This post explains a mechanism for the mouse to stick translation, that allows to reach a better precision and that is required to perform a 1:1 translation.
If you don’t want to read some math, then skip this post 
All values are considered positive so as to simplify the explanation.
Each 10 ms, the sixaxis emulator processes all pending events.
All mouse motion events are merged and as a result, a single mouse event is processed.
This mouse event contains cumulated x and y deltas: (dx1, dy1).
This motion vector is then translated into a stick position according to the following formula:
(1) pos1 = dz1 + mul * pow(mv1, exp)
pos1 = desired stick position (real)
dz1 = deadzone (real, function of dx1, dy1 and the deadzone shape)
mv1 = mouse value (integer), dx1 or dy1
mul = multiplier (real, constant)
exp = exponent (real, constant)
The result pos1 is a real. But the final stick position has to be an integer.
What emuclient was doing previously is rounding pos1 to the nearest integer.
This obviously results into a lower or higher speed than the desired one.
A much more precise solution to tackle this real to integer conversion is to truncate pos1, and compute a mouse vector remainder to be added next iteration.
The mouse event that corresponds to the applied stick position is: (dx2, dy2).
dx2 and dy2 are reals.
(2) pos2 = dz1 + mul * pow(mv2, exp)
pos2 = applied stick position (truncate(pos1), integer)
dz1 = deadzone (same value, the angle is considered to be the same)
mv2 = mouse value (real), dx2 or dy2
mul = multiplier (real, constant)
exp = exponent (real, constant)
Note: norm(mv1) >= norm(mv2)
(3) vrem = mv1 - mv2
vrem = mouse vector remainder
mv1 is known.
mv2 is computed from (2):
(4) mv2 = pow[(pos2 - dz1)/mul, 1/exp]
mv2 can be replaced in (3):
(5) vrem = mv1 - pow[(pos2 - dz1)/mul, 1/exp]
Conclusion:
- (5) gives the mouse vector remainder that has to be added the next iteration
- this emulates a subposition precision
Notes:
- this is useful with 8-bit precision for stick axes as the speed difference between two neighbor stick positions is high
- this is useless with 16-bit precision for stick axes