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Programming Basics (Pseudocode Tracing)

Real values are represented in IEEE 754 double precision (binary64), and each sum is rounded to the nearest representable value. What is the value of c after the following program runs?

real: x ← 0.0
integer: c ← 0
while (x < 1.0)
  x ← x + 0.1
  c ← c + 1
endwhile
  1. A9
  2. B10
  3. C11
Show answer and explanation

Answer: C. 11

0.1 is a repeating fraction in binary, and in double precision it is stored as an approximation slightly larger than 0.1. Each addition adds a rounding error, and the result of adding it 10 times is not 1.0 but 0.9999999999999999. This value satisfies x < 1.0, so an 11th addition is made, x becomes about 1.1 and the loop ends, giving c = 11. Count loop iterations with an integer variable, and avoid comparing real numbers with = or ≠. With the condition x ≠ 1.0, x never equals exactly 1.0 and the loop never ends.

  • A. After 9 additions the value is about 0.9, still below 1.0.
  • B. Without errors x would be 1.0 after 10 additions, but the actual 10th result is slightly below 1.0.
  • C. Correct. The sum after 10 additions is 0.9999999999999999, below 1.0, so the body runs 11 times.
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