diff --git a/hypothesis-python/RELEASE.rst b/hypothesis-python/RELEASE.rst
new file mode 100644
index 0000000000..dea9b4765b
--- /dev/null
+++ b/hypothesis-python/RELEASE.rst
@@ -0,0 +1,6 @@
+RELEASE_TYPE: patch
+
+This patch substantially improves our ability to avoid generating redundant
+inputs when choosing between a non-power-of-two number of alternatives.
+In certain circumstances, this was causing serious performance problems -
+see :issue:`1864`, :issue:`1982`, and :issue:`2027`.
diff --git a/hypothesis-python/src/hypothesis/extra/numpy.py b/hypothesis-python/src/hypothesis/extra/numpy.py
index 583d924f89..a00e459519 100644
--- a/hypothesis-python/src/hypothesis/extra/numpy.py
+++ b/hypothesis-python/src/hypothesis/extra/numpy.py
@@ -709,8 +709,8 @@ def valid_tuple_axes(ndim, min_size=0, max_size=None):
     check_valid_interval(max_size, ndim, "max_size", "ndim")
 
     # shrink axis values from negative to positive
-    axes = st.integers(0, max(0, 2 * ndim - 1)).map(
-        lambda x: x if x < ndim else x - 2 * ndim
+    axes = st.tuples(st.booleans(), st.integers(0, ndim - 1)).map(
+        lambda args: args[1] - ndim if args[0] else args[1]
     )
     return st.lists(axes, min_size, max_size, unique_by=lambda x: x % ndim).map(tuple)
 
diff --git a/hypothesis-python/src/hypothesis/internal/conjecture/utils.py b/hypothesis-python/src/hypothesis/internal/conjecture/utils.py
index 4180e02b32..6fa54ffb16 100644
--- a/hypothesis-python/src/hypothesis/internal/conjecture/utils.py
+++ b/hypothesis-python/src/hypothesis/internal/conjecture/utils.py
@@ -90,7 +90,6 @@ def integer_range(data, lower, upper, center=None):
     assert gap > 0
 
     bits = bit_length(gap)
-    probe = gap + 1
 
     if bits > 24 and data.draw_bits(3):
         # For large ranges, we combine the uniform random distribution from draw_bits
@@ -100,10 +99,17 @@ def integer_range(data, lower, upper, center=None):
         sizes = [8, 16, 32, 64, 128]
         bits = min(bits, sizes[idx])
 
-    while probe > gap:
-        data.start_example(INTEGER_RANGE_DRAW_LABEL)
-        probe = data.draw_bits(bits)
-        data.stop_example(discard=probe > gap)
+    data.start_example(INTEGER_RANGE_DRAW_LABEL)
+    probe = data.draw_bits(bits)
+    data.stop_example()
+    # Rejection sampling interacts badly with our prefix tree (see e.g. #1864),
+    # so we want to adjust this value to fit in the range *without* distorting
+    # the distribution.
+    high = (1 << bits) - 1
+    if high > gap:
+        leave = gap - (high - gap)
+        if probe > leave:
+            probe = leave + (probe - leave) // 2
 
     if above:
         result = center + probe
diff --git a/hypothesis-python/tests/cover/test_conjecture_engine.py b/hypothesis-python/tests/cover/test_conjecture_engine.py
index e345a2bd56..f758175881 100644
--- a/hypothesis-python/tests/cover/test_conjecture_engine.py
+++ b/hypothesis-python/tests/cover/test_conjecture_engine.py
@@ -43,7 +43,7 @@
 )
 from hypothesis.internal.conjecture.shrinker import Shrinker, block_program
 from hypothesis.internal.conjecture.shrinking import Float
-from hypothesis.internal.conjecture.utils import Sampler, calc_label_from_name
+from hypothesis.internal.conjecture.utils import calc_label_from_name
 from hypothesis.internal.entropy import deterministic_PRNG
 from tests.common.strategies import SLOW, HardToShrink
 from tests.common.utils import counts_calls, no_shrink
@@ -1111,27 +1111,6 @@ def accept(f):
     return accept
 
 
-def test_dependent_block_pairs_is_up_to_shrinking_integers():
-    # Unit test extracted from a failure in tests/nocover/test_integers.py
-    distribution = Sampler([4.0, 8.0, 1.0, 1.0, 0.5])
-
-    sizes = [8, 16, 32, 64, 128]
-
-    @shrinking_from(b"\x03\x01\x00\x00\x00\x00\x00\x01\x00\x02\x01")
-    def shrinker(data):
-        size = sizes[distribution.sample(data)]
-        result = data.draw_bits(size)
-        sign = (-1) ** (result & 1)
-        result = (result >> 1) * sign
-        cap = data.draw_bits(8)
-
-        if result >= 32768 and cap == 1:
-            data.mark_interesting()
-
-    shrinker.minimize_individual_blocks()
-    assert list(shrinker.shrink_target.buffer) == [1, 1, 0, 1, 0, 0, 1]
-
-
 def test_finding_a_minimal_balanced_binary_tree():
     # Tests iteration while the shape of the thing being iterated over can
     # change. In particular the current example can go from trivial to non
diff --git a/hypothesis-python/tests/cover/test_simple_characters.py b/hypothesis-python/tests/cover/test_simple_characters.py
index 360cc67ddc..be3cccb04b 100644
--- a/hypothesis-python/tests/cover/test_simple_characters.py
+++ b/hypothesis-python/tests/cover/test_simple_characters.py
@@ -130,7 +130,6 @@ def test_whitelisted_characters_override():
     assert_no_examples(st, lambda c: c not in good_characters + "0123456789")
 
 
-@pytest.mark.skip  # temporary skip due to 560 second (!) perf regression; see #1864
 def test_blacklisted_characters():
     bad_chars = u"te02тест49st"
     st = characters(
diff --git a/hypothesis-python/tests/nocover/test_sampled_from.py b/hypothesis-python/tests/nocover/test_sampled_from.py
index ab6a8c0413..3b37c3c1af 100644
--- a/hypothesis-python/tests/nocover/test_sampled_from.py
+++ b/hypothesis-python/tests/nocover/test_sampled_from.py
@@ -73,3 +73,14 @@ def test_unsat_sets_of_samples(x):
 @given(st.sets(st.sampled_from(range(50)), min_size=50))
 def test_efficient_sets_of_samples(x):
     assert x == set(range(50))
+
+
+def test_stops_quickly():
+    # https://github.com/HypothesisWorks/hypothesis/issues/2027
+    @given(st.sampled_from(range(3)))
+    def inner(x):
+        count[0] += 1
+
+    count = [0]
+    inner()
+    assert 2 < count[0] <= 4
diff --git a/hypothesis-python/tests/pandas/test_data_frame.py b/hypothesis-python/tests/pandas/test_data_frame.py
index 2376740732..1fef24db9a 100644
--- a/hypothesis-python/tests/pandas/test_data_frame.py
+++ b/hypothesis-python/tests/pandas/test_data_frame.py
@@ -211,7 +211,7 @@ def test_uniqueness_does_not_affect_other_rows_1():
             pdst.column("A", dtype=int, unique=True),
             pdst.column("B", dtype=int, unique=False),
         ],
-        rows=st.tuples(st.integers(0, 10), st.integers(0, 10)),
+        rows=st.tuples(st.integers(0, 3), st.integers(0, 3)),
         index=pdst.range_indexes(2, 2),
     )
     find_any(data_frames, lambda x: x["B"][0] == x["B"][1])
diff --git a/hypothesis-python/tests/quality/test_integers.py b/hypothesis-python/tests/quality/test_integers.py
index 9faa3819b0..27cef37d18 100644
--- a/hypothesis-python/tests/quality/test_integers.py
+++ b/hypothesis-python/tests/quality/test_integers.py
@@ -20,16 +20,7 @@
 from random import Random
 
 import hypothesis.strategies as st
-from hypothesis import (
-    HealthCheck,
-    Phase,
-    Verbosity,
-    assume,
-    example,
-    given,
-    reject,
-    settings,
-)
+from hypothesis import HealthCheck, Phase, Verbosity, assume, given, reject, settings
 from hypothesis.internal.compat import hbytes
 from hypothesis.internal.conjecture.data import ConjectureData, Status, StopTest
 from hypothesis.internal.conjecture.engine import ConjectureRunner
@@ -52,9 +43,6 @@ def problems(draw):
             pass
 
 
-@example((2, b"\x00\x00\n\x01"))
-@example((1, b"\x00\x00\x06\x01"))
-@example(problem=(32768, b"\x03\x01\x00\x00\x00\x00\x00\x01\x00\x02\x01"))
 @settings(
     suppress_health_check=HealthCheck.all(),
     deadline=None,