Chris Angelico <ros...@gmail.com>: > With a single abstract numeric type, what exactly does "inexact" mean, > where does it come from, and how does that affect the expected > behaviour and performance of numbers?
Not much is said in the standard: Thus inexactness is a contagious property of a number. If two implementations produce exact results for a computation that did not involve inexact intermediate results, the two ultimate results will be mathematically equivalent. This is generally not true of computations involving inexact numbers since approximate methods such as floating point arithmetic may be used, but it is the duty of each implementation to make the result as close as practical to the mathematically ideal result. <URL: http://www.schemers.org/Documents/Standards/R5RS/HTML/r5rs-Z-H-9.h tml#%_sec_6.2.2> Exactness should be considered a flag that expresses whether the result is an approximation. > Will an "Exact" non-integer be stored as Decimal or Fraction? How do > you know? They have vastly different semantics, and you should be able > to choose. The manual of the Scheme implementation would likely explain the particular properties of its number system implementation. At its simplest, you could have an implementation that only supports, say, signed 16-bit exact integers. All other numbers would be inexact floats. Implementations are encouraged, but not required, to 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 may cause an error later. An implementation may use floating point and other approximate representation strategies for inexact numbers. This report recommends, but does not require, that the IEEE 32-bit and 64-bit floating point standards be followed by implementations that use flonum representations, and that implementations using other representations should match or exceed the precision achievable using these floating point standards [12]. In particular, implementations that use flonum representations must follow these rules: A flonum result must be represented with at least as much precision as is used to express any of the inexact arguments to that operation. It is desirable (but not required) for potentially inexact operations such as sqrt, when applied to exact arguments, to produce exact answers whenever possible (for example the square root of an exact 4 ought to be an exact 2). <URL: http://www.schemers.org/Documents/Standards/R5RS/HTML/r5rs-Z-H-9.h tml#%_sec_6.2.3> Marko -- https://mail.python.org/mailman/listinfo/python-list
