diff --git a/lib/json_schemer/openapi30/meta.rb b/lib/json_schemer/openapi30/meta.rb
index 2a833f9..17e3604 100644
--- a/lib/json_schemer/openapi30/meta.rb
+++ b/lib/json_schemer/openapi30/meta.rb
@@ -4,8 +4,10 @@ module OpenAPI30
     BASE_URI = URI('json-schemer://openapi30/schema')
     # https://spec.openapis.org/oas/v3.0.3#data-types
     FORMATS = OpenAPI31::FORMATS.merge(
-      'byte' => proc { |instance, _value| ContentEncoding::BASE64.call(instance).first },
-      'binary' => proc { |instance, _value| instance.is_a?(String) && instance.encoding == Encoding::BINARY },
+      'int32' => proc { |instance, _format| !instance.is_a?(Integer) || instance.floor.bit_length <= 32 },
+      'int64' => proc { |instance, _format| !instance.is_a?(Integer) || instance.floor.bit_length <= 64 },
+      'byte' => proc { |instance, _value| !instance.is_a?(String) || ContentEncoding::BASE64.call(instance).first },
+      'binary' => proc { |instance, _value| !instance.is_a?(String) || instance.encoding == Encoding::BINARY },
       'date' => Format::DATE
     )
     SCHEMA = {
diff --git a/lib/json_schemer/openapi31/meta.rb b/lib/json_schemer/openapi31/meta.rb
index e9bb7ac..bb20db0 100644
--- a/lib/json_schemer/openapi31/meta.rb
+++ b/lib/json_schemer/openapi31/meta.rb
@@ -4,10 +4,16 @@ module OpenAPI31
     BASE_URI = URI('https://spec.openapis.org/oas/3.1/dialect/base')
     # https://spec.openapis.org/oas/v3.1.0#data-types
     FORMATS = {
-      'int32' => proc { |instance, _format| instance.is_a?(Integer) && instance.bit_length <= 32 },
-      'int64' => proc { |instance, _format| instance.is_a?(Integer) && instance.bit_length <= 64 },
-      'float' => proc { |instance, _format| instance.is_a?(Float) },
-      'double' => proc { |instance, _format| instance.is_a?(Float) },
+      'int32' => proc do |instance, _format|
+        valid_type = instance.is_a?(Numeric) && (instance.is_a?(Integer) || instance.floor == instance)
+        !valid_type || instance.floor.bit_length <= 32
+      end,
+      'int64' => proc do |instance, _format|
+        valid_type = instance.is_a?(Numeric) && (instance.is_a?(Integer) || instance.floor == instance)
+        !valid_type || instance.floor.bit_length <= 64
+      end,
+      'float' => proc { |instance, _format| !instance.is_a?(Numeric) || instance.is_a?(Float) },
+      'double' => proc { |instance, _format| !instance.is_a?(Numeric) || instance.is_a?(Float) },
       'password' => proc { |_instance, _format| true }
     }
     SCHEMA = {
diff --git a/test/open_api_test.rb b/test/open_api_test.rb
index fcdc36c..ebf1c78 100644
--- a/test/open_api_test.rb
+++ b/test/open_api_test.rb
@@ -714,16 +714,100 @@ def test_openapi31_formats
     schemer = JSONSchemer.schema(schema, :meta_schema => JSONSchemer.openapi31)
 
     assert(schemer.valid_schema?)
+    # int32
     assert(schemer.valid?({ 'a' => 2.pow(31) }))
+    assert(schemer.valid?({ 'a' => 2.pow(31).to_f }))
+    assert(schemer.valid?({ 'a' => 2.pow(31).to_s }))
     refute(schemer.valid?({ 'a' => 2.pow(32) }))
+    refute(schemer.valid?({ 'a' => 2.pow(32).to_f }))
+    # int64
     assert(schemer.valid?({ 'b' => 2.pow(63) }))
+    assert(schemer.valid?({ 'b' => 2.pow(63).to_f }))
+    assert(schemer.valid?({ 'a' => 2.pow(63).to_s }))
     refute(schemer.valid?({ 'b' => 2.pow(64) }))
+    refute(schemer.valid?({ 'b' => 2.pow(64).to_f }))
+    # float
     assert(schemer.valid?({ 'c' => 2.0 }))
+    assert(schemer.valid?({ 'c' => 2.to_s }))
     refute(schemer.valid?({ 'c' => 2 }))
+    # double
     assert(schemer.valid?({ 'd' => 2.0 }))
+    assert(schemer.valid?({ 'd' => 2.to_s }))
     refute(schemer.valid?({ 'd' => 2 }))
+    # password
+    assert(schemer.valid?({ 'e' => 'anything' }))
     assert(schemer.valid?({ 'e' => 2 }))
+  end
+
+  def test_openapi31_formats_with_type
+    schema = {
+      'properties' => {
+        'a' => { 'type' => 'integer', 'format' => 'int32' },
+        'b' => { 'type' => 'integer', 'format' => 'int64' },
+        'c' => { 'type' => 'number', 'format' => 'float' },
+        'd' => { 'type' => 'number', 'format' => 'double' },
+        'e' => { 'type' => 'string', 'format' => 'password' }
+      }
+    }
+
+    schemer = JSONSchemer.schema(schema, :meta_schema => JSONSchemer.openapi31)
+
+    assert(schemer.valid_schema?)
+    # int32
+    assert(schemer.valid?({ 'a' => 2.pow(31) }))
+    assert(schemer.valid?({ 'a' => 2.pow(31).to_f }))
+    refute(schemer.valid?({ 'a' => 2.pow(32) }))
+    refute(schemer.valid?({ 'a' => 2.pow(32).to_f }))
+    # int64
+    assert(schemer.valid?({ 'b' => 2.pow(63) }))
+    assert(schemer.valid?({ 'b' => 2.pow(63).to_f }))
+    refute(schemer.valid?({ 'b' => 2.pow(64) }))
+    refute(schemer.valid?({ 'b' => 2.pow(64).to_f }))
+    # float
+    assert(schemer.valid?({ 'c' => 2.0 }))
+    refute(schemer.valid?({ 'c' => 2 }))
+    # double
+    assert(schemer.valid?({ 'd' => 2.0 }))
+    refute(schemer.valid?({ 'd' => 2 }))
+    # password
     assert(schemer.valid?({ 'e' => 'anything' }))
+    refute(schemer.valid?({ 'e' => 2 }))
+  end
+
+  def test_openapi31_formats_with_multiple_types
+    schema = {
+      'properties' => {
+        'a' => { 'type' => ['integer', 'null'], 'format' => 'int32' },
+        'b' => { 'type' => ['integer', 'null'], 'format' => 'int64' },
+        'c' => { 'type' => ['number', 'null'], 'format' => 'float' },
+        'd' => { 'type' => ['number', 'null'], 'format' => 'double' },
+        'e' => { 'type' => ['string', 'null'], 'format' => 'password' }
+      }
+    }
+
+    schemer = JSONSchemer.schema(schema, :meta_schema => JSONSchemer.openapi31)
+
+    assert(schemer.valid_schema?)
+    # int32
+    assert(schemer.valid?({ 'a' => 2.pow(31) }))
+    assert(schemer.valid?({ 'a' => nil }))
+    refute(schemer.valid?({ 'a' => 2.pow(32) }))
+    # int64
+    assert(schemer.valid?({ 'b' => 2.pow(63) }))
+    assert(schemer.valid?({ 'b' => nil }))
+    refute(schemer.valid?({ 'b' => 2.pow(64) }))
+    # float
+    assert(schemer.valid?({ 'c' => 2.0 }))
+    assert(schemer.valid?({ 'c' => nil }))
+    refute(schemer.valid?({ 'c' => 2 }))
+    # double
+    assert(schemer.valid?({ 'd' => 2.0 }))
+    assert(schemer.valid?({ 'd' => nil }))
+    refute(schemer.valid?({ 'd' => 2 }))
+    # password
+    assert(schemer.valid?({ 'e' => 'anything' }))
+    assert(schemer.valid?({ 'e' => nil }))
+    refute(schemer.valid?({ 'e' => 2 }))
   end
 
   def test_openapi30_formats
@@ -744,24 +828,158 @@ def test_openapi30_formats
     schemer = JSONSchemer.schema(schema, :meta_schema => JSONSchemer.openapi30)
 
     assert(schemer.valid_schema?)
+    # int32
     assert(schemer.valid?({ 'a' => 2.pow(31) }))
+    assert(schemer.valid?({ 'a' => 2.pow(31).to_f }))
+    assert(schemer.valid?({ 'a' => 2.pow(31).to_s }))
+    assert(schemer.valid?({ 'a' => 2.pow(32).to_f }))
     refute(schemer.valid?({ 'a' => 2.pow(32) }))
+    # int64
     assert(schemer.valid?({ 'b' => 2.pow(63) }))
+    assert(schemer.valid?({ 'b' => 2.pow(63).to_f }))
+    assert(schemer.valid?({ 'a' => 2.pow(63).to_s }))
+    assert(schemer.valid?({ 'b' => 2.pow(64).to_f }))
     refute(schemer.valid?({ 'b' => 2.pow(64) }))
+    # float
     assert(schemer.valid?({ 'c' => 2.0 }))
+    assert(schemer.valid?({ 'c' => 2.to_s }))
     refute(schemer.valid?({ 'c' => 2 }))
+    # double
     assert(schemer.valid?({ 'd' => 2.0 }))
+    assert(schemer.valid?({ 'd' => 2.to_s }))
     refute(schemer.valid?({ 'd' => 2 }))
-    assert(schemer.valid?({ 'e' => 2 }))
+    # password
     assert(schemer.valid?({ 'e' => 'anything' }))
-    refute(schemer.valid?({ 'f' => '!' }))
+    assert(schemer.valid?({ 'e' => 2 }))
+    # byte
     assert(schemer.valid?({ 'f' => 'IQ==' }))
-    refute(schemer.valid?({ 'g' => '!' }))
+    assert(schemer.valid?({ 'f' => 123 }))
+    refute(schemer.valid?({ 'f' => '!' }))
+    # binary
     assert(schemer.valid?({ 'g' => '!'.b }))
+    assert(schemer.valid?({ 'g' => 123 }))
+    refute(schemer.valid?({ 'g' => '!' }))
+    # date
+    assert(schemer.valid?({ 'h' => '2001-02-03' }))
+    assert(schemer.valid?({ 'h' => 123 }))
     refute(schemer.valid?({ 'h' => '2001-02-03T04:05:06.123456789+07:00' }))
+    # date-time
+    assert(schemer.valid?({ 'i' => '2001-02-03T04:05:06.123456789+07:00' }))
+    assert(schemer.valid?({ 'i' => 123 }))
+    refute(schemer.valid?({ 'i' => '2001-02-03' }))
+  end
+
+  def test_openapi30_formats_with_type
+    schema = {
+      'properties' => {
+        'a' => { 'type' => 'integer', 'format' => 'int32' },
+        'b' => { 'type' => 'integer', 'format' => 'int64' },
+        'c' => { 'type' => 'number', 'format' => 'float' },
+        'd' => { 'type' => 'number', 'format' => 'double' },
+        'e' => { 'type' => 'string', 'format' => 'password' },
+        'f' => { 'type' => 'string', 'format' => 'byte' },
+        'g' => { 'type' => 'string', 'format' => 'binary' },
+        'h' => { 'type' => 'string', 'format' => 'date' },
+        'i' => { 'type' => 'string', 'format' => 'date-time' }
+      }
+    }
+
+    schemer = JSONSchemer.schema(schema, :meta_schema => JSONSchemer.openapi30)
+
+    assert(schemer.valid_schema?)
+    # int32
+    assert(schemer.valid?({ 'a' => 2.pow(31) }))
+    refute(schemer.valid?({ 'a' => 2.pow(31).to_s }))
+    refute(schemer.valid?({ 'a' => 2.pow(32) }))
+    refute(schemer.valid?({ 'a' => 2.pow(32).to_f }))
+    # int64
+    assert(schemer.valid?({ 'b' => 2.pow(63) }))
+    refute(schemer.valid?({ 'a' => 2.pow(63).to_s }))
+    refute(schemer.valid?({ 'b' => 2.pow(64) }))
+    refute(schemer.valid?({ 'b' => 2.pow(64).to_f }))
+    # float
+    assert(schemer.valid?({ 'c' => 2.0 }))
+    refute(schemer.valid?({ 'c' => 2 }))
+    refute(schemer.valid?({ 'c' => 2.to_s }))
+    # double
+    assert(schemer.valid?({ 'd' => 2.0 }))
+    refute(schemer.valid?({ 'd' => 2 }))
+    refute(schemer.valid?({ 'd' => 2.to_s }))
+    # password
+    assert(schemer.valid?({ 'e' => 'anything' }))
+    refute(schemer.valid?({ 'e' => 2 }))
+    # byte
+    assert(schemer.valid?({ 'f' => 'IQ==' }))
+    refute(schemer.valid?({ 'f' => '!' }))
+    refute(schemer.valid?({ 'f' => 123 }))
+    # binary
+    assert(schemer.valid?({ 'g' => '!'.b }))
+    refute(schemer.valid?({ 'g' => '!' }))
+    refute(schemer.valid?({ 'g' => 123 }))
+    # date
     assert(schemer.valid?({ 'h' => '2001-02-03' }))
+    refute(schemer.valid?({ 'h' => '2001-02-03T04:05:06.123456789+07:00' }))
+    refute(schemer.valid?({ 'h' => 123 }))
+    # date-time
+    assert(schemer.valid?({ 'i' => '2001-02-03T04:05:06.123456789+07:00' }))
     refute(schemer.valid?({ 'i' => '2001-02-03' }))
+    refute(schemer.valid?({ 'i' => 123 }))
+  end
+
+  def test_openapi30_nullable_formats
+    schema = {
+      'properties' => {
+        'a' => { 'type' => 'integer', 'format' => 'int32', 'nullable' => true },
+        'b' => { 'type' => 'integer', 'format' => 'int64', 'nullable' => true },
+        'c' => { 'type' => 'number', 'format' => 'float', 'nullable' => true },
+        'd' => { 'type' => 'number', 'format' => 'double', 'nullable' => true },
+        'e' => { 'type' => 'string', 'format' => 'password', 'nullable' => true },
+        'f' => { 'type' => 'string', 'format' => 'byte', 'nullable' => true },
+        'g' => { 'type' => 'string', 'format' => 'binary', 'nullable' => true },
+        'h' => { 'type' => 'string', 'format' => 'date', 'nullable' => true },
+        'i' => { 'type' => 'string', 'format' => 'date-time', 'nullable' => true }
+      }
+    }
+
+    schemer = JSONSchemer.schema(schema, :meta_schema => JSONSchemer.openapi30)
+
+    assert(schemer.valid_schema?)
+    # int32
+    assert(schemer.valid?({ 'a' => 2.pow(31) }))
+    assert(schemer.valid?({ 'a' => nil }))
+    refute(schemer.valid?({ 'a' => 2.pow(32) }))
+    # int64
+    assert(schemer.valid?({ 'b' => 2.pow(63) }))
+    assert(schemer.valid?({ 'b' => nil }))
+    refute(schemer.valid?({ 'b' => 2.pow(64) }))
+    # float
+    assert(schemer.valid?({ 'c' => 2.0 }))
+    assert(schemer.valid?({ 'c' => nil }))
+    refute(schemer.valid?({ 'c' => 2 }))
+    # double
+    assert(schemer.valid?({ 'd' => 2.0 }))
+    assert(schemer.valid?({ 'd' => nil }))
+    refute(schemer.valid?({ 'd' => 2 }))
+    # password
+    assert(schemer.valid?({ 'e' => 'anything' }))
+    assert(schemer.valid?({ 'e' => nil }))
+    refute(schemer.valid?({ 'e' => 2 }))
+    # byte
+    assert(schemer.valid?({ 'f' => 'IQ==' }))
+    assert(schemer.valid?({ 'f' => nil }))
+    refute(schemer.valid?({ 'f' => '!' }))
+    # binary
+    assert(schemer.valid?({ 'g' => '!'.b }))
+    assert(schemer.valid?({ 'g' => nil }))
+    refute(schemer.valid?({ 'g' => '!' }))
+    # date
+    assert(schemer.valid?({ 'h' => '2001-02-03' }))
+    assert(schemer.valid?({ 'h' => nil }))
+    refute(schemer.valid?({ 'h' => '2001-02-03T04:05:06.123456789+07:00' }))
+    # date-time
     assert(schemer.valid?({ 'i' => '2001-02-03T04:05:06.123456789+07:00' }))
+    assert(schemer.valid?({ 'i' => nil }))
+    refute(schemer.valid?({ 'i' => '2001-02-03' }))
   end
 
   def test_unsupported_openapi_version