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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.3628056667376623 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{if}\;b \le 4.284496431235526 \cdot 10^{-145}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{if}\;b \le 6.619815002390032 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{c \cdot \left(a \cdot 4\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{1}}\\ \end{array}\]

Derivation

Split input into 4 regimes
if b < -1.3628056667376623e+154
1. Initial program 60.9
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Taylor expanded around -inf 2.1
  \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}}\]
3. Applied simplify2.1
  \[\leadsto \color{blue}{\frac{-b}{a}}\]
if -1.3628056667376623e+154 < b < 4.284496431235526e-145
1. Initial program 10.7
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
if 4.284496431235526e-145 < b < 6.619815002390032e+34
1. Initial program 36.2
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Using strategy rm
3. Applied flip-+36.3
  \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]
4. Applied simplify17.0
  \[\leadsto \frac{\frac{\color{blue}{c \cdot \left(a \cdot 4\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
if 6.619815002390032e+34 < b
1. Initial program 55.5
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Taylor expanded around inf 43.1
  \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(b - 2 \cdot \frac{c \cdot a}{b}\right)}}{2 \cdot a}\]
3. Applied simplify4.3
  \[\leadsto \color{blue}{\frac{-c}{\frac{b}{1}}}\]
Recombined 4 regimes into one program.

Runtime

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))

Error

Derivation

`if b < -1.3628056667376623e+154`

`if -1.3628056667376623e+154 < b < 4.284496431235526e-145`

`if 4.284496431235526e-145 < b < 6.619815002390032e+34`

`if 6.619815002390032e+34 < b`

Runtime