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x86 floating-point follows the IEEE 754 standard. A number is stored as sign, exponent, and mantissa:

• Single (32-bit): 1 + 8 + 23
• Double (64-bit): 1 + 11 + 52
• Extended (80-bit): used internally by the x87 FPU

Most decimal fractions cannot be represented exactly in binary. For example 0.1 + 0.2 does not equal exactly 0.3. Accumulated rounding error is a core hazard in numerical code.

IEEE 754 defines four rounding modes:

• Round to nearest, ties to even (default)
• Round toward negative infinity (floor)
• Round toward positive infinity (ceil)
• Round toward zero (truncate)

The x87 control word holds the rounding-control (RC) bits. You read it with fstcw, modify, then load with fldcw.

SSE has its own control/status register, MXCSR. It holds rounding-control bits plus exception masks and flags. Manage it with ldmxcsr and stmxcsr.

The standard defines six exceptions:

• Invalid (e.g. 0/0, sqrt of negative)
• Denormal operand
• Divide by zero
• Overflow
• Underflow
• Inexact (rounding occurred)

Each exception can be masked. A masked exception produces a default result (like infinity or NaN) and sets a flag. An unmasked exception raises a CPU trap so your handler can intervene.

IEEE 754 reserves bit patterns for special values:

• +Inf / -Inf from overflow or divide-by-zero
• NaN (Not a Number) from invalid operations

Any comparison with NaN is false — even NaN == NaN.

Because NaN is unordered, you detect it with an unordered compare. In SSE, ucomiss sets the parity flag when an operand is NaN.

This self-contained C program shows that floating-point addition is not exact and detects a NaN.

To write robust floating-point code:

• Never compare floats with ==; use an epsilon tolerance
• Be aware of accumulation order in sums
• Keep the default round-to-nearest unless you have a reason
• Check status flags after risky operations

Test your floating-point knowledge.

You explored floating-point behavior:

• IEEE 754 defines single, double, and 80-bit extended formats
• Four rounding modes are configured via the x87 control word or MXCSR
• Six exceptions can be masked (default result + flag) or unmasked (trap)
• NaN and infinity are special values; NaN compares false to everything