Essays/IEEE Floating-Point Numbers
< Essays
Jump to navigation
Jump to search
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 0000000000000000 0 positive zero 8000000000000000 0 negative zero 0000000000000001 4.9406564584124654e_324 the smallest positive number 8000000000000001 _4.9406564584124654e_324 the largest negative number 000fffffffffffff 2.2250738585072009e_308 the largest subnormal positive number 800fffffffffffff _2.2250738585072009e_308 the smallest subnormal negative number 0010000000000000 2.2250738585072014e_308 the smallest normal positive number 8010000000000000 _2.2250738585072014e_308 the largest normal negative number 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 7ff8000000000000 _. quiet NaN
Contributed by Roger Hui.