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/.

            
            // SPDX-FileCopyrightText: The Eigen Authors

            
            // SPDX-License-Identifier: 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::isfinite)(float(half(65504.0f))));

            
              VERIFY((numext::isfinite)(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)))));


              VERIFY((numext::isnan)(float(NumTraits<half>::quiet_NaN())));

            
              VERIFY((numext::isinf)(float(NumTraits<half>::infinity())));

            
              VERIFY((numext::isinf)(float(-NumTraits<half>::infinity())));


              // 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))));


              VERIFY((numext::isnan)(NumTraits<half>::quiet_NaN()));

            
              VERIFY((numext::isinf)(NumTraits<half>::infinity()));

            
              VERIFY((numext::isinf)(-NumTraits<half>::infinity()));


              // 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)));


              // A double rounds to half once: just above the midpoint 1 + 2^-11 it rounds up, although its float rounding is

            
              // the midpoint itself, which ties down. The same holds below the midpoint and for tiny values.

            
              VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(half(1.0 + std::ldexp(1.0, -11) + std::ldexp(1.0, -40))), 0x3c01);

            
              VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(

            
                                  half(-(1.0 + std::ldexp(1.0, -10) + std::ldexp(1.0, -11) - std::ldexp(1.0, -40)))),

            
                              0xbc01);

            
              VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(half(std::ldexp(1.0, -25) + std::ldexp(1.0, -60))), 0x0001);

            
              VERIFY_IS_EQUAL(

            
                  numext::bit_cast<numext::uint16_t>(half(std::complex<double>(std::ldexp(1.0, -25) + std::ldexp(1.0, -60)))),

            
                  0x0001);

            
              // Beyond float's range a double overflows to infinity or underflows to zero, keeping its sign.

            
              VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(half(1e39)), 0x7c00);

            
              VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(half(-1e39)), 0xfc00);

            
              VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(half(1e-300)), 0x0000);

            
              VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(half(-1e-300)), 0x8000);

            
              VERIFY((numext::isnan)(half(std::numeric_limits<double>::quiet_NaN())));

            
              // Integers convert through float directly: those it rounds overflow half anyway.

            
              VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(half(int32_t(65519))), 0x7bff);

            
              VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(half(-int64_t(65520))), 0xfc00);

            
              VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(half((std::numeric_limits<uint64_t>::max)())), 0x7c00);

            
            }


            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);


              // The fp16 machine epsilon is 2^-10 = 0x1400, and NumTraits must agree with

            
              // std::numeric_limits (NumTraits used to return 2^-13, making every

            
              // epsilon-based threshold 8x too tight).

            
              VERIFY_HALF_BITS_EQUAL(NumTraits<half>::epsilon(), 0x1400);

            
              VERIFY_HALF_BITS_EQUAL(std::numeric_limits<half>::epsilon(), 0x1400);

            
              VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(NumTraits<half>::epsilon()),

            
                              numext::bit_cast<numext::uint16_t>(std::numeric_limits<half>::epsilon()));


              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);


              // fma rounds once: a * b = 2145/2048 is a midpoint of half, and c = 2^-24 rounds the sum up, not to even. A float

            
              // fma rounds the sum to that midpoint first.

            
              const half tiny = half(std::ldexp(1.0f, -24));

            
              VERIFY_IS_EQUAL(float(numext::fma(half(33.f / 32), half(65.f / 64), tiny)), 1.f + 49.f / 1024);

            
              VERIFY_IS_EQUAL(float(numext::fma(half(-33.f / 32), half(65.f / 64), -tiny)), -1.f - 49.f / 1024);


              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)));

            
            #if EIGEN_HAS_STD_HASH

            
              // Equal values hash alike.

            
              VERIFY_IS_EQUAL(std::hash<half>()(half(0.0f)), std::hash<half>()(half(-0.0f)));

            
            #endif


              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.

            
              VERIFY(!(NumTraits<half>::quiet_NaN() == NumTraits<half>::quiet_NaN()));

            
              VERIFY(NumTraits<half>::quiet_NaN() != NumTraits<half>::quiet_NaN());


              VERIFY(!(internal::random<half>() == NumTraits<half>::quiet_NaN()));

            
              VERIFY(!(internal::random<half>() < NumTraits<half>::quiet_NaN()));

            
              VERIFY(!(internal::random<half>() > NumTraits<half>::quiet_NaN()));

            
              VERIFY(!(internal::random<half>() <= NumTraits<half>::quiet_NaN()));

            
              VERIFY(!(internal::random<half>() >= NumTraits<half>::quiet_NaN()));

            
              VERIFY(internal::random<half>() != NumTraits<half>::quiet_NaN());


              VERIFY(!(NumTraits<half>::quiet_NaN() == internal::random<half>()));

            
              VERIFY(!(NumTraits<half>::quiet_NaN() < internal::random<half>()));

            
              VERIFY(!(NumTraits<half>::quiet_NaN() > internal::random<half>()));

            
              VERIFY(!(NumTraits<half>::quiet_NaN() <= internal::random<half>()));

            
              VERIFY(!(NumTraits<half>::quiet_NaN() >= internal::random<half>()));

            
              VERIFY(NumTraits<half>::quiet_NaN() != internal::random<half>());


              VERIFY(internal::random<half>() < NumTraits<half>::infinity());

            
              VERIFY(internal::random<half>() > -NumTraits<half>::infinity());

            
            }


            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>());

            
            }


            void test_nextafter() {

            
              VERIFY((numext::isnan)(numext::nextafter(std::numeric_limits<half>::quiet_NaN(), half(1.0f))));

            
              VERIFY((numext::isnan)(numext::nextafter(half(1.0f), std::numeric_limits<half>::quiet_NaN())));

            
              VERIFY(numext::nextafter(half(0.0f), half(0.0f)) == half(0.0f));

            
              VERIFY(numext::nextafter(half(1.0f), half(1.0f)) == half(1.0f));

            
              VERIFY(numext::nextafter(half(-1.0f), half(-1.0f)) == half(-1.0f));

            
              VERIFY(numext::nextafter(std::numeric_limits<half>::infinity(), std::numeric_limits<half>::infinity()) ==

            
                     std::numeric_limits<half>::infinity());

            
              VERIFY(numext::nextafter(std::numeric_limits<half>::infinity(), half(0.0f)) == (std::numeric_limits<half>::max)());

            
              VERIFY(numext::nextafter(-std::numeric_limits<half>::infinity(), half(0.0f)) == -(std::numeric_limits<half>::max)());

            
              VERIFY(numext::nextafter(half(1.0f), std::numeric_limits<half>::infinity()) ==

            
                     half(1.0f) + std::numeric_limits<half>::epsilon());

            
              VERIFY(numext::nextafter(half(1.0f), -std::numeric_limits<half>::infinity()) ==

            
                     half(1.0f) - std::numeric_limits<half>::epsilon() / half(2.0f));

            
              VERIFY(numext::nextafter(half(-1.0f), -std::numeric_limits<half>::infinity()) ==

            
                     half(-1.0f) - std::numeric_limits<half>::epsilon());

            
              VERIFY(numext::nextafter(half(-1.0f), std::numeric_limits<half>::infinity()) ==

            
                     half(-1.0f) + std::numeric_limits<half>::epsilon() / half(2.0f));

            
              VERIFY(numext::nextafter((std::numeric_limits<half>::max)(), std::numeric_limits<half>::infinity()) ==

            
                     std::numeric_limits<half>::infinity());

            
              VERIFY(numext::nextafter(-(std::numeric_limits<half>::max)(), -std::numeric_limits<half>::infinity()) ==

            
                     -std::numeric_limits<half>::infinity());

            
              // The neighbors of ±0 are the smallest subnormals with the sign of the

            
              // direction, matching std::nextafter (IEEE-754 nextUp/nextDown of zero).

            
              VERIFY_HALF_BITS_EQUAL(numext::nextafter(half(0.0f), half(1.0f)), 0x0001);

            
              VERIFY_HALF_BITS_EQUAL(numext::nextafter(half(-0.0f), half(1.0f)), 0x0001);

            
              VERIFY_HALF_BITS_EQUAL(numext::nextafter(half(0.0f), half(-1.0f)), 0x8001);

            
              VERIFY_HALF_BITS_EQUAL(numext::nextafter(half(-0.0f), half(-1.0f)), 0x8001);

            
              // from == to returns to, preserving the sign of zero.

            
              VERIFY_HALF_BITS_EQUAL(numext::nextafter(half(0.0f), half(-0.0f)), 0x8000);

            
              VERIFY_HALF_BITS_EQUAL(numext::nextafter(half(-0.0f), half(0.0f)), 0x0000);

            
              VERIFY_HALF_BITS_EQUAL(numext::nextafter(half(0.0f), half(0.0f)), 0x0000);

            
              VERIFY_HALF_BITS_EQUAL(numext::nextafter(half(-0.0f), half(-0.0f)), 0x8000);

            
            }


            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());

            
                CALL_SUBTEST(test_nextafter());

            
              }

            
            }
	// 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/.
	// SPDX-FileCopyrightText: The Eigen Authors
	// SPDX-License-Identifier: 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::isfinite)(float(half(65504.0f))));
	VERIFY((numext::isfinite)(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)))));

	VERIFY((numext::isnan)(float(NumTraits<half>::quiet_NaN())));
	VERIFY((numext::isinf)(float(NumTraits<half>::infinity())));
	VERIFY((numext::isinf)(float(-NumTraits<half>::infinity())));

	// 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))));

	VERIFY((numext::isnan)(NumTraits<half>::quiet_NaN()));
	VERIFY((numext::isinf)(NumTraits<half>::infinity()));
	VERIFY((numext::isinf)(-NumTraits<half>::infinity()));

	// 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)));

	// A double rounds to half once: just above the midpoint 1 + 2^-11 it rounds up, although its float rounding is
	// the midpoint itself, which ties down. The same holds below the midpoint and for tiny values.
	VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(half(1.0 + std::ldexp(1.0, -11) + std::ldexp(1.0, -40))), 0x3c01);
	VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(
	half(-(1.0 + std::ldexp(1.0, -10) + std::ldexp(1.0, -11) - std::ldexp(1.0, -40)))),
	0xbc01);
	VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(half(std::ldexp(1.0, -25) + std::ldexp(1.0, -60))), 0x0001);
	VERIFY_IS_EQUAL(
	numext::bit_cast<numext::uint16_t>(half(std::complex<double>(std::ldexp(1.0, -25) + std::ldexp(1.0, -60)))),
	0x0001);
	// Beyond float's range a double overflows to infinity or underflows to zero, keeping its sign.
	VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(half(1e39)), 0x7c00);
	VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(half(-1e39)), 0xfc00);
	VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(half(1e-300)), 0x0000);
	VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(half(-1e-300)), 0x8000);
	VERIFY((numext::isnan)(half(std::numeric_limits<double>::quiet_NaN())));
	// Integers convert through float directly: those it rounds overflow half anyway.
	VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(half(int32_t(65519))), 0x7bff);
	VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(half(-int64_t(65520))), 0xfc00);
	VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(half((std::numeric_limits<uint64_t>::max)())), 0x7c00);
	}

	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);

	// The fp16 machine epsilon is 2^-10 = 0x1400, and NumTraits must agree with
	// std::numeric_limits (NumTraits used to return 2^-13, making every
	// epsilon-based threshold 8x too tight).
	VERIFY_HALF_BITS_EQUAL(NumTraits<half>::epsilon(), 0x1400);
	VERIFY_HALF_BITS_EQUAL(std::numeric_limits<half>::epsilon(), 0x1400);
	VERIFY_IS_EQUAL(numext::bit_cast<numext::uint16_t>(NumTraits<half>::epsilon()),
	numext::bit_cast<numext::uint16_t>(std::numeric_limits<half>::epsilon()));

	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);

	// fma rounds once: a * b = 2145/2048 is a midpoint of half, and c = 2^-24 rounds the sum up, not to even. A float
	// fma rounds the sum to that midpoint first.
	const half tiny = half(std::ldexp(1.0f, -24));
	VERIFY_IS_EQUAL(float(numext::fma(half(33.f / 32), half(65.f / 64), tiny)), 1.f + 49.f / 1024);
	VERIFY_IS_EQUAL(float(numext::fma(half(-33.f / 32), half(65.f / 64), -tiny)), -1.f - 49.f / 1024);

	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)));
	#if EIGEN_HAS_STD_HASH
	// Equal values hash alike.
	VERIFY_IS_EQUAL(std::hash<half>()(half(0.0f)), std::hash<half>()(half(-0.0f)));
	#endif

	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.
	VERIFY(!(NumTraits<half>::quiet_NaN() == NumTraits<half>::quiet_NaN()));
	VERIFY(NumTraits<half>::quiet_NaN() != NumTraits<half>::quiet_NaN());

	VERIFY(!(internal::random<half>() == NumTraits<half>::quiet_NaN()));
	VERIFY(!(internal::random<half>() < NumTraits<half>::quiet_NaN()));
	VERIFY(!(internal::random<half>() > NumTraits<half>::quiet_NaN()));
	VERIFY(!(internal::random<half>() <= NumTraits<half>::quiet_NaN()));
	VERIFY(!(internal::random<half>() >= NumTraits<half>::quiet_NaN()));
	VERIFY(internal::random<half>() != NumTraits<half>::quiet_NaN());

	VERIFY(!(NumTraits<half>::quiet_NaN() == internal::random<half>()));
	VERIFY(!(NumTraits<half>::quiet_NaN() < internal::random<half>()));
	VERIFY(!(NumTraits<half>::quiet_NaN() > internal::random<half>()));
	VERIFY(!(NumTraits<half>::quiet_NaN() <= internal::random<half>()));
	VERIFY(!(NumTraits<half>::quiet_NaN() >= internal::random<half>()));
	VERIFY(NumTraits<half>::quiet_NaN() != internal::random<half>());

	VERIFY(internal::random<half>() < NumTraits<half>::infinity());
	VERIFY(internal::random<half>() > -NumTraits<half>::infinity());
	}

	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>());
	}

	void test_nextafter() {
	VERIFY((numext::isnan)(numext::nextafter(std::numeric_limits<half>::quiet_NaN(), half(1.0f))));
	VERIFY((numext::isnan)(numext::nextafter(half(1.0f), std::numeric_limits<half>::quiet_NaN())));
	VERIFY(numext::nextafter(half(0.0f), half(0.0f)) == half(0.0f));
	VERIFY(numext::nextafter(half(1.0f), half(1.0f)) == half(1.0f));
	VERIFY(numext::nextafter(half(-1.0f), half(-1.0f)) == half(-1.0f));
	VERIFY(numext::nextafter(std::numeric_limits<half>::infinity(), std::numeric_limits<half>::infinity()) ==
	std::numeric_limits<half>::infinity());
	VERIFY(numext::nextafter(std::numeric_limits<half>::infinity(), half(0.0f)) == (std::numeric_limits<half>::max)());
	VERIFY(numext::nextafter(-std::numeric_limits<half>::infinity(), half(0.0f)) == -(std::numeric_limits<half>::max)());
	VERIFY(numext::nextafter(half(1.0f), std::numeric_limits<half>::infinity()) ==
	half(1.0f) + std::numeric_limits<half>::epsilon());
	VERIFY(numext::nextafter(half(1.0f), -std::numeric_limits<half>::infinity()) ==
	half(1.0f) - std::numeric_limits<half>::epsilon() / half(2.0f));
	VERIFY(numext::nextafter(half(-1.0f), -std::numeric_limits<half>::infinity()) ==
	half(-1.0f) - std::numeric_limits<half>::epsilon());
	VERIFY(numext::nextafter(half(-1.0f), std::numeric_limits<half>::infinity()) ==
	half(-1.0f) + std::numeric_limits<half>::epsilon() / half(2.0f));
	VERIFY(numext::nextafter((std::numeric_limits<half>::max)(), std::numeric_limits<half>::infinity()) ==
	std::numeric_limits<half>::infinity());
	VERIFY(numext::nextafter(-(std::numeric_limits<half>::max)(), -std::numeric_limits<half>::infinity()) ==
	-std::numeric_limits<half>::infinity());
	// The neighbors of ±0 are the smallest subnormals with the sign of the
	// direction, matching std::nextafter (IEEE-754 nextUp/nextDown of zero).
	VERIFY_HALF_BITS_EQUAL(numext::nextafter(half(0.0f), half(1.0f)), 0x0001);
	VERIFY_HALF_BITS_EQUAL(numext::nextafter(half(-0.0f), half(1.0f)), 0x0001);
	VERIFY_HALF_BITS_EQUAL(numext::nextafter(half(0.0f), half(-1.0f)), 0x8001);
	VERIFY_HALF_BITS_EQUAL(numext::nextafter(half(-0.0f), half(-1.0f)), 0x8001);
	// from == to returns to, preserving the sign of zero.
	VERIFY_HALF_BITS_EQUAL(numext::nextafter(half(0.0f), half(-0.0f)), 0x8000);
	VERIFY_HALF_BITS_EQUAL(numext::nextafter(half(-0.0f), half(0.0f)), 0x0000);
	VERIFY_HALF_BITS_EQUAL(numext::nextafter(half(0.0f), half(0.0f)), 0x0000);
	VERIFY_HALF_BITS_EQUAL(numext::nextafter(half(-0.0f), half(-0.0f)), 0x8000);
	}

	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());
	CALL_SUBTEST(test_nextafter());
	}
	}