[Python-checkins] peps: Update PEP 485 from Chris Barker's edits (w/ minor formatting)
chris.angelico
python-checkins at python.org
Wed Feb 11 13:14:09 CET 2015
https://hg.python.org/peps/rev/6625f2ec2d3e
changeset: 5694:6625f2ec2d3e
user: Chris Angelico <rosuav at gmail.com>
date: Wed Feb 11 23:13:44 2015 +1100
summary:
Update PEP 485 from Chris Barker's edits (w/ minor formatting)
files:
pep-0485.txt | 190 ++++++++++++++++++++------------------
1 files changed, 101 insertions(+), 89 deletions(-)
diff --git a/pep-0485.txt b/pep-0485.txt
--- a/pep-0485.txt
+++ b/pep-0485.txt
@@ -16,16 +16,14 @@
This PEP proposes the addition of a function to the standard library
that determines whether one value is approximately equal or "close" to
-another value. It is also proposed that an assertion be added to the
-``unittest.TestCase`` class to provide easy access for those using
-unittest for testing.
+another value.
Rationale
=========
Floating point values contain limited precision, which results in
-their being unable to exactly represent some values, and for error to
+their being unable to exactly represent some values, and for errors to
accumulate with repeated computation. As a result, it is common
advice to only use an equality comparison in very specific situations.
Often a inequality comparison fits the bill, but there are times
@@ -49,8 +47,8 @@
* Is an absolute difference test. Often the measure of difference
requires, particularly for floating point numbers, a relative error,
- i.e "Are these two values within x% of each-other?", rather than an
- absolute error. Particularly when the magnatude of the values is
+ i.e. "Are these two values within x% of each-other?", rather than an
+ absolute error. Particularly when the magnitude of the values is
unknown a priori.
The numpy package has the ``allclose()`` and ``isclose()`` functions,
@@ -63,7 +61,7 @@
Overflow and other help sites.
Many other non-python systems provide such a test, including the Boost C++
-library and the APL language (reference?).
+library and the APL language [4]_.
These existing implementations indicate that this is a common need and
not trivial to write oneself, making it a candidate for the standard
@@ -73,28 +71,31 @@
Proposed Implementation
=======================
-NOTE: this PEP is the result of an extended discussion on the
+NOTE: this PEP is the result of extended discussions on the
python-ideas list [1]_.
-The new function will have the following signature::
+The new function will go into the math module, and have the following
+signature::
- is_close(a, b, rel_tolerance=1e-9, abs_tolerance=0.0)
+ isclose(a, b, rel_tol=1e-9, abs_tol=0.0)
``a`` and ``b``: are the two values to be tested to relative closeness
-``rel_tolerance``: is the relative tolerance -- it is the amount of
-error allowed, relative to the magnitude a and b. For example, to set
-a tolerance of 5%, pass tol=0.05. The default tolerance is 1e-8, which
-assures that the two values are the same within about 8 decimal
-digits.
+``rel_tol``: is the relative tolerance -- it is the amount of error
+allowed, relative to the larger absolute value of a or b. For example,
+to set a tolerance of 5%, pass tol=0.05. The default tolerance is 1e-9,
+which assures that the two values are the same within about 9 decimal
+digits. ``rel_tol`` must be greater than 0.0
-``abs_tolerance``: is an minimum absolute tolerance level -- useful for
+``abs_tol``: is an minimum absolute tolerance level -- useful for
comparisons near zero.
Modulo error checking, etc, the function will return the result of::
- abs(a-b) <= max( rel_tolerance * min(abs(a), abs(b), abs_tolerance )
+ abs(a-b) <= max( rel_tol * max(abs(a), abs(b)), abs_tol )
+The name, ``isclose``, is selected for consistency with the existing
+``isnan`` and ``isinf``.
Handling of non-finite numbers
------------------------------
@@ -111,17 +112,18 @@
The primary use-case is expected to be floating point numbers.
However, users may want to compare other numeric types similarly. In
theory, it should work for any type that supports ``abs()``,
-comparisons, and subtraction. The code will be written and tested to
-accommodate these types:
+multiplication, comparisons, and subtraction. The code will be written
+and tested to accommodate these types:
-* ``Decimal``: for Decimal, the tolerance must be set to a Decimal type.
+* ``Decimal``
* ``int``
* ``Fraction``
-* ``complex``: for complex, ``abs(z)`` will be used for scaling and
- comparison.
+* ``complex``: for complex, the absolute value of the complex values
+ will be used for scaling and comparison. If a complex tolerance is
+ passed in, the absolute value will be used as the tolerance.
Behavior near zero
@@ -130,9 +132,9 @@
Relative comparison is problematic if either value is zero. By
definition, no value is small relative to zero. And computationally,
if either value is zero, the difference is the absolute value of the
-other value, and the computed absolute tolerance will be rel_tolerance
-times that value. rel-tolerance is always less than one, so the
-difference will never be less than the tolerance.
+other value, and the computed absolute tolerance will be ``rel_tol``
+times that value. When ``rel_tol`` is less than one, the difference will
+never be less than the tolerance.
However, while mathematically correct, there are many use cases where
a user will need to know if a computed value is "close" to zero. This
@@ -146,50 +148,51 @@
if a is approximately equal to -b, then a and b will never be computed
as "close".
-To handle this case, an optional parameter, ``abs_tolerance`` can be
+To handle this case, an optional parameter, ``abs_tol`` can be
used to set a minimum tolerance used in the case of very small or zero
-computed absolute tolerance. That is, the values will be always be
-considered close if the difference between them is less than the
-abs_tolerance
+computed relative tolerance. That is, the values will be always be
+considered close if the difference between them is less than
+``abs_tol``
The default absolute tolerance value is set to zero because there is
no value that is appropriate for the general case. It is impossible to
know an appropriate value without knowing the likely values expected
for a given use case. If all the values tested are on order of one,
-then a value of about 1e-8 might be appropriate, but that would be far
-too large if expected values are on order of 1e-12 or smaller.
+then a value of about 1e-9 might be appropriate, but that would be far
+too large if expected values are on order of 1e-9 or smaller.
Any non-zero default might result in user's tests passing totally
-inappropriately. If, on the other hand a test against zero fails the
+inappropriately. If, on the other hand, a test against zero fails the
first time with defaults, a user will be prompted to select an
appropriate value for the problem at hand in order to get the test to
pass.
NOTE: that the author of this PEP has resolved to go back over many of
-his tests that use the numpy ``all_close()`` function, which provides
-a default abs_tolerance, and make sure that the default value is
+his tests that use the numpy ``allclose()`` function, which provides
+a default absolute tolerance, and make sure that the default value is
appropriate.
-If the user sets the rel_tolerance parameter to 0.0, then only the
+If the user sets the rel_tol parameter to 0.0, then only the
absolute tolerance will effect the result. While not the goal of the
function, it does allow it to be used as a purely absolute tolerance
check as well.
-unittest assertion
--------------------
-[need text here]
-
-implementation
+Implementation
--------------
A sample implementation is available (as of Jan 22, 2015) on gitHub:
-https://github.com/PythonCHB/close_pep/blob/master
+https://github.com/PythonCHB/close_pep/blob/master/is_close.py
This implementation has a flag that lets the user select which
relative tolerance test to apply -- this PEP does not suggest that
-that be retained, but rather than the strong test be selected.
+that be retained, but rather than the weak test be selected.
+
+There are also drafts of this PEP and test code, etc. there:
+
+https://github.com/PythonCHB/close_pep
+
Relative Difference
===================
@@ -215,7 +218,7 @@
4) The absolute value of the arithmetic mean of the two
-These lead to the following possibilities for determining if two
+These leads to the following possibilities for determining if two
values, a and b, are close to each other.
1) ``abs(a-b) <= tol*abs(a)``
@@ -228,9 +231,9 @@
NOTE: (2) and (3) can also be written as:
-2) ``(abs(a-b) <= tol*abs(a)) or (abs(a-b) <= tol*abs(a))``
+2) ``(abs(a-b) <= abs(tol*a)) or (abs(a-b) <= abs(tol*b))``
-3) ``(abs(a-b) <= tol*abs(a)) and (abs(a-b) <= tol*abs(a))``
+3) ``(abs(a-b) <= abs(tol*a)) and (abs(a-b) <= abs(tol*b))``
(Boost refers to these as the "weak" and "strong" formulations [3]_)
These can be a tiny bit more computationally efficient, and thus are
@@ -292,7 +295,7 @@
minimum tolerance ``tol * b == 0.1 * 9.0 == 0.9``
-delta = ``(1.0 - 0.9) * 0.1 = 0.1`` or ``tol**2 * a = 0.1**2 * 10 = .01``
+delta = ``(1.0 - 0.9) = 0.1`` or ``tol**2 * a = 0.1**2 * 10 = .1``
The absolute difference between the maximum and minimum tolerance
tests in this case could be substantial. However, the primary use
@@ -310,7 +313,7 @@
yield exactly the same results for all values of a and b.
In addition, in common use, tolerances are defined to 1 significant
-figure -- that is, 1e-8 is specifying about 8 decimal digits of
+figure -- that is, 1e-9 is specifying about 9 decimal digits of
accuracy. So the difference between the various possible tests is well
below the precision to which the tolerance is specified.
@@ -321,11 +324,11 @@
A relative comparison can be either symmetric or non-symmetric. For a
symmetric algorithm:
-``is_close_to(a,b)`` is always the same as ``is_close_to(b,a)``
+``isclose(a,b)`` is always the same as ``isclose(b,a)``
If a relative closeness test uses only one of the values (such as (1)
-above), then the result is asymmetric, i.e. is_close_to(a,b) is not
-necessarily the same as is_close_to(b,a).
+above), then the result is asymmetric, i.e. isclose(a,b) is not
+necessarily the same as isclose(b,a).
Which approach is most appropriate depends on what question is being
asked. If the question is: "are these two numbers close to each
@@ -369,18 +372,31 @@
the value be either added together before dividing by 2, which could
result in extra overflow to inf for very large numbers, or require
each value to be divided by two before being added together, which
-could result in underflow to -inf for very small numbers. This effect
+could result in underflow to zero for very small numbers. This effect
would only occur at the very limit of float values, but it was decided
-there as no benefit to the method worth reducing the range of
-functionality.
+there was no benefit to the method worth reducing the range of
+functionality or adding the complexity of checking values to determine
+the order of computation.
This leaves the boost "weak" test (2)-- or using the smaller value to
scale the tolerance, or the Boost "strong" (3) test, which uses the
smaller of the values to scale the tolerance. For small tolerance,
-they yield the same result, but this proposal uses the boost "strong"
-test case: it is symmetric and provides a slightly stricter criteria
-for tolerance.
+they yield the same result, but this proposal uses the boost "weak"
+test case: it is symmetric and provides a more useful result for very
+large tolerances.
+Large tolerances
+----------------
+
+The most common use case is expected to be small tolerances -- on order of the
+default 1e-9. However there may be use cases where a user wants to know if two
+fairly disparate values are within a particular range of each other: "is a
+within 200% (rel_tol = 2.0) of b? In this case, the string test would never
+indicate that two values are within that range of each other if one of them is
+zero. The strong case, however would use the larger (non-zero) value for the
+test, and thus return true if one value is zero. For example: is 0 within 200%
+of 10? 200% of ten is 20, so the range within 200% of ten is -10 to +30. Zero
+falls within that range, so it will return True.
Defaults
========
@@ -392,16 +408,15 @@
The relative tolerance required for two values to be considered
"close" is entirely use-case dependent. Nevertheless, the relative
-tolerance needs to be less than 1.0, and greater than 1e-16
-(approximate precision of a python float). The value of 1e-9 was
-selected because it is the largest relative tolerance for which the
-various possible methods will yield the same result, and it is also
-about half of the precision available to a python float. In the
-general case, a good numerical algorithm is not expected to lose more
-than about half of available digits of accuracy, and if a much larger
-tolerance is acceptable, the user should be considering the proper
-value in that case. Thus 1-e9 is expected to "just work" for many
-cases.
+tolerance needs to be greater than 1e-16 (approximate precision of a
+python float). The value of 1e-9 was selected because it is the
+largest relative tolerance for which the various possible methods will
+yield the same result, and it is also about half of the precision
+available to a python float. In the general case, a good numerical
+algorithm is not expected to lose more than about half of available
+digits of accuracy, and if a much larger tolerance is acceptable, the
+user should be considering the proper value in that case. Thus 1-e9 is
+expected to "just work" for many cases.
Absolute tolerance default
--------------------------
@@ -437,19 +452,16 @@
results computed near what I expect as a result?" This sort of test
may or may not be part of a formal unit testing suite. Such testing
could be used one-off at the command line, in an iPython notebook,
-part of doctests, or simple assets in an ``if __name__ == "__main__"``
+part of doctests, or simple asserts in an ``if __name__ == "__main__"``
block.
-The proposed unitest.TestCase assertion would have course be used in
-unit testing.
-
It would also be an appropriate function to use for the termination
criteria for a simple iterative solution to an implicit function::
guess = something
while True:
new_guess = implicit_function(guess, *args)
- if is_close(new_guess, guess):
+ if isclose(new_guess, guess):
break
guess = new_guess
@@ -481,13 +493,13 @@
This method is purely an absolute tolerance test, and does not address
the need for a relative tolerance test.
-numpy ``is_close()``
---------------------
+numpy ``isclose()``
+-------------------
http://docs.scipy.org/doc/numpy-dev/reference/generated/numpy.isclose.html
-The numpy package provides the vectorized functions is_close() and
-all_close, for similar use cases as this proposal:
+The numpy package provides the vectorized functions isclose() and
+allclose(), for similar use cases as this proposal:
``isclose(a, b, rtol=1e-05, atol=1e-08, equal_nan=False)``
@@ -522,9 +534,9 @@
-------------------------------
The Boost project ( [3]_ ) provides a floating point comparison
-function. Is is a symmetric approach, with both "weak" (larger of the
+function. It is a symmetric approach, with both "weak" (larger of the
two relative errors) and "strong" (smaller of the two relative errors)
-options. This proposal uses the Boost "strong" approach. There is no
+options. This proposal uses the Boost "weak" approach. There is no
need to complicate the API by providing the option to select different
methods when the results will be similar in most cases, and the user
is unlikely to know which to select in any case.
@@ -545,10 +557,6 @@
test to, at the very least, copy the function into their code base,
and select the comparison method to use.
-In addition, adding the function to the standard library allows it to
-be used in the ``unittest.TestCase.assertIsClose()`` method, providing
-a substantial convenience to those using unittest.
-
``zero_tol``
''''''''''''
@@ -568,16 +576,15 @@
'''''''''''''''''''''
Given the issues with comparing to zero, another possibility would
-have been to only provide a relative tolerance, and let every
-comparison to zero fail. In this case, the user would need to do a
-simple absolute test: `abs(val) < zero_tol` in the case where the
-comparison involved zero.
+have been to only provide a relative tolerance, and let comparison to
+zero fail. In this case, the user would need to do a simple absolute
+test: `abs(val) < zero_tol` in the case where the comparison involved
+zero.
However, this would not allow the same call to be used for a sequence
-of values, such as in a loop or comprehension, or in the
-``TestCase.assertClose()`` method. Making the function far less
-useful. It is noted that the default abs_tolerance=0.0 achieves the
-same effect if the default is not overidden.
+of values, such as in a loop or comprehension. Making the function far
+less useful. It is noted that the default abs_tol=0.0 achieves the
+same effect if the default is not overridden.
Other tests
''''''''''''
@@ -605,6 +612,11 @@
http://www.boost.org/doc/libs/1_35_0/libs/test/doc/components/test_tools/floating_point_comparison.html
+.. [4] 1976. R. H. Lathwell. APL comparison tolerance. Proceedings of
+ the eighth international conference on APL Pages 255 - 258
+
+ http://dl.acm.org/citation.cfm?doid=800114.803685
+
.. Bruce Dawson's discussion of floating point.
https://randomascii.wordpress.com/2012/02/25/comparing-floating-point-numbers-2012-edition/
--
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