I interpreted OP's "bit-cast to integer, strip few least significant bits and then compare for equality" message as suggesting this kind of comparison (Go):

  func equiv(x, y float32, ignoreBits int) bool {
      mask := uint32(0xFFFFFFFF) << ignoreBits
      xi, yi := math.Float32bits(x), math.Float32bits(y)
      return xi&mask == yi&mask
  }

with the sensitivity controlled by ignoreBits, higher values being less sensitive.

Supposing y is 1.0 and x is the predecessor of 1.0, the smallest value of ignoreBits for which equiv would return true is 24.

But a worst case example is found at the very next power of 2, 2.0 (bitwise 0x40000000), whose predecessor is quite different (bitwise 0x3FFFFFFF). In this case, you'd have to set ignoreBits to 31, and thus equivalence here is no better than checking that the two numbers have the same sign.