IEEE 754 Floating Point Standard Explained

What is IEEE 754?

IEEE 754 is the international standard for representing real numbers in binary computers. It defines bit layouts for single precision (32-bit), double precision (64-bit), and extended formats. Virtually every modern CPU, GPU, and programming language uses IEEE 754 for float and double types.

Single-Precision Layout (32 bits)

Field	Bits	Description
Sign	1	0 = positive, 1 = negative
Exponent	8	Biased by 127
Mantissa	23	Fractional part (implicit leading 1)

Value (normalized): (−1)^sign × 1.mantissa × 2^(exponent−127)

Example: Encode 3.14 (approximate)

Sign 0, exponent biased ~128, mantissa stores fractional bits. Exact representation: 0x40490FDB (single). Not all decimals are exactly representable — rounding occurs.

Special Values

Exponent	Mantissa	Meaning
All 0	0	±0
All 1	0	±Infinity
All 1	non-zero	NaN (Not a Number)

Double Precision (64 bits)

1 sign + 11 exponent (bias 1023) + 52 mantissa bits. Higher precision for scientific computing.

Why It Matters

• Rounding errors: 0.1 + 0.2 ≠ 0.3 in binary float — famous JavaScript example
• Comparisons: Never use == for floats; use epsilon tolerance
• Debugging: Inspecting raw hex bits reveals NaN sources

Convert with Numverto

The IEEE 754 Converter breaks down sign, exponent, and mantissa for any decimal input in 32-bit or 64-bit format.

Related: Floating Point Representation deep dive.

Frequently Asked Questions

Why can’t computers store 0.1 exactly?

0.1 has an infinite repeating binary fraction — like 1/3 in decimal.

What is NaN?

Not a Number — result of invalid ops like 0/0 or sqrt(-1).

What is subnormal (denormal) number?

Very small numbers with exponent all zeros and implicit leading 0 instead of 1.

Is IEEE 754 used in GPUs?

Yes — CUDA, OpenCL, and graphics shaders use IEEE 754 floats.

How do I compare two floats safely?

Check abs(a - b) < epsilon for small epsilon relative to magnitude.