Computer Arithmetic

What is the effect of not storing the leading 1 of the significand (the hidden bit) in IEEE 754 normalized numbers?

  1. AOmitting the leading 1 makes negative numbers representable
  2. BPrecision increases by one bit for the same number of stored bits
  3. COmitting the leading 1 makes results always exact
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Answer: B. Precision increases by one bit for the same number of stored bits

Normalized in binary, every nonzero number has the form 1.xxx × 2^exponent, so the leading digit is always 1. There is no need to store a known 1, so it is omitted and the bit is used for the fraction instead. Single precision thus stores 23 significand bits but has 24 bits of precision. Subnormals, with an all-zero exponent, treat the leading digit as 0. Negative numbers use the sign bit, and rounding error does not disappear.

  • A. Negative numbers are represented by the sign bit.
  • B. Correct. Omitting the known 1 gains one bit of precision.
  • C. Rounding error does not disappear.
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More Computer Arithmetic questions

  1. What is the hexadecimal bit pattern of decimal 1.0 in IEEE 754 single precision?
  2. Converting −6.25 to single precision, what is the sign–exponent–significand bit string written in hexadecimal?
  3. Which correctly describes representing decimal 0.1 as a binary floating-point number?
  4. In many programming languages, why is the double-precision comparison 0.1 + 0.2 == 0.3 false?
  5. What is the machine epsilon (the gap between 1 and the next larger representable number) in IEEE 754 single precision?