Derivation

Split input into 3 regimes
if b < -7.641765040703004e+148
1. Initial program 59.0
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Taylor expanded around -inf 2.0
  \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]
if -7.641765040703004e+148 < b < 2.34789448631425e-125
1. Initial program 11.0
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Taylor expanded around 0 11.0
  \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
if 2.34789448631425e-125 < b
1. Initial program 50.4
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Taylor expanded around 0 50.4
  \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
3. Taylor expanded around inf 11.8
  \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
4. Simplified11.8
  \[\leadsto \color{blue}{\frac{-c}{b}}\]
Recombined 3 regimes into one program.
Final simplification10.3
\[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.641765040703004 \cdot 10^{+148}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 2.34789448631425 \cdot 10^{-125}:\\ \;\;\;\;\frac{\sqrt{{b}^{2} - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Runtime

herbie shell --seed 2018349 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))

In
Out