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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -3.4568206269163544 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{c}{b}}{1} - \frac{b}{a}\\ \mathbf{if}\;b \le 9.942544796435686 \cdot 10^{-53}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-2}{2}\\ \end{array}\]

Derivation

Split input into 3 regimes
if b < -3.4568206269163544e+152
1. Initial program 60.4
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Taylor expanded around -inf 10.5
  \[\leadsto \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a}\]
3. Applied simplify2.1
  \[\leadsto \color{blue}{\frac{\frac{c}{b}}{1} - \frac{b}{a}}\]
if -3.4568206269163544e+152 < b < 9.942544796435686e-53
1. Initial program 12.5
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Using strategy rm
3. Applied div-inv12.7
  \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}}\]
if 9.942544796435686e-53 < b
1. Initial program 53.6
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Taylor expanded around inf 18.9
  \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c \cdot a}{b}}}{2 \cdot a}\]
3. Applied simplify8.0
  \[\leadsto \color{blue}{\frac{c}{b} \cdot \frac{-2}{2}}\]
Recombined 3 regimes into one program.

Runtime

herbie shell --seed '#(4187538376 1520029361 2786545844 1568991248 4244367449 2261141537)' 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))

Error

Derivation

`if b < -3.4568206269163544e+152`

`if -3.4568206269163544e+152 < b < 9.942544796435686e-53`

`if 9.942544796435686e-53 < b`

Runtime