Error

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.410267969009899e-98

   1. Initial program 52.1

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

   2. Initial simplification52.1

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

   4. Applied div-sub52.6

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

   5. Taylor expanded around -inf 10.0

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

   if -2.410267969009899e-98 < b_2 < 4.8539323749735836e+132

   1. Initial program 11.8

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

   2. Initial simplification11.8

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

   4. Applied div-sub11.8

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

   6. Applied div-inv11.9

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

   7. Applied add-cube-cbrt12.1

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{-b_2}{a}} \cdot \sqrt[3]{\frac{-b_2}{a}}\right) \cdot \sqrt[3]{\frac{-b_2}{a}}} - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \frac{1}{a}\]

   8. Applied prod-diff12.2

      \[\leadsto \color{blue}{(\left(\sqrt[3]{\frac{-b_2}{a}} \cdot \sqrt[3]{\frac{-b_2}{a}}\right) \cdot \left(\sqrt[3]{\frac{-b_2}{a}}\right) + \left(-\frac{1}{a} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right))_* + (\left(-\frac{1}{a}\right) \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right) + \left(\frac{1}{a} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right))_*}\]

   9. Simplified12.0

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

   10. Simplified11.9

       \[\leadsto (\left(\frac{-1}{a}\right) \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right) + \left(\frac{-b_2}{a}\right))_* + \color{blue}{0}\]

   if 4.8539323749735836e+132 < b_2

   1. Initial program 53.0

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

   2. Initial simplification53.0

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

   4. Applied div-sub53.0

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

   5. Taylor expanded around inf 2.2

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

   6. Simplified2.2

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

4. Final simplification9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.410267969009899 \cdot 10^{-98}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 4.8539323749735836 \cdot 10^{+132}:\\ \;\;\;\;(\left(\frac{-1}{a}\right) \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a}\right) + \left(-\frac{b_2}{a}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) - (\left(\frac{c}{b_2}\right) \cdot \frac{-1}{2} + \left(\frac{b_2}{a}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 29.9s)Debug logProfile

herbie shell --seed 2018354 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))