Error

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.5081953238138552e+114

   1. Initial program 48.5

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Simplified48.5

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

   3. Taylor expanded around -inf 3.2

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

   if -1.5081953238138552e+114 < b_2 < 1.270925720464309e-123

   1. Initial program 10.9

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Simplified10.9

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
   3. Using strategy rm

   4. Applied clear-num11.1

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\]

   if 1.270925720464309e-123 < b_2

   1. Initial program 51.5

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Simplified51.5

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

   3. Taylor expanded around inf 11.2

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
3. Recombined 3 regimes into one program.

4. Final simplification10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.5081953238138552 \cdot 10^{+114}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.270925720464309 \cdot 10^{-123}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019050 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))