What is the exponent bias used in the IEEE-754 single-precision floating-point…

Concept

IEEE-754 stores the exponent of a floating-point number as a biased (non-negative) value instead of a signed value, so exponent fields can be compared as plain unsigned integers. For a k-bit exponent field, the bias is defined as 2(k-1) - 1, and the stored exponent equals the true exponent plus this bias.

Application

1. Single-precision (32-bit) IEEE-754 layout: 1 sign bit + 8 exponent bits + 23 mantissa bits, so k = 8.

2. Half the exponent range is 2(k-1) = 27 = 128.

3. Bias = 2(k-1) - 1 = 128 - 1 = 127.

4. So a stored exponent field E encodes a true exponent of E - 127 (for example, a stored field of 127 represents a true exponent of 0, i.e. the leading power of two is 20 = 1).

Cross-check

Double-precision (64-bit) IEEE-754 uses an 11-bit exponent field (k = 11), giving bias = 2(11-1) - 1 = 210 - 1 = 1023, the well-known double-precision bias. This confirms the same 2(k-1) - 1 formula applied above for the 8-bit single-precision case. It also matches the reserved-code check: the maximum unbiased 8-bit code, 28 - 1 = 255, is split so that codes 0 and 255 are reserved for special values (zero/subnormals and infinity/NaN), leaving 127 as the exact midpoint bias for the usable exponent range.

So the exponent bias used by IEEE-754 single precision is 127.