Error

Try it out

Derivation

1. Split input into 4 regimes

2. if b < -5.4734139045519595e+113

   1. Initial program 48.5

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

   2. Initial simplification48.5

      \[\leadsto \frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}\]

   3. Taylor expanded around -inf 3.3

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

   if -5.4734139045519595e+113 < b < 1.9379739650628456e-78 or 2.5482541494853623e-62 < b < 2.387647669713383e-51

   1. Initial program 13.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

   2. Initial simplification13.4

      \[\leadsto \frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}\]

   3. Taylor expanded around inf 13.4

      \[\leadsto \frac{\sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}} - b}{2 \cdot a}\]

   if 1.9379739650628456e-78 < b < 2.5482541494853623e-62 or 1.378908863446963e-10 < b

   1. Initial program 54.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

   2. Initial simplification54.7

      \[\leadsto \frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}\]

   3. Taylor expanded around inf 54.7

      \[\leadsto \frac{\sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}} - b}{2 \cdot a}\]

   4. Taylor expanded around inf 6.1

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

   5. Simplified6.1

      \[\leadsto \color{blue}{\frac{-c}{b}}\]

   if 2.387647669713383e-51 < b < 1.378908863446963e-10

   1. Initial program 38.8

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

   2. Initial simplification38.8

      \[\leadsto \frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}\]
   3. Using strategy rm

   4. Applied flip--38.8

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b \cdot b}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}}}{2 \cdot a}\]

   5. Applied associate-/l/43.7

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b \cdot b}{\left(2 \cdot a\right) \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}}\]

   6. Simplified20.7

      \[\leadsto \frac{\color{blue}{\left(c \cdot -4\right) \cdot a}}{\left(2 \cdot a\right) \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.4734139045519595 \cdot 10^{+113}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.9379739650628456 \cdot 10^{-78}:\\ \;\;\;\;\frac{\sqrt{{b}^{2} - \left(c \cdot a\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 2.5482541494853623 \cdot 10^{-62}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.387647669713383 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{{b}^{2} - \left(c \cdot a\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.378908863446963 \cdot 10^{-10}:\\ \;\;\;\;\frac{a \cdot \left(-4 \cdot c\right)}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Runtime

Time bar (total: 30.6s)Debug logProfile

herbie shell --seed 2018354 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))