Basic Specular Shading

I am playing around with a simple code that generates a shaded sphere. I don't yet fully understand the math but I am figuring it out as I play with the code. I was wondering how one might implement specular shading into this based on this code. Any suggestions?

for (y=0;y<screenHeight;y++)
for (x=0;x<screenWidth;x++)
if (sqr((x-xcenter)*(x-xcenter)+(y-ycenter)*(y-ycenter))<radius)
 {
 vx=(x-xcenter);
 vy=(y-xcenter);
 vz=sqr(radius*radius-vx*vx-vy*vy);
 vl=sqr(vx*vx+vy*vy+vz*vz);
 co_angle=(lx*vx+ly*vy+lz*vz)/(ll*vl);
 pixel=co_angle*255;
 }

I was looking at this thread and the second image is what I am after. But I also don't fully understand the math there either: Trouble with Phong Shading

Thanks in advance.

To add reflection first you need mirror light vector by surface normal

all these are unit vectors:
l - (yellow) to light source
n - (aqua) surface normal
r - (green) reflected light direction
e - (orange) to eye/camera direction
points:
p - (red) - rendered pixel position
q - (magenta) - reflection midpoint
axises:
brown - reflection midpoint
yellow - to light
gray - to eye/camera

So how to do it?

n,p,e,l are knowns or can be easily computed
if light is directional then l is constant for all pixels
if light source is a point then l=light_pos-p; l/=|l|;
e=eye_pos-p; e/=|e|;
now you need to find point q I use dot product for this q=p+(n*dot(l,n));
now the r vector is easy r=(p+l)+2*(q-(p+l))-p=2*(q-p)-l;
hope I did not make a silly math mistake/typo somewhere but it should be clear how I obtain the equations

now you have reflection vector r

so add shading by multiplying pixel color by m=light_color*dot(r,e)+ambient_light;
to add specular spots you need add to m the maximal reflections which are present only near r
so ca=cos(ang)=dot(r,e); you do not need ang directly cosine is fine
now to limit the specular cone size do ca=pow(ca,5.0) can use any exponent
this will lower the values below one so it is much much less then raw cosine
the bigger the exponent the smaller spot cone size so now:
m=(light_color*0.5*(ca+dot(r,e)))+ambient_light;
you can also add mixing coefficients to change the ratio between specular and normal reflection light strength
now render pixel p with color=surface_color*m;