How can we calculate 1/3 by using fixed-point arithmetic?

Question

I'm interested in fixed-point arithmetic especially division operation. However many sites explain the way to do it only using power-of-two division, namely a bit shift.

What if the denominator is not power-of-two? How can we compute 1/3 in for example unsigned 16-bit arithmetic? Assuming the 4-right bits are fractional part.

1 is 0000 0000 0001 0000

3 is 0000 0000 0011 0000

Answer 1

If the denominator is not a power of two then the result is most likely to be an infinite series. For example:

Since you only have 4 fractional bits, the result will be:

You can calculate this by the following:

• Take a value with only the MSB set: 1000 0000 0000 0000
• Divide by the denominator
• Right shift by the number of integer bits minus 1. This is equivalent to converting the initial value to 1 in the fixed point representation

For the example above:

• [1000 0000 0000 0000] / 3 = 0010 1010 1010 1010
• [0010 1010 1010 1010] >> (12 - 1) = 0000 0000 0000 0101