By piecing the formulas from the following accounts I was able to
arrive at the proper defintions for the formula that I was using
without actuall deriving the formula!


http://www.hsu.edu/faculty/worthf/cubic.html
  http://www.hsu.edu/dept/mth/math.html

x^4 +ax^3 + bx^2 +cx +d =0
 z= x +a/4
z^4  +qz^2 +rz^ +s =0
q= -3/8 a^2 +b
r=  1/8 a^2 -ab/2 +b
s= -3/256 a^4 +b/16 a^2 -ac/4 +d


  (z^2 +kz +l)(x^2 -kx +m) =0
  k^6 +2qk^4 +(q^2-4s)k^2 -r^2 =0
  A= 1
  B= 2q
  C= q^2 -4s
  D= -r^2
  2m= q +k^2 +r/k
  2l= q +k^2 -r/k


http://www.sosmath.com/algebra/factor/fac12/fac12.html

resolvent cubic polynomial

http://www.utm/edu/~jschomme/cardano.htm   jschomme@utm.edu
  reduced:  z^4  +pz^2 +qz^ +r =0     z= x +b/4
        w^3 +2pw^2 +(p^2 -4r)z -q^2 =0 has root alpha^2
      z^2 +alpha*z +(p +alpha^2 -q/alpha)/2 =0   z1, z2
      z^2 -alpha*z +(p +alpha^2 +q/alpha)/2 =0   z3, z4
   The VNR Concise Encyclopedia of Math
   A History of Algebra van der Waerden
   Abstract Algebra Goldstein


http://www.netsrq.com/~hahn/quartic.html

http://forum.swarthmore.edu/dr.math/faq/faq.cubic.equations.html

posted question
 resolvent@juno.com

side or possibley remotely related issues:
approximation algorithms
  http://netlib.amass.ac.cn/a/catalog.html
  http://www.math.ucl.ac.be/~magnus/num1a/approx.html
map projections:
  http://www.cs.albany.edu/~amit/bib/mapproj.txt





q= -3/8 a^2 +b
r=  1/8 a^2 -ab/2 +b
s= -3/256 a^4 +b/16 a^2 -ac/4 +d

/*
x**4 + quart_a*x**3 + quart_b*x**2 + quart_c*x + quart_d = 0

red_d =  quart_a*quart_a
red_b = -3/8 quart_a*quart_a +quart_b
red_c =  1/8 quart_a**3 -quart_a*quart_b/2 + quart_c
red_d = -3/256 quart_a**4 + quart_a**2*quart_b/16 - quart_a*quart_c/4 + quart_d

cube_a = 1
cube_b = 0.5*red_b
cube_c = (red_b*red_b -4*red_d)/16
cube_d = red_c*red_c/64

alpha2 = 4*rl_0
alpha = Math.sqrt(rl_0)

quad_a = 1
quad_b = alpha
quad_c = (red_b +rl_0 -red_c/alpha)
   z1,z2

quad_a = 1
quad_b = -alpha
quad_c = (red_b +rl_0 +red_c/alpha)
   z3,z4

x = z - quart_a/4
*/

/*

red_b = 0.5 * cube_b;
red_c = Math.sqrt(cube_d);
red_d = 0.25(0.25 * cube_b * cube_b - cube_c);






red_c =  1/8 quart_a*quart_a + (1 -quart_a/2)*quart_b
quart_b = (red_c - quart_a*quart_a/8) / (1 - quart_a/2)
quart_b = red_b + 3*quart_a*quart_a/8

(red_b + 3*quart_a*quart_a/8)*(1 - quart_a/2) = (red_c - quart_a*quart_a/8)

   -3/8 quart_a**3  +3/8 quart_a**2  -red_b  quart_a  + red_b
                                        = red_c   - 1/8  quart_a**2

   -3/8 quart_a**3 + 1/2 quart_a**2 - red_b quart_a + red_b-red_c =0
*/