test/half_float.cpp - mirror - Git at Google

 // 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 "main.h"

 #define VERIFY_HALF_BITS_EQUAL(h, bits) \
   VERIFY_IS_EQUAL((numext::bit_cast<numext::uint16_t>(h)), (static_cast<numext::uint16_t>(bits)))

 // Make sure it's possible to forward declare Eigen::half
 namespace Eigen {
 struct half;
 }

 using Eigen::half;

 void test_conversion() {
   using Eigen::half_impl::__half_raw;

   // Round-trip bit-cast with uint16.
   VERIFY_IS_EQUAL(numext::bit_cast<half>(numext::bit_cast<numext::uint16_t>(half(1.0f))), half(1.0f));
   VERIFY_IS_EQUAL(numext::bit_cast<half>(numext::bit_cast<numext::uint16_t>(half(0.5f))), half(0.5f));
   VERIFY_IS_EQUAL(numext::bit_cast<half>(numext::bit_cast<numext::uint16_t>(half(-0.33333f))), half(-0.33333f));
   VERIFY_IS_EQUAL(numext::bit_cast<half>(numext::bit_cast<numext::uint16_t>(half(0.0f))), half(0.0f));

   // Conversion from float.
   VERIFY_HALF_BITS_EQUAL(half(1.0f), 0x3c00);
   VERIFY_HALF_BITS_EQUAL(half(0.5f), 0x3800);
   VERIFY_HALF_BITS_EQUAL(half(0.33333f), 0x3555);
   VERIFY_HALF_BITS_EQUAL(half(0.0f), 0x0000);
   VERIFY_HALF_BITS_EQUAL(half(-0.0f), 0x8000);
   VERIFY_HALF_BITS_EQUAL(half(65504.0f), 0x7bff);
   VERIFY_HALF_BITS_EQUAL(half(65536.0f), 0x7c00);  // Becomes infinity.

   // Denormals.
   VERIFY_HALF_BITS_EQUAL(half(-5.96046e-08f), 0x8001);
   VERIFY_HALF_BITS_EQUAL(half(5.96046e-08f), 0x0001);
   VERIFY_HALF_BITS_EQUAL(half(1.19209e-07f), 0x0002);

   // Verify round-to-nearest-even behavior.
   float val1 = float(half(__half_raw(0x3c00)));
   float val2 = float(half(__half_raw(0x3c01)));
   float val3 = float(half(__half_raw(0x3c02)));
   VERIFY_HALF_BITS_EQUAL(half(0.5f * (val1 + val2)), 0x3c00);
   VERIFY_HALF_BITS_EQUAL(half(0.5f * (val2 + val3)), 0x3c02);

   // Conversion from int.
   VERIFY_HALF_BITS_EQUAL(half(-1), 0xbc00);
   VERIFY_HALF_BITS_EQUAL(half(0), 0x0000);
   VERIFY_HALF_BITS_EQUAL(half(1), 0x3c00);
   VERIFY_HALF_BITS_EQUAL(half(2), 0x4000);
   VERIFY_HALF_BITS_EQUAL(half(3), 0x4200);

   // Conversion from bool.
   VERIFY_HALF_BITS_EQUAL(half(false), 0x0000);
   VERIFY_HALF_BITS_EQUAL(half(true), 0x3c00);

   // Conversion to float.
   VERIFY_IS_EQUAL(float(half(__half_raw(0x0000))), 0.0f);
   VERIFY_IS_EQUAL(float(half(__half_raw(0x3c00))), 1.0f);

   // Denormals.
   VERIFY_IS_APPROX(float(half(__half_raw(0x8001))), -5.96046e-08f);
   VERIFY_IS_APPROX(float(half(__half_raw(0x0001))), 5.96046e-08f);
   VERIFY_IS_APPROX(float(half(__half_raw(0x0002))), 1.19209e-07f);

   // NaNs and infinities.
   VERIFY(!(numext::isinf)(float(half(65504.0f))));  // Largest finite number.
   VERIFY(!(numext::isnan)(float(half(0.0f))));
   VERIFY((numext::isinf)(float(half(__half_raw(0xfc00)))));
   VERIFY((numext::isnan)(float(half(__half_raw(0xfc01)))));
   VERIFY((numext::isinf)(float(half(__half_raw(0x7c00)))));
   VERIFY((numext::isnan)(float(half(__half_raw(0x7c01)))));

 #if !EIGEN_COMP_MSVC
   // Visual Studio errors out on divisions by 0
   VERIFY((numext::isnan)(float(half(0.0 / 0.0))));
   VERIFY((numext::isinf)(float(half(1.0 / 0.0))));
   VERIFY((numext::isinf)(float(half(-1.0 / 0.0))));
 #endif

   // Exactly same checks as above, just directly on the half representation.
   VERIFY(!(numext::isinf)(half(__half_raw(0x7bff))));
   VERIFY(!(numext::isnan)(half(__half_raw(0x0000))));
   VERIFY((numext::isinf)(half(__half_raw(0xfc00))));
   VERIFY((numext::isnan)(half(__half_raw(0xfc01))));
   VERIFY((numext::isinf)(half(__half_raw(0x7c00))));
   VERIFY((numext::isnan)(half(__half_raw(0x7c01))));

 #if !EIGEN_COMP_MSVC
   // Visual Studio errors out on divisions by 0
   VERIFY((numext::isnan)(half(0.0 / 0.0)));
   VERIFY((numext::isinf)(half(1.0 / 0.0)));
   VERIFY((numext::isinf)(half(-1.0 / 0.0)));
 #endif

   // Conversion to bool
   VERIFY(!static_cast<bool>(half(0.0)));
   VERIFY(!static_cast<bool>(half(-0.0)));
   VERIFY(static_cast<bool>(half(__half_raw(0x7bff))));
   VERIFY(static_cast<bool>(half(-0.33333)));
   VERIFY(static_cast<bool>(half(1.0)));
   VERIFY(static_cast<bool>(half(-1.0)));
   VERIFY(static_cast<bool>(half(-5.96046e-08f)));
 }

 void test_numtraits() {
   std::cout << "epsilon       = " << NumTraits<half>::epsilon() << "  (0x" << std::hex
             << numext::bit_cast<numext::uint16_t>(NumTraits<half>::epsilon()) << ")" << std::endl;
   std::cout << "highest       = " << NumTraits<half>::highest() << "  (0x" << std::hex
             << numext::bit_cast<numext::uint16_t>(NumTraits<half>::highest()) << ")" << std::endl;
   std::cout << "lowest        = " << NumTraits<half>::lowest() << "  (0x" << std::hex
             << numext::bit_cast<numext::uint16_t>(NumTraits<half>::lowest()) << ")" << std::endl;
   std::cout << "min           = " << (std::numeric_limits<half>::min)() << "  (0x" << std::hex
             << numext::bit_cast<numext::uint16_t>(half((std::numeric_limits<half>::min)())) << ")" << std::endl;
   std::cout << "denorm min    = " << (std::numeric_limits<half>::denorm_min)() << "  (0x" << std::hex
             << numext::bit_cast<numext::uint16_t>(half((std::numeric_limits<half>::denorm_min)())) << ")" << std::endl;
   std::cout << "infinity      = " << NumTraits<half>::infinity() << "  (0x" << std::hex
             << numext::bit_cast<numext::uint16_t>(NumTraits<half>::infinity()) << ")" << std::endl;
   std::cout << "quiet nan     = " << NumTraits<half>::quiet_NaN() << "  (0x" << std::hex
             << numext::bit_cast<numext::uint16_t>(NumTraits<half>::quiet_NaN()) << ")" << std::endl;
   std::cout << "signaling nan = " << std::numeric_limits<half>::signaling_NaN() << "  (0x" << std::hex
             << numext::bit_cast<numext::uint16_t>(std::numeric_limits<half>::signaling_NaN()) << ")" << std::endl;

   VERIFY(NumTraits<half>::IsSigned);

   VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(std::numeric_limits<half>::infinity()),
                   numext::bit_cast<numext::uint16_t>(half(std::numeric_limits<float>::infinity())));
   // There is no guarantee that casting a 32-bit NaN to 16-bit has a precise
   // bit pattern.  We test that it is in fact a NaN, then test the signaling
   // bit (msb of significand is 1 for quiet, 0 for signaling).
   const numext::uint16_t HALF_QUIET_BIT = 0x0200;
   VERIFY((numext::isnan)(std::numeric_limits<half>::quiet_NaN()) &&
          (numext::isnan)(half(std::numeric_limits<float>::quiet_NaN())) &&
          ((numext::bit_cast<numext::uint16_t>(std::numeric_limits<half>::quiet_NaN()) & HALF_QUIET_BIT) > 0) &&
          ((numext::bit_cast<numext::uint16_t>(half(std::numeric_limits<float>::quiet_NaN())) & HALF_QUIET_BIT) > 0));
   // After a cast to half, a signaling NaN may become non-signaling
   // (e.g. in the case of casting float to native __fp16). Thus, we check that
   // both are NaN, and that only the `numeric_limits` version is signaling.
   VERIFY((numext::isnan)(std::numeric_limits<half>::signaling_NaN()) &&
          (numext::isnan)(half(std::numeric_limits<float>::signaling_NaN())) &&
          ((numext::bit_cast<numext::uint16_t>(std::numeric_limits<half>::signaling_NaN()) & HALF_QUIET_BIT) == 0));

   VERIFY((std::numeric_limits<half>::min)() > half(0.f));
   VERIFY((std::numeric_limits<half>::denorm_min)() > half(0.f));
   VERIFY((std::numeric_limits<half>::min)() / half(2) > half(0.f));
   VERIFY_IS_EQUAL((std::numeric_limits<half>::denorm_min)() / half(2), half(0.f));

   // Test to see that we are able to link against the symbols for digits and
   // digits10.
   volatile const int& digits10 = std::numeric_limits<half>::digits10;
   volatile const int& digits = std::numeric_limits<half>::digits;
   VERIFY((digits10) != (digits));
 }

 void test_arithmetic() {
   VERIFY_IS_EQUAL(float(half(2) + half(2)), 4);
   VERIFY_IS_EQUAL(float(half(2) + half(-2)), 0);
   VERIFY_IS_APPROX(float(half(0.33333f) + half(0.66667f)), 1.0f);
   VERIFY_IS_EQUAL(float(half(2.0f) * half(-5.5f)), -11.0f);
   VERIFY_IS_APPROX(float(half(1.0f) / half(3.0f)), 0.33333f);
   VERIFY_IS_EQUAL(float(-half(4096.0f)), -4096.0f);
   VERIFY_IS_EQUAL(float(-half(-4096.0f)), 4096.0f);

   half x(3);
   half y = ++x;
   VERIFY_IS_EQUAL(x, half(4));
   VERIFY_IS_EQUAL(y, half(4));
   y = --x;
   VERIFY_IS_EQUAL(x, half(3));
   VERIFY_IS_EQUAL(y, half(3));
   y = x++;
   VERIFY_IS_EQUAL(x, half(4));
   VERIFY_IS_EQUAL(y, half(3));
   y = x--;
   VERIFY_IS_EQUAL(x, half(3));
   VERIFY_IS_EQUAL(y, half(4));
 }

 void test_comparison() {
   VERIFY(half(1.0f) > half(0.5f));
   VERIFY(half(0.5f) < half(1.0f));
   VERIFY(!(half(1.0f) < half(0.5f)));
   VERIFY(!(half(0.5f) > half(1.0f)));

   VERIFY(!(half(4.0f) > half(4.0f)));
   VERIFY(!(half(4.0f) < half(4.0f)));

   VERIFY(!(half(0.0f) < half(-0.0f)));
   VERIFY(!(half(-0.0f) < half(0.0f)));
   VERIFY(!(half(0.0f) > half(-0.0f)));
   VERIFY(!(half(-0.0f) > half(0.0f)));

   VERIFY(half(0.2f) > half(-1.0f));
   VERIFY(half(-1.0f) < half(0.2f));
   VERIFY(half(-16.0f) < half(-15.0f));

   VERIFY(half(1.0f) == half(1.0f));
   VERIFY(half(1.0f) != half(2.0f));

   // Comparisons with NaNs and infinities.
 #if !EIGEN_COMP_MSVC
   // Visual Studio errors out on divisions by 0
   VERIFY(!(half(0.0 / 0.0) == half(0.0 / 0.0)));
   VERIFY(half(0.0 / 0.0) != half(0.0 / 0.0));

   VERIFY(!(half(1.0) == half(0.0 / 0.0)));
   VERIFY(!(half(1.0) < half(0.0 / 0.0)));
   VERIFY(!(half(1.0) > half(0.0 / 0.0)));
   VERIFY(half(1.0) != half(0.0 / 0.0));

   VERIFY(half(1.0) < half(1.0 / 0.0));
   VERIFY(half(1.0) > half(-1.0 / 0.0));
 #endif
 }

 void test_basic_functions() {
   constexpr float PI = static_cast<float>(EIGEN_PI);

   VERIFY_IS_EQUAL(float(numext::abs(half(3.5f))), 3.5f);
   VERIFY_IS_EQUAL(float(abs(half(3.5f))), 3.5f);
   VERIFY_IS_EQUAL(float(numext::abs(half(-3.5f))), 3.5f);
   VERIFY_IS_EQUAL(float(abs(half(-3.5f))), 3.5f);

   VERIFY_IS_EQUAL(float(numext::floor(half(3.5f))), 3.0f);
   VERIFY_IS_EQUAL(float(floor(half(3.5f))), 3.0f);
   VERIFY_IS_EQUAL(float(numext::floor(half(-3.5f))), -4.0f);
   VERIFY_IS_EQUAL(float(floor(half(-3.5f))), -4.0f);

   VERIFY_IS_EQUAL(float(numext::ceil(half(3.5f))), 4.0f);
   VERIFY_IS_EQUAL(float(ceil(half(3.5f))), 4.0f);
   VERIFY_IS_EQUAL(float(numext::ceil(half(-3.5f))), -3.0f);
   VERIFY_IS_EQUAL(float(ceil(half(-3.5f))), -3.0f);

   VERIFY_IS_APPROX(float(numext::sqrt(half(0.0f))), 0.0f);
   VERIFY_IS_APPROX(float(sqrt(half(0.0f))), 0.0f);
   VERIFY_IS_APPROX(float(numext::sqrt(half(4.0f))), 2.0f);
   VERIFY_IS_APPROX(float(sqrt(half(4.0f))), 2.0f);

   VERIFY_IS_APPROX(float(numext::pow(half(0.0f), half(1.0f))), 0.0f);
   VERIFY_IS_APPROX(float(pow(half(0.0f), half(1.0f))), 0.0f);
   VERIFY_IS_APPROX(float(numext::pow(half(2.0f), half(2.0f))), 4.0f);
   VERIFY_IS_APPROX(float(pow(half(2.0f), half(2.0f))), 4.0f);

   VERIFY_IS_EQUAL(float(numext::exp(half(0.0f))), 1.0f);
   VERIFY_IS_EQUAL(float(exp(half(0.0f))), 1.0f);
   VERIFY_IS_APPROX(float(numext::exp(half(PI))), 20.f + PI);
   VERIFY_IS_APPROX(float(exp(half(PI))), 20.f + PI);

   VERIFY_IS_EQUAL(float(numext::expm1(half(0.0f))), 0.0f);
   VERIFY_IS_EQUAL(float(expm1(half(0.0f))), 0.0f);
   VERIFY_IS_APPROX(float(numext::expm1(half(2.0f))), 6.3890561f);
   VERIFY_IS_APPROX(float(expm1(half(2.0f))), 6.3890561f);

   VERIFY_IS_EQUAL(float(numext::log(half(1.0f))), 0.0f);
   VERIFY_IS_EQUAL(float(log(half(1.0f))), 0.0f);
   VERIFY_IS_APPROX(float(numext::log(half(10.0f))), 2.30273f);
   VERIFY_IS_APPROX(float(log(half(10.0f))), 2.30273f);

   VERIFY_IS_EQUAL(float(numext::log1p(half(0.0f))), 0.0f);
   VERIFY_IS_EQUAL(float(log1p(half(0.0f))), 0.0f);
   VERIFY_IS_APPROX(float(numext::log1p(half(10.0f))), 2.3978953f);
   VERIFY_IS_APPROX(float(log1p(half(10.0f))), 2.3978953f);

   VERIFY_IS_APPROX(numext::fmod(half(5.3f), half(2.0f)), half(1.3f));
   VERIFY_IS_APPROX(fmod(half(5.3f), half(2.0f)), half(1.3f));
   VERIFY_IS_APPROX(numext::fmod(half(-18.5f), half(-4.2f)), half(-1.7f));
   VERIFY_IS_APPROX(fmod(half(-18.5f), half(-4.2f)), half(-1.7f));
 }

 void test_trigonometric_functions() {
   constexpr float PI = static_cast<float>(EIGEN_PI);
   VERIFY_IS_APPROX(numext::cos(half(0.0f)), half(cosf(0.0f)));
   VERIFY_IS_APPROX(cos(half(0.0f)), half(cosf(0.0f)));
   VERIFY_IS_APPROX(numext::cos(half(PI)), half(cosf(PI)));
   // VERIFY_IS_APPROX(numext::cos(half(PI/2)), half(cosf(PI/2)));
   // VERIFY_IS_APPROX(numext::cos(half(3*PI/2)), half(cosf(3*PI/2)));
   VERIFY_IS_APPROX(numext::cos(half(3.5f)), half(cosf(3.5f)));

   VERIFY_IS_APPROX(numext::sin(half(0.0f)), half(sinf(0.0f)));
   VERIFY_IS_APPROX(sin(half(0.0f)), half(sinf(0.0f)));
   //  VERIFY_IS_APPROX(numext::sin(half(PI)), half(sinf(PI)));
   VERIFY_IS_APPROX(numext::sin(half(PI / 2)), half(sinf(PI / 2)));
   VERIFY_IS_APPROX(numext::sin(half(3 * PI / 2)), half(sinf(3 * PI / 2)));
   VERIFY_IS_APPROX(numext::sin(half(3.5f)), half(sinf(3.5f)));

   VERIFY_IS_APPROX(numext::tan(half(0.0f)), half(tanf(0.0f)));
   VERIFY_IS_APPROX(tan(half(0.0f)), half(tanf(0.0f)));
   //  VERIFY_IS_APPROX(numext::tan(half(PI)), half(tanf(PI)));
   //  VERIFY_IS_APPROX(numext::tan(half(PI/2)), half(tanf(PI/2)));
   // VERIFY_IS_APPROX(numext::tan(half(3*PI/2)), half(tanf(3*PI/2)));
   VERIFY_IS_APPROX(numext::tan(half(3.5f)), half(tanf(3.5f)));
 }

 void test_array() {
   typedef Array<half, 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, half(2) * a1);
   VERIFY((a1.abs() >= half(0)).all());
   VERIFY_IS_APPROX((a1 * a1).sqrt(), a1.abs());

   VERIFY(((a1.min)(a2) <= (a1.max)(a2)).all());
   a1(i) = half(-10.);
   VERIFY_IS_EQUAL(a1.minCoeff(), half(-10.));
   a1(i) = half(10.);
   VERIFY_IS_EQUAL(a1.maxCoeff(), half(10.));

   std::stringstream ss;
   ss << a1;
 }

 void test_product() {
   typedef Matrix<half, 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<half>());
 }

 EIGEN_DECLARE_TEST(half_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());
   }
 }
	// 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 "main.h"

	#define VERIFY_HALF_BITS_EQUAL(h, bits) \
	VERIFY_IS_EQUAL((numext::bit_cast<numext::uint16_t>(h)), (static_cast<numext::uint16_t>(bits)))

	// Make sure it's possible to forward declare Eigen::half
	namespace Eigen {
	struct half;
	}

	using Eigen::half;

	void test_conversion() {
	using Eigen::half_impl::__half_raw;

	// Round-trip bit-cast with uint16.
	VERIFY_IS_EQUAL(numext::bit_cast<half>(numext::bit_cast<numext::uint16_t>(half(1.0f))), half(1.0f));
	VERIFY_IS_EQUAL(numext::bit_cast<half>(numext::bit_cast<numext::uint16_t>(half(0.5f))), half(0.5f));
	VERIFY_IS_EQUAL(numext::bit_cast<half>(numext::bit_cast<numext::uint16_t>(half(-0.33333f))), half(-0.33333f));
	VERIFY_IS_EQUAL(numext::bit_cast<half>(numext::bit_cast<numext::uint16_t>(half(0.0f))), half(0.0f));

	// Conversion from float.
	VERIFY_HALF_BITS_EQUAL(half(1.0f), 0x3c00);
	VERIFY_HALF_BITS_EQUAL(half(0.5f), 0x3800);
	VERIFY_HALF_BITS_EQUAL(half(0.33333f), 0x3555);
	VERIFY_HALF_BITS_EQUAL(half(0.0f), 0x0000);
	VERIFY_HALF_BITS_EQUAL(half(-0.0f), 0x8000);
	VERIFY_HALF_BITS_EQUAL(half(65504.0f), 0x7bff);
	VERIFY_HALF_BITS_EQUAL(half(65536.0f), 0x7c00); // Becomes infinity.

	// Denormals.
	VERIFY_HALF_BITS_EQUAL(half(-5.96046e-08f), 0x8001);
	VERIFY_HALF_BITS_EQUAL(half(5.96046e-08f), 0x0001);
	VERIFY_HALF_BITS_EQUAL(half(1.19209e-07f), 0x0002);

	// Verify round-to-nearest-even behavior.
	float val1 = float(half(__half_raw(0x3c00)));
	float val2 = float(half(__half_raw(0x3c01)));
	float val3 = float(half(__half_raw(0x3c02)));
	VERIFY_HALF_BITS_EQUAL(half(0.5f * (val1 + val2)), 0x3c00);
	VERIFY_HALF_BITS_EQUAL(half(0.5f * (val2 + val3)), 0x3c02);

	// Conversion from int.
	VERIFY_HALF_BITS_EQUAL(half(-1), 0xbc00);
	VERIFY_HALF_BITS_EQUAL(half(0), 0x0000);
	VERIFY_HALF_BITS_EQUAL(half(1), 0x3c00);
	VERIFY_HALF_BITS_EQUAL(half(2), 0x4000);
	VERIFY_HALF_BITS_EQUAL(half(3), 0x4200);

	// Conversion from bool.
	VERIFY_HALF_BITS_EQUAL(half(false), 0x0000);
	VERIFY_HALF_BITS_EQUAL(half(true), 0x3c00);

	// Conversion to float.
	VERIFY_IS_EQUAL(float(half(__half_raw(0x0000))), 0.0f);
	VERIFY_IS_EQUAL(float(half(__half_raw(0x3c00))), 1.0f);

	// Denormals.
	VERIFY_IS_APPROX(float(half(__half_raw(0x8001))), -5.96046e-08f);
	VERIFY_IS_APPROX(float(half(__half_raw(0x0001))), 5.96046e-08f);
	VERIFY_IS_APPROX(float(half(__half_raw(0x0002))), 1.19209e-07f);

	// NaNs and infinities.
	VERIFY(!(numext::isinf)(float(half(65504.0f)))); // Largest finite number.
	VERIFY(!(numext::isnan)(float(half(0.0f))));
	VERIFY((numext::isinf)(float(half(__half_raw(0xfc00)))));
	VERIFY((numext::isnan)(float(half(__half_raw(0xfc01)))));
	VERIFY((numext::isinf)(float(half(__half_raw(0x7c00)))));
	VERIFY((numext::isnan)(float(half(__half_raw(0x7c01)))));

	#if !EIGEN_COMP_MSVC
	// Visual Studio errors out on divisions by 0
	VERIFY((numext::isnan)(float(half(0.0 / 0.0))));
	VERIFY((numext::isinf)(float(half(1.0 / 0.0))));
	VERIFY((numext::isinf)(float(half(-1.0 / 0.0))));
	#endif

	// Exactly same checks as above, just directly on the half representation.
	VERIFY(!(numext::isinf)(half(__half_raw(0x7bff))));
	VERIFY(!(numext::isnan)(half(__half_raw(0x0000))));
	VERIFY((numext::isinf)(half(__half_raw(0xfc00))));
	VERIFY((numext::isnan)(half(__half_raw(0xfc01))));
	VERIFY((numext::isinf)(half(__half_raw(0x7c00))));
	VERIFY((numext::isnan)(half(__half_raw(0x7c01))));

	#if !EIGEN_COMP_MSVC
	// Visual Studio errors out on divisions by 0
	VERIFY((numext::isnan)(half(0.0 / 0.0)));
	VERIFY((numext::isinf)(half(1.0 / 0.0)));
	VERIFY((numext::isinf)(half(-1.0 / 0.0)));
	#endif

	// Conversion to bool
	VERIFY(!static_cast<bool>(half(0.0)));
	VERIFY(!static_cast<bool>(half(-0.0)));
	VERIFY(static_cast<bool>(half(__half_raw(0x7bff))));
	VERIFY(static_cast<bool>(half(-0.33333)));
	VERIFY(static_cast<bool>(half(1.0)));
	VERIFY(static_cast<bool>(half(-1.0)));
	VERIFY(static_cast<bool>(half(-5.96046e-08f)));
	}

	void test_numtraits() {
	std::cout << "epsilon = " << NumTraits<half>::epsilon() << " (0x" << std::hex
	<< numext::bit_cast<numext::uint16_t>(NumTraits<half>::epsilon()) << ")" << std::endl;
	std::cout << "highest = " << NumTraits<half>::highest() << " (0x" << std::hex
	<< numext::bit_cast<numext::uint16_t>(NumTraits<half>::highest()) << ")" << std::endl;
	std::cout << "lowest = " << NumTraits<half>::lowest() << " (0x" << std::hex
	<< numext::bit_cast<numext::uint16_t>(NumTraits<half>::lowest()) << ")" << std::endl;
	std::cout << "min = " << (std::numeric_limits<half>::min)() << " (0x" << std::hex
	<< numext::bit_cast<numext::uint16_t>(half((std::numeric_limits<half>::min)())) << ")" << std::endl;
	std::cout << "denorm min = " << (std::numeric_limits<half>::denorm_min)() << " (0x" << std::hex
	<< numext::bit_cast<numext::uint16_t>(half((std::numeric_limits<half>::denorm_min)())) << ")" << std::endl;
	std::cout << "infinity = " << NumTraits<half>::infinity() << " (0x" << std::hex
	<< numext::bit_cast<numext::uint16_t>(NumTraits<half>::infinity()) << ")" << std::endl;
	std::cout << "quiet nan = " << NumTraits<half>::quiet_NaN() << " (0x" << std::hex
	<< numext::bit_cast<numext::uint16_t>(NumTraits<half>::quiet_NaN()) << ")" << std::endl;
	std::cout << "signaling nan = " << std::numeric_limits<half>::signaling_NaN() << " (0x" << std::hex
	<< numext::bit_cast<numext::uint16_t>(std::numeric_limits<half>::signaling_NaN()) << ")" << std::endl;

	VERIFY(NumTraits<half>::IsSigned);

	VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(std::numeric_limits<half>::infinity()),
	numext::bit_cast<numext::uint16_t>(half(std::numeric_limits<float>::infinity())));
	// There is no guarantee that casting a 32-bit NaN to 16-bit has a precise
	// bit pattern. We test that it is in fact a NaN, then test the signaling
	// bit (msb of significand is 1 for quiet, 0 for signaling).
	const numext::uint16_t HALF_QUIET_BIT = 0x0200;
	VERIFY((numext::isnan)(std::numeric_limits<half>::quiet_NaN()) &&
	(numext::isnan)(half(std::numeric_limits<float>::quiet_NaN())) &&
	((numext::bit_cast<numext::uint16_t>(std::numeric_limits<half>::quiet_NaN()) & HALF_QUIET_BIT) > 0) &&
	((numext::bit_cast<numext::uint16_t>(half(std::numeric_limits<float>::quiet_NaN())) & HALF_QUIET_BIT) > 0));
	// After a cast to half, a signaling NaN may become non-signaling
	// (e.g. in the case of casting float to native __fp16). Thus, we check that
	// both are NaN, and that only the `numeric_limits` version is signaling.
	VERIFY((numext::isnan)(std::numeric_limits<half>::signaling_NaN()) &&
	(numext::isnan)(half(std::numeric_limits<float>::signaling_NaN())) &&
	((numext::bit_cast<numext::uint16_t>(std::numeric_limits<half>::signaling_NaN()) & HALF_QUIET_BIT) == 0));

	VERIFY((std::numeric_limits<half>::min)() > half(0.f));
	VERIFY((std::numeric_limits<half>::denorm_min)() > half(0.f));
	VERIFY((std::numeric_limits<half>::min)() / half(2) > half(0.f));
	VERIFY_IS_EQUAL((std::numeric_limits<half>::denorm_min)() / half(2), half(0.f));

	// Test to see that we are able to link against the symbols for digits and
	// digits10.
	volatile const int& digits10 = std::numeric_limits<half>::digits10;
	volatile const int& digits = std::numeric_limits<half>::digits;
	VERIFY((digits10) != (digits));
	}

	void test_arithmetic() {
	VERIFY_IS_EQUAL(float(half(2) + half(2)), 4);
	VERIFY_IS_EQUAL(float(half(2) + half(-2)), 0);
	VERIFY_IS_APPROX(float(half(0.33333f) + half(0.66667f)), 1.0f);
	VERIFY_IS_EQUAL(float(half(2.0f) * half(-5.5f)), -11.0f);
	VERIFY_IS_APPROX(float(half(1.0f) / half(3.0f)), 0.33333f);
	VERIFY_IS_EQUAL(float(-half(4096.0f)), -4096.0f);
	VERIFY_IS_EQUAL(float(-half(-4096.0f)), 4096.0f);

	half x(3);
	half y = ++x;
	VERIFY_IS_EQUAL(x, half(4));
	VERIFY_IS_EQUAL(y, half(4));
	y = --x;
	VERIFY_IS_EQUAL(x, half(3));
	VERIFY_IS_EQUAL(y, half(3));
	y = x++;
	VERIFY_IS_EQUAL(x, half(4));
	VERIFY_IS_EQUAL(y, half(3));
	y = x--;
	VERIFY_IS_EQUAL(x, half(3));
	VERIFY_IS_EQUAL(y, half(4));
	}

	void test_comparison() {
	VERIFY(half(1.0f) > half(0.5f));
	VERIFY(half(0.5f) < half(1.0f));
	VERIFY(!(half(1.0f) < half(0.5f)));
	VERIFY(!(half(0.5f) > half(1.0f)));

	VERIFY(!(half(4.0f) > half(4.0f)));
	VERIFY(!(half(4.0f) < half(4.0f)));

	VERIFY(!(half(0.0f) < half(-0.0f)));
	VERIFY(!(half(-0.0f) < half(0.0f)));
	VERIFY(!(half(0.0f) > half(-0.0f)));
	VERIFY(!(half(-0.0f) > half(0.0f)));

	VERIFY(half(0.2f) > half(-1.0f));
	VERIFY(half(-1.0f) < half(0.2f));
	VERIFY(half(-16.0f) < half(-15.0f));

	VERIFY(half(1.0f) == half(1.0f));
	VERIFY(half(1.0f) != half(2.0f));

	// Comparisons with NaNs and infinities.
	#if !EIGEN_COMP_MSVC
	// Visual Studio errors out on divisions by 0
	VERIFY(!(half(0.0 / 0.0) == half(0.0 / 0.0)));
	VERIFY(half(0.0 / 0.0) != half(0.0 / 0.0));

	VERIFY(!(half(1.0) == half(0.0 / 0.0)));
	VERIFY(!(half(1.0) < half(0.0 / 0.0)));
	VERIFY(!(half(1.0) > half(0.0 / 0.0)));
	VERIFY(half(1.0) != half(0.0 / 0.0));

	VERIFY(half(1.0) < half(1.0 / 0.0));
	VERIFY(half(1.0) > half(-1.0 / 0.0));
	#endif
	}

	void test_basic_functions() {
	constexpr float PI = static_cast<float>(EIGEN_PI);

	VERIFY_IS_EQUAL(float(numext::abs(half(3.5f))), 3.5f);
	VERIFY_IS_EQUAL(float(abs(half(3.5f))), 3.5f);
	VERIFY_IS_EQUAL(float(numext::abs(half(-3.5f))), 3.5f);
	VERIFY_IS_EQUAL(float(abs(half(-3.5f))), 3.5f);

	VERIFY_IS_EQUAL(float(numext::floor(half(3.5f))), 3.0f);
	VERIFY_IS_EQUAL(float(floor(half(3.5f))), 3.0f);
	VERIFY_IS_EQUAL(float(numext::floor(half(-3.5f))), -4.0f);
	VERIFY_IS_EQUAL(float(floor(half(-3.5f))), -4.0f);

	VERIFY_IS_EQUAL(float(numext::ceil(half(3.5f))), 4.0f);
	VERIFY_IS_EQUAL(float(ceil(half(3.5f))), 4.0f);
	VERIFY_IS_EQUAL(float(numext::ceil(half(-3.5f))), -3.0f);
	VERIFY_IS_EQUAL(float(ceil(half(-3.5f))), -3.0f);

	VERIFY_IS_APPROX(float(numext::sqrt(half(0.0f))), 0.0f);
	VERIFY_IS_APPROX(float(sqrt(half(0.0f))), 0.0f);
	VERIFY_IS_APPROX(float(numext::sqrt(half(4.0f))), 2.0f);
	VERIFY_IS_APPROX(float(sqrt(half(4.0f))), 2.0f);

	VERIFY_IS_APPROX(float(numext::pow(half(0.0f), half(1.0f))), 0.0f);
	VERIFY_IS_APPROX(float(pow(half(0.0f), half(1.0f))), 0.0f);
	VERIFY_IS_APPROX(float(numext::pow(half(2.0f), half(2.0f))), 4.0f);
	VERIFY_IS_APPROX(float(pow(half(2.0f), half(2.0f))), 4.0f);

	VERIFY_IS_EQUAL(float(numext::exp(half(0.0f))), 1.0f);
	VERIFY_IS_EQUAL(float(exp(half(0.0f))), 1.0f);
	VERIFY_IS_APPROX(float(numext::exp(half(PI))), 20.f + PI);
	VERIFY_IS_APPROX(float(exp(half(PI))), 20.f + PI);

	VERIFY_IS_EQUAL(float(numext::expm1(half(0.0f))), 0.0f);
	VERIFY_IS_EQUAL(float(expm1(half(0.0f))), 0.0f);
	VERIFY_IS_APPROX(float(numext::expm1(half(2.0f))), 6.3890561f);
	VERIFY_IS_APPROX(float(expm1(half(2.0f))), 6.3890561f);

	VERIFY_IS_EQUAL(float(numext::log(half(1.0f))), 0.0f);
	VERIFY_IS_EQUAL(float(log(half(1.0f))), 0.0f);
	VERIFY_IS_APPROX(float(numext::log(half(10.0f))), 2.30273f);
	VERIFY_IS_APPROX(float(log(half(10.0f))), 2.30273f);

	VERIFY_IS_EQUAL(float(numext::log1p(half(0.0f))), 0.0f);
	VERIFY_IS_EQUAL(float(log1p(half(0.0f))), 0.0f);
	VERIFY_IS_APPROX(float(numext::log1p(half(10.0f))), 2.3978953f);
	VERIFY_IS_APPROX(float(log1p(half(10.0f))), 2.3978953f);

	VERIFY_IS_APPROX(numext::fmod(half(5.3f), half(2.0f)), half(1.3f));
	VERIFY_IS_APPROX(fmod(half(5.3f), half(2.0f)), half(1.3f));
	VERIFY_IS_APPROX(numext::fmod(half(-18.5f), half(-4.2f)), half(-1.7f));
	VERIFY_IS_APPROX(fmod(half(-18.5f), half(-4.2f)), half(-1.7f));
	}

	void test_trigonometric_functions() {
	constexpr float PI = static_cast<float>(EIGEN_PI);
	VERIFY_IS_APPROX(numext::cos(half(0.0f)), half(cosf(0.0f)));
	VERIFY_IS_APPROX(cos(half(0.0f)), half(cosf(0.0f)));
	VERIFY_IS_APPROX(numext::cos(half(PI)), half(cosf(PI)));
	// VERIFY_IS_APPROX(numext::cos(half(PI/2)), half(cosf(PI/2)));
	// VERIFY_IS_APPROX(numext::cos(half(3PI/2)), half(cosf(3PI/2)));
	VERIFY_IS_APPROX(numext::cos(half(3.5f)), half(cosf(3.5f)));

	VERIFY_IS_APPROX(numext::sin(half(0.0f)), half(sinf(0.0f)));
	VERIFY_IS_APPROX(sin(half(0.0f)), half(sinf(0.0f)));
	// VERIFY_IS_APPROX(numext::sin(half(PI)), half(sinf(PI)));
	VERIFY_IS_APPROX(numext::sin(half(PI / 2)), half(sinf(PI / 2)));
	VERIFY_IS_APPROX(numext::sin(half(3 * PI / 2)), half(sinf(3 * PI / 2)));
	VERIFY_IS_APPROX(numext::sin(half(3.5f)), half(sinf(3.5f)));

	VERIFY_IS_APPROX(numext::tan(half(0.0f)), half(tanf(0.0f)));
	VERIFY_IS_APPROX(tan(half(0.0f)), half(tanf(0.0f)));
	// VERIFY_IS_APPROX(numext::tan(half(PI)), half(tanf(PI)));
	// VERIFY_IS_APPROX(numext::tan(half(PI/2)), half(tanf(PI/2)));
	// VERIFY_IS_APPROX(numext::tan(half(3PI/2)), half(tanf(3PI/2)));
	VERIFY_IS_APPROX(numext::tan(half(3.5f)), half(tanf(3.5f)));
	}

	void test_array() {
	typedef Array<half, 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, half(2) * a1);
	VERIFY((a1.abs() >= half(0)).all());
	VERIFY_IS_APPROX((a1 * a1).sqrt(), a1.abs());

	VERIFY(((a1.min)(a2) <= (a1.max)(a2)).all());
	a1(i) = half(-10.);
	VERIFY_IS_EQUAL(a1.minCoeff(), half(-10.));
	a1(i) = half(10.);
	VERIFY_IS_EQUAL(a1.maxCoeff(), half(10.));

	std::stringstream ss;
	ss << a1;
	}

	void test_product() {
	typedef Matrix<half, 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<half>());
	}

	EIGEN_DECLARE_TEST(half_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());
	}
	}