19.4s)Debug log