Subnormals

Subnormal numbers, also called denormals, are a class of floating-point values used to represent numbers closer to zero than the smallest normal numbers. In IEEE 754 binary formats, subnormals occur when the exponent field is zero. In this case the hidden leading bit of the significand is zero (not one, as in normal numbers), and the value is sign × (0.fraction) × 2^(1−bias). This allows a continuous underflow from zero up to the smallest normal number, at the cost of reduced precision and possible performance penalties.

For binary32 (single precision), the smallest normal is 2^−126 ≈ 1.17549435 × 10^−38, while the smallest subnormal is 2^−149 ≈ 1.40129846 × 10^−45, and the largest subnormal is (2^−126) − 2^−149 ≈ 1.17549421 × 10^−38. Similar ranges exist for other formats, such as binary64 (double precision), where the smallest normal is 2^−1022 and the smallest subnormal is 2^−1074. In general, a subnormal uses an exponent of zero with a leading zero in the significand, extending the dynamic range toward zero.

Subnormals enable gradual underflow and improve relative accuracy near zero, but arithmetic on them may be slower on some hardware. Some implementations support a flush-to-zero option, which treats subnormals as zero for performance, reducing precision near zero. Subnormals are part of the IEEE 754 standard and are supported in most modern floating-point formats, including half-precision and extended precisions. They help preserve a meaningful zero neighborhood and enhance numerical robustness in certain computations.