10.1.2  Representation by hardware floats

A real is represented by a floating number d, that is

If α>1−2¹⁰, then m ≥ 1/2, and d is a normalized floating point number, otherwise d is denormalized (α=1−2¹⁰). The special exponent 2¹⁰ is used to represent plus or minus infinity and NaN (Not a Number). A hardware float is made of 64 bits:

• the first bit is for the sign of d (0 for ’+’ and 1 for ’-’)
• the 11 following bits represents the exponent, more precisely if α denotes the integer given by the 11 bits, the exponent is α+2¹⁰−1,
• the 52 last bits codes the mantissa m, more precisely if M denotes the integer given by the 52 bits, then m=1/2+M/2⁵³ for normalized floats and m=M/2⁵³ for denormalized floats.

Examples of representations of the exponent:

• α=0 is coded by 011 1111 1111
• α=1 is coded by 100 0000 0000
• α=4 is coded by 100 0000 0011
• α=5 is coded by 100 0000 0100
• α=−1 is coded by 011 1111 1110
• α=−4 is coded by 011 1111 1011
• α=−5 is coded by 011 1111 1010
• α=2¹⁰ is coded by 111 1111 1111
• α=2⁻¹⁰−1 is coded by 000 0000 000

Remark: 2⁻⁵²=0.2220446049250313e−15