Error

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.463485767056463e+98

   1. Initial program 44.9

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

   2. Initial simplification44.9

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

   4. Applied div-inv45.0

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

   5. Taylor expanded around inf 45.0

      \[\leadsto \left(\sqrt{\color{blue}{{b_2}^{2} - a \cdot c}} - b_2\right) \cdot \frac{1}{a}\]

   6. Taylor expanded around -inf 4.4

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

   7. Simplified4.4

      \[\leadsto \color{blue}{(-2 \cdot \left(\frac{b_2}{a}\right) + \left(\frac{c}{\frac{b_2}{\frac{1}{2}}}\right))_*}\]

   if -1.463485767056463e+98 < b_2 < 5.571024132141775e-260

   1. Initial program 8.9

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

   2. Initial simplification8.9

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

   4. Applied div-inv9.1

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

   5. Taylor expanded around inf 9.1

      \[\leadsto \left(\sqrt{\color{blue}{{b_2}^{2} - a \cdot c}} - b_2\right) \cdot \frac{1}{a}\]
   6. Using strategy rm

   7. Applied un-div-inv8.9

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

   if 5.571024132141775e-260 < b_2 < 1.6382797079886894e+28

   1. Initial program 30.0

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

   2. Initial simplification30.0

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

   4. Applied div-inv30.0

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

   6. Applied flip--30.1

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

   7. Applied associate-*l/30.2

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

   8. Simplified17.5

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

   if 1.6382797079886894e+28 < b_2

   1. Initial program 56.2

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

   2. Initial simplification56.2

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

   4. Applied div-inv56.2

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

   5. Taylor expanded around inf 4.2

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

4. Final simplification8.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.463485767056463 \cdot 10^{+98}:\\ \;\;\;\;(-2 \cdot \left(\frac{b_2}{a}\right) + \left(\frac{c}{\frac{b_2}{\frac{1}{2}}}\right))_*\\ \mathbf{elif}\;b_2 \le 5.571024132141775 \cdot 10^{-260}:\\ \;\;\;\;\frac{\sqrt{{b_2}^{2} - c \cdot a} - b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.6382797079886894 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{(a \cdot \left(-c\right) + 0)_*}{a}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Runtime

Time bar (total: 32.0s)Debug logProfile

herbie shell --seed 2018252 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))