(2025-06-16) The hunt for an optimal sine
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First, I've got an announcement to make. I'm setting up a Nex station and its
contents will not duplicate or mirror this Gopherhole in general, and
updates are going to be irregular there (no scheduled posting like here).
The station is self-hosted on my Orange Pi Zero (tunneled from this VPS) and
you can access it via nex://hoi.st, nex://piratezeppel.in or
nex://luxferre.top, whatever fits your preference. Foo the Gopher postings,
I'm taking a break until the next month, I think. Don't worry, I'll still be
here. Just a bit tired again.
Second, I've got a promise to keep: tell you about calculating transcendental
functions on an abacus or adding machines in general. To start with
something, let's talk about sine. I've already mentioned one of the formulas
in the previous post and it gets you rather good results. But what if we can
do better?
Of course, we can ALWAYS do better by introducing more digits into
coefficients, but remember that our aim is to be able to calculate sine as
manually as we can, and the abacus has limited capacity too, so ideally we
don't want to handle over 5 _significant_ digits at a time. Let's first
recap what the general form of polynomial sine approximation looks like.
Here's the general algorithm:
x = orig_x * PM{deg|rad}
xx = x * x
sin(orig_x) = x * (A + xx * (B + xx * (C + xx * D)))
And the errors are calculated as:
MAE = |oursin - truesin|
MRE = |MAE / truesin|
Here, PM_{deg|rad} is a pre-multiplier to use when the argument is in degrees
or radians, MAE is maximum absolute error and MRE is maximum relative error.
Now, here are some manually found sets of coefficients for the first quadrant
(the range from 0 to 90 degrees) with their error values:
Now, it looks like we've got an absolute champion, but keep in mind the
coefficients can be further adjusted for this specific range. Let's do this
for some of the "poorest" performing polynomials in an attempt to bring the
MAE to <5E-5 (so that we know that our sine is accurate to 4 decimal places):
PM_deg|PM_rad|coeff. A |coeff. B |coeff. C |coeff. D | MAE | MRE
------|------|---------|----------|---------|---------|--------|--------
1/360 |1/2*pi|6.2813 |-41.0957 |73.5846 |0 |6.67E-5 |3E-4
1/90 |2/pi |1.57065 |-0.64323 |0.0727 |0 |8.63E-5 |1.22E-4
pi/180|1 |1 |-1/6 |1/127 |0 |1.01E-4 |1.43E-4
1/90 |2/pi |1.57 |-0.64 |0.07 |0 |1.17E-4 |5.07E-4
No dice. But what if we try further reducing the range? Remember that we have
some formulas for double-angle and triple-angle sines. Of course they will
require more elementary calculations than just introducing the fourth
coefficient, but at least they were relatively easy to find with a
bruteforce script. So, here we go:
From here, we can deduce that we have two basic approaches that give the best
results:
1) convert the argument into radians and just use a 7-degree Taylor
polynomial with the last coefficient being 1/5125 instead of 1/5040 (use it
for the 45 degrees range and then just extend it with the double-angle
formula or something),
2) pre-divide the argument by 360deg or 2pi rad and use a 5-degree polynomial
with the coefficients 6.283, -41.245 and 71.127, then use the triple-angle
formula to extend the range.
Now, I am absolutely sure there exists a 7-degree polynomial with all four
coefficients having up to just 3 decimal places (making it no more than 5
significant digits to deal with on an abacus) that will give a much more
accurate result for the 30-degree range than this 5-degree polynomial, but
it will take forever to find with just a bruteforce script (considering I
spent almost a day running it to find one with 2 decimals). I will keep you
posted if I manage to find a better approach or a set of coefficients,
however, these two will (theoretically) give you a reliable sine result
accurate to four decimals no matter what.
When I return in July, we'll continue looking at the other transcendentals
like arctangents, logarithms or exponents. For now though, keep checking out
my Nex and stay tuned!
