Error

Derivation

1. Split input into 3 regimes

2. if b < 9.861384713817287e-111

   1. Initial program 20.0

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

   2. Applied simplify20.0

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

   4. Applied div-sub20.0

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

   if 9.861384713817287e-111 < b < 3.1792309455891456e+54

   1. Initial program 39.4

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

   2. Applied simplify39.4

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

   4. Applied associate-/r*39.4

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

   6. Applied clear-num39.4

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

   7. Taylor expanded around 0 26.3

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

   8. Applied simplify25.9

      \[\leadsto \color{blue}{\frac{\frac{c}{-2}}{b}}\]

   if 3.1792309455891456e+54 < b

   1. Initial program 56.5

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

   2. Applied simplify56.6

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

   4. Applied associate-/r*56.6

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

   6. Applied clear-num56.6

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

   7. Taylor expanded around 0 4.7

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

   8. Applied simplify3.8

      \[\leadsto \color{blue}{\frac{\frac{c}{-2}}{b}}\]
3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))