47th BCS Technical Written (CSE-971)প্রশ্ন ও সমাধান

Floating-point numbers are stored in IEEE-754 format as sign, exponent, and a finite-length mantissa — since only a fixed number of mantissa bits are available, most real numbers cannot be represented exactly and must be rounded to the nearest representable value.

Format	Total bits	Mantissa bits	Approx. decimal precision
Single precision	32	23	~7 digits
Double precision	64	52	~15-16 digits

Why rounding errors occur

• Many decimal fractions (e.g., 0.1) are repeating in binary, so they are truncated/rounded to fit the mantissa — the stored value is only an approximation.
• Every arithmetic operation (add, multiply, divide) can introduce a fresh rounding step, since the true result may need more bits than the mantissa holds.
• Error accumulation: repeated operations (e.g., summing millions of values in a loop) compound these small errors.
• Catastrophic cancellation: subtracting two nearly equal floating-point numbers cancels the significant leading digits and leaves mostly rounding noise.

Worked example:

0.1 in binary = 0.0001100110011... (repeating, truncated to 23/52 bits)
0.1 + 0.2 in double precision = 0.30000000000000004   (not exactly 0.3)

This happens because both 0.1 and 0.2 are stored as slightly-off approximations, and their sum inherits both errors.

Exam takeaway: Never compare floats with ==; use a tolerance (|a-b| < ε) and avoid subtracting nearly-equal large numbers to reduce cancellation error.