Modern Computer Architecture and Organization

Modern Computer Architecture and Organization, by Jim Ledin. Published by Packt Publishing.

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Chapter 9, Exercise 1

Using a programming language that allows access to the byte representation of floating-point data types (such as C or C++), write a function that accepts a 32-bit single-precision value as input. Extract the sign, exponent, and mantissa from the bytes of the floating-point value and display them. Remove the bias term from the exponent before displaying its value and display the mantissa as a decimal number. Test the program with the values 0, -0, 1, -1, 6.674e-11, 1.0e38, 1.0e39, 1.0e-38, and 1.0e-39. The numeric values listed here containing “e” are in the C/C++ text format for floating-point numbers. For example, 6.674e-11 means 6.674 × 10^-11.

Answer

See the C++ file Ex__1_float_format.cpp.

This is the output of the program:

  Type  |   Number  |       Bytes      | Sign | Exponent | Mantissa
 -------|-----------|------------------|------|----------|---------
 Float  |         0 |     00000000     |   0  |   -126   | 0.000000
 Float  |         0 |     00000000     |   0  |   -126   | 0.000000
 Float  |         1 |     3F800000     |   0  |      0   | 1.000000
 Float  |        -1 |     BF800000     |   1  |      0   | 1.000000
 Float  | 6.674e-11 |     2E92C348     |   0  |    -34   | 1.146585
 Float  |     1e+38 |     7E967699     |   0  |    126   | 1.175494
 Float  |     1e-38 |     006CE3EE     |   0  |   -126   | 0.850706
 Float  |     1e-39 |     000AE398     |   0  |   -126   | 0.085071

These are some notes about the results:

• Zero in IEEE 754 can have a positive or negative sign. The zero passed to the print_float function in the second row of the table is preceded with a minus sign, but the sign is ignored during the conversion to floating-point.
• The value 1.0e39f is not shown because using it causes a compile-time error: floating constant is out of range.
• Zero is represented as a mantissa of zero and a biased exponent of zero.
• The last two rows contain numbers that cannot be represented with an implicit leading 1 bit because the exponent would underflow. These numbers are called subnormals, and contain the special biased exponent of 0. Subnormals have reduced precision because not all bits of the mantissa contain meaningful digits.
• Numerically, subnormal floats actually use a biased exponent of 1, which translates to an unbiased exponent of -126.

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