diff --git a/Lib/test/test_float.py b/Lib/test/test_float.py
index ff9cfee..a8b29fc 100644
--- a/Lib/test/test_float.py
+++ b/Lib/test/test_float.py
@@ -8,6 +8,7 @@
 import random
 import fractions
 import sys
+import re
 
 INF = float("inf")
 NAN = float("nan")
@@ -21,6 +22,74 @@
 test_dir = os.path.dirname(__file__) or os.curdir
 format_testfile = os.path.join(test_dir, 'formatfloat_testcases.txt')
 
+finite_decimal_parser = re.compile(r"""    # A numeric string consists of:
+    (?P<sign>[-+])?          # an optional sign, followed by
+    (?=\d|\.\d)              # a number with at least one digit
+    (?P<int>\d*)             # having a (possibly empty) integer part
+    (?:\.(?P<frac>\d*))?     # followed by an optional fractional part
+    (?:E(?P<exp>[-+]?\d+))?  # and an optional exponent
+    \Z
+""", re.VERBOSE | re.IGNORECASE | re.UNICODE).match
+
+# Pure Python version of correctly rounded string->float conversion.
+# Avoids any use of floating-point by returning the result as a hex string.
+def strtod(s, mant_dig=53, min_exp = -1021, max_exp = 1024):
+    """Convert a finite decimal string to a hex string representing an
+    IEEE 754 binary64 float.  Return 'inf' or '-inf' on overflow.
+    This function makes no use of floating-point arithmetic at any
+    stage."""
+
+    # parse string into a pair of integers 'a' and 'b' such that
+    # abs(decimal value) = a/b, and a boolean 'negative'.
+    m = finite_decimal_parser(s)
+    if m is None:
+        raise ValueError('invalid numeric string')
+    fraction = m.group('frac') or ''
+    intpart = int(m.group('int') + fraction)
+    exp = int(m.group('exp') or '0') - len(fraction)
+    negative = m.group('sign') == '-'
+    a, b = intpart*10**max(exp, 0), 10**max(0, -exp)
+
+    # quick return for zeros
+    if not a:
+        return '-0x0.0p+0' if negative else '0x0.0p+0'
+
+    # compute exponent e for result; may be one too small in the case
+    # that the rounded value of a/b lies in a different binade from a/b
+    d = a.bit_length() - b.bit_length()
+    d += (a >> d if d >= 0 else a << -d) >= b
+    e = max(d, min_exp) - mant_dig
+
+    # approximate a/b by number of the form q * 2**e; adjust e if necessary
+    a, b = a << max(-e, 0), b << max(e, 0)
+    q, r = divmod(a, b)
+    if 2*r > b or 2*r == b and q & 1:
+        q += 1
+        if q.bit_length() == mant_dig+1:
+            q //= 2
+            e += 1
+
+    # double check that (q, e) has the right form
+    assert q.bit_length() <= mant_dig and e >= min_exp - mant_dig
+    assert q.bit_length() == mant_dig or e == min_exp - mant_dig
+
+    # check for overflow and underflow
+    if e + q.bit_length() > max_exp:
+        return '-inf' if negative else 'inf'
+    if not q:
+        return '-0x0.0p+0' if negative else '0x0.0p+0'
+
+    # for hex representation, shift so # bits after point is a multiple of 4
+    hexdigs = 1 + (mant_dig-2)//4
+    shift = 3 - (mant_dig-2)%4
+    q, e = q << shift, e - shift
+    return '{}0x{:x}.{:0{}x}p{:+d}'.format(
+        '-' if negative else '',
+        q // 16**hexdigs,
+        q % 16**hexdigs,
+        hexdigs,
+        e + 4*hexdigs)
+
 class GeneralFloatCases(unittest.TestCase):
 
     def test_float(self):
@@ -1299,6 +1368,38 @@
             else:
                 self.identical(x, fromHex(toHex(x)))
 
+class StrtodTestCase(unittest.TestCase):
+    def check_string(self, s):
+        expected = strtod(s)
+        try:
+            fs = float(s)
+        except OverflowError:
+            got = '-inf' if s[0] == '-' else 'inf'
+        else:
+            got = fs.hex()
+        self.assertEqual(expected, got,
+                         "Incorrectly rounded str->float conversion for "
+                         "{}: expected {}, got {}".format(s, expected, got))
+
+    @unittest.skipUnless(getattr(sys, 'float_repr_style', '') == 'short',
+                         "applies only when using short float repr style")
+    def test_bug7632(self):
+        # check a few particular values that gave incorrectly rounded
+        # results with previous versions of dtoa.c
+        test_strings = [
+            '94393431193180696942841837085033647913224148539854e-358',
+            '12579816049008305546974391768996369464963024663104e-357',
+            '17489628565202117263145367596028389348922981857013e-357',
+            '18487398785991994634182916638542680759613590482273e-357',
+            '32002864200581033134358724675198044527469366773928e-358',
+            '73608278998966969345824653500136787876436005957953e-358',
+            '64774478836417299491718435234611299336288082136054e-358',
+            '13704940134126574534878641876947980878824688451169e-357',
+            '46697445774047060960624497964425416610480524760471e-358',
+            ]
+        for s in test_strings:
+            self.check_string(s)
+
 
 def test_main():
     test_support.run_unittest(
@@ -1310,6 +1411,7 @@
         RoundTestCase,
         InfNanTest,
         HexFloatTestCase,
+        StrtodTestCase,
         )
 
 if __name__ == '__main__':
diff --git a/Misc/NEWS b/Misc/NEWS
index 9fc17dc..d2b75b1 100644
--- a/Misc/NEWS
+++ b/Misc/NEWS
@@ -14,7 +14,8 @@
 
 - Issue #7632: Fix a crash in dtoa.c that occurred in debug builds
   when parsing certain long numeric strings corresponding to subnormal
-  values.
+  values.  Also fix a number of bugs in dtoa.c that could lead to
+  incorrectly rounded results when converting strings to floats.
 
 - Issue #7319: Silence DeprecationWarning by default.
 
diff --git a/Python/dtoa.c b/Python/dtoa.c
index 4d64be5..1fe20f4 100644
--- a/Python/dtoa.c
+++ b/Python/dtoa.c
@@ -1130,6 +1130,26 @@
     return q;
 }
 
+/* version of ulp(x) that takes bc.scale into account.
+
+   Assuming that x is finite and nonzero, and x / 2^bc.scale is exactly
+   representable as a double, sulp(x) is equivalent to 2^bc.scale * ulp(x /
+   2^bc.scale). */
+
+static double
+sulp(U *x, BCinfo *bc)
+{
+    U u;
+
+    if (bc->scale && 2*P + 1 - ((word0(x) & Exp_mask) >> Exp_shift) > 0) {
+        /* rv/2^bc->scale is subnormal */
+        word0(&u) = (P+2)*Exp_msk1;
+        word1(&u) = 0;
+        return u.d;
+    }
+    else
+        return ulp(x);
+}
 
 /* return 0 on success, -1 on failure */
 
@@ -1289,12 +1309,12 @@
     else if (dd < 0) {
         if (!dsign)     /* does not happen for round-near */
           retlow1:
-            dval(rv) -= ulp(rv);
+            dval(rv) -= sulp(rv, bc);
     }
     else if (dd > 0) {
         if (dsign) {
           rethi1:
-            dval(rv) += ulp(rv);
+            dval(rv) += sulp(rv, bc);
         }
     }
     else {