numpy
diff --git a/‎doc/release/1.17.0-notes.rst
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diff --git a/‎numpy/core/src/npymath/halffloat.c
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diff --git a/‎numpy/core/tests/test_half.py
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@@ -30,6 +30,13 @@ Expired deprecations
 Compatibility notes
 ===================
 
+float16 subnormal rounding
+--------------------------
+Casting from a different floating point precision to float16 used incorrect
+rounding in some edge cases. This means in rare cases, subnormal results will
+now be rounded up instead of down, changing the last bit (ULP) of the result.
+
+
 
 C API changes
 =============
 
@@ -301,15 +301,23 @@ npy_uint16 npy_floatbits_to_halfbits(npy_uint32 f)
             npy_set_floatstatus_underflow();
         }
 #endif
+        /*
+         * Usually the significand is shifted by 13. For subnormals an
+         * additional shift needs to occur. This shift is one for the largest
+         * exponent giving a subnormal `f_exp = 0x38000000 >> 23 = 112`, which
+         * offsets the new first bit. At most the shift can be 1+10 bits.
+         */
         f_sig >>= (113 - f_exp);
         /* Handle rounding by adding 1 to the bit beyond half precision */
 #if NPY_HALF_ROUND_TIES_TO_EVEN
         /*
          * If the last bit in the half significand is 0 (already even), and
          * the remaining bit pattern is 1000...0, then we do not add one
-         * to the bit after the half significand.  In all other cases, we do.
+         * to the bit after the half significand. However, the (113 - f_exp)
+         * shift can lose up to 11 bits, so the || checks them in the original.
+         * In all other cases, we can just add one.
          */
-        if ((f_sig&0x00003fffu) != 0x00001000u) {
+        if (((f_sig&0x00003fffu) != 0x00001000u) || (f&0x000007ffu)) {
             f_sig += 0x00001000u;
         }
 #else
@@ -416,21 +424,30 @@ npy_uint16 npy_doublebits_to_halfbits(npy_uint64 d)
             npy_set_floatstatus_underflow();
         }
 #endif
-        d_sig >>= (1009 - d_exp);
+        /*
+         * Unlike floats, doubles have enough room to shift left to align
+         * the subnormal significand leading to no loss of the last bits.
+         * The smallest possible exponent giving a subnormal is:
+         * `d_exp = 0x3e60000000000000 >> 52 = 998`. All larger subnormals are
+         * shifted with respect to it. This adds a shift of 10+1 bits the final
+         * right shift when comparing it to the one in the normal branch.
+         */
+        assert(d_exp - 998 >= 0);
+        d_sig <<= (d_exp - 998);
         /* Handle rounding by adding 1 to the bit beyond half precision */
 #if NPY_HALF_ROUND_TIES_TO_EVEN
         /*
          * If the last bit in the half significand is 0 (already even), and
          * the remaining bit pattern is 1000...0, then we do not add one
          * to the bit after the half significand.  In all other cases, we do.
          */
-        if ((d_sig&0x000007ffffffffffULL) != 0x0000020000000000ULL) {
-            d_sig += 0x0000020000000000ULL;
+        if ((d_sig&0x003fffffffffffffULL) != 0x0010000000000000ULL) {
+            d_sig += 0x0010000000000000ULL;
         }
 #else
-        d_sig += 0x0000020000000000ULL;
+        d_sig += 0x0010000000000000ULL;
 #endif
-        h_sig = (npy_uint16) (d_sig >> 42);
+        h_sig = (npy_uint16) (d_sig >> 53);
         /*
          * If the rounding causes a bit to spill into h_exp, it will
          * increment h_exp from zero to one and h_sig will be zero.
 
@@ -69,6 +69,85 @@ def test_half_conversions(self):
         j = np.array(i_f16, dtype=int)
         assert_equal(i_int, j)
 
+    @pytest.mark.parametrize("offset", [None, "up", "down"])
+    @pytest.mark.parametrize("shift", [None, "up", "down"])
+    @pytest.mark.parametrize("float_t", [np.float32, np.float64])
+    def test_half_conversion_rounding(self, float_t, shift, offset):
+        # Assumes that round to even is used during casting.
+        max_pattern = np.float16(np.finfo(np.float16).max).view(np.uint16)
+
+        # Test all (positive) finite numbers, denormals are most interesting
+        # however:
+        f16s_patterns = np.arange(0, max_pattern+1, dtype=np.uint16)
+        f16s_float = f16s_patterns.view(np.float16).astype(float_t)
+
+        # Shift the values by half a bit up or a down (or do not shift),
+        if shift == "up":
+            f16s_float = 0.5 * (f16s_float[:-1] + f16s_float[1:])[1:]
+        elif shift == "down":
+            f16s_float = 0.5 * (f16s_float[:-1] + f16s_float[1:])[:-1]
+        else:
+            f16s_float = f16s_float[1:-1]
+
+        # Increase the float by a minimal value:
+        if offset == "up":
+            f16s_float = np.nextafter(f16s_float, float_t(1e50))
+        elif offset == "down":
+            f16s_float = np.nextafter(f16s_float, float_t(-1e50))
+
+        # Convert back to float16 and its bit pattern:
+        res_patterns = f16s_float.astype(np.float16).view(np.uint16)
+
+        # The above calculations tries the original values, or the exact
+        # mid points between the float16 values. It then further offsets them
+        # by as little as possible. If no offset occurs, "round to even"
+        # logic will be necessary, an arbitrarily small offset should cause
+        # normal up/down rounding always.
+
+        # Calculate the expecte pattern:
+        cmp_patterns = f16s_patterns[1:-1].copy()
+
+        if shift == "down" and offset != "up":
+            shift_pattern = -1
+        elif shift == "up" and offset != "down":
+            shift_pattern = 1
+        else:
+            # There cannot be a shift, either shift is None, so all rounding
+            # will go back to original, or shift is reduced by offset too much.
+            shift_pattern = 0
+
+        # If rounding occurs, is it normal rounding or round to even?
+        if offset is None:
+            # Round to even occurs, modify only non-even, cast to allow + (-1)
+            cmp_patterns[0::2].view(np.int16)[...] += shift_pattern
+        else:
+            cmp_patterns.view(np.int16)[...] += shift_pattern
+
+        assert_equal(res_patterns, cmp_patterns)
+
+    @pytest.mark.parametrize(["float_t", "uint_t", "bits"],
+                             [(np.float32, np.uint32, 23),
+                              (np.float64, np.uint64, 52)])
+    def test_half_conversion_denormal_round_even(self, float_t, uint_t, bits):
+        # Test specifically that all bits are considered when deciding
+        # whether round to even should occur (i.e. no bits are lost at the
+        # end. Compare also gh-12721. The most bits can get lost for the
+        # smallest denormal:
+        smallest_value = np.uint16(1).view(np.float16).astype(float_t)
+        assert smallest_value == 2**-24
+
+        # Will be rounded to zero based on round to even rule:
+        rounded_to_zero = smallest_value / float_t(2)
+        assert rounded_to_zero.astype(np.float16) == 0
+
+        # The significand will be all 0 for the float_t, test that we do not
+        # lose the lower ones of these:
+        for i in range(bits):
+            # slightly increasing the value should make it round up:
+            larger_pattern = rounded_to_zero.view(uint_t) | uint_t(1 << i)
+            larger_value = larger_pattern.view(float_t)
+            assert larger_value.astype(np.float16) == smallest_value
+
     def test_nans_infs(self):
         with np.errstate(all='ignore'):
             # Check some of the ufuncs