The class Math contains methods for performing basic
numeric operations such as the elementary exponential, logarithm,
square root, and trigonometric functions.
Unlike some of the numeric methods of class
StrictMath, all implementations of the equivalent
functions of class Math are not defined to return the
bit-for-bit same results. This relaxation permits
better-performing implementations where strict reproducibility is
not required.
By default many of the Math methods simply call
the equivalent method in StrictMath for their
implementation. Code generators are encouraged to use
platform-specific native libraries or microprocessor instructions,
where available, to provide higher-performance implementations of
Math methods. Such higher-performance
implementations still must conform to the specification for
Math.
The quality of implementation specifications concern two
properties, accuracy of the returned result and monotonicity of the
method. Accuracy of the floating-point Math methods is
measured in terms of ulps, units in the last place. For a
given floating-point format, an ulp of a
specific real number value is the distance between the two
floating-point values bracketing that numerical value. When
discussing the accuracy of a method as a whole rather than at a
specific argument, the number of ulps cited is for the worst-case
error at any argument. If a method always has an error less than
0.5 ulps, the method always returns the floating-point number
nearest the exact result; such a method is correctly
rounded. A correctly rounded method is generally the best a
floating-point approximation can be; however, it is impractical for
many floating-point methods to be correctly rounded. Instead, for
the Math class, a larger error bound of 1 or 2 ulps is
allowed for certain methods. Informally, with a 1 ulp error bound,
when the exact result is a representable number, the exact result
should be returned as the computed result; otherwise, either of the
two floating-point values which bracket the exact result may be
returned. For exact results large in magnitude, one of the
endpoints of the bracket may be infinite. Besides accuracy at
individual arguments, maintaining proper relations between the
method at different arguments is also important. Therefore, most
methods with more than 0.5 ulp errors are required to be
semi-monotonic: whenever the mathematical function is
non-decreasing, so is the floating-point approximation, likewise,
whenever the mathematical function is non-increasing, so is the
floating-point approximation. Not all approximations that have 1
ulp accuracy will automatically meet the monotonicity requirements.
The platform uses signed two's complement integer arithmetic with
int and long primitive types. The developer should choose
the primitive type to ensure that arithmetic operations consistently
produce correct results, which in some cases means the operations
will not overflow the range of values of the computation.
The best practice is to choose the primitive type and algorithm to avoid
overflow. In cases where the size is int or long and
overflow errors need to be detected, the methods addExact,
subtractExact, multiplyExact, and toIntExact
throw an ArithmeticException when the results overflow.
For other arithmetic operations such as divide, absolute value,
increment by one, decrement by one, and negation, overflow occurs only with
a specific minimum or maximum value and should be checked against
the minimum or maximum as appropriate.