I had to remove the roots from an an old grape vine with a 4" diameter stump. The stump had already been cut down to just above ground level.
It would have been better to leave some limbs sticking out a foot or so to get the cable will not slip off.
So, I dug down around the stump to get down to one horizontal root so I could get a cable under it. This might not be necessary if the top was wide enough to keep the cable from slipping off.
I let water stand in the hole overnight so the soil was really damp.

The trick is to use a farm jack (the same as a jeep Hi-Lift jack).
They are available at Harbor Freight. I haven't seen them at Home Depot or Lowes.

Most attach a chain to the jack, but I used a 7' bike security cable with both ends wrapped around the stump.
If it is not too big you can do it with the jack alone, but it tends to pull to one side, so a tripod with a couple of 2x4's bolted to the top is better.

The jack was rated at 2 1/2 tons for winching and test to 3 1/2 tons and I had to use most of my weight to break the 2 inch taproot a foot down.
Another option would have been to dig a wider hole down a foot so you could get under the stump and saw the tap root off.

A taproot is a large, central, and dominant root from which other roots sprout laterally. Typically a taproot is somewhat straight and very thick, is tapering in shape, and grows directly downward. A carrot is the taproot of a carrot plant. Dandelions have one thick taproot that can extend more than a foot into the ground.
In most plants it is replaced later in the plant's development by a fibrous root system.
Most trees start with a tap root but the main roots spred out horizontally once they are established. Trees such as walnuts, pecans, oaks, elms, pines and firs, maintain a taproot, but most fruit trees, almonds and many shade trees grow a fibrous root system. This is composed of a number of horizontal roots radiating out from the trunk.
Soil characteristics strongly influence the architecture of taproots; for example, deep rich soils favour the development of vertical taproots.

One of the videos below got the roots of a 30' cyprus out by getting the jack under a couple of the large horizontal roots.

If wood is left you may want to treat for termites.
Termites in tree stump: how the stump is attracting termites?
Subterranean and Other Termites Management Guidelines--UC Integrated Pest Management (IPM) says

The primary methods of controlling these termites are insecticides, either applied to the soil adjacent to the structure, directly to nests via shelter tubes, or through bait stations.
Liquid applications of pesticides are most often used for subterranean termite control and applied to the soil either in drenches or by injection. There are no reliable over-the-counter termite control products available for the public in California; all effective products are for professional use only.

You may want to protect your house with a bait system.
Advance termite bait system: instruction and pros and cons
Small shrub with farm jack by itself
Large 30' cyprus stump
Small stump with braced farm jack

How to remove a plant stump - Google Search

last updated 28 July 2019