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6.2.3 Implementation restrictions

Implementations of Scheme are not required to implement the whole tower of subtypes given in Numerical types, but they must implement a coherent subset consistent with both the purposes of the implementation and the spirit of the Scheme language. For example, implementations in which all numbers are real, or in which non-real numbers are always inexact, or in which exact numbers are always integer, are still quite useful.

Implementations may also support only a limited range of numbers of any type, subject to the requirements of this section. The supported range for exact numbers of any type may be different from the supported range for inexact numbers of that type. For example, an implementation that uses IEEE binary double-precision floating-point numbers to represent all its inexact real numbers may also support a practically unbounded range of exact integers and rationals while limiting the range of inexact reals (and therefore the range of inexact integers and rationals) to the dynamic range of the IEEE binary double format. Furthermore, the gaps between the representable inexact integers and rationals are likely to be very large in such an implementation as the limits of this range are approached.

An implementation of Scheme must support exact integers throughout the range of numbers permitted as indexes of lists, vectors, bytevectors, and strings or that result from computing the length of one of these. The length, vector-length, bytevector-length, and string-length procedures must return an exact integer, and it is an error to use anything but an exact integer as an index. Furthermore, any integer constant within the index range, if expressed by an exact integer syntax, must be read as an exact integer, regardless of any implementation restrictions that apply outside this range. Finally, the procedures listed below will always return exact integer results provided all their arguments are exact integers and the mathematically expected results are representable as exact integers within the implementation:

It is recommended, but not required, that implementations support exact integers and exact rationals of practically unlimited size and precision, and to implement the above procedures and the / procedure in such a way that they always return exact results when given exact arguments. If one of these procedures is unable to deliver an exact result when given exact arguments, then it may either report a violation of an implementation restriction or it may silently coerce its result to an inexact number; such a coercion can cause an error later. Nevertheless, implementations that do not provide exact rational numbers should return inexact rational numbers rather than reporting an implementation restriction.

An implementation may use floating-point and other approximate representation strategies for inexact numbers. This report recommends, but does not require, that implementations that use floating-point representations follow the IEEE 754 standard, and that implementations using other representations should match or exceed the precision achievable using these floating-point standards [IEEE]. In particular, the description of transcendental functions in IEEE 754-2008 should be followed by such implementations, particularly with respect to infinities and NaNs.

Although Scheme allows a variety of written notations for numbers, any particular implementation may support only some of them. For example, an implementation in which all numbers are real need not support the rectangular and polar notations for complex numbers. If an implementation encounters an exact numerical constant that it cannot represent as an exact number, then it may either report a violation of an implementation restriction or it may silently represent the constant by an inexact number.

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