diff --git a/benchmark/histogram_filling.cpp b/benchmark/histogram_filling.cpp
index dbe8bf1b..470aafcf 100644
--- a/benchmark/histogram_filling.cpp
+++ b/benchmark/histogram_filling.cpp
@@ -31,7 +31,33 @@ using DStore = boost::histogram::adaptive_storage<>;
 #endif
 
 using namespace boost::histogram;
+namespace op = boost::histogram::axis::option;
 using reg = axis::regular<>;
+using reg_closed =
+    axis::regular<double, boost::use_default, boost::use_default, op::none_t>;
+
+class reg_closed_unsafe {
+public:
+  reg_closed_unsafe(axis::index_type n, double start, double stop)
+      : min_{start}, delta_{stop - start}, size_{n} {}
+
+  axis::index_type index(double x) const noexcept {
+    // Runs in hot loop, please measure impact of changes
+    auto z = (x - min_) / delta_;
+    // assume that z < 0 and z > 1 never happens, promised by inclusive()
+    if (z == 1) return size() - 1;
+    return static_cast<axis::index_type>(z * size());
+  }
+
+  axis::index_type size() const noexcept { return size_; }
+
+  static constexpr bool inclusive() { return true; }
+
+private:
+  double min_;
+  double delta_;
+  axis::index_type size_;
+};
 
 template <class Distribution, class Tag, class Storage = SStore>
 static void fill_1d(benchmark::State& state) {
@@ -41,6 +67,22 @@ static void fill_1d(benchmark::State& state) {
   state.SetItemsProcessed(state.iterations());
 }
 
+template <class Distribution, class Tag, class Storage = SStore>
+static void fill_1d_closed(benchmark::State& state) {
+  auto h = make_s(Tag(), Storage(), reg_closed(100, 0, 1));
+  auto gen = generator<Distribution>();
+  for (auto _ : state) benchmark::DoNotOptimize(h(gen()));
+  state.SetItemsProcessed(state.iterations());
+}
+
+template <class Distribution, class Tag, class Storage = SStore>
+static void fill_1d_closed_unsafe(benchmark::State& state) {
+  auto h = make_s(Tag(), Storage(), reg_closed_unsafe(100, 0, 1));
+  auto gen = generator<Distribution>();
+  for (auto _ : state) benchmark::DoNotOptimize(h(gen()));
+  state.SetItemsProcessed(state.iterations());
+}
+
 template <class Distribution, class Tag, class Storage = SStore>
 static void fill_n_1d(benchmark::State& state) {
   auto h = make_s(Tag(), Storage(), reg(100, 0, 1));
@@ -49,6 +91,22 @@ static void fill_n_1d(benchmark::State& state) {
   state.SetItemsProcessed(state.iterations() * gen.size());
 }
 
+template <class Distribution, class Tag, class Storage = SStore>
+static void fill_n_1d_closed(benchmark::State& state) {
+  auto h = make_s(Tag(), Storage(), reg_closed(100, 0, 1));
+  auto gen = generator<Distribution>();
+  for (auto _ : state) h.fill(gen);
+  state.SetItemsProcessed(state.iterations() * gen.size());
+}
+
+template <class Distribution, class Tag, class Storage = SStore>
+static void fill_n_1d_closed_unsafe(benchmark::State& state) {
+  auto h = make_s(Tag(), Storage(), reg_closed_unsafe(100, 0, 1));
+  auto gen = generator<Distribution>();
+  for (auto _ : state) h.fill(gen);
+  state.SetItemsProcessed(state.iterations() * gen.size());
+}
+
 template <class Distribution, class Tag, class Storage = SStore>
 static void fill_2d(benchmark::State& state) {
   auto h = make_s(Tag(), Storage(), reg(100, 0, 1), reg(100, 0, 1));
@@ -111,6 +169,8 @@ BENCHMARK_TEMPLATE(fill_1d, uniform, dynamic_tag);
 // BENCHMARK_TEMPLATE(fill_1d, uniform, dynamic_tag, DStore);
 BENCHMARK_TEMPLATE(fill_1d, normal, dynamic_tag);
 // BENCHMARK_TEMPLATE(fill_1d, normal, dynamic_tag, DStore);
+BENCHMARK_TEMPLATE(fill_1d_closed, uniform, static_tag);
+BENCHMARK_TEMPLATE(fill_1d_closed_unsafe, uniform, static_tag);
 
 BENCHMARK_TEMPLATE(fill_n_1d, uniform, static_tag);
 // BENCHMARK_TEMPLATE(fill_n_1d, uniform, static_tag, DStore);
@@ -120,6 +180,8 @@ BENCHMARK_TEMPLATE(fill_n_1d, uniform, dynamic_tag);
 // BENCHMARK_TEMPLATE(fill_n_1d, uniform, dynamic_tag, DStore);
 BENCHMARK_TEMPLATE(fill_n_1d, normal, dynamic_tag);
 // BENCHMARK_TEMPLATE(fill_n_1d, normal, dynamic_tag, DStore);
+BENCHMARK_TEMPLATE(fill_n_1d_closed, uniform, static_tag);
+BENCHMARK_TEMPLATE(fill_n_1d_closed_unsafe, uniform, static_tag);
 
 BENCHMARK_TEMPLATE(fill_2d, uniform, static_tag);
 // BENCHMARK_TEMPLATE(fill_2d, uniform, static_tag, DStore);
diff --git a/include/boost/histogram/axis/category.hpp b/include/boost/histogram/axis/category.hpp
index 9871e9b7..bcaececc 100644
--- a/include/boost/histogram/axis/category.hpp
+++ b/include/boost/histogram/axis/category.hpp
@@ -26,22 +26,21 @@ namespace boost {
 namespace histogram {
 namespace axis {
 
-/**
-  Maps at a set of unique values to bin indices.
-
-  The axis maps a set of values to bins, following the order of arguments in the
-  constructor. The optional overflow bin for this axis counts input values that
-  are not part of the set. Binning has O(N) complexity, but with a very small
-  factor. For small N (the typical use case) it beats other kinds of lookup.
-
-  @tparam Value input value type, must be equal-comparable.
-  @tparam MetaData type to store meta data.
-  @tparam Options see boost::histogram::axis::option.
-  @tparam Allocator allocator to use for dynamic memory management.
-
-  The options `underflow` and `circular` are not allowed. The options `growth`
-  and `overflow` are mutually exclusive.
-*/
+/** Maps at a set of unique values to bin indices.
+
+   The axis maps a set of values to bins, following the order of arguments in the
+   constructor. The optional overflow bin for this axis counts input values that
+   are not part of the set. Binning has O(N) complexity, but with a very small
+   factor. For small N (the typical use case) it beats other kinds of lookup.
+
+   @tparam Value input value type, must be equal-comparable.
+   @tparam MetaData type to store meta data.
+   @tparam Options see boost::histogram::axis::option.
+   @tparam Allocator allocator to use for dynamic memory management.
+
+   The options `underflow` and `circular` are not allowed. The options `growth`
+   and `overflow` are mutually exclusive.
+ */
 template <class Value, class MetaData, class Options, class Allocator>
 class category : public iterator_mixin<category<Value, MetaData, Options, Allocator>>,
                  public metadata_base_t<MetaData> {
@@ -66,12 +65,12 @@ class category : public iterator_mixin<category<Value, MetaData, Options, Alloca
   explicit category(allocator_type alloc) : vec_(alloc) {}
 
   /** Construct from iterator range of unique values.
-   *
-   * @param begin    begin of category range of unique values.
-   * @param end      end of category range of unique values.
-   * @param meta     description of the axis (optional).
-   * @param options  see boost::histogram::axis::option (optional).
-   * @param alloc    allocator instance to use (optional).
+
+     @param begin    begin of category range of unique values.
+     @param end      end of category range of unique values.
+     @param meta     description of the axis (optional).
+     @param options  see boost::histogram::axis::option (optional).
+     @param alloc    allocator instance to use (optional).
    */
   template <class It, class = detail::requires_iterator<It>>
   category(It begin, It end, metadata_type meta = {}, options_type options = {},
@@ -91,11 +90,11 @@ class category : public iterator_mixin<category<Value, MetaData, Options, Alloca
       : category(begin, end, std::move(meta), {}, std::move(alloc)) {}
 
   /** Construct axis from iterable sequence of unique values.
-   *
-   * @param iterable sequence of unique values.
-   * @param meta     description of the axis.
-   * @param options  see boost::histogram::axis::option (optional).
-   * @param alloc    allocator instance to use.
+
+     @param iterable sequence of unique values.
+     @param meta     description of the axis.
+     @param options  see boost::histogram::axis::option (optional).
+     @param alloc    allocator instance to use.
    */
   template <class C, class = detail::requires_iterable<C>>
   category(const C& iterable, metadata_type meta = {}, options_type options = {},
@@ -110,11 +109,11 @@ class category : public iterator_mixin<category<Value, MetaData, Options, Alloca
                  std::move(alloc)) {}
 
   /** Construct axis from an initializer list of unique values.
-   *
-   * @param list     `std::initializer_list` of unique values.
-   * @param meta     description of the axis.
-   * @param options  see boost::histogram::axis::option (optional).
-   * @param alloc    allocator instance to use.
+
+     @param list     `std::initializer_list` of unique values.
+     @param meta     description of the axis.
+     @param options  see boost::histogram::axis::option (optional).
+     @param alloc    allocator instance to use.
    */
   template <class U>
   category(std::initializer_list<U> list, metadata_type meta = {},
diff --git a/include/boost/histogram/axis/integer.hpp b/include/boost/histogram/axis/integer.hpp
index 4110875c..d61dfdd8 100644
--- a/include/boost/histogram/axis/integer.hpp
+++ b/include/boost/histogram/axis/integer.hpp
@@ -29,14 +29,13 @@ namespace boost {
 namespace histogram {
 namespace axis {
 
-/**
-  Axis for an interval of integer values with unit steps.
+/** Axis for an interval of integer values with unit steps.
 
-  Binning is a O(1) operation. This axis bins faster than a regular axis.
+   Binning is a O(1) operation. This axis bins faster than a regular axis.
 
-  @tparam Value     input value type. Must be integer or floating point.
-  @tparam MetaData  type to store meta data.
-  @tparam Options   see boost::histogram::axis::option.
+   @tparam Value     input value type. Must be integer or floating point.
+   @tparam MetaData  type to store meta data.
+   @tparam Options   see boost::histogram::axis::option.
  */
 template <class Value, class MetaData, class Options>
 class integer : public iterator_mixin<integer<Value, MetaData, Options>>,
@@ -72,11 +71,11 @@ class integer : public iterator_mixin<integer<Value, MetaData, Options>>,
   constexpr integer() = default;
 
   /** Construct over semi-open integer interval [start, stop).
-   *
-   * @param start    first integer of covered range.
-   * @param stop     one past last integer of covered range.
-   * @param meta     description of the axis (optional).
-   * @param options  see boost::histogram::axis::option (optional).
+
+     @param start    first integer of covered range.
+     @param stop     one past last integer of covered range.
+     @param meta     description of the axis (optional).
+     @param options  see boost::histogram::axis::option (optional).
    */
   integer(value_type start, value_type stop, metadata_type meta = {},
           options_type options = {})
diff --git a/include/boost/histogram/axis/regular.hpp b/include/boost/histogram/axis/regular.hpp
index 3f7df9e0..403931f8 100644
--- a/include/boost/histogram/axis/regular.hpp
+++ b/include/boost/histogram/axis/regular.hpp
@@ -165,15 +165,22 @@ step_type<T> step(T t) {
   return step_type<T>{t};
 }
 
-/**
-  Axis for equidistant intervals on the real line.
+/** Axis for equidistant intervals on the real line.
+
+   The most common binning strategy. Very fast. Binning is a O(1) operation.
 
-  The most common binning strategy. Very fast. Binning is a O(1) operation.
+   If the axis has an overflow bin (the default), a value on the upper edge of the last
+   bin is put in the overflow bin. The axis range represents a semi-open interval.
 
-  @tparam Value input value type, must be floating point.
-  @tparam Transform builtin or user-defined transform type.
-  @tparam MetaData type to store meta data.
-  @tparam Options see boost::histogram::axis::option.
+   If the overflow bin is deactivated, then a value on the upper edge of the last bin is
+   still counted towards the last bin. The axis range represents a closed interval. This
+   is the desired behavior for random numbers drawn from a bounded interval, which is
+   usually closed.
+
+   @tparam Value input value type, must be floating point.
+   @tparam Transform builtin or user-defined transform type.
+   @tparam MetaData type to store meta data.
+   @tparam Options see boost::histogram::axis::option.
  */
 template <class Value, class Transform, class MetaData, class Options>
 class regular : public iterator_mixin<regular<Value, Transform, MetaData, Options>>,
@@ -207,13 +214,13 @@ class regular : public iterator_mixin<regular<Value, Transform, MetaData, Option
   constexpr regular() = default;
 
   /** Construct n bins over real transformed range [start, stop).
-   *
-   * @param trans    transform instance to use.
-   * @param n        number of bins.
-   * @param start    low edge of first bin.
-   * @param stop     high edge of last bin.
-   * @param meta     description of the axis (optional).
-   * @param options  see boost::histogram::axis::option (optional).
+
+     @param trans    transform instance to use.
+     @param n        number of bins.
+     @param start    low edge of first bin.
+     @param stop     high edge of last bin.
+     @param meta     description of the axis (optional).
+     @param options  see boost::histogram::axis::option (optional).
    */
   regular(transform_type trans, unsigned n, value_type start, value_type stop,
           metadata_type meta = {}, options_type options = {})
@@ -232,30 +239,30 @@ class regular : public iterator_mixin<regular<Value, Transform, MetaData, Option
   }
 
   /** Construct n bins over real range [start, stop).
-   *
-   * @param n        number of bins.
-   * @param start    low edge of first bin.
-   * @param stop     high edge of last bin.
-   * @param meta     description of the axis (optional).
-   * @param options  see boost::histogram::axis::option (optional).
+
+     @param n        number of bins.
+     @param start    low edge of first bin.
+     @param stop     high edge of last bin.
+     @param meta     description of the axis (optional).
+     @param options  see boost::histogram::axis::option (optional).
    */
   regular(unsigned n, value_type start, value_type stop, metadata_type meta = {},
           options_type options = {})
       : regular({}, n, start, stop, std::move(meta), options) {}
 
   /** Construct bins with the given step size over real transformed range
-   * [start, stop).
-   *
-   * @param trans    transform instance to use.
-   * @param step     width of a single bin.
-   * @param start    low edge of first bin.
-   * @param stop     upper limit of high edge of last bin (see below).
-   * @param meta     description of the axis (optional).
-   * @param options  see boost::histogram::axis::option (optional).
-   *
-   * The axis computes the number of bins as n = abs(stop - start) / step,
-   * rounded down. This means that stop is an upper limit to the actual value
-   * (start + n * step).
+     [start, stop).
+
+     @param trans    transform instance to use.
+     @param step     width of a single bin.
+     @param start    low edge of first bin.
+     @param stop     upper limit of high edge of last bin (see below).
+     @param meta     description of the axis (optional).
+     @param options  see boost::histogram::axis::option (optional).
+
+     The axis computes the number of bins as n = abs(stop - start) / step,
+     rounded down. This means that stop is an upper limit to the actual value
+     (start + n * step).
    */
   template <class T>
   regular(transform_type trans, step_type<T> step, value_type start, value_type stop,
@@ -267,16 +274,16 @@ class regular : public iterator_mixin<regular<Value, Transform, MetaData, Option
                 std::move(meta), options) {}
 
   /** Construct bins with the given step size over real range [start, stop).
-   *
-   * @param step     width of a single bin.
-   * @param start    low edge of first bin.
-   * @param stop     upper limit of high edge of last bin (see below).
-   * @param meta     description of the axis (optional).
-   * @param options  see boost::histogram::axis::option (optional).
-   *
-   * The axis computes the number of bins as n = abs(stop - start) / step,
-   * rounded down. This means that stop is an upper limit to the actual value
-   * (start + n * step).
+
+     @param step     width of a single bin.
+     @param start    low edge of first bin.
+     @param stop     upper limit of high edge of last bin (see below).
+     @param meta     description of the axis (optional).
+     @param options  see boost::histogram::axis::option (optional).
+
+     The axis computes the number of bins as n = abs(stop - start) / step,
+     rounded down. This means that stop is an upper limit to the actual value
+     (start + n * step).
    */
   template <class T>
   regular(step_type<T> step, value_type start, value_type stop, metadata_type meta = {},
@@ -311,6 +318,8 @@ class regular : public iterator_mixin<regular<Value, Transform, MetaData, Option
         else
           return -1;
       }
+      // upper edge of last bin is inclusive if overflow bin is not present
+      if (!options_type::test(option::overflow) && z == 1) return size() - 1;
     }
     return size(); // also returned if x is NaN
   }
diff --git a/include/boost/histogram/axis/variable.hpp b/include/boost/histogram/axis/variable.hpp
index f80e54c1..d34df5ed 100644
--- a/include/boost/histogram/axis/variable.hpp
+++ b/include/boost/histogram/axis/variable.hpp
@@ -34,17 +34,24 @@ namespace boost {
 namespace histogram {
 namespace axis {
 
-/**
-  Axis for non-equidistant bins on the real line.
+/** Axis for non-equidistant bins on the real line.
 
   Binning is a O(log(N)) operation. If speed matters and the problem domain
   allows it, prefer a regular axis, possibly with a transform.
 
+  If the axis has an overflow bin (the default), a value on the upper edge of the last
+  bin is put in the overflow bin. The axis range represents a semi-open interval.
+
+  If the overflow bin is deactivated, then a value on the upper edge of the last bin is
+  still counted towards the last bin. The axis range represents a closed interval. This
+  is the desired behavior for random numbers drawn from a bounded interval, which is
+  usually closed.
+
   @tparam Value     input value type, must be floating point.
   @tparam MetaData  type to store meta data.
   @tparam Options   see boost::histogram::axis::option.
   @tparam Allocator allocator to use for dynamic memory management.
- */
+*/
 template <class Value, class MetaData, class Options, class Allocator>
 class variable : public iterator_mixin<variable<Value, MetaData, Options, Allocator>>,
                  public metadata_base_t<MetaData> {
@@ -72,12 +79,12 @@ class variable : public iterator_mixin<variable<Value, MetaData, Options, Alloca
   explicit variable(allocator_type alloc) : vec_(alloc) {}
 
   /** Construct from iterator range of bin edges.
-   *
-   * @param begin   begin of edge sequence.
-   * @param end     end of edge sequence.
-   * @param meta    description of the axis (optional).
-   * @param options see boost::histogram::axis::option (optional).
-   * @param alloc   allocator instance to use (optional).
+
+     @param begin   begin of edge sequence.
+     @param end     end of edge sequence.
+     @param meta    description of the axis (optional).
+     @param options see boost::histogram::axis::option (optional).
+     @param alloc   allocator instance to use (optional).
    */
   template <class It, class = detail::requires_iterator<It>>
   variable(It begin, It end, metadata_type meta = {}, options_type options = {},
@@ -106,11 +113,11 @@ class variable : public iterator_mixin<variable<Value, MetaData, Options, Alloca
       : variable(begin, end, std::move(meta), {}, std::move(alloc)) {}
 
   /** Construct variable axis from iterable range of bin edges.
-   *
-   * @param iterable iterable range of bin edges.
-   * @param meta     description of the axis (optional).
-   * @param options  see boost::histogram::axis::option (optional).
-   * @param alloc    allocator instance to use (optional).
+
+     @param iterable iterable range of bin edges.
+     @param meta     description of the axis (optional).
+     @param options  see boost::histogram::axis::option (optional).
+     @param alloc    allocator instance to use (optional).
    */
   template <class U, class = detail::requires_iterable<U>>
   variable(const U& iterable, metadata_type meta = {}, options_type options = {},
@@ -125,11 +132,11 @@ class variable : public iterator_mixin<variable<Value, MetaData, Options, Alloca
                  std::move(alloc)) {}
 
   /** Construct variable axis from initializer list of bin edges.
-   *
-   * @param list     `std::initializer_list` of bin edges.
-   * @param meta     description of the axis (optional).
-   * @param options  see boost::histogram::axis::option (optional).
-   * @param alloc    allocator instance to use (optional).
+
+     @param list     `std::initializer_list` of bin edges.
+     @param meta     description of the axis (optional).
+     @param options  see boost::histogram::axis::option (optional).
+     @param alloc    allocator instance to use (optional).
    */
   template <class U>
   variable(std::initializer_list<U> list, metadata_type meta = {},
@@ -159,6 +166,8 @@ class variable : public iterator_mixin<variable<Value, MetaData, Options, Alloca
       const auto b = vec_[size()];
       x -= std::floor((x - a) / (b - a)) * (b - a);
     }
+    // upper edge of last bin is inclusive if overflow bin is not present
+    if (!options_type::test(option::overflow) && x == vec_.back()) return size() - 1;
     return static_cast<index_type>(std::upper_bound(vec_.begin(), vec_.end(), x) -
                                    vec_.begin() - 1);
   }
diff --git a/test/axis_regular_test.cpp b/test/axis_regular_test.cpp
index 2fece7f2..3c3102e2 100644
--- a/test/axis_regular_test.cpp
+++ b/test/axis_regular_test.cpp
@@ -20,6 +20,7 @@ int main() {
   using namespace boost::histogram;
   using def = use_default;
   namespace tr = axis::transform;
+  namespace op = axis::option;
 
   BOOST_TEST(std::is_nothrow_move_assignable<axis::regular<>>::value);
   BOOST_TEST(std::is_nothrow_move_constructible<axis::regular<>>::value);
@@ -202,7 +203,7 @@ int main() {
   // with growth
   {
     using pii_t = std::pair<axis::index_type, axis::index_type>;
-    axis::regular<double, def, def, axis::option::growth_t> a{1, 0, 1};
+    axis::regular<double, def, def, op::growth_t> a{1, 0, 1};
     BOOST_TEST_EQ(a.size(), 1);
     BOOST_TEST_EQ(a.update(0), pii_t(0, 0));
     BOOST_TEST_EQ(a.size(), 1);
@@ -223,11 +224,32 @@ int main() {
     BOOST_TEST_EQ(a.update(-std::numeric_limits<double>::infinity()), pii_t(-1, 0));
   }
 
+  // axis with overflow bin represents open interval
+  {
+    axis::regular<double, def, def, op::overflow_t> a{2, 0, 1};
+    BOOST_TEST_EQ(a.index(0), 0);
+    BOOST_TEST_EQ(a.index(0.49), 0);
+    BOOST_TEST_EQ(a.index(0.50), 1);
+    BOOST_TEST_EQ(a.index(0.99), 1);
+    BOOST_TEST_EQ(a.index(1), 2);   // overflow bin
+    BOOST_TEST_EQ(a.index(1.1), 2); // overflow bin
+  }
+
+  // axis without overflow bin represents a closed interval
+  {
+    axis::regular<double, def, def, op::none_t> a{2, 0, 1};
+    BOOST_TEST_EQ(a.index(0), 0);
+    BOOST_TEST_EQ(a.index(0.49), 0);
+    BOOST_TEST_EQ(a.index(0.50), 1);
+    BOOST_TEST_EQ(a.index(0.99), 1);
+    BOOST_TEST_EQ(a.index(1), 1);   // last ordinary bin
+    BOOST_TEST_EQ(a.index(1.1), 2); // out of range
+  }
+
   // iterators
   {
     test_axis_iterator(axis::regular<>(5, 0, 1), 0, 5);
-    test_axis_iterator(axis::regular<double, def, def, axis::option::none_t>(5, 0, 1), 0,
-                       5);
+    test_axis_iterator(axis::regular<double, def, def, op::none_t>(5, 0, 1), 0, 5);
     test_axis_iterator(axis::circular<>(5, 0, 1), 0, 5);
   }
 
diff --git a/test/axis_variable_test.cpp b/test/axis_variable_test.cpp
index b4c51ba1..5b205c71 100644
--- a/test/axis_variable_test.cpp
+++ b/test/axis_variable_test.cpp
@@ -18,6 +18,7 @@
 #include "utility_str.hpp"
 
 using namespace boost::histogram;
+namespace op = boost::histogram::axis::option;
 
 int main() {
   constexpr auto inf = std::numeric_limits<double>::infinity();
@@ -104,7 +105,7 @@ int main() {
 
   // axis::variable circular
   {
-    axis::variable<double, axis::null_type, axis::option::circular_t> a{-1, 1, 2};
+    axis::variable<double, axis::null_type, op::circular_t> a{-1, 1, 2};
     BOOST_TEST_EQ(a.value(-2), -4);
     BOOST_TEST_EQ(a.value(-1), -2);
     BOOST_TEST_EQ(a.value(0), -1);
@@ -125,7 +126,7 @@ int main() {
   // axis::regular with growth
   {
     using pii_t = std::pair<axis::index_type, axis::index_type>;
-    axis::variable<double, axis::null_type, axis::option::growth_t> a{0, 1};
+    axis::variable<double, axis::null_type, op::growth_t> a{0, 1};
     BOOST_TEST_EQ(a.size(), 1);
     BOOST_TEST_EQ(a.update(0), pii_t(0, 0));
     BOOST_TEST_EQ(a.size(), 1);
@@ -149,11 +150,33 @@ int main() {
     BOOST_TEST_EQ(a.update(nan), pii_t(a.size(), 0));
   }
 
+  // axis with overflow bin represents open interval
+  {
+    axis::variable<double, boost::use_default, op::overflow_t> a{0.0, 0.5, 1.0};
+    BOOST_TEST_EQ(a.index(0), 0);
+    BOOST_TEST_EQ(a.index(0.49), 0);
+    BOOST_TEST_EQ(a.index(0.50), 1);
+    BOOST_TEST_EQ(a.index(0.99), 1);
+    BOOST_TEST_EQ(a.index(1), 2);   // overflow bin
+    BOOST_TEST_EQ(a.index(1.1), 2); // overflow bin
+  }
+
+  // axis without overflow bin represents a closed interval
+  {
+    axis::variable<double, boost::use_default, op::none_t> a{0.0, 0.5, 1.0};
+    BOOST_TEST_EQ(a.index(0), 0);
+    BOOST_TEST_EQ(a.index(0.49), 0);
+    BOOST_TEST_EQ(a.index(0.50), 1);
+    BOOST_TEST_EQ(a.index(0.99), 1);
+    BOOST_TEST_EQ(a.index(1), 1);   // last ordinary bin
+    BOOST_TEST_EQ(a.index(1.1), 2); // out of range
+  }
+
   // iterators
   {
     test_axis_iterator(axis::variable<>{1, 2, 3}, 0, 2);
-    test_axis_iterator(
-        axis::variable<double, axis::null_type, axis::option::circular_t>{1, 2, 3}, 0, 2);
+    test_axis_iterator(axis::variable<double, axis::null_type, op::circular_t>{1, 2, 3},
+                       0, 2);
   }
 
   // shrink and rebin
@@ -176,7 +199,7 @@ int main() {
 
   // shrink and rebin with circular option
   {
-    using A = axis::variable<double, axis::null_type, axis::option::circular_t>;
+    using A = axis::variable<double, axis::null_type, op::circular_t>;
     auto a = A({1, 2, 3, 4, 5});
     BOOST_TEST_THROWS(A(a, 1, 4, 1), std::invalid_argument);
     BOOST_TEST_THROWS(A(a, 0, 3, 1), std::invalid_argument);