Essays/IEEE Floating-Point Numbers

64-bit floating-point numbers in the IEEE 754 standard have 1 bit for the sign, 11 bits for the exponent with a bias of 1023, and 52 bits for the mantissa. The hex representation and its inverse (3!:3 and 3!:2) facilitate explorations with these numbers.

unhex=: 3!:2
hex  =: 2&(3!:3)
cons =: unhex @ ((}: hex 0.5)&,) " 1

   hex 1.5
e200000000000000
0000000000000008
0000000000000001
0000000000000000
3ff8000000000000
   unhex hex 1.5
1.5
   cons '3ff8000011112222'
1.5
   ":!.18 cons '3ff8000011112222'
1.5000000635792579

`x`	`":!.18 cons x`	Description
`0000000000000001`	`4.9406564584124654e_324`	the smallest positive number
`8000000000000001`	`_4.9406564584124654e_324`	the largest negative number
`0000000000000000`	`0`	0
`3fefffffffffffff`	`0.99999999999999989`	the largest number less than 1
`3ff0000000000000`	`1`	1
`3ff0000000000001`	`1.0000000000000002`	the smallest number greater than 1
`3feffffffffffe00`	`0.99999999999994316`	the smallest number equal to 1 with a tolerance of `2^_44`
`3ff0000000000100`	`1.0000000000000568`	the largest number equal to 1 with a tolerance of `2^_44`
`4005bf0a8b145769`	`2.7182818284590451`	Euler's number
`400921fb54442d18`	`3.1415926535897931`	pi
`4024000000000000`	`10`	10
`4340000000000000`	`9007199254740992`	the smallest positive `x` such that `x` and `x+1` have the same representation (`2^53`)
`baabaabaabaabaaa`	`_4.4698305231135437e_26`	sheepish number
`deadbeefdeadbeef`	`_1.2785010645440762e148`	hamburger
`7fefffffffffffff`	`1.7976931348623157e308`	the largest finite number
`ffefffffffffffff`	`_1.7976931348623157e308`	the smallest finite number
`7ff0000000000000`	`_`	positive infinity
`fff0000000000000`	`__`	negative infinity

Contributed by RogerHui.