Error

Derivation

1. Split input into 4 regimes

2. if b_2 < -6.269787584020147e+84

   1. Initial program 57.9

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

   2. Initial simplification57.9

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

   4. Applied fma-neg57.9

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

   5. Taylor expanded around -inf 2.7

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

   if -6.269787584020147e+84 < b_2 < 2.56767878635075e-309

   1. Initial program 30.2

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

   2. Initial simplification30.2

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

   4. Applied flip--30.3

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

   5. Applied associate-/l/35.0

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

   6. Simplified21.2

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

   8. Applied times-frac8.5

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

   9. Simplified8.5

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

   10. Simplified8.5

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

   if 2.56767878635075e-309 < b_2 < 1.3586174053149808e+85

   1. Initial program 9.6

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

   2. Initial simplification9.6

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

   4. Applied fma-neg9.6

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

   6. Applied *-un-lft-identity9.6

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

   7. Applied associate-/l*9.8

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

   if 1.3586174053149808e+85 < b_2

   1. Initial program 42.0

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

   2. Initial simplification42.0

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

   4. Applied fma-neg42.0

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

   5. Taylor expanded around inf 4.7

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

   6. Simplified4.7

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

4. Final simplification6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -6.269787584020147 \cdot 10^{+84}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.56767878635075 \cdot 10^{-309}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 1.3586174053149808 \cdot 10^{+85}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{(b_2 \cdot b_2 + \left(\left(-c\right) \cdot a\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;(-2 \cdot \left(\frac{b_2}{a}\right) + \left(\frac{c}{\frac{b_2}{\frac{1}{2}}}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 23.9s)Debug logProfile

herbie shell --seed 2018274 +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))