Floating-Point Numbers

IEEE 754 standard.

2 types:

• single precision
• double precision

Single Precision

Uses $32$ bits.

• sign bit - $1$ bit
• exponent - $8$ bit
• mantissa - $23$ bit

Single-Precision Sign Bit

$0$ if positive or zero. $1$ if negative.

Single-Precision Exponent

Exponent field range - $[0,255]$. In this range $[1,254]$ is defined for normal numbers. $0$ and $255$ are reserved for subnormal, infinite, signed zeros and NaN.

To support negative exponents, $127$ (half of $254$) is subtracted from this range. $[-126,127]$. This range is the representable range.

Single-Precision Mantissa

In scientific notation, the part that doesn’t contain the base and the power.

In binary scientific notation, there will always be exactly one $1$ bit before the dot. So the inital $1$ is not included in the mantissa.

Double Precision

Uses $64$ bits.

• sign bit - $1$ bit
• exponent - $11$ bit
• mantissa - $52$ bit

Double Precision Sign Bit

$0$ if positive or zero. $1$ if negative.

Double Precision Exponent

Exponent field range - $[0,2047]$. In this range $[1,2046]$ is defined for normal numbers. $0$ and $2047$ are reserved for subnormal, infinite, signed zeros and NaN.

To support negative exponents, $1023$ (half of $2046$) is subtracted from this range. $[-1022,1023]$. This range is the representable range.

Double Precision Mantissa

In scientific notation, the part that doesn’t contain the base and the power.

In binary scientific notation, there will always be exactly one $1$ bit before the dot. So we don’t include that one.