How to convert -1x10^200 to IEEE 754 double precision

Question

So like above, I want to find the IEEE 754 representation of -1x10^200.

I know we can get the sign be to be 1, as we have a negative number. However I am unsure of how to find the mantissa/exponent. My initial idea was to convert 10^200 to 2^x. However x is not a whole number. So I figure we need to get a fraction somehow by separating the 10^200 somehow. Theoretically one could use very long devision, but I am looking for a more elegant answer that can be done without a high precision calculator.

Answer 1

−10²⁰⁰ cannot be represented in IEEE-754 basic 64-bit binary format. The closest number that can be represented is -99999999999999996973312221251036165947450327545502362648241750950346848435554075534196338404706251868027512415973882408182135734368278484639385041047239877871023591066789981811181813306167128854888448.

The encoding of this in the 64-bit format is 0xe974e718d7d7625a. It has a sign of − (encoded as 1 in bit 63), an exponent of 664 (encoded as 1687 or 0x697 in bits 62 to 52), and a significand of 0x1.4e718d7d7625a (encoded as 0x4e718d7d7625a in bits 51 to 0).

Given that the exponent is 664, you can find the significand by dividing 10²⁰⁰ by 2⁶⁶⁴, writing the result in binary, and rounding after 52 bits after the radix point. Alternately, after dividing by 2⁶⁶⁴, multiply by 2⁵² and round to an integer.