Single-precision floating-point format Range

Question

Sign = 1 bit, Biased Exponent = 8 bits, Mantissa = 23 bits

What is the positive and negative possible range? My teacher told me the following range for ieee 754:

-0.5*2^-128 to -(1-2^-24)*2^127 (for negative floating point numbers)

0.5*2^-128 to (1-2^-24)*2^127 (for positive floating point numbers)

But I don't find this range correct because I am not able to understand how to store 0.5 * 2^-128 into this format. Please explain.

Answer 1

Firstly, the floating-point number format is symmetric for positive and negative numbers. So we will only look at the positive case.

The maximum positive number has the maximum mantissa 1.11111111111111111111111₂ and maximum non-infinite exponent 127. Thus 1.11111111111111111111111₂ × 2¹²⁷ = (2 − 2⁻²³) × 2¹²⁷ ≈ 3.402 × 10³⁸ ≈ 2¹²⁸.

The minimum positive number has the non-zero mantissa 0.00000000000000000000001₂ and the minimum exponent −126 for subnormal/denormalized numbers. Thus 0.00000000000000000000001₂ × 2⁻¹²⁶ = 2⁻²³ × 2⁻¹²⁶ = 2⁻¹⁴⁹ ≈ 1.401 × 10⁻⁴⁵.

Further reading: https://en.wikipedia.org/wiki/Single-precision_floating-point_format