计算机存储浮点数

计算机存储浮点数

A computer stores floating-point numbers using a standardized format called IEEE 754. This format is designed to represent real numbers in a way that balances range and precision. Here's how it works:

Basic Structure of IEEE 754 Floating-Point Numbers

A floating-point number in a computer is typically represented by three components:

1. Sign bit (S): This determines whether the number is positive (0) or negative (1).
2. Exponent (E): This stores the exponent value, which determines the range of the number (i.e., how large or small it can be).
3. Mantissa (or Significand) (M): This holds the significant digits of the number, representing its precision.

The general formula for a floating-point number is:

\[(-1)^{S} \times 1.M \times 2^{(E - \text{bias})} \]

Where:

• S is the sign bit (0 for positive, 1 for negative).
• M is the mantissa (or significand), typically in normalized form (starting with a leading 1).
• E is the exponent, adjusted by a bias.

Common Floating-Point Formats

The two most common floating-point formats are single precision (32-bit) and double precision (64-bit).

1. Single Precision (32-bit Floating-Point):

• 1 bit for sign (S)
• 8 bits for exponent (E)
• 23 bits for mantissa (M)

A 32-bit floating-point number has the following layout:

| S |  E (8 bits)  |       M (23 bits)         |

• Range of exponent: The exponent is stored with a bias of 127 (i.e., ( E - 127 )), meaning the actual exponent is calculated as E - 127.
• Mantissa: The 23 bits store the fractional part. The number is assumed to have a leading 1. (known as implicit leading 1), which is not stored explicitly. For example, a mantissa of 001 would be interpreted as 1.001.

2. Double Precision (64-bit Floating-Point):

• 1 bit for sign (S)
• 11 bits for exponent (E)
• 52 bits for mantissa (M)

A 64-bit floating-point number has the following layout:

| S |      E (11 bits)     |               M (52 bits)               |

• Range of exponent: The exponent is stored with a bias of 1023 (i.e., ( E - 1023 )).
• Mantissa: The 52 bits store the fractional part, with an implicit leading 1.

Example of Single-Precision Float Representation

Suppose we want to store the number -6.75 as a 32-bit float:

1. Convert to binary:

   • 6.75 in decimal is 110.11 in binary (6 = 110, and .75 = .11 in binary).

2. Normalize the number:

   • In scientific notation, this is ( -1.1011 \times 2^2 ). This shows the sign bit is 1, the exponent is 2, and the mantissa is 1.1011.

3. Set the components:

   • Sign bit: 1 (since the number is negative)
   • Exponent: 2 + 127 = 129 in decimal, which is 10000001 in binary.
   • Mantissa: The 1. is implicit, so we only store 1011, padded to 23 bits: 10110000000000000000000.

Thus, the 32-bit representation of -6.75 is:

1 10000001 10110000000000000000000

Precision and Limitations

• Precision: The more bits in the mantissa, the more precise the number. Single-precision floats are accurate to about 7 decimal digits, while double-precision floats are accurate to about 15-16 decimal digits.
• Range: The exponent allows floating-point numbers to represent a vast range, from very small numbers (close to zero) to very large ones.

Special Values

IEEE 754 also defines special cases:

• Zero: Represented by all bits in the exponent and mantissa being zero.
• Infinity: Represented by all bits in the exponent being 1, and the mantissa being all 0.
• NaN (Not a Number): Represented by all bits in the exponent being 1, and the mantissa containing non-zero bits.

Summary

• Floating-point numbers are stored in three parts: sign, exponent, and mantissa.
• Single precision uses 32 bits, while double precision uses 64 bits.
• The IEEE 754 standard defines how these components are laid out and how the numbers are calculated, enabling computers to store a wide range of real numbers with a trade-off between precision and range.