Error

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.9382684235936148e+96

   1. Initial program 59.1

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

   2. Initial simplification59.1

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

   4. Applied div-inv59.1

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

   5. Taylor expanded around 0 59.1

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

   6. Taylor expanded around -inf 2.4

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

   if -1.9382684235936148e+96 < b_2 < 5.455479100180077e-247

   1. Initial program 29.7

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

   2. Initial simplification29.7

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

   4. Applied div-inv29.8

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

   6. Applied flip--29.9

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

   7. Applied associate-*l/29.9

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

   8. Simplified15.7

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

   if 5.455479100180077e-247 < b_2 < 2.3215983058813906e+131

   1. Initial program 8.0

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

   2. Initial simplification8.0

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

   4. Applied div-sub8.0

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

   if 2.3215983058813906e+131 < b_2

   1. Initial program 52.7

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

   2. Initial simplification52.7

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

   4. Applied div-inv52.7

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

   5. Taylor expanded around 0 52.7

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

   6. Taylor expanded around inf 3.4

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

   7. Simplified3.4

      \[\leadsto \color{blue}{(-2 \cdot \left(\frac{b_2}{a}\right) + \left(\frac{c}{\frac{b_2}{\frac{1}{2}}}\right))_*}\]
3. Recombined 4 regimes into one program.

4. Final simplification8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.9382684235936148 \cdot 10^{+96}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.455479100180077 \cdot 10^{-247}:\\ \;\;\;\;\frac{\frac{c \cdot a}{a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} + \left(-b_2\right)}\\ \mathbf{elif}\;b_2 \le 2.3215983058813906 \cdot 10^{+131}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;(-2 \cdot \left(\frac{b_2}{a}\right) + \left(\frac{c}{\frac{b_2}{\frac{1}{2}}}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 55.7s)Debug logProfile

herbie shell --seed 2018234 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))