Error

Try it out

Derivation

1. Split input into 4 regimes

2. if b_2 < -5.18597177717162e+27

   1. Initial program 35.2

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

   2. Simplified35.2

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

   3. Taylor expanded around -inf 6.4

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

   if -5.18597177717162e+27 < b_2 < -3.4927265156327194e-305

   1. Initial program 10.7

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

   2. Simplified10.7

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

   4. Applied div-inv10.8

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

   if -3.4927265156327194e-305 < b_2 < 2.65285615146505e+68

   1. Initial program 30.3

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

   2. Simplified30.3

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

   4. Applied div-inv30.4

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

   6. Applied flip--30.5

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

   7. Applied associate-*l/30.5

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

   8. Simplified16.0

      \[\leadsto \frac{\color{blue}{\frac{\left(-a \cdot c\right) + 0}{a}}}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}\]

   9. Taylor expanded around 0 9.3

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

   10. Simplified9.3

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

   if 2.65285615146505e+68 < b_2

   1. Initial program 56.9

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

   2. Simplified56.9

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

   4. Applied div-inv56.9

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

   5. Taylor expanded around inf 3.6

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
3. Recombined 4 regimes into one program.

4. Final simplification7.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.18597177717162 \cdot 10^{+27}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le -3.4927265156327194 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)\\ \mathbf{elif}\;b_2 \le 2.65285615146505 \cdot 10^{+68}:\\ \;\;\;\;\frac{-c}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 1137770271 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

Details

Time bar (total: 28.2s)Debug log