Issue #1869 (and 4707, 5118, 5473, 1456775): use the new

string <-> float conversion routines to make round(x, n) correctly
rounded for floats x, so that it always agrees with format(x, '.<n>f').

Also fix some other round nuisances, like round(123.456, 1-2**31) giving
an integer rather than a float.

Objects/floatobject.c

@@ -899,43 +899,161 @@ float_trunc(PyObject *v)
 	return PyLong_FromDouble(wholepart);
 }
 
+/* double_round: rounds a finite double to the closest multiple of
+   10**-ndigits; here ndigits is within reasonable bounds (typically, -308 <=
+   ndigits <= 323).  Returns a Python float, or sets a Python error and
+   returns NULL on failure (OverflowError and memory errors are possible). */
+
+#ifndef PY_NO_SHORT_FLOAT_REPR
+/* version of double_round that uses the correctly-rounded string<->double
+   conversions from Python/dtoa.c */
+
+static PyObject *
+double_round(double x, int ndigits) {
+
+	double rounded;
+	Py_ssize_t buflen, mybuflen=100;
+	char *buf, *buf_end, shortbuf[100], *mybuf=shortbuf;
+	int decpt, sign;
+	PyObject *result = NULL;
+
+	/* round to a decimal string */
+	buf = _Py_dg_dtoa(x, 3, ndigits, &decpt, &sign, &buf_end);
+	if (buf == NULL) {
+		PyErr_NoMemory();
+		return NULL;
+	}
+
+	/* Get new buffer if shortbuf is too small.  Space needed <= buf_end -
+	buf + 8: (1 extra for '0', 1 for sign, 5 for exp, 1 for '\0').  */
+	buflen = buf_end - buf;
+	if (buflen + 8 > mybuflen) {
+		mybuflen = buflen+8;
+		mybuf = (char *)PyMem_Malloc(mybuflen);
+		if (mybuf == NULL) {
+			PyErr_NoMemory();
+			goto exit;
+		}
+	}
+	/* copy buf to mybuf, adding exponent, sign and leading 0 */
+	PyOS_snprintf(mybuf, mybuflen, "%s0%se%d", (sign ? "-" : ""),
+		      buf, decpt - (int)buflen);
+
+	/* and convert the resulting string back to a double */
+	errno = 0;
+	rounded = _Py_dg_strtod(mybuf, NULL);
+	if (errno == ERANGE && fabs(rounded) >= 1.)
+		PyErr_SetString(PyExc_OverflowError,
+				"rounded value too large to represent");
+	else
+		result = PyFloat_FromDouble(rounded);
+
+	/* done computing value;  now clean up */
+	if (mybuf != shortbuf)
+		PyMem_Free(mybuf);
+  exit:
+	_Py_dg_freedtoa(buf);
+	return result;
+}
+
+#else /* PY_NO_SHORT_FLOAT_REPR */
+
+/* fallback version, to be used when correctly rounded binary<->decimal
+   conversions aren't available */
+
+static PyObject *
+double_round(double x, int ndigits) {
+	double pow1, pow2, y, z;
+	if (ndigits >= 0) {
+		if (ndigits > 22) {
+			/* pow1 and pow2 are each safe from overflow, but
+			   pow1*pow2 ~= pow(10.0, ndigits) might overflow */
+			pow1 = pow(10.0, (double)(ndigits-22));
+			pow2 = 1e22;
+		}
+		else {
+			pow1 = pow(10.0, (double)ndigits);
+			pow2 = 1.0;
+		}
+		y = (x*pow1)*pow2;
+		/* if y overflows, then rounded value is exactly x */
+		if (!Py_IS_FINITE(y))
+			return PyFloat_FromDouble(x);
+	}
+	else {
+		pow1 = pow(10.0, (double)-ndigits);
+		pow2 = 1.0; /* unused; silences a gcc compiler warning */
+		y = x / pow1;
+	}
+
+	z = round(y);
+	if (fabs(y-z) == 0.5)
+		/* halfway between two integers; use round-half-even */
+		z = 2.0*round(y/2.0);
+
+	if (ndigits >= 0)
+		z = (z / pow2) / pow1;
+	else
+		z *= pow1;
+
+	/* if computation resulted in overflow, raise OverflowError */
+	if (!Py_IS_FINITE(z)) {
+		PyErr_SetString(PyExc_OverflowError,
+				"overflow occurred during round");
+		return NULL;
+	}
+
+	return PyFloat_FromDouble(z);
+}
+
+#endif /* PY_NO_SHORT_FLOAT_REPR */
+
+/* round a Python float v to the closest multiple of 10**-ndigits */
+
 static PyObject *
 float_round(PyObject *v, PyObject *args)
 {
-#define UNDEF_NDIGITS (-0x7fffffff) /* Unlikely ndigits value */
-	double x;
-	double f = 1.0;
-	double flr, cil;
-	double rounded;
-	int ndigits = UNDEF_NDIGITS;
-
-	if (!PyArg_ParseTuple(args, "|i", &ndigits))
-		return NULL;
+	double x, rounded;
+	PyObject *o_ndigits = NULL;
+	Py_ssize_t ndigits;
 
 	x = PyFloat_AsDouble(v);
-
-	if (ndigits != UNDEF_NDIGITS) {
-		f = pow(10.0, ndigits);
-		x *= f;
+	if (!PyArg_ParseTuple(args, "|O", &o_ndigits))
+		return NULL;
+	if (o_ndigits == NULL) {
+		/* single-argument round: round to nearest integer */
+		rounded = round(x);
+		if (fabs(x-rounded) == 0.5)
+			/* halfway case: round to even */
+			rounded = 2.0*round(x/2.0);
+		return PyLong_FromDouble(rounded);
 	}
 
-	flr = floor(x);
-	cil = ceil(x);
+	/* interpret second argument as a Py_ssize_t; clips on overflow */
+	ndigits = PyNumber_AsSsize_t(o_ndigits, NULL);
+	if (ndigits == -1 && PyErr_Occurred())
+		return NULL;
 
-	if (x-flr > 0.5)
-		rounded = cil;
-	else if (x-flr == 0.5)
-		rounded = fmod(flr, 2) == 0 ? flr : cil;
+	/* nans and infinities round to themselves */
+	if (!Py_IS_FINITE(x))
+		return PyFloat_FromDouble(x);
+
+	/* Deal with extreme values for ndigits. For ndigits > NDIGITS_MAX, x
+	   always rounds to itself.  For ndigits < NDIGITS_MIN, x always
+	   rounds to +-0.0.  Here 0.30103 is an upper bound for log10(2). */
+#define NDIGITS_MAX ((int)((DBL_MANT_DIG-DBL_MIN_EXP) * 0.30103))
+#define NDIGITS_MIN (-(int)((DBL_MAX_EXP + 1) * 0.30103))
+	if (ndigits > NDIGITS_MAX)
+		/* return x */
+		return PyFloat_FromDouble(x);
+	else if (ndigits < NDIGITS_MIN)
+		/* return 0.0, but with sign of x */
+		return PyFloat_FromDouble(0.0*x);
 	else
-		rounded = flr;
-
-	if (ndigits != UNDEF_NDIGITS) {
-		rounded /= f;
-		return PyFloat_FromDouble(rounded);
-	}
-
-	return PyLong_FromDouble(rounded);
-#undef UNDEF_NDIGITS
+		/* finite x, and ndigits is not unreasonably large */
+		return double_round(x, (int)ndigits);
+#undef NDIGITS_MAX
+#undef NDIGITS_MIN
 }
 
 static PyObject *