Error

Try it out

Derivation

1. Split input into 4 regimes

2. if b_2 < -3.6860937275891304e+120 or -9.565134490438525e-14 < b_2 < -1.4327714371069e-133

   1. Initial program 51.6

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

   2. Taylor expanded around inf 51.6

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

   3. Taylor expanded around -inf 11.4

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

   if -3.6860937275891304e+120 < b_2 < -9.565134490438525e-14

   1. Initial program 46.3

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied flip--46.4

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

   4. Applied associate-/l/48.1

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

   5. Simplified14.1

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

   if -1.4327714371069e-133 < b_2 < 1.6367105644834092e+113

   1. Initial program 11.5

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

   2. Taylor expanded around inf 11.5

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

   if 1.6367105644834092e+113 < b_2

   1. Initial program 46.8

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity46.8

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

   4. Applied associate-/l*46.8

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

   5. Taylor expanded around 0 2.8

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}}\]
3. Recombined 4 regimes into one program.

4. Final simplification10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.6860937275891304 \cdot 10^{+120}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -9.565134490438525 \cdot 10^{-14}:\\ \;\;\;\;\frac{c \cdot a}{a \cdot \left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}\\ \mathbf{elif}\;b_2 \le -1.4327714371069 \cdot 10^{-133}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.6367105644834092 \cdot 10^{+113}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{{b_2}^{2} - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

Details

Time bar (total: 15.7s)Debug log