Error

Derivation

1. Split input into 4 regimes

2. if b_2 < -9.474748217008791e+27

   1. Initial program 55.6

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

   2. Initial simplification55.6

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

   4. Applied fma-neg55.6

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

   5. Taylor expanded around -inf 5.4

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

   if -9.474748217008791e+27 < b_2 < -8.936387808061156e-86

   1. Initial program 40.4

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

   2. Initial simplification40.4

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

   4. Applied flip--40.5

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

   5. Applied associate-/l/42.9

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

   6. Simplified20.4

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

   if -8.936387808061156e-86 < b_2 < 5.044669390308597e+80

   1. Initial program 12.5

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

   2. Initial simplification12.5

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

   4. Applied fma-neg12.5

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

   if 5.044669390308597e+80 < b_2

   1. Initial program 40.9

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

   2. Initial simplification40.9

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

   4. Applied fma-neg40.9

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

   5. Taylor expanded around inf 4.2

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

   6. Simplified4.2

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

4. Final simplification9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -9.474748217008791 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -8.936387808061156 \cdot 10^{-86}:\\ \;\;\;\;\frac{c \cdot a}{a \cdot \left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}\\ \mathbf{elif}\;b_2 \le 5.044669390308597 \cdot 10^{+80}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{(b_2 \cdot b_2 + \left(c \cdot \left(-a\right)\right))_*}}{a}\\ \mathbf{else}:\\ \;\;\;\;(-2 \cdot \left(\frac{b_2}{a}\right) + \left(\frac{c}{\frac{b_2}{\frac{1}{2}}}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 28.6s)Debug logProfile

herbie shell --seed 2018339 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))