Support BFloat16 in Eigen

6 files changed, 404 insertions, 0 deletions

diff --git a/test/CMakeLists.txt b/test/CMakeLists.txt
index 06c7144ee..b0950d081 100644
--- a/test/CMakeLists.txt
+++ b/test/CMakeLists.txt
@@ -286,6 +286,7 @@ ei_add_test(ctorleak)
 ei_add_test(mpl2only)
 ei_add_test(inplace_decomposition)
 ei_add_test(half_float)
+ei_add_test(bfloat16_float)
 ei_add_test(array_of_string)
 ei_add_test(num_dimensions)
 ei_add_test(stl_iterators)
diff --git a/test/bfloat16_float.cpp b/test/bfloat16_float.cpp
new file mode 100644
index 000000000..eb55f7d45
--- /dev/null
+++ b/test/bfloat16_float.cpp
@@ -0,0 +1,399 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+#include <sstream>
+#include <memory>
+#include <math.h>
+
+#include "main.h"
+
+#include <Eigen/src/Core/arch/Default/BFloat16.h>
+
+// Make sure it's possible to forward declare Eigen::bfloat16
+namespace Eigen {
+struct bfloat16;
+}
+
+using Eigen::bfloat16;
+
+float BinaryToFloat(uint32_t sign, uint32_t exponent, uint32_t high_mantissa,
+                    uint32_t low_mantissa) {
+  float dest;
+  uint32_t src = (sign << 31) + (exponent << 23) + (high_mantissa << 16) + low_mantissa;
+  memcpy(static_cast<void*>(&dest),
+         static_cast<const void*>(&src), sizeof(dest));
+  return dest;
+}
+
+void test_truncate(float input, float expected_truncation, float expected_rounding){
+  bfloat16 truncated = Eigen::bfloat16_impl::truncate_to_bfloat16(input);
+  bfloat16 rounded = Eigen::bfloat16_impl::float_to_bfloat16_rtne(input);
+  if ((numext::isnan)(input)){
+    VERIFY((numext::isnan)(static_cast<float>(truncated)) || (numext::isinf)(static_cast<float>(truncated)));
+    VERIFY((numext::isnan)(static_cast<float>(rounded)) || (numext::isinf)(static_cast<float>(rounded)));
+    return;
+  }
+  VERIFY_IS_EQUAL(expected_truncation, static_cast<float>(truncated));
+  VERIFY_IS_EQUAL(expected_rounding, static_cast<float>(rounded));
+}
+
+void test_conversion()
+{
+  using Eigen::bfloat16_impl::__bfloat16_raw;
+
+  // Conversion from float.
+  VERIFY_IS_EQUAL(bfloat16(1.0f).value, 0x3f80);
+  VERIFY_IS_EQUAL(bfloat16(0.5f).value, 0x3f00);
+  VERIFY_IS_EQUAL(bfloat16(0.33333f).value, 0x3eab);
+  VERIFY_IS_EQUAL(bfloat16(3.38e38f).value, 0x7f7e);
+  VERIFY_IS_EQUAL(bfloat16(3.40e38f).value, 0x7f80);  // Becomes infinity.
+
+  // Verify round-to-nearest-even behavior.
+  float val1 = static_cast<float>(bfloat16(__bfloat16_raw(0x3c00)));
+  float val2 = static_cast<float>(bfloat16(__bfloat16_raw(0x3c01)));
+  float val3 = static_cast<float>(bfloat16(__bfloat16_raw(0x3c02)));
+  VERIFY_IS_EQUAL(bfloat16(0.5f * (val1 + val2)).value, 0x3c00);
+  VERIFY_IS_EQUAL(bfloat16(0.5f * (val2 + val3)).value, 0x3c02);
+
+  // Conversion from int.
+  VERIFY_IS_EQUAL(bfloat16(-1).value, 0xbf80);
+  VERIFY_IS_EQUAL(bfloat16(0).value, 0x0000);
+  VERIFY_IS_EQUAL(bfloat16(1).value, 0x3f80);
+  VERIFY_IS_EQUAL(bfloat16(2).value, 0x4000);
+  VERIFY_IS_EQUAL(bfloat16(3).value, 0x4040);
+  VERIFY_IS_EQUAL(bfloat16(12).value, 0x4140);
+
+  // Conversion from bool.
+  VERIFY_IS_EQUAL(bfloat16(false).value, 0x0000);
+  VERIFY_IS_EQUAL(bfloat16(true).value, 0x3f80);
+
+  // Conversion to float.
+  VERIFY_IS_EQUAL(static_cast<float>(bfloat16(__bfloat16_raw(0x0000))), 0.0f);
+  VERIFY_IS_EQUAL(static_cast<float>(bfloat16(__bfloat16_raw(0x3f80))), 1.0f);
+
+  // Zero representations
+  VERIFY_IS_EQUAL(bfloat16(0.0f), bfloat16(0.0f));
+  VERIFY_IS_EQUAL(bfloat16(-0.0f), bfloat16(0.0f));
+  VERIFY_IS_EQUAL(bfloat16(-0.0f), bfloat16(-0.0f));
+  VERIFY_IS_EQUAL(bfloat16(0.0f).value, 0x0000);
+  VERIFY_IS_EQUAL(bfloat16(-0.0f).value, 0x8000);
+
+  // Flush denormals to zero
+  for (float denorm = -std::numeric_limits<float>::denorm_min();
+       denorm < std::numeric_limits<float>::denorm_min();
+       denorm = nextafterf(denorm, 1.0f)) {
+    bfloat16 bf_trunc = Eigen::bfloat16_impl::truncate_to_bfloat16(denorm);
+    VERIFY_IS_EQUAL(static_cast<float>(bf_trunc), 0.0f);
+    if (std::signbit(denorm)) {
+      VERIFY_IS_EQUAL(bf_trunc.value, 0x8000);
+    } else {
+      VERIFY_IS_EQUAL(bf_trunc.value, 0x0000);
+    }
+    bfloat16 bf_round = Eigen::bfloat16_impl::float_to_bfloat16_rtne(denorm);
+    VERIFY_IS_EQUAL(static_cast<float>(bf_round), 0.0f);
+    if (std::signbit(denorm)) {
+      VERIFY_IS_EQUAL(bf_round.value, 0x8000);
+    } else {
+      VERIFY_IS_EQUAL(bf_round.value, 0x0000);
+    }
+  }
+
+  // Default is zero
+  VERIFY_IS_EQUAL(static_cast<float>(bfloat16()), 0.0f);
+
+  // Representable floats round trip via bfloat16
+  VERIFY_IS_EQUAL(static_cast<float>(static_cast<bfloat16>(-std::numeric_limits<float>::infinity())), -std::numeric_limits<float>::infinity());
+  VERIFY_IS_EQUAL(static_cast<float>(static_cast<bfloat16>(std::numeric_limits<float>::infinity())), std::numeric_limits<float>::infinity());
+  VERIFY_IS_EQUAL(static_cast<float>(static_cast<bfloat16>(-1.0f)), -1.0f);
+  VERIFY_IS_EQUAL(static_cast<float>(static_cast<bfloat16>(-0.5f)), -0.5f);
+  VERIFY_IS_EQUAL(static_cast<float>(static_cast<bfloat16>(-0.0f)), -0.0f);
+  VERIFY_IS_EQUAL(static_cast<float>(static_cast<bfloat16>(1.0f)), 1.0f);
+  VERIFY_IS_EQUAL(static_cast<float>(static_cast<bfloat16>(0.5f)), 0.5f);
+  VERIFY_IS_EQUAL(static_cast<float>(static_cast<bfloat16>(0.0f)), 0.0f);
+
+  // Truncate test
+  test_truncate(
+      BinaryToFloat(0, 0x80, 0x48, 0xf5c3),
+      BinaryToFloat(0, 0x80, 0x48, 0x0000),
+      BinaryToFloat(0, 0x80, 0x49, 0x0000));
+  test_truncate(
+      BinaryToFloat(1, 0x80, 0x48, 0xf5c3),
+      BinaryToFloat(1, 0x80, 0x48, 0x0000),
+      BinaryToFloat(1, 0x80, 0x49, 0x0000));
+  test_truncate(
+      BinaryToFloat(0, 0x80, 0x48, 0x8000),
+      BinaryToFloat(0, 0x80, 0x48, 0x0000),
+      BinaryToFloat(0, 0x80, 0x48, 0x0000));
+  test_truncate(
+      BinaryToFloat(0, 0xff, 0x00, 0x0001),
+      BinaryToFloat(0, 0xff, 0x40, 0x0000),
+      BinaryToFloat(0, 0xff, 0x40, 0x0000));
+  test_truncate(
+      BinaryToFloat(0, 0xff, 0x7f, 0xffff),
+      BinaryToFloat(0, 0xff, 0x40, 0x0000),
+      BinaryToFloat(0, 0xff, 0x40, 0x0000));
+  test_truncate(
+      BinaryToFloat(1, 0x80, 0x48, 0xc000),
+      BinaryToFloat(1, 0x80, 0x48, 0x0000),
+      BinaryToFloat(1, 0x80, 0x49, 0x0000));
+  test_truncate(
+      BinaryToFloat(0, 0x80, 0x48, 0x0000),
+      BinaryToFloat(0, 0x80, 0x48, 0x0000),
+      BinaryToFloat(0, 0x80, 0x48, 0x0000));
+  test_truncate(
+      BinaryToFloat(0, 0x80, 0x48, 0x4000),
+      BinaryToFloat(0, 0x80, 0x48, 0x0000),
+      BinaryToFloat(0, 0x80, 0x48, 0x0000));
+  test_truncate(
+      BinaryToFloat(0, 0x80, 0x48, 0x8000),
+      BinaryToFloat(0, 0x80, 0x48, 0x0000),
+      BinaryToFloat(0, 0x80, 0x48, 0x0000));
+  test_truncate(
+      BinaryToFloat(0, 0x00, 0x48, 0x8000),
+      BinaryToFloat(0, 0x00, 0x00, 0x0000),
+      BinaryToFloat(0, 0x00, 0x00, 0x0000));
+  test_truncate(
+      BinaryToFloat(0, 0x00, 0x7f, 0xc000),
+      BinaryToFloat(0, 0x00, 0x00, 0x0000),
+      BinaryToFloat(0, 0x00, 0x00, 0x0000));
+
+  // Conversion
+  Array<float,1,100> a;
+  for (int i = 0; i < 100; i++) a(i) = i + 1.25;
+  Array<bfloat16,1,100> b = a.cast<bfloat16>();
+  Array<float,1,100> c = b.cast<float>();
+  for (int i = 0; i < 100; ++i) {
+    VERIFY_LE(numext::abs(c(i) - a(i)), a(i) / 128);
+  }
+
+  // Epsilon
+  VERIFY_LE(1.0f, static_cast<float>((std::numeric_limits<bfloat16>::epsilon)() + bfloat16(1.0f)));
+  VERIFY_IS_EQUAL(1.0f, static_cast<float>((std::numeric_limits<bfloat16>::epsilon)() / bfloat16(2.0f) + bfloat16(1.0f)));
+
+  // Negate
+  VERIFY_IS_EQUAL(static_cast<float>(-bfloat16(3.0f)), -3.0f);
+  VERIFY_IS_EQUAL(static_cast<float>(-bfloat16(-4.5f)), 4.5f);
+
+
+#if !EIGEN_COMP_MSVC
+  // Visual Studio errors out on divisions by 0
+  VERIFY((numext::isnan)(static_cast<float>(bfloat16(0.0 / 0.0))));
+  VERIFY((numext::isinf)(static_cast<float>(bfloat16(1.0 / 0.0))));
+  VERIFY((numext::isinf)(static_cast<float>(bfloat16(-1.0 / 0.0))));
+
+  // Visual Studio errors out on divisions by 0
+  VERIFY((numext::isnan)(bfloat16(0.0 / 0.0)));
+  VERIFY((numext::isinf)(bfloat16(1.0 / 0.0)));
+  VERIFY((numext::isinf)(bfloat16(-1.0 / 0.0)));
+#endif
+
+  // NaNs and infinities.
+  VERIFY(!(numext::isinf)(static_cast<float>(bfloat16(3.38e38f))));  // Largest finite number.
+  VERIFY(!(numext::isnan)(static_cast<float>(bfloat16(0.0f))));
+  VERIFY((numext::isinf)(static_cast<float>(bfloat16(__bfloat16_raw(0xff80)))));
+  VERIFY((numext::isnan)(static_cast<float>(bfloat16(__bfloat16_raw(0xffc0)))));
+  VERIFY((numext::isinf)(static_cast<float>(bfloat16(__bfloat16_raw(0x7f80)))));
+  VERIFY((numext::isnan)(static_cast<float>(bfloat16(__bfloat16_raw(0x7fc0)))));
+
+  // Exactly same checks as above, just directly on the bfloat16 representation.
+  VERIFY(!(numext::isinf)(bfloat16(__bfloat16_raw(0x7bff))));
+  VERIFY(!(numext::isnan)(bfloat16(__bfloat16_raw(0x0000))));
+  VERIFY((numext::isinf)(bfloat16(__bfloat16_raw(0xff80))));
+  VERIFY((numext::isnan)(bfloat16(__bfloat16_raw(0xffc0))));
+  VERIFY((numext::isinf)(bfloat16(__bfloat16_raw(0x7f80))));
+  VERIFY((numext::isnan)(bfloat16(__bfloat16_raw(0x7fc0))));
+}
+
+void test_numtraits()
+{
+  std::cout << "epsilon       = " << NumTraits<bfloat16>::epsilon() << "  (0x" << std::hex << NumTraits<bfloat16>::epsilon().value << ")" << std::endl;
+  std::cout << "highest       = " << NumTraits<bfloat16>::highest() << "  (0x" << std::hex << NumTraits<bfloat16>::highest().value << ")" << std::endl;
+  std::cout << "lowest        = " << NumTraits<bfloat16>::lowest() << "  (0x" << std::hex << NumTraits<bfloat16>::lowest().value << ")" << std::endl;
+  std::cout << "min           = " << (std::numeric_limits<bfloat16>::min)() << "  (0x" << std::hex << (std::numeric_limits<bfloat16>::min)().value << ")" << std::endl;
+  std::cout << "denorm min    = " << (std::numeric_limits<bfloat16>::denorm_min)() << "  (0x" << std::hex << (std::numeric_limits<bfloat16>::denorm_min)().value << ")" << std::endl;
+  std::cout << "infinity      = " << NumTraits<bfloat16>::infinity() << "  (0x" << std::hex << NumTraits<bfloat16>::infinity().value << ")" << std::endl;
+  std::cout << "quiet nan     = " << NumTraits<bfloat16>::quiet_NaN() << "  (0x" << std::hex << NumTraits<bfloat16>::quiet_NaN().value << ")" << std::endl;
+  std::cout << "signaling nan = " << std::numeric_limits<bfloat16>::signaling_NaN() << "  (0x" << std::hex << std::numeric_limits<bfloat16>::signaling_NaN().value << ")" << std::endl;
+
+  VERIFY(NumTraits<bfloat16>::IsSigned);
+
+  VERIFY_IS_EQUAL( std::numeric_limits<bfloat16>::infinity().value, bfloat16(std::numeric_limits<float>::infinity()).value );
+  VERIFY_IS_EQUAL( std::numeric_limits<bfloat16>::quiet_NaN().value, bfloat16(std::numeric_limits<float>::quiet_NaN()).value );
+  VERIFY( (std::numeric_limits<bfloat16>::min)() > bfloat16(0.f) );
+  VERIFY( (std::numeric_limits<bfloat16>::denorm_min)() > bfloat16(0.f) );
+  VERIFY_IS_EQUAL( (std::numeric_limits<bfloat16>::denorm_min)()/bfloat16(2), bfloat16(0.f) );
+}
+
+void test_arithmetic()
+{
+  VERIFY_IS_EQUAL(static_cast<float>(bfloat16(2) + bfloat16(2)), 4);
+  VERIFY_IS_EQUAL(static_cast<float>(bfloat16(2) + bfloat16(-2)), 0);
+  VERIFY_IS_APPROX(static_cast<float>(bfloat16(0.33333f) + bfloat16(0.66667f)), 1.0f);
+  VERIFY_IS_EQUAL(static_cast<float>(bfloat16(2.0f) * bfloat16(-5.5f)), -11.0f);
+  VERIFY_IS_APPROX(static_cast<float>(bfloat16(1.0f) / bfloat16(3.0f)), 0.3339f);
+  VERIFY_IS_EQUAL(static_cast<float>(-bfloat16(4096.0f)), -4096.0f);
+  VERIFY_IS_EQUAL(static_cast<float>(-bfloat16(-4096.0f)), 4096.0f);
+}
+
+void test_comparison()
+{
+  VERIFY(bfloat16(1.0f) > bfloat16(0.5f));
+  VERIFY(bfloat16(0.5f) < bfloat16(1.0f));
+  VERIFY(!(bfloat16(1.0f) < bfloat16(0.5f)));
+  VERIFY(!(bfloat16(0.5f) > bfloat16(1.0f)));
+
+  VERIFY(!(bfloat16(4.0f) > bfloat16(4.0f)));
+  VERIFY(!(bfloat16(4.0f) < bfloat16(4.0f)));
+
+  VERIFY(!(bfloat16(0.0f) < bfloat16(-0.0f)));
+  VERIFY(!(bfloat16(-0.0f) < bfloat16(0.0f)));
+  VERIFY(!(bfloat16(0.0f) > bfloat16(-0.0f)));
+  VERIFY(!(bfloat16(-0.0f) > bfloat16(0.0f)));
+
+  VERIFY(bfloat16(0.2f) > bfloat16(-1.0f));
+  VERIFY(bfloat16(-1.0f) < bfloat16(0.2f));
+  VERIFY(bfloat16(-16.0f) < bfloat16(-15.0f));
+
+  VERIFY(bfloat16(1.0f) == bfloat16(1.0f));
+  VERIFY(bfloat16(1.0f) != bfloat16(2.0f));
+
+  // Comparisons with NaNs and infinities.
+#if !EIGEN_COMP_MSVC
+  // Visual Studio errors out on divisions by 0
+  VERIFY(!(bfloat16(0.0 / 0.0) == bfloat16(0.0 / 0.0)));
+  VERIFY(bfloat16(0.0 / 0.0) != bfloat16(0.0 / 0.0));
+
+  VERIFY(!(bfloat16(1.0) == bfloat16(0.0 / 0.0)));
+  VERIFY(!(bfloat16(1.0) < bfloat16(0.0 / 0.0)));
+  VERIFY(!(bfloat16(1.0) > bfloat16(0.0 / 0.0)));
+  VERIFY(bfloat16(1.0) != bfloat16(0.0 / 0.0));
+
+  VERIFY(bfloat16(1.0) < bfloat16(1.0 / 0.0));
+  VERIFY(bfloat16(1.0) > bfloat16(-1.0 / 0.0));
+#endif
+}
+
+void test_basic_functions()
+{
+  VERIFY_IS_EQUAL(static_cast<float>(numext::abs(bfloat16(3.5f))), 3.5f);
+  VERIFY_IS_EQUAL(static_cast<float>(abs(bfloat16(3.5f))), 3.5f);
+  VERIFY_IS_EQUAL(static_cast<float>(numext::abs(bfloat16(-3.5f))), 3.5f);
+  VERIFY_IS_EQUAL(static_cast<float>(abs(bfloat16(-3.5f))), 3.5f);
+
+  VERIFY_IS_EQUAL(static_cast<float>(numext::floor(bfloat16(3.5f))), 3.0f);
+  VERIFY_IS_EQUAL(static_cast<float>(floor(bfloat16(3.5f))), 3.0f);
+  VERIFY_IS_EQUAL(static_cast<float>(numext::floor(bfloat16(-3.5f))), -4.0f);
+  VERIFY_IS_EQUAL(static_cast<float>(floor(bfloat16(-3.5f))), -4.0f);
+
+  VERIFY_IS_EQUAL(static_cast<float>(numext::ceil(bfloat16(3.5f))), 4.0f);
+  VERIFY_IS_EQUAL(static_cast<float>(ceil(bfloat16(3.5f))), 4.0f);
+  VERIFY_IS_EQUAL(static_cast<float>(numext::ceil(bfloat16(-3.5f))), -3.0f);
+  VERIFY_IS_EQUAL(static_cast<float>(ceil(bfloat16(-3.5f))), -3.0f);
+
+  VERIFY_IS_APPROX(static_cast<float>(numext::sqrt(bfloat16(0.0f))), 0.0f);
+  VERIFY_IS_APPROX(static_cast<float>(sqrt(bfloat16(0.0f))), 0.0f);
+  VERIFY_IS_APPROX(static_cast<float>(numext::sqrt(bfloat16(4.0f))), 2.0f);
+  VERIFY_IS_APPROX(static_cast<float>(sqrt(bfloat16(4.0f))), 2.0f);
+
+  VERIFY_IS_APPROX(static_cast<float>(numext::pow(bfloat16(0.0f), bfloat16(1.0f))), 0.0f);
+  VERIFY_IS_APPROX(static_cast<float>(pow(bfloat16(0.0f), bfloat16(1.0f))), 0.0f);
+  VERIFY_IS_APPROX(static_cast<float>(numext::pow(bfloat16(2.0f), bfloat16(2.0f))), 4.0f);
+  VERIFY_IS_APPROX(static_cast<float>(pow(bfloat16(2.0f), bfloat16(2.0f))), 4.0f);
+
+  VERIFY_IS_EQUAL(static_cast<float>(numext::exp(bfloat16(0.0f))), 1.0f);
+  VERIFY_IS_EQUAL(static_cast<float>(exp(bfloat16(0.0f))), 1.0f);
+  VERIFY_IS_APPROX(static_cast<float>(numext::exp(bfloat16(EIGEN_PI))), 20.f + static_cast<float>(EIGEN_PI));
+  VERIFY_IS_APPROX(static_cast<float>(exp(bfloat16(EIGEN_PI))), 20.f + static_cast<float>(EIGEN_PI));
+
+  VERIFY_IS_EQUAL(static_cast<float>(numext::expm1(bfloat16(0.0f))), 0.0f);
+  VERIFY_IS_EQUAL(static_cast<float>(expm1(bfloat16(0.0f))), 0.0f);
+  VERIFY_IS_APPROX(static_cast<float>(numext::expm1(bfloat16(2.0f))), 6.375f);
+  VERIFY_IS_APPROX(static_cast<float>(expm1(bfloat16(2.0f))), 6.375f);
+
+  VERIFY_IS_EQUAL(static_cast<float>(numext::log(bfloat16(1.0f))), 0.0f);
+  VERIFY_IS_EQUAL(static_cast<float>(log(bfloat16(1.0f))), 0.0f);
+  VERIFY_IS_APPROX(static_cast<float>(numext::log(bfloat16(10.0f))), 2.296875f);
+  VERIFY_IS_APPROX(static_cast<float>(log(bfloat16(10.0f))), 2.296875f);
+
+  VERIFY_IS_EQUAL(static_cast<float>(numext::log1p(bfloat16(0.0f))), 0.0f);
+  VERIFY_IS_EQUAL(static_cast<float>(log1p(bfloat16(0.0f))), 0.0f);
+  VERIFY_IS_APPROX(static_cast<float>(numext::log1p(bfloat16(10.0f))), 2.390625f);
+  VERIFY_IS_APPROX(static_cast<float>(log1p(bfloat16(10.0f))), 2.390625f);
+}
+
+void test_trigonometric_functions()
+{
+  VERIFY_IS_APPROX(numext::cos(bfloat16(0.0f)), bfloat16(cosf(0.0f)));
+  VERIFY_IS_APPROX(cos(bfloat16(0.0f)), bfloat16(cosf(0.0f)));
+  VERIFY_IS_APPROX(numext::cos(bfloat16(EIGEN_PI)), bfloat16(cosf(EIGEN_PI)));
+  // VERIFY_IS_APPROX(numext::cos(bfloat16(EIGEN_PI/2)), bfloat16(cosf(EIGEN_PI/2)));
+  // VERIFY_IS_APPROX(numext::cos(bfloat16(3*EIGEN_PI/2)), bfloat16(cosf(3*EIGEN_PI/2)));
+  VERIFY_IS_APPROX(numext::cos(bfloat16(3.5f)), bfloat16(cosf(3.5f)));
+
+  VERIFY_IS_APPROX(numext::sin(bfloat16(0.0f)), bfloat16(sinf(0.0f)));
+  VERIFY_IS_APPROX(sin(bfloat16(0.0f)), bfloat16(sinf(0.0f)));
+  // VERIFY_IS_APPROX(numext::sin(bfloat16(EIGEN_PI)), bfloat16(sinf(EIGEN_PI)));
+  VERIFY_IS_APPROX(numext::sin(bfloat16(EIGEN_PI/2)), bfloat16(sinf(EIGEN_PI/2)));
+  VERIFY_IS_APPROX(numext::sin(bfloat16(3*EIGEN_PI/2)), bfloat16(sinf(3*EIGEN_PI/2)));
+  VERIFY_IS_APPROX(numext::sin(bfloat16(3.5f)), bfloat16(sinf(3.5f)));
+
+  VERIFY_IS_APPROX(numext::tan(bfloat16(0.0f)), bfloat16(tanf(0.0f)));
+  VERIFY_IS_APPROX(tan(bfloat16(0.0f)), bfloat16(tanf(0.0f)));
+  // VERIFY_IS_APPROX(numext::tan(bfloat16(EIGEN_PI)), bfloat16(tanf(EIGEN_PI)));
+  // VERIFY_IS_APPROX(numext::tan(bfloat16(EIGEN_PI/2)), bfloat16(tanf(EIGEN_PI/2)));
+  // VERIFY_IS_APPROX(numext::tan(bfloat16(3*EIGEN_PI/2)), bfloat16(tanf(3*EIGEN_PI/2)));
+  VERIFY_IS_APPROX(numext::tan(bfloat16(3.5f)), bfloat16(tanf(3.5f)));
+}
+
+void test_array()
+{
+  typedef Array<bfloat16,1,Dynamic> ArrayXh;
+  Index size = internal::random<Index>(1,10);
+  Index i = internal::random<Index>(0,size-1);
+  ArrayXh a1 = ArrayXh::Random(size), a2 = ArrayXh::Random(size);
+  VERIFY_IS_APPROX( a1+a1, bfloat16(2)*a1 );
+  VERIFY( (a1.abs() >= bfloat16(0)).all() );
+  VERIFY_IS_APPROX( (a1*a1).sqrt(), a1.abs() );
+
+  VERIFY( ((a1.min)(a2) <= (a1.max)(a2)).all() );
+  a1(i) = bfloat16(-10.);
+  VERIFY_IS_EQUAL( a1.minCoeff(), bfloat16(-10.) );
+  a1(i) = bfloat16(10.);
+  VERIFY_IS_EQUAL( a1.maxCoeff(), bfloat16(10.) );
+
+  std::stringstream ss;
+  ss << a1;
+}
+
+void test_product()
+{
+  typedef Matrix<bfloat16,Dynamic,Dynamic> MatrixXh;
+  Index rows  = internal::random<Index>(1,EIGEN_TEST_MAX_SIZE);
+  Index cols  = internal::random<Index>(1,EIGEN_TEST_MAX_SIZE);
+  Index depth = internal::random<Index>(1,EIGEN_TEST_MAX_SIZE);
+  MatrixXh Ah = MatrixXh::Random(rows,depth);
+  MatrixXh Bh = MatrixXh::Random(depth,cols);
+  MatrixXh Ch = MatrixXh::Random(rows,cols);
+  MatrixXf Af = Ah.cast<float>();
+  MatrixXf Bf = Bh.cast<float>();
+  MatrixXf Cf = Ch.cast<float>();
+  VERIFY_IS_APPROX(Ch.noalias()+=Ah*Bh, (Cf.noalias()+=Af*Bf).cast<bfloat16>());
+}
+
+EIGEN_DECLARE_TEST(bfloat16_float)
+{
+  CALL_SUBTEST(test_numtraits());
+  for(int i = 0; i < g_repeat; i++) {
+    CALL_SUBTEST(test_conversion());
+    CALL_SUBTEST(test_arithmetic());
+    CALL_SUBTEST(test_comparison());
+    CALL_SUBTEST(test_basic_functions());
+    CALL_SUBTEST(test_trigonometric_functions());
+    CALL_SUBTEST(test_array());
+    CALL_SUBTEST(test_product());
+  }
+}
diff --git a/test/main.h b/test/main.h
index 54553f742..19e6f959d 100644
--- a/test/main.h
+++ b/test/main.h
@@ -435,6 +435,7 @@ EIGEN_TEST_SCALAR_TEST_OVERLOAD(unsigned long long)
 EIGEN_TEST_SCALAR_TEST_OVERLOAD(float)
 EIGEN_TEST_SCALAR_TEST_OVERLOAD(double)
 EIGEN_TEST_SCALAR_TEST_OVERLOAD(half)
+EIGEN_TEST_SCALAR_TEST_OVERLOAD(bfloat16)
 
 #undef EIGEN_TEST_SCALAR_TEST_OVERLOAD
 
diff --git a/test/numext.cpp b/test/numext.cpp
index 8c6447d40..ff4d13ff3 100644
--- a/test/numext.cpp
+++ b/test/numext.cpp
@@ -45,6 +45,7 @@ EIGEN_DECLARE_TEST(numext) {
   CALL_SUBTEST( check_abs<long>() );
   CALL_SUBTEST( check_abs<unsigned long>() );
   CALL_SUBTEST( check_abs<half>() );
+  CALL_SUBTEST( check_abs<bfloat16>() );
   CALL_SUBTEST( check_abs<float>() );
   CALL_SUBTEST( check_abs<double>() );
   CALL_SUBTEST( check_abs<long double>() );
diff --git a/test/packetmath.cpp b/test/packetmath.cpp
index 64ba28741..a82b2b87a 100644
--- a/test/packetmath.cpp
+++ b/test/packetmath.cpp
@@ -836,6 +836,7 @@ EIGEN_DECLARE_TEST(packetmath)
 #ifdef EIGEN_PACKET_MATH_SSE_H
     CALL_SUBTEST_14(( packetmath<bool,internal::packet_traits<bool>::type>() ));
 #endif
+    CALL_SUBTEST_15(( packetmath<bfloat16,internal::packet_traits<bfloat16>::type>() ));
     g_first_pass = false;
   }
 }
diff --git a/test/packetmath_test_shared.h b/test/packetmath_test_shared.h
index 04f719f96..5be10997a 100644
--- a/test/packetmath_test_shared.h
+++ b/test/packetmath_test_shared.h
@@ -50,6 +50,7 @@ T apply_bit_op(Bits a, Bits b, Func f) {
   EIGEN_TEST_MAKE_BITWISE2(OP,FUNC,float)                 \
   EIGEN_TEST_MAKE_BITWISE2(OP,FUNC,double)                \
   EIGEN_TEST_MAKE_BITWISE2(OP,FUNC,half)                  \
+  EIGEN_TEST_MAKE_BITWISE2(OP,FUNC,bfloat16)              \
   EIGEN_TEST_MAKE_BITWISE2(OP,FUNC,std::complex<float>)   \
   EIGEN_TEST_MAKE_BITWISE2(OP,FUNC,std::complex<double>)