Error

Derivation

1. Split input into 4 regimes

2. if b_2 < -2.2624163142142334e+27 or -2.014594234406352e-62 < b_2 < -1.1320615158913861e-146

   1. Initial program 51.7

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

   2. Taylor expanded around inf 51.7

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

   3. Taylor expanded around -inf 10.6

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

   if -2.2624163142142334e+27 < b_2 < -2.014594234406352e-62

   1. Initial program 39.9

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

   2. Taylor expanded around inf 39.9

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

   4. Applied clear-num39.9

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

   6. Applied div-inv39.9

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

   7. Applied associate-/r*39.9

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

   9. Applied flip--40.0

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

   10. Applied associate-/r/40.0

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

   11. Applied div-inv40.0

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

   12. Applied times-frac42.9

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

   13. Simplified17.8

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

   14. Simplified17.8

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

   if -1.1320615158913861e-146 < b_2 < 2.8358752990862995e+119

   1. Initial program 11.3

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

   2. Taylor expanded around inf 11.3

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

   if 2.8358752990862995e+119 < b_2

   1. Initial program 49.2

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

   2. Taylor expanded around inf 49.2

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

   3. Taylor expanded around inf 2.7

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

   4. Simplified2.7

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

4. Final simplification10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.2624163142142334 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -2.014594234406352 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{1}{a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2} \cdot \left(c \cdot a\right)\\ \mathbf{elif}\;b_2 \le -1.1320615158913861 \cdot 10^{-146}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.8358752990862995 \cdot 10^{+119}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{{b_2}^{2} - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;(-2 \cdot \left(\frac{b_2}{a}\right) + \left(\frac{c}{\frac{b_2}{\frac{1}{2}}}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 44.3s)Debug logProfile

herbie shell --seed 2018352 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))