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The JVM's floating-point support adheres to the IEEE-754 1985 floating-point standard. This standard defines the format of 32-bit and 64-bit floating-point numbers and defines the operations upon those numbers. In the JVM, floating-point arithmetic is performed on 32-bit floats and 64-bit doubles. For each bytecode that performs arithmetic on floats, there is a corresponding bytecode that performs the same operation on doubles.
A floating-point number has four parts -- a sign, a mantissa, a radix, and an exponent. The sign is either a 1 or -1. The mantissa, always a positive number, holds the significant digits of the floating-point number. The exponent indicates the positive or negative power of the radix that the mantissa and sign should be multiplied by. The four components are combined as follows to get the floating-point value:
Floating-point numbers have multiple representations, because one can always multiply the mantissa of any floating-point number by some power of the radix and change the exponent to get the original number. For example, the number -5 can be represented equally by any of the following forms in radix 10:
For each floating-point number there is one representation that is said to be normalized. A floating-point number is normalized if its mantissa is within the range defined by the following relation:
A normalized radix 10 floating-point number has its decimal point just to the left of the first non-zero digit in the mantissa. The normalized floating-point representation of -5 is -1 * 0.5 * 10 1. In other words, a normalized floating-point number's mantissa has no non-zero digits to the left of the decimal point and a non-zero digit just to the right of the decimal point. Any floating-point number that doesn't fit into this category is said to be denormalized. Note that the number zero has no normalized representation, because it has no non-zero digit to put just to the right of the decimal point. "Why be normalized?" is a common exclamation among zeros.
Floating-point numbers in the JVM use a radix of two. Floating-point numbers in the JVM, therefore, have the following form:
The mantissa of a floating-point number in the JVM is expressed as a binary number. A normalized mantissa has its binary point (the base-two equivalent of a decimal point) just to the left of the most significant non-zero digit. Because the binary number system has just two digits -- zero and one -- the most significant digit of a normalized mantissa is always a one.