28.6 Exact Floating-Point Arithmetic

As long as the numbers are exactly representable (fractions whose denominator is a power of 2), and intermediate results do not require rounding, then floating-point arithmetic is exact. It is easy to predict how many digits are needed for the results of arithmetic operations:

• addition and subtraction of two n-digit values with the same exponent require at most n + 1 digits, but when the exponents differ, many more digits may be needed;
• multiplication of two n-digit values requires exactly 2 n digits;
• although integer division produces a quotient and a remainder of no more than n-digits, floating-point remainder and square root may require an unbounded number of digits, and the quotient can need many more digits than can be stored.

Whenever a result requires more than n digits, rounding is needed.