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

Sign bit

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

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.

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.

Example

Take $31.3125$. $$

• In binary: $11111.0101_2$
• In binary scientific notation: $1.11110101_2 \times 2^4$
• Add $127$ to exponent: $131$
• Convert exponent to binary $10000011$
• Write the final result: $0\;10000011\;11110101000000000000000$

Take $-0.125$. $$

• In binary: $-0.001_2$
• In binary scientific notation: $-1.0_2 \times 2^{-3}$
• Add $127$ to exponent: $124$
• Convert exponent to binary $01111100$
• Write the final result: $1\;01111100\;00000000000000000000000$

Double precision

Uses $64$ bits. $$

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

Sign bit

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

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.

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. $$

Example

Take $31.3125$. $$

• In binary: $11111.0101_2$
• In binary scientific notation: $1.11110101_2 \times 2^4$
• Add $1023$ to exponent: $1027$
• Convert exponent to binary: $10000000011$
• Write the final result: $0\;10000000011\;11110101000000000000000000000000000000000000000000000$

Take $0.125$. $$

• In binary: $-0.001_2$
• In binary scientific notation: $-1.0_2 \times 2^-3$
• Add $1023$ to exponent: $1020$
• Convert exponent to binary: $1111111100$
• Write the final result: $1\;1111111100\;00000000000000000000000000000000000000000000000000000$