Why does floor lose precision, and how is it affecting transitivity of equality?

Question

I begin by defining a large integer n:

Prelude> let n = 5705979550618670446308578858542675373983
Prelude> n :: Integer
5705979550618670446308578858542675373983

Next I looked at the behavior of s1 and s2:

Prelude> let s1 = (sqrt (fromIntegral n))^2
Prelude> let s2 = (floor(sqrt(fromIntegral n)))^2

Prelude> s1 == fromIntegral n
True
Prelude> s1 == fromIntegral s2
True
Prelude> (fromIntegral n) == (fromIntegral s2)
False

Since any fractional part might be discarded, equality on the last 2 expressions was not expected. However, I didn't expect equality to be intransitive (e.g. n == s1, s1 == s2, but n != s2.)

Furthermore, floor appears to lose precision on the integer part, despite retaining 40 significant digits.

Prelude> s1
5.70597955061867e39

Prelude> s2
5705979550618669899723442048678773129216

This lost precision becomes obvious when testing subtraction:

Prelude> (fromIntegral n) - s1
0.0

Prelude> (fromIntegral n) - (fromIntegral s2)
546585136809863902244767

Why does floor lose precision, and how is this violating transitivity of equality (if at all)?

What is the best approach to computing floor . sqrt without loss of precision?

Answer 1

It’s not floor that is losing precision, but the conversion from Integer (an arbitrary-precision integer) to Double (a floating-point value, which has a limited precision). Accordingly, fromIntegral n :: Double is no longer the same value as n.

Double has a 53-bit mantissa (52 explicitly stored, the leading one implicit), which is approximately equivalent to 16 decimal digits. Thus, only the (approx.) 16 most significant digits of the result are valid. The rest is just noise.

Finally, your first two comparisons compare Doubles; and n converted into Double, s2 converted into Double, and s1 are all equal. In the third comparison, however, n and s2 are both Integers; they can be compared as Integers, so calling fromIntegral on them is a no-op, and their non-converted integer values are different. If you force conversion to Double, the values become equal again:

Prelude> ((fromIntegral n) :: Double) == ((fromIntegral s2) :: Double)
True