denormals

Denormals, also called subnormal numbers, are a class of floating-point numbers that occur in IEEE 754 formats when the exponent field is zero but the significand is nonzero. They lie between zero and the smallest normal positive value and enable gradual underflow, preserving information about very small magnitudes that would otherwise disappear when underflow occurs.

In a binary floating-point format, normal numbers have an implicit leading 1 in the significand, while subnormals do not. For subnormals, the exponent is set to zero and the value is given by (-1)^sign × (0.fraction) × 2^(1 − bias). In IEEE 754 single precision (bias 127, 23 fraction bits), the smallest normal positive is 2^-126, and the largest subnormal is (1 − 2^-23) × 2^-126. The smallest positive subnormal is 2^-149. Thus there is a gap between zero and the smallest normal, bridged by subnormals.

Denormals serve to represent very small magnitudes and maintain continuous scale near zero, but they come with trade-offs. Precision degrades as values approach zero, and arithmetic with subnormals can be slower on some hardware. Many implementations provide a flush-to-zero mode or allow disabling denormals for performance, in which case results that would be subnormal are rounded to zero.

Denormal numbers occur in all IEEE 754 binary formats, including double precision and extended precisions, with analogous definitions and values. They are an important aspect of the underflow behavior of floating-point systems and are specified to ensure a smoother transition to zero in numerical computations.