What is the largest and smallest Hex Floating Point Literal value that can be represented in IEEE 754-2008 64-bit binary (radix 2) format?

Question

I know that in IEEE 754-2008 64-bit binary (radix 2) format the largest Decimal Floating Point Number that can be represented is 1.7976931348623157E+308 and the smallest is 4.94065645841246544E-325. So the values that results from converting this to hex is not the largest and smallest Hex Floating Point value?

Answer 1

The largest finite value has an exponent of 11111111110 (because the largest one is reserved for INFs and NaNs) which means , representing 2^{2046 - 1023} = 2⁺¹⁰²³ , and a significand of all ones:

0 11111111110 11111111111111111111111111111111111111111111111111112

which is 0x7FEFFFFFFFFFFFFF in raw hex form and 0x1.fffffffffffffp+1023. You can check this with float.exposed or floating-point-converter

It can also be calculated like this: the significand of binary64 contains 52 explicit bits and a hidden 1 bit, so the largest significand is 1.fffffffffffff. There are 11 exponent bits, allowing for the largest exponent of 2¹¹ - 1 - 1023 - 1 = 1023. That means the largest value is 0x1.fffffffffffffp+1023

Similarly the smallest non-zero normalized number has a exponent pattern of 00000000001, representing 2^{1 - 1023} = 2^-1022 and significand = 0, so

0 00000000001 00000000000000000000000000000000000000000000000000002
= 0x0000000000000001
= 0x1p-1022

The smallest non-zero subnormal number has exponent of all zeros and significand = 1:

0 00000000000 00000000000000000000000000000000000000000000000000012
= 0x0010000000000000
= 0x1p-1074