diff --git a/README.md b/README.md
index d7492da3..42d057a3 100644
--- a/README.md
+++ b/README.md
@@ -1557,4 +1557,6 @@ $ cmake --build .
 $ ctest --output-on-failure
 ```
 
-For more information, have a look at the file [.travis.yml](https://github.com/nlohmann/json/blob/master/.travis.yml).
+Note that during the `ctest` stage, several JSON test files are downloaded from an [external repository](https://github.com/nlohmann/json_test_data). If policies forbid downloading artifacts during testing, you can download the files yourself and pass the directory with the test files via `-DJSON_TestDataDirectory=path` to CMake. Then, no Internet connectivity is required. See [issue #2189](https://github.com/nlohmann/json/issues/2189) for more information.
+
+In case you have downloaded the library rather than checked out the code via Git, test `cmake_fetch_content_configure`. Please execute `ctest -LE git_required` to skip these tests. See [issue #2189](https://github.com/nlohmann/json/issues/2189) for more information.
diff --git a/cmake/download_test_data.cmake b/cmake/download_test_data.cmake
index fe95235f..a3f3f199 100644
--- a/cmake/download_test_data.cmake
+++ b/cmake/download_test_data.cmake
@@ -1,17 +1,22 @@
-find_package(Git)
-
 set(JSON_TEST_DATA_URL     https://github.com/nlohmann/json_test_data)
-set(JSON_TEST_DATA_VERSION 2.0.0)
+set(JSON_TEST_DATA_VERSION 3.0.0)
 
-# target to download test data
-add_custom_target(download_test_data
-    COMMAND test -d json_test_data || ${GIT_EXECUTABLE} clone -c advice.detachedHead=false --branch v${JSON_TEST_DATA_VERSION} ${JSON_TEST_DATA_URL}.git --quiet --depth 1
-    COMMENT "Downloading test data from ${JSON_TEST_DATA_URL} (v${JSON_TEST_DATA_VERSION})"
-    WORKING_DIRECTORY ${CMAKE_BINARY_DIR}
-)
-
-# create a header with the path to the downloaded test data
-file(WRITE ${CMAKE_BINARY_DIR}/include/test_data.hpp "#define TEST_DATA_DIRECTORY \"${CMAKE_BINARY_DIR}/json_test_data\"\n")
+# if variable is set, use test data from given directory rather than downloading them
+if(JSON_TestDataDirectory)
+    message(STATUS "Using test data in ${JSON_TestDataDirectory}.")
+    add_custom_target(download_test_data)
+    file(WRITE ${CMAKE_BINARY_DIR}/include/test_data.hpp "#define TEST_DATA_DIRECTORY \"${JSON_TestDataDirectory}\"\n")
+else()
+    find_package(Git)
+    # target to download test data
+    add_custom_target(download_test_data
+        COMMAND test -d json_test_data || ${GIT_EXECUTABLE} clone -c advice.detachedHead=false --branch v${JSON_TEST_DATA_VERSION} ${JSON_TEST_DATA_URL}.git --quiet --depth 1
+        COMMENT "Downloading test data from ${JSON_TEST_DATA_URL} (v${JSON_TEST_DATA_VERSION})"
+        WORKING_DIRECTORY ${CMAKE_BINARY_DIR}
+    )
+    # create a header with the path to the downloaded test data
+    file(WRITE ${CMAKE_BINARY_DIR}/include/test_data.hpp "#define TEST_DATA_DIRECTORY \"${CMAKE_BINARY_DIR}/json_test_data\"\n")
+endif()
 
 # determine the operating system (for debug and support purposes)
 find_program(UNAME_COMMAND uname)
diff --git a/doc/mkdocs/docs/features/binary_formats/messagepack.md b/doc/mkdocs/docs/features/binary_formats/messagepack.md
index ed061056..3e041bb7 100644
--- a/doc/mkdocs/docs/features/binary_formats/messagepack.md
+++ b/doc/mkdocs/docs/features/binary_formats/messagepack.md
@@ -31,7 +31,8 @@ number_unsigned | 128..255                          | uint 8           | 0xCC
 number_unsigned | 256..65535                        | uint 16          | 0xCD
 number_unsigned | 65536..4294967295                 | uint 32          | 0xCE
 number_unsigned | 4294967296..18446744073709551615  | uint 64          | 0xCF
-number_float    | *any value*                       | float 64         | 0xCB
+number_float    | *any value representable by a float*     | float 32  | 0xCA
+number_float    | *any value NOT representable by a float* | float 64  | 0xCB
 string          | *length*: 0..31                   | fixstr           | 0xA0..0xBF
 string          | *length*: 32..255                 | str 8            | 0xD9
 string          | *length*: 256..65535              | str 16           | 0xDA
@@ -61,10 +62,6 @@ binary          | *size*: 65536..4294967295         | bin 32           | 0xC6
 	  - arrays with more than 4294967295 elements
 	  - objects with more than 4294967295 elements
 
-!!! info "Unused MessagePack types"
-
-	The following MessagePack types are not used in the conversion: float 32 (0xCA)
-
 !!! info "NaN/infinity handling"
 
 	If NaN or Infinity are stored inside a JSON number, they are serialized properly. function which serializes NaN or Infinity to `null`.
diff --git a/include/nlohmann/detail/output/binary_writer.hpp b/include/nlohmann/detail/output/binary_writer.hpp
index 269df0d5..7289da3d 100644
--- a/include/nlohmann/detail/output/binary_writer.hpp
+++ b/include/nlohmann/detail/output/binary_writer.hpp
@@ -28,6 +28,7 @@ class binary_writer
 {
     using string_t = typename BasicJsonType::string_t;
     using binary_t = typename BasicJsonType::binary_t;
+    using number_float_t = typename BasicJsonType::number_float_t;
 
   public:
     /*!
@@ -194,18 +195,7 @@ class binary_writer
                 }
                 else
                 {
-                    if (static_cast<double>(j.m_value.number_float) >= static_cast<double>(std::numeric_limits<float>::lowest()) and
-                            static_cast<double>(j.m_value.number_float) <= static_cast<double>((std::numeric_limits<float>::max)()) and
-                            static_cast<double>(static_cast<float>(j.m_value.number_float)) == static_cast<double>(j.m_value.number_float))
-                    {
-                        oa->write_character(get_cbor_float_prefix(static_cast<float>(j.m_value.number_float)));
-                        write_number(static_cast<float>(j.m_value.number_float));
-                    }
-                    else
-                    {
-                        oa->write_character(get_cbor_float_prefix(j.m_value.number_float));
-                        write_number(j.m_value.number_float);
-                    }
+                    write_compact_float(j.m_value.number_float, detail::input_format_t::cbor);
                 }
                 break;
             }
@@ -504,8 +494,7 @@ class binary_writer
 
             case value_t::number_float:
             {
-                oa->write_character(get_msgpack_float_prefix(j.m_value.number_float));
-                write_number(j.m_value.number_float);
+                write_compact_float(j.m_value.number_float, detail::input_format_t::msgpack);
                 break;
             }
 
@@ -1518,6 +1507,26 @@ class binary_writer
         oa->write_characters(vec.data(), sizeof(NumberType));
     }
 
+    void write_compact_float(const number_float_t n, detail::input_format_t format)
+    {
+        if (static_cast<double>(n) >= static_cast<double>(std::numeric_limits<float>::lowest()) and
+                static_cast<double>(n) <= static_cast<double>((std::numeric_limits<float>::max)()) and
+                static_cast<double>(static_cast<float>(n)) == static_cast<double>(n))
+        {
+            oa->write_character(format == detail::input_format_t::cbor
+                                ? get_cbor_float_prefix(static_cast<float>(n))
+                                : get_msgpack_float_prefix(static_cast<float>(n)));
+            write_number(static_cast<float>(n));
+        }
+        else
+        {
+            oa->write_character(format == detail::input_format_t::cbor
+                                ? get_cbor_float_prefix(n)
+                                : get_msgpack_float_prefix(n));
+            write_number(n);
+        }
+    }
+
   public:
     // The following to_char_type functions are implement the conversion
     // between uint8_t and CharType. In case CharType is not unsigned,
diff --git a/include/nlohmann/json.hpp b/include/nlohmann/json.hpp
index 8698a9bb..f4fa590c 100644
--- a/include/nlohmann/json.hpp
+++ b/include/nlohmann/json.hpp
@@ -7061,7 +7061,8 @@ class basic_json
     number_unsigned | 256..65535                        | uint 16          | 0xCD
     number_unsigned | 65536..4294967295                 | uint 32          | 0xCE
     number_unsigned | 4294967296..18446744073709551615  | uint 64          | 0xCF
-    number_float    | *any value*                       | float 64         | 0xCB
+    number_float    | *any value representable by a float*     | float 32 | 0xCA
+    number_float    | *any value NOT representable by a float* | float 64 | 0xCB
     string          | *length*: 0..31                   | fixstr           | 0xA0..0xBF
     string          | *length*: 32..255                 | str 8            | 0xD9
     string          | *length*: 256..65535              | str 16           | 0xDA
@@ -7085,9 +7086,6 @@ class basic_json
           - arrays with more than 4294967295 elements
           - objects with more than 4294967295 elements
 
-    @note The following MessagePack types are not used in the conversion:
-          - float 32 (0xCA)
-
     @note Any MessagePack output created @ref to_msgpack can be successfully
           parsed by @ref from_msgpack.
 
diff --git a/single_include/nlohmann/json.hpp b/single_include/nlohmann/json.hpp
index c341d40d..f4fe8642 100644
--- a/single_include/nlohmann/json.hpp
+++ b/single_include/nlohmann/json.hpp
@@ -12329,6 +12329,7 @@ class binary_writer
 {
     using string_t = typename BasicJsonType::string_t;
     using binary_t = typename BasicJsonType::binary_t;
+    using number_float_t = typename BasicJsonType::number_float_t;
 
   public:
     /*!
@@ -12495,18 +12496,7 @@ class binary_writer
                 }
                 else
                 {
-                    if (static_cast<double>(j.m_value.number_float) >= static_cast<double>(std::numeric_limits<float>::lowest()) and
-                            static_cast<double>(j.m_value.number_float) <= static_cast<double>((std::numeric_limits<float>::max)()) and
-                            static_cast<double>(static_cast<float>(j.m_value.number_float)) == static_cast<double>(j.m_value.number_float))
-                    {
-                        oa->write_character(get_cbor_float_prefix(static_cast<float>(j.m_value.number_float)));
-                        write_number(static_cast<float>(j.m_value.number_float));
-                    }
-                    else
-                    {
-                        oa->write_character(get_cbor_float_prefix(j.m_value.number_float));
-                        write_number(j.m_value.number_float);
-                    }
+                    write_compact_float(j.m_value.number_float, detail::input_format_t::cbor);
                 }
                 break;
             }
@@ -12805,8 +12795,7 @@ class binary_writer
 
             case value_t::number_float:
             {
-                oa->write_character(get_msgpack_float_prefix(j.m_value.number_float));
-                write_number(j.m_value.number_float);
+                write_compact_float(j.m_value.number_float, detail::input_format_t::msgpack);
                 break;
             }
 
@@ -13819,6 +13808,26 @@ class binary_writer
         oa->write_characters(vec.data(), sizeof(NumberType));
     }
 
+    void write_compact_float(const number_float_t n, detail::input_format_t format)
+    {
+        if (static_cast<double>(n) >= static_cast<double>(std::numeric_limits<float>::lowest()) and
+                static_cast<double>(n) <= static_cast<double>((std::numeric_limits<float>::max)()) and
+                static_cast<double>(static_cast<float>(n)) == static_cast<double>(n))
+        {
+            oa->write_character(format == detail::input_format_t::cbor
+                                ? get_cbor_float_prefix(static_cast<float>(n))
+                                : get_msgpack_float_prefix(static_cast<float>(n)));
+            write_number(static_cast<float>(n));
+        }
+        else
+        {
+            oa->write_character(format == detail::input_format_t::cbor
+                                ? get_cbor_float_prefix(n)
+                                : get_msgpack_float_prefix(n));
+            write_number(n);
+        }
+    }
+
   public:
     // The following to_char_type functions are implement the conversion
     // between uint8_t and CharType. In case CharType is not unsigned,
@@ -22933,7 +22942,8 @@ class basic_json
     number_unsigned | 256..65535                        | uint 16          | 0xCD
     number_unsigned | 65536..4294967295                 | uint 32          | 0xCE
     number_unsigned | 4294967296..18446744073709551615  | uint 64          | 0xCF
-    number_float    | *any value*                       | float 64         | 0xCB
+    number_float    | *any value representable by a float*     | float 32 | 0xCA
+    number_float    | *any value NOT representable by a float* | float 64 | 0xCB
     string          | *length*: 0..31                   | fixstr           | 0xA0..0xBF
     string          | *length*: 32..255                 | str 8            | 0xD9
     string          | *length*: 256..65535              | str 16           | 0xDA
@@ -22957,9 +22967,6 @@ class basic_json
           - arrays with more than 4294967295 elements
           - objects with more than 4294967295 elements
 
-    @note The following MessagePack types are not used in the conversion:
-          - float 32 (0xCA)
-
     @note Any MessagePack output created @ref to_msgpack can be successfully
           parsed by @ref from_msgpack.
 
diff --git a/test/cmake_fetch_content/CMakeLists.txt b/test/cmake_fetch_content/CMakeLists.txt
index 6d92d149..0b0d9f65 100644
--- a/test/cmake_fetch_content/CMakeLists.txt
+++ b/test/cmake_fetch_content/CMakeLists.txt
@@ -11,8 +11,10 @@ if (${CMAKE_VERSION} VERSION_GREATER "3.11.0")
   )
   set_tests_properties(cmake_fetch_content_configure PROPERTIES
     FIXTURES_SETUP cmake_fetch_content
+    LABELS git_required
   )
   set_tests_properties(cmake_fetch_content_build PROPERTIES
     FIXTURES_REQUIRED cmake_fetch_content
+    LABELS git_required
   )
 endif()
diff --git a/test/cmake_fetch_content/project/CMakeLists.txt b/test/cmake_fetch_content/project/CMakeLists.txt
index 742a112b..fd8fbdd5 100644
--- a/test/cmake_fetch_content/project/CMakeLists.txt
+++ b/test/cmake_fetch_content/project/CMakeLists.txt
@@ -4,9 +4,8 @@ project(DummyImport CXX)
 
 include(FetchContent)
 
-FetchContent_Declare(json
-  GIT_REPOSITORY ${CMAKE_CURRENT_SOURCE_DIR}/../../..
-  GIT_TAG HEAD)
+get_filename_component(GIT_REPOSITORY_DIRECTORY ${CMAKE_CURRENT_SOURCE_DIR}/../../.. ABSOLUTE)
+FetchContent_Declare(json GIT_REPOSITORY ${GIT_REPOSITORY_DIRECTORY} GIT_TAG HEAD)
 
 FetchContent_GetProperties(json)
 if(NOT json_POPULATED)
diff --git a/test/src/unit-msgpack.cpp b/test/src/unit-msgpack.cpp
index 2744a67b..5baecf02 100644
--- a/test/src/unit-msgpack.cpp
+++ b/test/src/unit-msgpack.cpp
@@ -783,6 +783,40 @@ TEST_CASE("MessagePack")
                     CHECK(json::from_msgpack(result) == v);
                     CHECK(json::from_msgpack(result, true, false) == j);
                 }
+
+                SECTION("1.0")
+                {
+                    double v = 1.0;
+                    json j = v;
+                    std::vector<uint8_t> expected =
+                    {
+                        0xca, 0x3f, 0x80, 0x00, 0x00
+                    };
+                    const auto result = json::to_msgpack(j);
+                    CHECK(result == expected);
+
+                    // roundtrip
+                    CHECK(json::from_msgpack(result) == j);
+                    CHECK(json::from_msgpack(result) == v);
+                    CHECK(json::from_msgpack(result, true, false) == j);
+                }
+
+                SECTION("128.128")
+                {
+                    double v = 128.1280059814453125;
+                    json j = v;
+                    std::vector<uint8_t> expected =
+                    {
+                        0xca, 0x43, 0x00, 0x20, 0xc5
+                    };
+                    const auto result = json::to_msgpack(j);
+                    CHECK(result == expected);
+
+                    // roundtrip
+                    CHECK(json::from_msgpack(result) == j);
+                    CHECK(json::from_msgpack(result) == v);
+                    CHECK(json::from_msgpack(result, true, false) == j);
+                }
             }
         }