8.1.2 Representation by hardware floats
A real is represented by a floating number d, that is
d=2(1+m), 0<m<1, −2^{10} < α < 2^{10} 
If α>1−2^{10}, then m ≥ 1/2, and d is
a normalized floating point number, otherwise
d is denormalized (α=1−2^{10}). The special exponent 2^{10}
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 from the 11 bits,
the exponent is α+2^{10}−1,
 the 52 last bits codes the mantissa m, more precisely if
M denotes the integer from the 52 bits, then
m=1/2+M/2^{53} for normalized floats and m=M/2^{53} 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=2^{10} is coded by 111 1111 1111
 α=2^{−10}−1 is coded by 000 0000 000
Remark: 2^{−52}=0.2220446049250313e−15