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The most significant bit of a float or double is its sign bit. The mantissa occupies the 23 least significant bits of a float and the 52 least significant bits of a double. The exponent, 8 bits in a float and 11 bits in a double, sits between the sign and mantissa. The format of a float is shown below. The sign bit is shown as an "s," the exponent bits are shown as "e," and the mantissa bits are shown as "m":
| s eeeeeeee mmmmmmmmmmmmmmmmmmmmmmm |
A sign bit of zero indicates a positive number and a sign bit of one indicates a negative number. The mantissa is always interpreted as a positive base-two number. It is not a twos-complement number. If the sign bit is one, the floating-point value is negative, but the mantissa is still interpreted as a positive number that must be multiplied by -1.
The exponent field is interpreted in one of three ways. An exponent of all ones indicates the floating-point number has one of the special values of plus or minus infinity, or "not a number" (NaN). NaN is the result of certain operations, such as the division of zero by zero. An exponent of all zeros indicates a denormalized floating-point number. Any other exponent indicates a normalized floating-point number.
The mantissa contains one extra bit of precision beyond those that appear in the mantissa bits. The mantissa of a float, which occupies only 23 bits, has 24 bits of precision. The mantissa of a double, which occupies 52 bits, has 53 bits of precision. The most significant mantissa bit is predictable, and is therefore not included, because the exponent of floating-point numbers in the JVM indicates whether or not the number is normalized. If the exponent is all zeros, the floating-point number is denormalized and the most significant bit of the mantissa is known to be a zero. Otherwise, the floating-point number is normalized and the most significant bit of the mantissa is known to be one.
The JVM throws no exceptions as a result of any floating-point operations. Special values, such as positive and negative infinity or NaN, are returned as the result of suspicious operations such as division by zero. An exponent of all ones indicates a special floating-point value. An exponent of all ones with a mantissa whose bits are all zero indicates an infinity. The sign of the infinity is indicated by the sign bit. An exponent of all ones with any other mantissa is interpreted to mean "not a number" (NaN). The JVM always produces the same mantissa for NaN, which is all zeros except for the most significant mantissa bit that appears in the number. These values are shown for a float below:
| Value | Float bits (sign exponent mantissa) |
|---|---|
| +Infinity | 0 11111111 00000000000000000000000 |
| -Infinity | 1 11111111 00000000000000000000000 |
| NaN | 1 11111111 10000000000000000000000 |
Exponents that are neither all ones nor all zeros indicate the power of two by which to multiply the normalized mantissa. The power of two can be determined by interpreting the exponent bits as a positive number, and then subtracting a bias from the positive number. For a float, the bias is 126. For a double, the bias is 1023. For example, an exponent field in a float of 00000001 yields a power of two by subtracting the bias (126) from the exponent field interpreted as a positive integer (1). The power of two, therefore, is 1 - 126, which is -125. This is the smallest possible power of two for a float. At the other extreme, an exponent field of 11111110 yields a power of two of (254 - 126) or 128. The number 128 is the largest power of two available to a float. Several examples of normalized floats are shown in the following table: