Error

Derivation

1. Split input into 4 regimes

2. if b < -1.343140870848858e+154

   1. Initial program 60.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

   2. Initial simplification60.8

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

   4. Applied flip--62.4

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

   5. Applied associate-/l/62.4

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

   6. Simplified62.5

      \[\leadsto \frac{\color{blue}{\left(-c\right) \cdot \left(3 \cdot a\right)}}{\left(3 \cdot a\right) \cdot \left(\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b\right)}\]
   7. Using strategy rm

   8. Applied distribute-lft-neg-out62.5

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

   9. Applied distribute-frac-neg62.5

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

   10. Simplified62.4

       \[\leadsto -\color{blue}{\frac{c}{b + \sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}}\]
   11. Using strategy rm

   12. Applied div-inv62.4

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

   13. Taylor expanded around -inf 24.0

       \[\leadsto -c \cdot \color{blue}{\left(\frac{2}{3} \cdot \frac{b}{a \cdot c} - \frac{1}{2} \cdot \frac{1}{b}\right)}\]

   14. Simplified15.9

       \[\leadsto -c \cdot \color{blue}{\left(\frac{\frac{\frac{2}{3}}{c}}{\frac{a}{b}} - \frac{\frac{1}{2}}{b}\right)}\]

   if -1.343140870848858e+154 < b < -4.331754688795972e-296

   1. Initial program 8.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

   2. Initial simplification8.4

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

   4. Applied associate-/r*8.4

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

   if -4.331754688795972e-296 < b < 2.452006489977816e+127

   1. Initial program 33.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

   2. Initial simplification33.9

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

   4. Applied flip--34.0

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

   5. Applied associate-/l/38.0

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

   6. Simplified20.7

      \[\leadsto \frac{\color{blue}{\left(-c\right) \cdot \left(3 \cdot a\right)}}{\left(3 \cdot a\right) \cdot \left(\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b\right)}\]
   7. Using strategy rm

   8. Applied distribute-lft-neg-out20.7

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

   9. Applied distribute-frac-neg20.7

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

   10. Simplified8.8

       \[\leadsto -\color{blue}{\frac{c}{b + \sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}}\]
   11. Using strategy rm

   12. Applied div-inv9.0

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

   if 2.452006489977816e+127 < b

   1. Initial program 60.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

   2. Initial simplification60.7

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

   4. Applied flip--60.7

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

   5. Applied associate-/l/60.8

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

   6. Simplified35.5

      \[\leadsto \frac{\color{blue}{\left(-c\right) \cdot \left(3 \cdot a\right)}}{\left(3 \cdot a\right) \cdot \left(\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b\right)}\]
   7. Using strategy rm

   8. Applied distribute-lft-neg-out35.5

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

   9. Applied distribute-frac-neg35.5

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

   10. Simplified34.6

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

   11. Taylor expanded around 0 2.2

       \[\leadsto -\frac{c}{b + \color{blue}{b}}\]
3. Recombined 4 regimes into one program.

4. Final simplification8.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.343140870848858 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{\frac{\frac{2}{3}}{c}}{\frac{a}{b}} - \frac{\frac{1}{2}}{b}\right) \cdot \left(-c\right)\\ \mathbf{elif}\;b \le -4.331754688795972 \cdot 10^{-296}:\\ \;\;\;\;\frac{\frac{\sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{3}}{a}\\ \mathbf{elif}\;b \le 2.452006489977816 \cdot 10^{+127}:\\ \;\;\;\;\left(-c\right) \cdot \frac{1}{b + \sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b + b}\\ \end{array}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018242 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))