An old example revisited with some FPU assembler. Using the FPU stack is, in theory, more efficient than loading variables from RAM.
This algorithm is quite complicated but it is highly efficient and converges on the answer with very few iterations.
'SUCCESSIVE APPROXIMATION USING NEGATIVE FEEDBACK
'================================================
uses console
'
macro ff(a)
'==========
'
'NULL EXPRESSIONS
'
'0
'1
'a^5-2 'roots of 2
'(a*a*a) + (a*a) -3 'compound roots
'a*a*a-2 'cube root of 2
abs (tan(a/4)-1) 'pi
'abs((1/a)-a+1) 'phi
'tan(rad(a))-2 'spherical angle of icosahedron facet edge
end macro
function approx(float e1,e2) as extended
========================================
dim as double a,a1,a2,b1,b2,fbk
dim as long i
a1=e1 ' first estimate of answer
a2=e2 ' second estimate of answer
b1=ff(a1) ' first result
b2=ff(a2) ' second result
print i tab a2 cr
do
if i>=1000 then exit do 'restrict iterations
if a1=a2 then exit do 'input precsion limit
if b1=b2 then exit do 'feedback precision limit
'
'fbk=(a2-a1)/(b2-b1) 'feedback
'a1=a2 'new a1 input
'a2=a2-b2*fbk 'new a2 input
'b1=b2 'new b1 output
fld qword a2
fld st0
fld st0
fsub qword a1
fld qword b2
fld st0
fld st0
fsub qword b1
fxch st1
fstp qword b1
fdivp 'fbk
fmulp 'qword b2
fsubp 'qword a2
fstp qword a2
fstp qword a1
b2=ff(a2) 'new b2 output
i++
'
print i tab a2 cr
end do
return a2
end function
'r=approx(-2,-1) ' lower est, upper est
r=approx(3.1414,3.1415) ' lower est, upper est
'print str(r)
waitkey
