The class
Math
contains methods for performing basic
numeric operations such as the elementary exponential, logarithm,
square root, and trigonometric functions.
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.