if you want to solve non-linear equations in a uC using real math and not
FloPo, my suggestion would be to consider a logarithmic number
system (LNS). either consider all your data in a fixed point binary LNS
(choose radix point to suit your application, 8b and 16b implementations
are practical: possibly an 8b as {sign,3-int,4-frac}), or implement
conversions.

why would you want to do this? to get multiply and divide in a single ADD
instruction, and to get square-root and square in a single-bit right- or
left-shift, negation as a single bit NOT on the sign, and reciprocation as
(n-1)bit NOT on the word. this also means you can often get
reciprocal-roots, negated quotients, etc., in a single instruction in a
single cycle.

all these operations are useful in solving systems, and are usually all
very expensive operations in a uC (wait and see how long a reciprocal
square-root will take...)

sounds too good to be true? the catch is addition and subtraction, for
which you need a lookup table at 8b, and (depending on memory and word
format) possibly a small linear interpolation at 16b.

unsure how applicable this is without more details of your application,
but i think i've given you enough information to evaluate the LNS as a
solution. its also useful for anything with a preponderance of
multiplicative operations (filters et cetera...)

further points of joy - greater dynamic range than floating point, higher
average resolution, constant resolution over full dynamic range, better
round-off noise characteristics c.f. FloPo due to 'representation exact'
results for mul,div,sqr,sqrt, etc. and no microcoded iteration for any
operation.

ok, that is my axe well and truely ground, :)

ed c

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