Error

Try it out

Derivation

1. Split input into 4 regimes

2. if b_2 < -0.023046085366440986

   1. Initial program 55.6

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

   2. Taylor expanded around -inf 16.3

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

   3. Applied simplify5.3

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

   if -0.023046085366440986 < b_2 < -2.5319625255424875e-142

   1. Initial program 32.1

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

   3. Applied flip--32.2

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

   4. Applied simplify16.9

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

   5. Applied simplify16.9

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

   7. Applied add-cube-cbrt17.6

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

   8. Applied times-frac13.6

      \[\leadsto \frac{\color{blue}{\frac{c}{\sqrt[3]{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2} \cdot \sqrt[3]{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}} \cdot \frac{a}{\sqrt[3]{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}}}{a}\]

   if -2.5319625255424875e-142 < b_2 < 1.156979912524877e+82

   1. Initial program 11.8

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

   3. Applied clear-num12.0

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

   if 1.156979912524877e+82 < b_2

   1. Initial program 41.3

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

   2. Taylor expanded around inf 4.2

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2020178 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))