Issue #19543: Implementation of isclose as per PEP 485

For details, see:
PEP 0485 -- A Function for testing approximate equality

Functions added: math.isclose() and cmath.isclose().

Original code by Chris Barker. Patch by Tal Einat.

Doc/library/cmath.rst

@@ -207,6 +207,38 @@ Classification functions
    and ``False`` otherwise.
 
 
+.. function:: isclose(a, b, *, rel_tol=1e-09, abs_tol=0.0)
+
+   Return ``True`` if the values *a* and *b* are close to each other and
+   ``False`` otherwise.
+
+   Whether or not two values are considered close is determined according to
+   given absolute and relative tolerances.
+
+   *rel_tol* is the relative tolerance -- it is the maximum allowed difference
+   between *a* and *b*, relative to the larger absolute value of *a* or *b*.
+   For example, to set a tolerance of 5%, pass ``rel_tol=0.05``.  The default
+   tolerance is ``1e-09``, which assures that the two values are the same
+   within about 9 decimal digits.  *rel_tol* must be greater than zero.
+
+   *abs_tol* is the minimum absolute tolerance -- useful for comparisons near
+   zero. *abs_tol* must be at least zero.
+
+   If no errors occur, the result will be:
+   ``abs(a-b) <= max(rel_tol * max(abs(a), abs(b)), abs_tol)``.
+
+   The IEEE 754 special values of ``NaN``, ``inf``, and ``-inf`` will be
+   handled according to IEEE rules.  Specifically, ``NaN`` is not considered
+   close to any other value, including ``NaN``.  ``inf`` and ``-inf`` are only
+   considered close to themselves.
+
+   .. versionadded:: 3.5
+
+   .. seealso::
+
+      :pep:`485` -- A function for testing approximate equality
+
+
 Constants
 ---------
 

Doc/library/math.rst

@@ -110,6 +110,38 @@ Number-theoretic and representation functions
    .. versionadded:: 3.5
 
 
+.. function:: isclose(a, b, *, rel_tol=1e-09, abs_tol=0.0)
+
+   Return ``True`` if the values *a* and *b* are close to each other and
+   ``False`` otherwise.
+
+   Whether or not two values are considered close is determined according to
+   given absolute and relative tolerances.
+
+   *rel_tol* is the relative tolerance -- it is the maximum allowed difference
+   between *a* and *b*, relative to the larger absolute value of *a* or *b*.
+   For example, to set a tolerance of 5%, pass ``rel_tol=0.05``.  The default
+   tolerance is ``1e-09``, which assures that the two values are the same
+   within about 9 decimal digits.  *rel_tol* must be greater than zero.
+
+   *abs_tol* is the minimum absolute tolerance -- useful for comparisons near
+   zero. *abs_tol* must be at least zero.
+
+   If no errors occur, the result will be:
+   ``abs(a-b) <= max(rel_tol * max(abs(a), abs(b)), abs_tol)``.
+
+   The IEEE 754 special values of ``NaN``, ``inf``, and ``-inf`` will be
+   handled according to IEEE rules.  Specifically, ``NaN`` is not considered
+   close to any other value, including ``NaN``.  ``inf`` and ``-inf`` are only
+   considered close to themselves.
+
+   .. versionadded:: 3.5
+
+   .. seealso::
+
+      :pep:`485` -- A function for testing approximate equality
+
+
 .. function:: isfinite(x)
 
    Return ``True`` if *x* is neither an infinity nor a NaN, and

Doc/whatsnew/3.5.rst

@@ -285,6 +285,18 @@ rather than being restricted to ASCII.
 
    :pep:`488` -- Multi-phase extension module initialization
 
+PEP 485: A function for testing approximate equality
+----------------------------------------------------
+
+:pep:`485` adds the :func:`math.isclose` and :func:`cmath.isclose`
+functions which tell whether two values are approximately equal or
+"close" to each other.  Whether or not two values are considered
+close is determined according to given absolute and relative tolerances.
+
+.. seealso::
+
+   :pep:`485` -- A function for testing approximate equality
+
 Other Language Changes
 ======================
 
@@ -346,6 +358,13 @@ cgi
 * :class:`~cgi.FieldStorage` now supports the context management protocol.
   (Contributed by Berker Peksag in :issue:`20289`.)
 
+cmath
+-----
+
+* :func:`cmath.isclose` function added.
+  (Contributed by Chris Barker and Tal Einat in :issue:`24270`.)
+
+
 code
 ----
 
@@ -578,6 +597,8 @@ math
 
 * :data:`math.inf` and :data:`math.nan` constants added.  (Contributed by Mark
   Dickinson in :issue:`23185`.)
+* :func:`math.isclose` function added.
+  (Contributed by Chris Barker and Tal Einat in :issue:`24270`.)
 
 shutil
 ------

Lib/test/test_cmath.py

@@ -1,5 +1,6 @@
 from test.support import requires_IEEE_754
 from test.test_math import parse_testfile, test_file
+import test.test_math as test_math
 import unittest
 import cmath, math
 from cmath import phase, polar, rect, pi
@@ -529,5 +530,46 @@ class CMathTests(unittest.TestCase):
             self.assertComplexIdentical(cmath.atanh(z), z)
 
 
+class IsCloseTests(test_math.IsCloseTests):
+    isclose = cmath.isclose
+
+    def test_reject_complex_tolerances(self):
+        with self.assertRaises(TypeError):
+            self.isclose(1j, 1j, rel_tol=1j)
+
+        with self.assertRaises(TypeError):
+            self.isclose(1j, 1j, abs_tol=1j)
+
+        with self.assertRaises(TypeError):
+            self.isclose(1j, 1j, rel_tol=1j, abs_tol=1j)
+
+    def test_complex_values(self):
+        # test complex values that are close to within 12 decimal places
+        complex_examples = [(1.0+1.0j, 1.000000000001+1.0j),
+                            (1.0+1.0j, 1.0+1.000000000001j),
+                            (-1.0+1.0j, -1.000000000001+1.0j),
+                            (1.0-1.0j, 1.0-0.999999999999j),
+                            ]
+
+        self.assertAllClose(complex_examples, rel_tol=1e-12)
+        self.assertAllNotClose(complex_examples, rel_tol=1e-13)
+
+    def test_complex_near_zero(self):
+        # test values near zero that are near to within three decimal places
+        near_zero_examples = [(0.001j, 0),
+                              (0.001, 0),
+                              (0.001+0.001j, 0),
+                              (-0.001+0.001j, 0),
+                              (0.001-0.001j, 0),
+                              (-0.001-0.001j, 0),
+                              ]
+
+        self.assertAllClose(near_zero_examples, abs_tol=1.5e-03)
+        self.assertAllNotClose(near_zero_examples, abs_tol=0.5e-03)
+
+        self.assertIsClose(0.001-0.001j, 0.001+0.001j, abs_tol=2e-03)
+        self.assertIsNotClose(0.001-0.001j, 0.001+0.001j, abs_tol=1e-03)
+
+
 if __name__ == "__main__":
     unittest.main()

Lib/test/test_math.py

@@ -1166,10 +1166,131 @@ class MathTests(unittest.TestCase):
                       '\n  '.join(failures))
 
 
+class IsCloseTests(unittest.TestCase):
+    isclose = math.isclose # sublcasses should override this
+
+    def assertIsClose(self, a, b, *args, **kwargs):
+        self.assertTrue(self.isclose(a, b, *args, **kwargs),
+                        msg="%s and %s should be close!" % (a, b))
+
+    def assertIsNotClose(self, a, b, *args, **kwargs):
+        self.assertFalse(self.isclose(a, b, *args, **kwargs),
+                         msg="%s and %s should not be close!" % (a, b))
+
+    def assertAllClose(self, examples, *args, **kwargs):
+        for a, b in examples:
+            self.assertIsClose(a, b, *args, **kwargs)
+
+    def assertAllNotClose(self, examples, *args, **kwargs):
+        for a, b in examples:
+            self.assertIsNotClose(a, b, *args, **kwargs)
+
+    def test_negative_tolerances(self):
+        # ValueError should be raised if either tolerance is less than zero
+        with self.assertRaises(ValueError):
+            self.assertIsClose(1, 1, rel_tol=-1e-100)
+        with self.assertRaises(ValueError):
+            self.assertIsClose(1, 1, rel_tol=1e-100, abs_tol=-1e10)
+
+    def test_identical(self):
+        # identical values must test as close
+        identical_examples = [(2.0, 2.0),
+                              (0.1e200, 0.1e200),
+                              (1.123e-300, 1.123e-300),
+                              (12345, 12345.0),
+                              (0.0, -0.0),
+                              (345678, 345678)]
+        self.assertAllClose(identical_examples, rel_tol=0.0, abs_tol=0.0)
+
+    def test_eight_decimal_places(self):
+        # examples that are close to 1e-8, but not 1e-9
+        eight_decimal_places_examples = [(1e8, 1e8 + 1),
+                                         (-1e-8, -1.000000009e-8),
+                                         (1.12345678, 1.12345679)]
+        self.assertAllClose(eight_decimal_places_examples, rel_tol=1e-8)
+        self.assertAllNotClose(eight_decimal_places_examples, rel_tol=1e-9)
+
+    def test_near_zero(self):
+        # values close to zero
+        near_zero_examples = [(1e-9, 0.0),
+                              (-1e-9, 0.0),
+                              (-1e-150, 0.0)]
+        # these should not be close to any rel_tol
+        self.assertAllNotClose(near_zero_examples, rel_tol=0.9)
+        # these should be close to abs_tol=1e-8
+        self.assertAllClose(near_zero_examples, abs_tol=1e-8)
+
+    def test_identical_infinite(self):
+        # these are close regardless of tolerance -- i.e. they are equal
+        self.assertIsClose(INF, INF)
+        self.assertIsClose(INF, INF, abs_tol=0.0)
+        self.assertIsClose(NINF, NINF)
+        self.assertIsClose(NINF, NINF, abs_tol=0.0)
+
+    def test_inf_ninf_nan(self):
+        # these should never be close (following IEEE 754 rules for equality)
+        not_close_examples = [(NAN, NAN),
+                              (NAN, 1e-100),
+                              (1e-100, NAN),
+                              (INF, NAN),
+                              (NAN, INF),
+                              (INF, NINF),
+                              (INF, 1.0),
+                              (1.0, INF),
+                              (INF, 1e308),
+                              (1e308, INF)]
+        # use largest reasonable tolerance
+        self.assertAllNotClose(not_close_examples, abs_tol=0.999999999999999)
+
+    def test_zero_tolerance(self):
+        # test with zero tolerance
+        zero_tolerance_close_examples = [(1.0, 1.0),
+                                         (-3.4, -3.4),
+                                         (-1e-300, -1e-300)]
+        self.assertAllClose(zero_tolerance_close_examples, rel_tol=0.0)
+
+        zero_tolerance_not_close_examples = [(1.0, 1.000000000000001),
+                                             (0.99999999999999, 1.0),
+                                             (1.0e200, .999999999999999e200)]
+        self.assertAllNotClose(zero_tolerance_not_close_examples, rel_tol=0.0)
+
+    def test_assymetry(self):
+        # test the assymetry example from PEP 485
+        self.assertAllClose([(9, 10), (10, 9)], rel_tol=0.1)
+
+    def test_integers(self):
+        # test with integer values
+        integer_examples = [(100000001, 100000000),
+                            (123456789, 123456788)]
+
+        self.assertAllClose(integer_examples, rel_tol=1e-8)
+        self.assertAllNotClose(integer_examples, rel_tol=1e-9)
+
+    def test_decimals(self):
+        # test with Decimal values
+        from decimal import Decimal
+
+        decimal_examples = [(Decimal('1.00000001'), Decimal('1.0')),
+                            (Decimal('1.00000001e-20'), Decimal('1.0e-20')),
+                            (Decimal('1.00000001e-100'), Decimal('1.0e-100'))]
+        self.assertAllClose(decimal_examples, rel_tol=1e-8)
+        self.assertAllNotClose(decimal_examples, rel_tol=1e-9)
+
+    def test_fractions(self):
+        # test with Fraction values
+        from fractions import Fraction
+
+        # could use some more examples here!
+        fraction_examples = [(Fraction(1, 100000000) + 1, Fraction(1))]
+        self.assertAllClose(fraction_examples, rel_tol=1e-8)
+        self.assertAllNotClose(fraction_examples, rel_tol=1e-9)
+
+
 def test_main():
     from doctest import DocFileSuite
     suite = unittest.TestSuite()
     suite.addTest(unittest.makeSuite(MathTests))
+    suite.addTest(unittest.makeSuite(IsCloseTests))
     suite.addTest(DocFileSuite("ieee754.txt"))
     run_unittest(suite)
 

Misc/NEWS

@@ -273,6 +273,9 @@ Library
 - Issue #24298: Fix inspect.signature() to correctly unwrap wrappers
   around bound methods.
 
+- Issue #24270: Add math.isclose() and cmath.isclose() functions as per PEP 485.
+  Contributed by Chris Barker and Tal Einat.
+
 IDLE
 ----
 

Modules/clinic/cmathmodule.c.h

@@ -806,4 +806,55 @@ cmath_isinf(PyModuleDef *module, PyObject *arg)
 exit:
     return return_value;
 }
-/*[clinic end generated code: output=274f59792cf4f418 input=a9049054013a1b77]*/
+
+PyDoc_STRVAR(cmath_isclose__doc__,
+"isclose($module, /, a, b, *, rel_tol=1e-09, abs_tol=0.0)\n"
+"--\n"
+"\n"
+"Determine whether two complex numbers are close in value.\n"
+"\n"
+"  rel_tol\n"
+"    maximum difference for being considered \"close\", relative to the\n"
+"    magnitude of the input values\n"
+"  abs_tol\n"
+"    maximum difference for being considered \"close\", regardless of the\n"
+"    magnitude of the input values\n"
+"\n"
+"Return True if a is close in value to b, and False otherwise.\n"
+"\n"
+"For the values to be considered close, the difference between them must be\n"
+"smaller than at least one of the tolerances.\n"
+"\n"
+"-inf, inf and NaN behave similarly to the IEEE 754 Standard. That is, NaN is\n"
+"not close to anything, even itself. inf and -inf are only close to themselves.");
+
+#define CMATH_ISCLOSE_METHODDEF    \
+    {"isclose", (PyCFunction)cmath_isclose, METH_VARARGS|METH_KEYWORDS, cmath_isclose__doc__},
+
+static int
+cmath_isclose_impl(PyModuleDef *module, Py_complex a, Py_complex b,
+                   double rel_tol, double abs_tol);
+
+static PyObject *
+cmath_isclose(PyModuleDef *module, PyObject *args, PyObject *kwargs)
+{
+    PyObject *return_value = NULL;
+    static char *_keywords[] = {"a", "b", "rel_tol", "abs_tol", NULL};
+    Py_complex a;
+    Py_complex b;
+    double rel_tol = 1e-09;
+    double abs_tol = 0.0;
+    int _return_value;
+
+    if (!PyArg_ParseTupleAndKeywords(args, kwargs, "DD|$dd:isclose", _keywords,
+        &a, &b, &rel_tol, &abs_tol))
+        goto exit;
+    _return_value = cmath_isclose_impl(module, a, b, rel_tol, abs_tol);
+    if ((_return_value == -1) && PyErr_Occurred())
+        goto exit;
+    return_value = PyBool_FromLong((long)_return_value);
+
+exit:
+    return return_value;
+}
+/*[clinic end generated code: output=229e9c48c9d27362 input=a9049054013a1b77]*/

Modules/cmathmodule.c

@@ -1114,6 +1114,73 @@ cmath_isinf_impl(PyModuleDef *module, Py_complex z)
                            Py_IS_INFINITY(z.imag));
 }
 
+/*[clinic input]
+cmath.isclose -> bool
+
+    a: Py_complex
+    b: Py_complex
+    *
+    rel_tol: double = 1e-09
+        maximum difference for being considered "close", relative to the
+        magnitude of the input values
+    abs_tol: double = 0.0
+        maximum difference for being considered "close", regardless of the
+        magnitude of the input values
+
+Determine whether two complex numbers are close in value.
+
+Return True if a is close in value to b, and False otherwise.
+
+For the values to be considered close, the difference between them must be
+smaller than at least one of the tolerances.
+
+-inf, inf and NaN behave similarly to the IEEE 754 Standard. That is, NaN is
+not close to anything, even itself. inf and -inf are only close to themselves.
+[clinic start generated code]*/
+
+static int
+cmath_isclose_impl(PyModuleDef *module, Py_complex a, Py_complex b,
+                   double rel_tol, double abs_tol)
+/*[clinic end generated code: output=da0c535fb54e2310 input=df9636d7de1d4ac3]*/
+{
+    double diff;
+
+    /* sanity check on the inputs */
+    if (rel_tol < 0.0 || abs_tol < 0.0 ) {
+        PyErr_SetString(PyExc_ValueError,
+                        "tolerances must be non-negative");
+        return -1;
+    }
+
+    if ( (a.real == b.real) && (a.imag == b.imag) ) {
+        /* short circuit exact equality -- needed to catch two infinities of
+           the same sign. And perhaps speeds things up a bit sometimes.
+        */
+        return 1;
+    }
+
+    /* This catches the case of two infinities of opposite sign, or
+       one infinity and one finite number. Two infinities of opposite
+       sign would otherwise have an infinite relative tolerance.
+       Two infinities of the same sign are caught by the equality check
+       above.
+    */
+
+    if (Py_IS_INFINITY(a.real) || Py_IS_INFINITY(a.imag) ||
+        Py_IS_INFINITY(b.real) || Py_IS_INFINITY(b.imag)) {
+        return 0;
+    }
+
+    /* now do the regular computation
+       this is essentially the "weak" test from the Boost library
+    */
+
+    diff = _Py_c_abs(_Py_c_diff(a, b));
+
+    return (((diff <= rel_tol * _Py_c_abs(b)) ||
+             (diff <= rel_tol * _Py_c_abs(a))) ||
+            (diff <= abs_tol));
+}
 
 PyDoc_STRVAR(module_doc,
 "This module is always available. It provides access to mathematical\n"
@@ -1129,6 +1196,7 @@ static PyMethodDef cmath_methods[] = {
     CMATH_COS_METHODDEF
     CMATH_COSH_METHODDEF
     CMATH_EXP_METHODDEF
+    CMATH_ISCLOSE_METHODDEF
     CMATH_ISFINITE_METHODDEF
     CMATH_ISINF_METHODDEF
     CMATH_ISNAN_METHODDEF

Modules/mathmodule.c

@@ -1990,6 +1990,83 @@ PyDoc_STRVAR(math_isinf_doc,
 "isinf(x) -> bool\n\n\
 Return True if x is a positive or negative infinity, and False otherwise.");
 
+static PyObject *
+math_isclose(PyObject *self, PyObject *args, PyObject *kwargs)
+{
+    double a, b;
+    double rel_tol = 1e-9;
+    double abs_tol = 0.0;
+    double diff = 0.0;
+    long result = 0;
+
+    static char *keywords[] = {"a", "b", "rel_tol", "abs_tol", NULL};
+
+
+    if (!PyArg_ParseTupleAndKeywords(args, kwargs, "dd|$dd:isclose",
+                                     keywords,
+                                     &a, &b, &rel_tol, &abs_tol
+                                     ))
+        return NULL;
+
+    /* sanity check on the inputs */
+    if (rel_tol < 0.0 || abs_tol < 0.0 ) {
+        PyErr_SetString(PyExc_ValueError,
+                        "tolerances must be non-negative");
+        return NULL;
+    }
+
+    if ( a == b ) {
+        /* short circuit exact equality -- needed to catch two infinities of
+           the same sign. And perhaps speeds things up a bit sometimes.
+        */
+        Py_RETURN_TRUE;
+    }
+
+    /* This catches the case of two infinities of opposite sign, or
+       one infinity and one finite number. Two infinities of opposite
+       sign would otherwise have an infinite relative tolerance.
+       Two infinities of the same sign are caught by the equality check
+       above.
+    */
+
+    if (Py_IS_INFINITY(a) || Py_IS_INFINITY(b)) {
+        Py_RETURN_FALSE;
+    }
+
+    /* now do the regular computation
+       this is essentially the "weak" test from the Boost library
+    */
+
+    diff = fabs(b - a);
+
+    result = (((diff <= fabs(rel_tol * b)) ||
+               (diff <= fabs(rel_tol * a))) ||
+              (diff <= abs_tol));
+
+    return PyBool_FromLong(result);
+}
+
+PyDoc_STRVAR(math_isclose_doc,
+"is_close(a, b, *, rel_tol=1e-09, abs_tol=0.0) -> bool\n"
+"\n"
+"Determine whether two floating point numbers are close in value.\n"
+"\n"
+"   rel_tol\n"
+"       maximum difference for being considered \"close\", relative to the\n"
+"       magnitude of the input values\n"
+"    abs_tol\n"
+"       maximum difference for being considered \"close\", regardless of the\n"
+"       magnitude of the input values\n"
+"\n"
+"Return True if a is close in value to b, and False otherwise.\n"
+"\n"
+"For the values to be considered close, the difference between them\n"
+"must be smaller than at least one of the tolerances.\n"
+"\n"
+"-inf, inf and NaN behave similarly to the IEEE 754 Standard.  That\n"
+"is, NaN is not close to anything, even itself.  inf and -inf are\n"
+"only close to themselves.");
+
 static PyMethodDef math_methods[] = {
     {"acos",            math_acos,      METH_O,         math_acos_doc},
     {"acosh",           math_acosh,     METH_O,         math_acosh_doc},
@@ -2016,6 +2093,8 @@ static PyMethodDef math_methods[] = {
     {"gamma",           math_gamma,     METH_O,         math_gamma_doc},
     {"gcd",             math_gcd,       METH_VARARGS,   math_gcd_doc},
     {"hypot",           math_hypot,     METH_VARARGS,   math_hypot_doc},
+    {"isclose", (PyCFunction) math_isclose, METH_VARARGS | METH_KEYWORDS,
+    math_isclose_doc},
     {"isfinite",        math_isfinite,  METH_O,         math_isfinite_doc},
     {"isinf",           math_isinf,     METH_O,         math_isinf_doc},
     {"isnan",           math_isnan,     METH_O,         math_isnan_doc},