Index: Objects/floatobject.c
===================================================================
--- Objects/floatobject.c	(revision 64857)
+++ Objects/floatobject.c	(working copy)
@@ -10,6 +10,11 @@
 #include <ctype.h>
 #include <float.h>
 
+#undef MAX
+#undef MIN
+#define MAX(x, y) ((x) < (y) ? (y) : (x))
+#define MIN(x, y) ((x) < (y) ? (x) : (y))
+
 #ifdef HAVE_IEEEFP_H
 #include <ieeefp.h>
 #endif
@@ -1109,7 +1114,312 @@
 	return v;
 }
 
+/* turn ASCII hex characters into integer values and vice versa */
+
+static char
+char_from_hex(int x)
+{
+	assert(0 <= x && x < 16);
+	if (x < 10)
+		return '0' + x;
+	else
+		return 'a' + (x-10);
+}
+
+static int
+hex_from_char(char c) {
+	assert(isxdigit(c));
+	return ('0' <= c && c <= '9' ? (int)(c - '0') :
+		'a' <= c && c <= 'f' ? (int)(c - 'a') + 10 :
+		(int)(c - 'A') + 10);
+}
+
+/* convert a float to a hexadecimal string */
+
+/* TOHEX_NBITS is DBL_MANT_DIG rounded up to the next integer
+   of the form 4k+1. */
+#define TOHEX_NBITS DBL_MANT_DIG + 3 - (DBL_MANT_DIG+2)%4
+
 static PyObject *
+float_hex(PyObject *v)
+{
+	double x, m;
+	int e, shift, i, si;
+	/* Space for 1+(TOHEX_NBITS-1)/4 digits, a decimal point, and the
+	   trailing NUL byte. */
+	char s[(TOHEX_NBITS-1)/4+3];
+
+	CONVERT_TO_DOUBLE(v, x);
+
+	if (Py_IS_NAN(x) || Py_IS_INFINITY(x))
+		return float_str((PyFloatObject *)v);
+
+	if (x == 0.0) {
+		if(copysign(1.0, x) == -1.0)
+			return PyString_FromString("-0x0p0");
+		else
+			return PyString_FromString("0x0p0");
+	}
+
+	m = frexp(fabs(x), &e);
+	shift = 1 - MAX(DBL_MIN_EXP - e, 0);
+	m = ldexp(m, shift);
+	e -= shift;
+
+	si = 0;
+	s[si++] = char_from_hex((int)m);
+	m -= (int)m;
+	s[si++] = '.';
+	for (i=0; i < (TOHEX_NBITS-1)/4; i++) {
+		m *= 16.0;
+		s[si++] = char_from_hex((int)m);
+		m -= (int)m;
+	}
+	s[si++] = '\0';
+
+	if (x < 0.0)
+		return PyString_FromFormat("-0x%sp%d", s, e);
+	else
+		return PyString_FromFormat("0x%sp%d", s, e);
+}
+
+/* Convert a hexadecimal string to a float. */
+
+static PyObject *
+float_fromhex(PyObject *cls, PyObject *arg)
+{
+	PyObject *result_as_float, *result;
+	double x;
+	long exp, top_exp, lsb, key_digit;
+	char *s, *coeff_start, *dot_pos, *coeff_end, *exp_start, *s_end;
+	int half_eps, digit, round_up, sign=1;
+	Py_ssize_t length, ndigits, fdigits, i;
+
+	/*
+	 * For the sake of simplicity and correctness, we impose an artificial
+	 * limit on ndigits, the total number of hex digits in the coefficient
+	 * The limit is chosen to ensure that, writing exp for the exponent,
+	 *
+	 *   (1) if exp > LONG_MAX/2 then the value of the hex string is
+	 *   guaranteed to overflow (provided it's nonzero)
+	 *
+	 *   (2) if exp < LONG_MIN/2 then the value of the hex string is
+	 *   guaranteed to underflow to 0.
+	 *
+	 *   (3) if LONG_MIN/2 <= exp <= LONG_MAX/2 then there's no danger of
+	 *   overflow in the calculation of exp and top_exp below.
+	 *
+	 * More specifically, ndigits is assumed to satisfy the following
+	 * inequalities:
+	 *
+	 *   4*ndigits <= DBL_MIN_EXP - DBL_MANT_DIG - LONG_MIN/2
+	 *   4*ndigits <= LONG_MAX/2 + 1 - DBL_MAX_EXP
+	 *
+	 * If either of these inequalities is not satisfied, a ValueError is
+	 * raised.  Otherwise, write x for the value of the hex string, and
+	 * assume x is nonzero.  Then
+	 *
+	 *   2**(exp-4*ndigits) <= |x| < 2**(exp+4*ndigits).
+	 *
+	 * Now if exp > LONG_MAX/2 then:
+	 *
+	 *   exp - 4*ndigits >= LONG_MAX/2 + 1 - (LONG_MAX/2 + 1 - DBL_MAX_EXP)
+	 *                    = DBL_MAX_EXP
+	 *
+	 * so |x| >= 2**DBL_MAX_EXP, which is too large to be stored in C
+	 * double, so overflows.  If exp < LONG_MIN/2, then
+	 *
+	 *   exp + 4*ndigits <= LONG_MIN/2 - 1 + (
+	 *                      DBL_MIN_EXP - DBL_MANT_DIG - LONG_MIN/2)
+	 *                    = DBL_MIN_EXP - DBL_MANT_DIG - 1
+	 *
+	 * and so |x| < 2**(DBL_MIN_EXP-DBL_MANT_DIG-1), hence underflows to 0
+	 * when converted to a C double.
+	 *
+	 * It's easy to show that if LONG_MIN/2 <= exp <= LONG_MAX/2 then both
+	 * exp+4*ndigits and exp-4*ndigits are within the range of a long.
+	 */
+
+	if (PyString_AsStringAndSize(arg, &s, &length))
+		return NULL;
+	s_end = s + length;
+
+	/********************
+	 * Parse the string *
+	 ********************/
+
+	/* leading whitespace and optional sign */
+	while (isspace(*s))
+		s++;
+	if (*s == '-') {
+		s++;
+		sign = -1;
+	}
+	else if (*s == '+')
+		s++;
+
+	/* infinities and nans */
+	if (PyOS_mystrnicmp(s, "nan", 4) == 0) {
+		x = Py_NAN;
+		goto finished;
+	}
+	if (PyOS_mystrnicmp(s, "inf", 4) == 0 ||
+	    PyOS_mystrnicmp(s, "infinity", 9) == 0) {
+		x = sign*Py_HUGE_VAL;
+		goto finished;
+	}
+
+	/* 0x <integer> [. <fraction>] */
+	if (*s++ != '0' || (tolower(*s++) != (int)'x'))
+		goto parse_error;
+	coeff_start = s;
+	while (isxdigit(*s))
+		s++;
+	dot_pos = s;
+	if (*s == '.') {
+		s++;
+		while (isxdigit(*s))
+			s++;
+		coeff_end = s-1;
+	}
+	else
+		coeff_end = s;
+
+	/* ndigits = total # of hex digits; fdigits = # after point */
+	ndigits = coeff_end - coeff_start;
+	fdigits = coeff_end - dot_pos;
+	if (ndigits == 0)
+		goto parse_error;
+	if (ndigits > MIN(DBL_MIN_EXP - DBL_MANT_DIG - LONG_MIN/2,
+			  LONG_MAX/2 + 1 - DBL_MAX_EXP)/4)
+		goto insane_length_error;
+
+	/* [p <exponent>] */
+	if (tolower(*s) == (int)'p') {
+		s++;
+		exp_start = s;
+		if (*s == '-' || *s == '+')
+			s++;
+		if (!isdigit(*s++))
+			goto parse_error;
+		while (isdigit(*s))
+			s++;
+		exp = strtol(exp_start, NULL, 10);
+	}
+	else
+		exp = 0;
+
+	/* optional trailing whitespace leading to the end of the string */
+	while (isspace(*s))
+		s++;
+	if (s != s_end)
+		goto parse_error;
+
+/* for 0 <= j < ndigits, HEX_DIGIT(j) gives the jth most significant digit */
+#define HEX_DIGIT(j) hex_from_char(*((j) < fdigits ?		\
+				     coeff_end-(j) :			\
+				     coeff_end-1-(j)))
+
+	/*******************************************
+	 * Compute rounded value of the hex string *
+	 *******************************************/
+
+	/* Discard leading zeros, and catch extreme overflow and underflow */
+	while (ndigits > 0 && HEX_DIGIT(ndigits-1) == 0)
+		ndigits--;
+	if (ndigits == 0 || exp < LONG_MIN/2) {
+		x = sign * 0.0;
+		goto finished;
+	}
+	if (exp > LONG_MAX/2)
+		goto overflow_error;
+
+	/* Adjust exponent for fractional part. */
+	exp = exp - 4*((long)fdigits);
+
+	/* top_exp = 1 more than exponent of most sig. bit of coefficient */
+	top_exp = exp + 4*((long)ndigits - 1);
+	for (digit = HEX_DIGIT(ndigits-1); digit != 0; digit /= 2)
+		top_exp += 1;
+
+	/* catch almost all nonextreme cases of overflow and underflow here */
+	if (top_exp < DBL_MIN_EXP - DBL_MANT_DIG) {
+		x = sign * 0.0;
+		goto finished;
+	}
+	if (top_exp > DBL_MAX_EXP)
+		goto overflow_error;
+
+	/* lsb = exponent of least significant bit of the *rounded* value.
+	   This is top_exp - DBL_MANT_DIG unless result is subnormal. */
+	lsb = MAX(top_exp, (long)DBL_MIN_EXP) - DBL_MANT_DIG;
+
+	x = 0.0;
+	if (exp >= lsb) {
+		/* no rounding required */
+		for (i = ndigits-1; i >= 0; i--)
+			x = 16.0*x + HEX_DIGIT(i);
+		x = sign * ldexp(x, (int)(exp));
+		goto finished;
+	}
+	/* rounding required.  key_digit is the index of the hex digit
+	   containing the first bit to be rounded away. */
+	half_eps = 1 << (int)((lsb - exp - 1) % 4);
+	key_digit = (lsb - exp - 1) / 4;
+	for (i = ndigits-1; i > key_digit; i--)
+		x = 16.0*x + HEX_DIGIT(i);
+	digit = HEX_DIGIT(key_digit);
+	x = 16.0*x + (double)(digit & (16-2*half_eps));
+
+	/* round-half-even: round up if bit lsb-1 is 1 and at least one of
+	   bits lsb, lsb-2, lsb-3, lsb-4, ... is 1. */
+	if ((digit & half_eps) != 0) {
+		round_up = 0;
+		if ((digit & (3*half_eps-1)) != 0 ||
+		    (half_eps == 8 && (HEX_DIGIT(key_digit+1) & 1) != 0))
+			round_up = 1;
+		else
+			for (i = key_digit-1; i >= 0; i--)
+				if (HEX_DIGIT(i) != 0) {
+					round_up = 1;
+					break;
+				}
+		if (round_up == 1) {
+			x += 2*half_eps;
+			if (top_exp == DBL_MAX_EXP &&
+			    /* overflow corner case: pre-rounded value is <
+			       2**DBL_MAX_EXP; rounded == 2**DBL_MAX_EXP. */
+			    x == ldexp((double)(2*half_eps), DBL_MANT_DIG))
+				goto overflow_error;
+		}
+	}
+	x = sign * ldexp(x, (int)(exp+4*key_digit));
+
+  finished:
+	result_as_float = Py_BuildValue("(d)", x);
+	if (result_as_float == NULL)
+		return NULL;
+	result = PyObject_CallObject(cls, result_as_float);
+	Py_DECREF(result_as_float);
+	return result;
+
+  overflow_error:
+	PyErr_SetString(PyExc_OverflowError,
+			"hexadecimal value too large to represent as a float");
+	return NULL;
+
+  parse_error:
+	PyErr_SetString(PyExc_ValueError,
+			"invalid hexadecimal floating-point string");
+	return NULL;
+
+  insane_length_error:
+	PyErr_SetString(PyExc_ValueError,
+			"hexadecimal string too long to convert");
+	return NULL;
+}
+
+static PyObject *
 float_as_integer_ratio(PyObject *v, PyObject *unused)
 {
 	double self;
@@ -1433,6 +1743,10 @@
          "Returns the Integral closest to x between 0 and x."},
 	{"as_integer_ratio", (PyCFunction)float_as_integer_ratio, METH_NOARGS,
 	 float_as_integer_ratio_doc},
+	{"fromhex", (PyCFunction)float_fromhex, METH_O|METH_CLASS,
+	 "Create a float from a hexadecimal string."},
+	{"hex", (PyCFunction)float_hex, METH_NOARGS,
+	 "Give represention of a float as a hexadecimal string."},
 	{"is_integer",	(PyCFunction)float_is_integer,	METH_NOARGS,
 	 "Returns True if the float is an integer."},
 #if 0
Index: Doc/whatsnew/2.6.rst
===================================================================
--- Doc/whatsnew/2.6.rst	(revision 64857)
+++ Doc/whatsnew/2.6.rst	(working copy)
@@ -1397,6 +1397,10 @@
   :func:`isnan`, return true if their floating-point argument is
   infinite or Not A Number.  (:issue:`1640`)
 
+  The float type has new methods :meth:`float.hex` and
+  :meth:`float.fromhex` to convert floating-point numbers to and from
+  hexadecimal strings, respectively.  (:issue:`3008`)
+
 * The :mod:`math` module has a number of new functions, and the existing
   functions have been improved to give more consistent behaviour
   across platforms, especially with respect to handling of
Index: Doc/library/stdtypes.rst
===================================================================
--- Doc/library/stdtypes.rst	(revision 64857)
+++ Doc/library/stdtypes.rst	(working copy)
@@ -448,6 +448,60 @@
 
 .. _typeiter:
 
+
+Additional Methods on Float
+---------------------------
+
+The float type has some additional methods to support conversion to
+and from hexadecimal strings.  Since Python's floats are stored
+internally as binary numbers, converting a float to or from a
+*decimal* string usually involves a small rounding error.  In
+contrast, hexadecimal strings allow exact representation and
+specification of floating-point numbers.  This can be useful when
+debugging, and in numerical work.
+
+
+.. method:: float.hex()
+
+   Return a representation of a floating-point number as a hexadecimal string.
+
+
+.. method:: float.fromhex(s)
+
+   Return the float represented by a hexadecimal string *s*.  The
+   string *s* may have leading and trailing whitespace.
+
+
+A hexadecimal string takes the form::
+
+   [sign] '0x' integer ['.' fraction] ['p' exponent]
+
+where the optional ``sign`` may by either '+' or '-', ``integer`` and
+``fraction`` are strings of hexadecimal digits, and ``exponent`` is a
+decimal integer with an optional leading sign.  Case is not
+significant, and there must be at least one hexadecimal digit in
+either the integer or the fraction.  (This syntax is identical to the
+syntax specified in section 6.4.4.2 of the C99 standard, and also to
+the syntax used in Java 1.5 onwards.)
+
+Note that the exponent is written in decimal rather than hexadecimal,
+and that it gives the power of 2 by which to multiply the coefficient.
+
+For example, the hexadecimal string ``0x3.a7p10`` represents the
+floating-point number ``(3 + 10./16 + 7./16**2) * 2.0**10``, or
+``3740.0``::
+
+   >>> float.fromhex('0x3.a7p10')
+   3740.0
+
+
+Applying the reverse conversion to ``3740.0`` gives a different
+hexadecimal string representing the same number::
+
+   >>> float.hex(3740.0)
+   '0x1.d380000000000p11'
+
+
 Iterator Types
 ==============
 
Index: Lib/test/test_float.py
===================================================================
--- Lib/test/test_float.py	(revision 64857)
+++ Lib/test/test_float.py	(working copy)
@@ -3,7 +3,7 @@
 import os
 from test import test_support
 import math
-from math import isinf, isnan
+from math import isinf, isnan, copysign, ldexp
 import operator
 
 INF = float("inf")
@@ -343,7 +343,387 @@
         self.failIf(NAN.is_inf())
         self.failIf((0.).is_inf())
 
+fromHex = float.fromhex
+toHex = float.hex
+class HexFloatTestCase(unittest.TestCase):
+    MAX = fromHex('0x.fffffffffffff8p+1024')  # max normal
+    MIN = fromHex('0x1p-1022')                # min normal
+    TINY = fromHex('0x0.0000000000001p-1022') # min subnormal
+    EPS = fromHex('0x0.0000000000001p0') # diff between 1.0 and next float up
 
+    def identical(self, x, y):
+        # check that floats x and y are identical, or that both
+        # are NaNs
+        if isnan(x) or isnan(y):
+            if isnan(x) == isnan(y):
+                return
+        elif x == y and (x != 0.0 or copysign(1.0, x) == copysign(1.0, y)):
+            return
+        self.fail('%r not identical to %r' % (x, y))
+
+    def test_ends(self):
+        self.identical(self.MIN, 2.**-1022)
+        self.identical(self.TINY, 2.**-1074)
+        self.identical(self.EPS, 2.**-52)
+        self.identical(self.MAX, 2.*(2.**1023 - 2.**970))
+
+    def test_invalid_inputs(self):
+        invalid_inputs = [
+            'infi',   # misspelt infinities and nans
+            '-Infinit',
+            '++inf',
+            '-+Inf',
+            '--nan',
+            '+-NaN',
+            'snan',
+            'NaNs',
+            'nna',
+            '0xnan',
+            '',
+            ' ',
+            '1.0p0',  # no hex indicator
+            '3',
+            '2.0',
+            'x1.0p0',
+            '0xX1.0p0',
+            '+ 0x1.0p0', # internal whitespace
+            '- 0x1.0p0',
+            '0 x1.0p0',
+            '0x 1.0p0',
+            '0x1 2.0p0',
+            '+0x1 .0p0',
+            '0x1. 0p0',
+            '-0x1.0 1p0',
+            '-0x1.0 p0',
+            '+0x1.0p +0',
+            '0x1.0p -0',
+            '0x1.0p 0',
+            '+0x1.0p+ 0',
+            '-0x1.0p- 0',
+            '++0x1.0p-0', # double signs
+            '--0x1.0p0',
+            '+-0x1.0p+0',
+            '-+0x1.0p0',
+            '0x1.0p++0',
+            '+0x1.0p+-0',
+            '-0x1.0p-+0',
+            '0x1.0p--0',
+            '0x1.0.p0',
+            '0x.p0', # no hex digits before or after point
+            '0x1,p0', # wrong decimal point character
+            '0x1pa',
+            u'0x1p\uff10',  # fullwidth Unicode digits
+            u'\uff10x1p0',
+            u'0x\uff11p0',
+            u'0x1.\uff10p0',
+            '0x1p0 \n 0x2p0',
+            '0x1p0\0 0x1p0',  # embedded null byte is not end of string
+            ]
+        for x in invalid_inputs:
+            self.assertRaises(ValueError, fromHex, x)
+
+
+    def test_from_hex(self):
+        MIN = self.MIN;
+        MAX = self.MAX;
+        TINY = self.TINY;
+        EPS = self.EPS;
+
+        # two spellings of infinity, with optional signs; case-insensitive
+        self.identical(fromHex('inf'), INF)
+        self.identical(fromHex('+Inf'), INF)
+        self.identical(fromHex('-INF'), -INF)
+        self.identical(fromHex('iNf'), INF)
+        self.identical(fromHex('Infinity'), INF)
+        self.identical(fromHex('+INFINITY'), INF)
+        self.identical(fromHex('-infinity'), -INF)
+        self.identical(fromHex('-iNFiNitY'), -INF)
+
+        # nans with optional sign; case insensitive
+        self.identical(fromHex('nan'), NAN)
+        self.identical(fromHex('+NaN'), NAN)
+        self.identical(fromHex('-NaN'), NAN)
+        self.identical(fromHex('-nAN'), NAN)
+
+        # variations in input format
+        self.identical(fromHex('0x1'), 1.0)    
+        self.identical(fromHex('0x1.'), 1.0)
+        self.identical(fromHex('0x1.0'), 1.0)
+        self.identical(fromHex('+0x1.0'), 1.0)
+        self.identical(fromHex('0x1p0'), 1.0)
+        self.identical(fromHex('0X1p0'), 1.0)
+        self.identical(fromHex('0X1P0'), 1.0)
+        self.identical(fromHex('0x1P0'), 1.0)
+        self.identical(fromHex('0x1.p0'), 1.0)
+        self.identical(fromHex('0x1.0p0'), 1.0)
+        self.identical(fromHex('0x.1p4'), 1.0)
+        self.identical(fromHex('0x.1p04'), 1.0)
+        self.identical(fromHex('0x.1p004'), 1.0)
+        self.identical(fromHex('0x1p+0'), 1.0)
+        self.identical(fromHex('0x1P-0'), 1.0)
+        self.identical(fromHex('+0x1p0'), 1.0)
+        self.identical(fromHex('0x01p0'), 1.0)
+        self.identical(fromHex('0x1p00'), 1.0)
+        self.identical(fromHex(u'0x1p0'), 1.0)
+        self.identical(fromHex(' 0x1p0 '), 1.0)
+        self.identical(fromHex('\n 0x1p0'), 1.0)
+        self.identical(fromHex('0x1p0 \t'), 1.0)
+        self.identical(fromHex('0xap0'), 10.0)
+        self.identical(fromHex('0xAp0'), 10.0)
+        self.identical(fromHex('0xaP0'), 10.0)
+        self.identical(fromHex('0xAP0'), 10.0)
+        self.identical(fromHex('0xbep0'), 190.0)
+        self.identical(fromHex('0xBep0'), 190.0)
+        self.identical(fromHex('0xbEp0'), 190.0)
+        self.identical(fromHex('0XBE0P-4'), 190.0)
+        self.identical(fromHex('0xBEp0'), 190.0)
+        self.identical(fromHex('0xB.Ep4'), 190.0)
+        self.identical(fromHex('0x.BEp8'), 190.0)
+        self.identical(fromHex('0x.0BEp12'), 190.0)
+
+        # moving the point around
+        pi = fromHex('0x1.921fb54442d18p1')
+        self.identical(fromHex('0x.006487ed5110b46p11'), pi)
+        self.identical(fromHex('0x.00c90fdaa22168cp10'), pi)
+        self.identical(fromHex('0x.01921fb54442d18p9'), pi)
+        self.identical(fromHex('0x.03243f6a8885a3p8'), pi)
+        self.identical(fromHex('0x.06487ed5110b46p7'), pi)
+        self.identical(fromHex('0x.0c90fdaa22168cp6'), pi)
+        self.identical(fromHex('0x.1921fb54442d18p5'), pi)
+        self.identical(fromHex('0x.3243f6a8885a3p4'), pi)
+        self.identical(fromHex('0x.6487ed5110b46p3'), pi)
+        self.identical(fromHex('0x.c90fdaa22168cp2'), pi)
+        self.identical(fromHex('0x1.921fb54442d18p1'), pi)
+        self.identical(fromHex('0x3.243f6a8885a3p0'), pi)
+        self.identical(fromHex('0x6.487ed5110b46p-1'), pi)
+        self.identical(fromHex('0xc.90fdaa22168cp-2'), pi)
+        self.identical(fromHex('0x19.21fb54442d18p-3'), pi)
+        self.identical(fromHex('0x32.43f6a8885a3p-4'), pi)
+        self.identical(fromHex('0x64.87ed5110b46p-5'), pi)
+        self.identical(fromHex('0xc9.0fdaa22168cp-6'), pi)
+        self.identical(fromHex('0x192.1fb54442d18p-7'), pi)
+        self.identical(fromHex('0x324.3f6a8885a3p-8'), pi)
+        self.identical(fromHex('0x648.7ed5110b46p-9'), pi)
+        self.identical(fromHex('0xc90.fdaa22168cp-10'), pi)
+        self.identical(fromHex('0x1921.fb54442d18p-11'), pi)
+        # ...
+        self.identical(fromHex('0x1921fb54442d1.8p-47'), pi)
+        self.identical(fromHex('0x3243f6a8885a3p-48'), pi)
+        self.identical(fromHex('0x6487ed5110b46p-49'), pi)
+        self.identical(fromHex('0xc90fdaa22168cp-50'), pi)
+        self.identical(fromHex('0x1921fb54442d18p-51'), pi)
+        self.identical(fromHex('0x3243f6a8885a30p-52'), pi)
+        self.identical(fromHex('0x6487ed5110b460p-53'), pi)
+        self.identical(fromHex('0xc90fdaa22168c0p-54'), pi)
+        self.identical(fromHex('0x1921fb54442d180p-55'), pi)
+
+
+        # results that should overflow...
+        self.assertRaises(OverflowError, fromHex, '-0x1p1024')
+        self.assertRaises(OverflowError, fromHex, '0x1p+1025')
+        self.assertRaises(OverflowError, fromHex, '+0X1p1030')
+        self.assertRaises(OverflowError, fromHex, '-0x1p+1100')
+        self.assertRaises(OverflowError, fromHex, '0X1p123456789123456789')
+        self.assertRaises(OverflowError, fromHex, '+0X.8p+1025')
+        self.assertRaises(OverflowError, fromHex, '+0x0.8p1025')
+        self.assertRaises(OverflowError, fromHex, '-0x0.4p1026')
+        self.assertRaises(OverflowError, fromHex, '0X2p+1023')
+        self.assertRaises(OverflowError, fromHex, '0x2.p1023')
+        self.assertRaises(OverflowError, fromHex, '-0x2.0p+1023')
+        self.assertRaises(OverflowError, fromHex, '+0X4p+1022')
+        self.assertRaises(OverflowError, fromHex, '0x1.ffffffffffffffp+1023')
+        self.assertRaises(OverflowError, fromHex, '-0X1.fffffffffffff9p1023')
+        self.assertRaises(OverflowError, fromHex, '0X1.fffffffffffff8p1023')
+        self.assertRaises(OverflowError, fromHex, '+0x3.fffffffffffffp1022')
+        self.assertRaises(OverflowError, fromHex, '0x3fffffffffffffp+970')
+        self.assertRaises(OverflowError, fromHex, '0x10000000000000000p960')
+        self.assertRaises(OverflowError, fromHex, '-0Xffffffffffffffffp960')
+
+        # ...and those that round to +-max float
+        self.identical(fromHex('+0x1.fffffffffffffp+1023'), MAX)
+        self.identical(fromHex('-0X1.fffffffffffff7p1023'), -MAX)
+        self.identical(fromHex('0X1.fffffffffffff7fffffffffffffp1023'), MAX)
+
+        # zeros
+        self.identical(fromHex('0x0p0'), 0.0)
+        self.identical(fromHex('0x0p1000'), 0.0)
+        self.identical(fromHex('-0x0p1023'), -0.0)
+        self.identical(fromHex('0X0p1024'), 0.0)
+        self.identical(fromHex('-0x0p1025'), -0.0)
+        self.identical(fromHex('0X0p2000'), 0.0)
+        self.identical(fromHex('0x0p123456789123456789'), 0.0)
+        self.identical(fromHex('-0X0p-0'), -0.0)
+        self.identical(fromHex('-0X0p-1000'), -0.0)
+        self.identical(fromHex('0x0p-1023'), 0.0)
+        self.identical(fromHex('-0X0p-1024'), -0.0)
+        self.identical(fromHex('-0x0p-1025'), -0.0)
+        self.identical(fromHex('-0x0p-1072'), -0.0)
+        self.identical(fromHex('0X0p-1073'), 0.0)
+        self.identical(fromHex('-0x0p-1074'), -0.0)
+        self.identical(fromHex('0x0p-1075'), 0.0)
+        self.identical(fromHex('0X0p-1076'), 0.0)
+        self.identical(fromHex('-0X0p-2000'), -0.0)
+        self.identical(fromHex('-0x0p-123456789123456789'), -0.0)
+
+        # values that should underflow to 0
+        self.identical(fromHex('0X1p-1075'), 0.0)
+        self.identical(fromHex('-0X1p-1075'), -0.0)
+        self.identical(fromHex('-0x1p-123456789123456789'), -0.0)
+        self.identical(fromHex('0x1.00000000000000001p-1075'), TINY)
+        self.identical(fromHex('-0x1.1p-1075'), -TINY)
+        self.identical(fromHex('0x1.fffffffffffffffffp-1075'), TINY)
+
+        # check round-half-even is working correctly near 0 ...
+        self.identical(fromHex('0x1p-1076'), 0.0)
+        self.identical(fromHex('0X2p-1076'), 0.0)
+        self.identical(fromHex('0X3p-1076'), TINY)
+        self.identical(fromHex('0x4p-1076'), TINY)
+        self.identical(fromHex('0X5p-1076'), TINY)
+        self.identical(fromHex('0X6p-1076'), 2*TINY)
+        self.identical(fromHex('0x7p-1076'), 2*TINY)
+        self.identical(fromHex('0X8p-1076'), 2*TINY)
+        self.identical(fromHex('0X9p-1076'), 2*TINY)
+        self.identical(fromHex('0xap-1076'), 2*TINY)
+        self.identical(fromHex('0Xbp-1076'), 3*TINY)
+        self.identical(fromHex('0xcp-1076'), 3*TINY)
+        self.identical(fromHex('0Xdp-1076'), 3*TINY)
+        self.identical(fromHex('0Xep-1076'), 4*TINY)
+        self.identical(fromHex('0xfp-1076'), 4*TINY)
+        self.identical(fromHex('0x10p-1076'), 4*TINY)
+        self.identical(fromHex('-0x1p-1076'), -0.0)
+        self.identical(fromHex('-0X2p-1076'), -0.0)
+        self.identical(fromHex('-0x3p-1076'), -TINY)
+        self.identical(fromHex('-0X4p-1076'), -TINY)
+        self.identical(fromHex('-0x5p-1076'), -TINY)
+        self.identical(fromHex('-0x6p-1076'), -2*TINY)
+        self.identical(fromHex('-0X7p-1076'), -2*TINY)
+        self.identical(fromHex('-0X8p-1076'), -2*TINY)
+        self.identical(fromHex('-0X9p-1076'), -2*TINY)
+        self.identical(fromHex('-0Xap-1076'), -2*TINY)
+        self.identical(fromHex('-0xbp-1076'), -3*TINY)
+        self.identical(fromHex('-0xcp-1076'), -3*TINY)
+        self.identical(fromHex('-0Xdp-1076'), -3*TINY)
+        self.identical(fromHex('-0xep-1076'), -4*TINY)
+        self.identical(fromHex('-0Xfp-1076'), -4*TINY)
+        self.identical(fromHex('-0X10p-1076'), -4*TINY)
+
+        # ... and near MIN ...
+        self.identical(fromHex('0x0.ffffffffffffd6p-1022'), MIN-3*TINY)
+        self.identical(fromHex('0x0.ffffffffffffd8p-1022'), MIN-2*TINY)
+        self.identical(fromHex('0x0.ffffffffffffdap-1022'), MIN-2*TINY)
+        self.identical(fromHex('0x0.ffffffffffffdcp-1022'), MIN-2*TINY)
+        self.identical(fromHex('0x0.ffffffffffffdep-1022'), MIN-2*TINY)
+        self.identical(fromHex('0x0.ffffffffffffe0p-1022'), MIN-2*TINY)
+        self.identical(fromHex('0x0.ffffffffffffe2p-1022'), MIN-2*TINY)
+        self.identical(fromHex('0x0.ffffffffffffe4p-1022'), MIN-2*TINY)
+        self.identical(fromHex('0x0.ffffffffffffe6p-1022'), MIN-2*TINY)
+        self.identical(fromHex('0x0.ffffffffffffe8p-1022'), MIN-2*TINY)
+        self.identical(fromHex('0x0.ffffffffffffeap-1022'), MIN-TINY)
+        self.identical(fromHex('0x0.ffffffffffffecp-1022'), MIN-TINY)
+        self.identical(fromHex('0x0.ffffffffffffeep-1022'), MIN-TINY)
+        self.identical(fromHex('0x0.fffffffffffff0p-1022'), MIN-TINY)
+        self.identical(fromHex('0x0.fffffffffffff2p-1022'), MIN-TINY)
+        self.identical(fromHex('0x0.fffffffffffff4p-1022'), MIN-TINY)
+        self.identical(fromHex('0x0.fffffffffffff6p-1022'), MIN-TINY)
+        self.identical(fromHex('0x0.fffffffffffff8p-1022'), MIN)
+        self.identical(fromHex('0x0.fffffffffffffap-1022'), MIN)
+        self.identical(fromHex('0x0.fffffffffffffcp-1022'), MIN)
+        self.identical(fromHex('0x0.fffffffffffffep-1022'), MIN)
+        self.identical(fromHex('0x1.00000000000000p-1022'), MIN)
+        self.identical(fromHex('0x1.00000000000002p-1022'), MIN)
+        self.identical(fromHex('0x1.00000000000004p-1022'), MIN)
+        self.identical(fromHex('0x1.00000000000006p-1022'), MIN)
+        self.identical(fromHex('0x1.00000000000008p-1022'), MIN)
+        self.identical(fromHex('0x1.0000000000000ap-1022'), MIN+TINY)
+        self.identical(fromHex('0x1.0000000000000cp-1022'), MIN+TINY)
+        self.identical(fromHex('0x1.0000000000000ep-1022'), MIN+TINY)
+        self.identical(fromHex('0x1.00000000000010p-1022'), MIN+TINY)
+        self.identical(fromHex('0x1.00000000000012p-1022'), MIN+TINY)
+        self.identical(fromHex('0x1.00000000000014p-1022'), MIN+TINY)
+        self.identical(fromHex('0x1.00000000000016p-1022'), MIN+TINY)
+        self.identical(fromHex('0x1.00000000000018p-1022'), MIN+2*TINY)
+
+        # ... and near 1.0.
+        self.identical(fromHex('0x0.fffffffffffff0p0'), 1.0-EPS)
+        self.identical(fromHex('0x0.fffffffffffff1p0'), 1.0-EPS)
+        self.identical(fromHex('0X0.fffffffffffff2p0'), 1.0-EPS)
+        self.identical(fromHex('0x0.fffffffffffff3p0'), 1.0-EPS)
+        self.identical(fromHex('0X0.fffffffffffff4p0'), 1.0-EPS)
+        self.identical(fromHex('0X0.fffffffffffff5p0'), 1.0-EPS/2)
+        self.identical(fromHex('0X0.fffffffffffff6p0'), 1.0-EPS/2)
+        self.identical(fromHex('0x0.fffffffffffff7p0'), 1.0-EPS/2)
+        self.identical(fromHex('0x0.fffffffffffff8p0'), 1.0-EPS/2)
+        self.identical(fromHex('0X0.fffffffffffff9p0'), 1.0-EPS/2)
+        self.identical(fromHex('0X0.fffffffffffffap0'), 1.0-EPS/2)
+        self.identical(fromHex('0x0.fffffffffffffbp0'), 1.0-EPS/2)
+        self.identical(fromHex('0X0.fffffffffffffcp0'), 1.0)
+        self.identical(fromHex('0x0.fffffffffffffdp0'), 1.0)
+        self.identical(fromHex('0X0.fffffffffffffep0'), 1.0)
+        self.identical(fromHex('0x0.ffffffffffffffp0'), 1.0)
+        self.identical(fromHex('0X1.00000000000000p0'), 1.0)
+        self.identical(fromHex('0X1.00000000000001p0'), 1.0)
+        self.identical(fromHex('0x1.00000000000002p0'), 1.0)
+        self.identical(fromHex('0X1.00000000000003p0'), 1.0)
+        self.identical(fromHex('0x1.00000000000004p0'), 1.0)
+        self.identical(fromHex('0X1.00000000000005p0'), 1.0)
+        self.identical(fromHex('0X1.00000000000006p0'), 1.0)
+        self.identical(fromHex('0X1.00000000000007p0'), 1.0)
+        self.identical(fromHex('0x1.00000000000007ffffffffffffffffffffp0'),
+                       1.0)
+        self.identical(fromHex('0x1.00000000000008p0'), 1.0)
+        self.identical(fromHex('0x1.00000000000008000000000000000001p0'),
+                       1+EPS)
+        self.identical(fromHex('0X1.00000000000009p0'), 1.0+EPS)
+        self.identical(fromHex('0x1.0000000000000ap0'), 1.0+EPS)
+        self.identical(fromHex('0x1.0000000000000bp0'), 1.0+EPS)
+        self.identical(fromHex('0X1.0000000000000cp0'), 1.0+EPS)
+        self.identical(fromHex('0x1.0000000000000dp0'), 1.0+EPS)
+        self.identical(fromHex('0x1.0000000000000ep0'), 1.0+EPS)
+        self.identical(fromHex('0X1.0000000000000fp0'), 1.0+EPS)
+        self.identical(fromHex('0x1.00000000000010p0'), 1.0+EPS)
+        self.identical(fromHex('0X1.00000000000011p0'), 1.0+EPS)
+        self.identical(fromHex('0x1.00000000000012p0'), 1.0+EPS)
+        self.identical(fromHex('0X1.00000000000013p0'), 1.0+EPS)
+        self.identical(fromHex('0X1.00000000000014p0'), 1.0+EPS)
+        self.identical(fromHex('0x1.00000000000015p0'), 1.0+EPS)
+        self.identical(fromHex('0x1.00000000000016p0'), 1.0+EPS)
+        self.identical(fromHex('0X1.00000000000017p0'), 1.0+EPS)
+        self.identical(fromHex('0x1.00000000000017ffffffffffffffffffffp0'),
+                       1.0+EPS)
+        self.identical(fromHex('0x1.00000000000018p0'), 1.0+2*EPS)
+        self.identical(fromHex('0X1.00000000000018000000000000000001p0'),
+                       1.0+2*EPS)
+        self.identical(fromHex('0x1.00000000000019p0'), 1.0+2*EPS)
+        self.identical(fromHex('0X1.0000000000001ap0'), 1.0+2*EPS)
+        self.identical(fromHex('0X1.0000000000001bp0'), 1.0+2*EPS)
+        self.identical(fromHex('0x1.0000000000001cp0'), 1.0+2*EPS)
+        self.identical(fromHex('0x1.0000000000001dp0'), 1.0+2*EPS)
+        self.identical(fromHex('0x1.0000000000001ep0'), 1.0+2*EPS)
+        self.identical(fromHex('0X1.0000000000001fp0'), 1.0+2*EPS)
+        self.identical(fromHex('0x1.00000000000020p0'), 1.0+2*EPS)
+
+    def test_roundtrip(self):
+        def roundtrip(x):
+            return fromHex(toHex(x))
+
+        for x in [NAN, INF, self.MAX, self.MIN, self.MIN-self.TINY, self.TINY, 0.0]:
+            self.identical(x, roundtrip(x))
+            self.identical(-x, roundtrip(-x))
+
+        # fromHex(toHex(x)) should exactly recover x, for any non-NaN float x.
+        import random
+        for i in xrange(10000):
+            e = random.randrange(-1200, 1200)
+            m = random.random()
+            s = random.choice([1.0, -1.0])
+            try:
+                x = s*ldexp(m, e)
+            except OverflowError:
+                pass
+            else:
+                self.identical(x, fromHex(toHex(x)))
+
+
 def test_main():
     test_support.run_unittest(
         FormatFunctionsTestCase,
@@ -351,6 +731,7 @@
         IEEEFormatTestCase,
         ReprTestCase,
         InfNanTest,
+        HexFloatTestCase,
         )
 
 if __name__ == '__main__':