cart-elc

Source code for CART-ELC
git clone git://git.laack.co/cart-elc.git
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bfloat16_float.cpp (17931B)


      1 // This file is part of Eigen, a lightweight C++ template library
      2 // for linear algebra.
      3 //
      4 // This Source Code Form is subject to the terms of the Mozilla
      5 // Public License v. 2.0. If a copy of the MPL was not distributed
      6 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
      7 
      8 #include <sstream>
      9 #include <memory>
     10 #include <math.h>
     11 
     12 #include "main.h"
     13 
     14 #include <Eigen/src/Core/arch/Default/BFloat16.h>
     15 
     16 #define VERIFY_BFLOAT16_BITS_EQUAL(h, bits) \
     17   VERIFY_IS_EQUAL((numext::bit_cast<numext::uint16_t>(h)), (static_cast<numext::uint16_t>(bits)))
     18 
     19 // Make sure it's possible to forward declare Eigen::bfloat16
     20 namespace Eigen {
     21 struct bfloat16;
     22 }
     23 
     24 using Eigen::bfloat16;
     25 
     26 float BinaryToFloat(uint32_t sign, uint32_t exponent, uint32_t high_mantissa,
     27                     uint32_t low_mantissa) {
     28   float dest;
     29   uint32_t src = (sign << 31) + (exponent << 23) + (high_mantissa << 16) + low_mantissa;
     30   memcpy(static_cast<void*>(&dest),
     31          static_cast<const void*>(&src), sizeof(dest));
     32   return dest;
     33 }
     34 
     35 template<typename T>
     36  void test_roundtrip() {
     37   // Representable T round trip via bfloat16
     38   VERIFY_IS_EQUAL((internal::cast<bfloat16,T>(internal::cast<T,bfloat16>(-std::numeric_limits<T>::infinity()))), -std::numeric_limits<T>::infinity());
     39   VERIFY_IS_EQUAL((internal::cast<bfloat16,T>(internal::cast<T,bfloat16>(std::numeric_limits<T>::infinity()))), std::numeric_limits<T>::infinity());
     40   VERIFY_IS_EQUAL((internal::cast<bfloat16,T>(internal::cast<T,bfloat16>(T(-1.0)))), T(-1.0));
     41   VERIFY_IS_EQUAL((internal::cast<bfloat16,T>(internal::cast<T,bfloat16>(T(-0.5)))), T(-0.5));
     42   VERIFY_IS_EQUAL((internal::cast<bfloat16,T>(internal::cast<T,bfloat16>(T(-0.0)))), T(-0.0));
     43   VERIFY_IS_EQUAL((internal::cast<bfloat16,T>(internal::cast<T,bfloat16>(T(1.0)))), T(1.0));
     44   VERIFY_IS_EQUAL((internal::cast<bfloat16,T>(internal::cast<T,bfloat16>(T(0.5)))), T(0.5));
     45   VERIFY_IS_EQUAL((internal::cast<bfloat16,T>(internal::cast<T,bfloat16>(T(0.0)))), T(0.0));
     46 }
     47 
     48 void test_conversion()
     49 {
     50   using Eigen::bfloat16_impl::__bfloat16_raw;
     51 
     52   // Round-trip casts
     53   VERIFY_IS_EQUAL(
     54     numext::bit_cast<bfloat16>(numext::bit_cast<numext::uint16_t>(bfloat16(1.0f))),
     55     bfloat16(1.0f));
     56   VERIFY_IS_EQUAL(
     57     numext::bit_cast<bfloat16>(numext::bit_cast<numext::uint16_t>(bfloat16(0.5f))),
     58     bfloat16(0.5f));
     59   VERIFY_IS_EQUAL(
     60     numext::bit_cast<bfloat16>(numext::bit_cast<numext::uint16_t>(bfloat16(-0.33333f))),
     61     bfloat16(-0.33333f));
     62    VERIFY_IS_EQUAL(
     63     numext::bit_cast<bfloat16>(numext::bit_cast<numext::uint16_t>(bfloat16(0.0f))),
     64     bfloat16(0.0f));
     65 
     66   // Conversion from float.
     67   VERIFY_BFLOAT16_BITS_EQUAL(bfloat16(1.0f), 0x3f80);
     68   VERIFY_BFLOAT16_BITS_EQUAL(bfloat16(0.5f), 0x3f00);
     69   VERIFY_BFLOAT16_BITS_EQUAL(bfloat16(0.33333f), 0x3eab);
     70   VERIFY_BFLOAT16_BITS_EQUAL(bfloat16(3.38e38f), 0x7f7e);
     71   VERIFY_BFLOAT16_BITS_EQUAL(bfloat16(3.40e38f), 0x7f80);  // Becomes infinity.
     72 
     73   // Verify round-to-nearest-even behavior.
     74   float val1 = static_cast<float>(bfloat16(__bfloat16_raw(0x3c00)));
     75   float val2 = static_cast<float>(bfloat16(__bfloat16_raw(0x3c01)));
     76   float val3 = static_cast<float>(bfloat16(__bfloat16_raw(0x3c02)));
     77   VERIFY_BFLOAT16_BITS_EQUAL(bfloat16(0.5f * (val1 + val2)), 0x3c00);
     78   VERIFY_BFLOAT16_BITS_EQUAL(bfloat16(0.5f * (val2 + val3)), 0x3c02);
     79 
     80   // Conversion from int.
     81   VERIFY_BFLOAT16_BITS_EQUAL(bfloat16(-1), 0xbf80);
     82   VERIFY_BFLOAT16_BITS_EQUAL(bfloat16(0), 0x0000);
     83   VERIFY_BFLOAT16_BITS_EQUAL(bfloat16(1), 0x3f80);
     84   VERIFY_BFLOAT16_BITS_EQUAL(bfloat16(2), 0x4000);
     85   VERIFY_BFLOAT16_BITS_EQUAL(bfloat16(3), 0x4040);
     86   VERIFY_BFLOAT16_BITS_EQUAL(bfloat16(12), 0x4140);
     87 
     88   // Conversion from bool.
     89   VERIFY_BFLOAT16_BITS_EQUAL(bfloat16(false), 0x0000);
     90   VERIFY_BFLOAT16_BITS_EQUAL(bfloat16(true), 0x3f80);
     91 
     92   // Conversion to bool
     93   VERIFY_IS_EQUAL(static_cast<bool>(bfloat16(3)), true);
     94   VERIFY_IS_EQUAL(static_cast<bool>(bfloat16(0.33333f)), true);
     95   VERIFY_IS_EQUAL(bfloat16(-0.0), false);
     96   VERIFY_IS_EQUAL(static_cast<bool>(bfloat16(0.0)), false);
     97 
     98   // Explicit conversion to float.
     99   VERIFY_IS_EQUAL(static_cast<float>(bfloat16(__bfloat16_raw(0x0000))), 0.0f);
    100   VERIFY_IS_EQUAL(static_cast<float>(bfloat16(__bfloat16_raw(0x3f80))), 1.0f);
    101 
    102   // Implicit conversion to float
    103   VERIFY_IS_EQUAL(bfloat16(__bfloat16_raw(0x0000)), 0.0f);
    104   VERIFY_IS_EQUAL(bfloat16(__bfloat16_raw(0x3f80)), 1.0f);
    105 
    106   // Zero representations
    107   VERIFY_IS_EQUAL(bfloat16(0.0f), bfloat16(0.0f));
    108   VERIFY_IS_EQUAL(bfloat16(-0.0f), bfloat16(0.0f));
    109   VERIFY_IS_EQUAL(bfloat16(-0.0f), bfloat16(-0.0f));
    110   VERIFY_BFLOAT16_BITS_EQUAL(bfloat16(0.0f), 0x0000);
    111   VERIFY_BFLOAT16_BITS_EQUAL(bfloat16(-0.0f), 0x8000);
    112 
    113   // Default is zero
    114   VERIFY_IS_EQUAL(static_cast<float>(bfloat16()), 0.0f);
    115 
    116   // Representable floats round trip via bfloat16
    117   test_roundtrip<float>();
    118   test_roundtrip<double>();
    119   test_roundtrip<std::complex<float> >();
    120   test_roundtrip<std::complex<double> >();
    121 
    122   // Conversion
    123   Array<float,1,100> a;
    124   for (int i = 0; i < 100; i++) a(i) = i + 1.25;
    125   Array<bfloat16,1,100> b = a.cast<bfloat16>();
    126   Array<float,1,100> c = b.cast<float>();
    127   for (int i = 0; i < 100; ++i) {
    128     VERIFY_LE(numext::abs(c(i) - a(i)), a(i) / 128);
    129   }
    130 
    131   // Epsilon
    132   VERIFY_LE(1.0f, static_cast<float>((std::numeric_limits<bfloat16>::epsilon)() + bfloat16(1.0f)));
    133   VERIFY_IS_EQUAL(1.0f, static_cast<float>((std::numeric_limits<bfloat16>::epsilon)() / bfloat16(2.0f) + bfloat16(1.0f)));
    134 
    135   // Negate
    136   VERIFY_IS_EQUAL(static_cast<float>(-bfloat16(3.0f)), -3.0f);
    137   VERIFY_IS_EQUAL(static_cast<float>(-bfloat16(-4.5f)), 4.5f);
    138 
    139 
    140 #if !EIGEN_COMP_MSVC
    141   // Visual Studio errors out on divisions by 0
    142   VERIFY((numext::isnan)(static_cast<float>(bfloat16(0.0 / 0.0))));
    143   VERIFY((numext::isinf)(static_cast<float>(bfloat16(1.0 / 0.0))));
    144   VERIFY((numext::isinf)(static_cast<float>(bfloat16(-1.0 / 0.0))));
    145 
    146   // Visual Studio errors out on divisions by 0
    147   VERIFY((numext::isnan)(bfloat16(0.0 / 0.0)));
    148   VERIFY((numext::isinf)(bfloat16(1.0 / 0.0)));
    149   VERIFY((numext::isinf)(bfloat16(-1.0 / 0.0)));
    150 #endif
    151 
    152   // NaNs and infinities.
    153   VERIFY(!(numext::isinf)(static_cast<float>(bfloat16(3.38e38f))));  // Largest finite number.
    154   VERIFY(!(numext::isnan)(static_cast<float>(bfloat16(0.0f))));
    155   VERIFY((numext::isinf)(static_cast<float>(bfloat16(__bfloat16_raw(0xff80)))));
    156   VERIFY((numext::isnan)(static_cast<float>(bfloat16(__bfloat16_raw(0xffc0)))));
    157   VERIFY((numext::isinf)(static_cast<float>(bfloat16(__bfloat16_raw(0x7f80)))));
    158   VERIFY((numext::isnan)(static_cast<float>(bfloat16(__bfloat16_raw(0x7fc0)))));
    159 
    160   // Exactly same checks as above, just directly on the bfloat16 representation.
    161   VERIFY(!(numext::isinf)(bfloat16(__bfloat16_raw(0x7bff))));
    162   VERIFY(!(numext::isnan)(bfloat16(__bfloat16_raw(0x0000))));
    163   VERIFY((numext::isinf)(bfloat16(__bfloat16_raw(0xff80))));
    164   VERIFY((numext::isnan)(bfloat16(__bfloat16_raw(0xffc0))));
    165   VERIFY((numext::isinf)(bfloat16(__bfloat16_raw(0x7f80))));
    166   VERIFY((numext::isnan)(bfloat16(__bfloat16_raw(0x7fc0))));
    167 
    168   VERIFY_BFLOAT16_BITS_EQUAL(bfloat16(BinaryToFloat(0x0, 0xff, 0x40, 0x0)), 0x7fc0);
    169   VERIFY_BFLOAT16_BITS_EQUAL(bfloat16(BinaryToFloat(0x1, 0xff, 0x40, 0x0)), 0xffc0);
    170 }
    171 
    172 void test_numtraits()
    173 {
    174   std::cout << "epsilon       = " << NumTraits<bfloat16>::epsilon() << "  (0x" << std::hex << numext::bit_cast<numext::uint16_t>(NumTraits<bfloat16>::epsilon()) << ")" << std::endl;
    175   std::cout << "highest       = " << NumTraits<bfloat16>::highest() << "  (0x" << std::hex << numext::bit_cast<numext::uint16_t>(NumTraits<bfloat16>::highest()) << ")" << std::endl;
    176   std::cout << "lowest        = " << NumTraits<bfloat16>::lowest() << "  (0x" << std::hex << numext::bit_cast<numext::uint16_t>(NumTraits<bfloat16>::lowest()) << ")" << std::endl;
    177   std::cout << "min           = " << (std::numeric_limits<bfloat16>::min)() << "  (0x" << std::hex << numext::bit_cast<numext::uint16_t>((std::numeric_limits<bfloat16>::min)()) << ")" << std::endl;
    178   std::cout << "denorm min    = " << (std::numeric_limits<bfloat16>::denorm_min)() << "  (0x" << std::hex << numext::bit_cast<numext::uint16_t>((std::numeric_limits<bfloat16>::denorm_min)()) << ")" << std::endl;
    179   std::cout << "infinity      = " << NumTraits<bfloat16>::infinity() << "  (0x" << std::hex << numext::bit_cast<numext::uint16_t>(NumTraits<bfloat16>::infinity()) << ")" << std::endl;
    180   std::cout << "quiet nan     = " << NumTraits<bfloat16>::quiet_NaN() << "  (0x" << std::hex << numext::bit_cast<numext::uint16_t>(NumTraits<bfloat16>::quiet_NaN()) << ")" << std::endl;
    181   std::cout << "signaling nan = " << std::numeric_limits<bfloat16>::signaling_NaN() << "  (0x" << std::hex << numext::bit_cast<numext::uint16_t>(std::numeric_limits<bfloat16>::signaling_NaN()) << ")" << std::endl;
    182 
    183   VERIFY(NumTraits<bfloat16>::IsSigned);
    184 
    185   VERIFY_IS_EQUAL(
    186     numext::bit_cast<numext::uint16_t>(std::numeric_limits<bfloat16>::infinity()),
    187     numext::bit_cast<numext::uint16_t>(bfloat16(std::numeric_limits<float>::infinity())) );
    188   // There is no guarantee that casting a 32-bit NaN to bfloat16 has a precise
    189   // bit pattern.  We test that it is in fact a NaN, then test the signaling
    190   // bit (msb of significand is 1 for quiet, 0 for signaling).
    191   const numext::uint16_t BFLOAT16_QUIET_BIT = 0x0040;
    192   VERIFY(
    193     (numext::isnan)(std::numeric_limits<bfloat16>::quiet_NaN())
    194     && (numext::isnan)(bfloat16(std::numeric_limits<float>::quiet_NaN()))
    195     && ((numext::bit_cast<numext::uint16_t>(std::numeric_limits<bfloat16>::quiet_NaN()) & BFLOAT16_QUIET_BIT) > 0)
    196     && ((numext::bit_cast<numext::uint16_t>(bfloat16(std::numeric_limits<float>::quiet_NaN())) & BFLOAT16_QUIET_BIT) > 0) );
    197   // After a cast to bfloat16, a signaling NaN may become non-signaling. Thus,
    198   // we check that both are NaN, and that only the `numeric_limits` version is
    199   // signaling.
    200   VERIFY(
    201     (numext::isnan)(std::numeric_limits<bfloat16>::signaling_NaN())
    202     && (numext::isnan)(bfloat16(std::numeric_limits<float>::signaling_NaN()))
    203     && ((numext::bit_cast<numext::uint16_t>(std::numeric_limits<bfloat16>::signaling_NaN()) & BFLOAT16_QUIET_BIT) == 0) );
    204 
    205   VERIFY( (std::numeric_limits<bfloat16>::min)() > bfloat16(0.f) );
    206   VERIFY( (std::numeric_limits<bfloat16>::denorm_min)() > bfloat16(0.f) );
    207   VERIFY_IS_EQUAL( (std::numeric_limits<bfloat16>::denorm_min)()/bfloat16(2), bfloat16(0.f) );
    208 }
    209 
    210 void test_arithmetic()
    211 {
    212   VERIFY_IS_EQUAL(static_cast<float>(bfloat16(2) + bfloat16(2)), 4);
    213   VERIFY_IS_EQUAL(static_cast<float>(bfloat16(2) + bfloat16(-2)), 0);
    214   VERIFY_IS_APPROX(static_cast<float>(bfloat16(0.33333f) + bfloat16(0.66667f)), 1.0f);
    215   VERIFY_IS_EQUAL(static_cast<float>(bfloat16(2.0f) * bfloat16(-5.5f)), -11.0f);
    216   VERIFY_IS_APPROX(static_cast<float>(bfloat16(1.0f) / bfloat16(3.0f)), 0.3339f);
    217   VERIFY_IS_EQUAL(static_cast<float>(-bfloat16(4096.0f)), -4096.0f);
    218   VERIFY_IS_EQUAL(static_cast<float>(-bfloat16(-4096.0f)), 4096.0f);
    219 }
    220 
    221 void test_comparison()
    222 {
    223   VERIFY(bfloat16(1.0f) > bfloat16(0.5f));
    224   VERIFY(bfloat16(0.5f) < bfloat16(1.0f));
    225   VERIFY(!(bfloat16(1.0f) < bfloat16(0.5f)));
    226   VERIFY(!(bfloat16(0.5f) > bfloat16(1.0f)));
    227 
    228   VERIFY(!(bfloat16(4.0f) > bfloat16(4.0f)));
    229   VERIFY(!(bfloat16(4.0f) < bfloat16(4.0f)));
    230 
    231   VERIFY(!(bfloat16(0.0f) < bfloat16(-0.0f)));
    232   VERIFY(!(bfloat16(-0.0f) < bfloat16(0.0f)));
    233   VERIFY(!(bfloat16(0.0f) > bfloat16(-0.0f)));
    234   VERIFY(!(bfloat16(-0.0f) > bfloat16(0.0f)));
    235 
    236   VERIFY(bfloat16(0.2f) > bfloat16(-1.0f));
    237   VERIFY(bfloat16(-1.0f) < bfloat16(0.2f));
    238   VERIFY(bfloat16(-16.0f) < bfloat16(-15.0f));
    239 
    240   VERIFY(bfloat16(1.0f) == bfloat16(1.0f));
    241   VERIFY(bfloat16(1.0f) != bfloat16(2.0f));
    242 
    243   // Comparisons with NaNs and infinities.
    244 #if !EIGEN_COMP_MSVC
    245   // Visual Studio errors out on divisions by 0
    246   VERIFY(!(bfloat16(0.0 / 0.0) == bfloat16(0.0 / 0.0)));
    247   VERIFY(bfloat16(0.0 / 0.0) != bfloat16(0.0 / 0.0));
    248 
    249   VERIFY(!(bfloat16(1.0) == bfloat16(0.0 / 0.0)));
    250   VERIFY(!(bfloat16(1.0) < bfloat16(0.0 / 0.0)));
    251   VERIFY(!(bfloat16(1.0) > bfloat16(0.0 / 0.0)));
    252   VERIFY(bfloat16(1.0) != bfloat16(0.0 / 0.0));
    253 
    254   VERIFY(bfloat16(1.0) < bfloat16(1.0 / 0.0));
    255   VERIFY(bfloat16(1.0) > bfloat16(-1.0 / 0.0));
    256 #endif
    257 }
    258 
    259 void test_basic_functions()
    260 {
    261   VERIFY_IS_EQUAL(static_cast<float>(numext::abs(bfloat16(3.5f))), 3.5f);
    262   VERIFY_IS_EQUAL(static_cast<float>(abs(bfloat16(3.5f))), 3.5f);
    263   VERIFY_IS_EQUAL(static_cast<float>(numext::abs(bfloat16(-3.5f))), 3.5f);
    264   VERIFY_IS_EQUAL(static_cast<float>(abs(bfloat16(-3.5f))), 3.5f);
    265 
    266   VERIFY_IS_EQUAL(static_cast<float>(numext::floor(bfloat16(3.5f))), 3.0f);
    267   VERIFY_IS_EQUAL(static_cast<float>(floor(bfloat16(3.5f))), 3.0f);
    268   VERIFY_IS_EQUAL(static_cast<float>(numext::floor(bfloat16(-3.5f))), -4.0f);
    269   VERIFY_IS_EQUAL(static_cast<float>(floor(bfloat16(-3.5f))), -4.0f);
    270 
    271   VERIFY_IS_EQUAL(static_cast<float>(numext::ceil(bfloat16(3.5f))), 4.0f);
    272   VERIFY_IS_EQUAL(static_cast<float>(ceil(bfloat16(3.5f))), 4.0f);
    273   VERIFY_IS_EQUAL(static_cast<float>(numext::ceil(bfloat16(-3.5f))), -3.0f);
    274   VERIFY_IS_EQUAL(static_cast<float>(ceil(bfloat16(-3.5f))), -3.0f);
    275 
    276   VERIFY_IS_APPROX(static_cast<float>(numext::sqrt(bfloat16(0.0f))), 0.0f);
    277   VERIFY_IS_APPROX(static_cast<float>(sqrt(bfloat16(0.0f))), 0.0f);
    278   VERIFY_IS_APPROX(static_cast<float>(numext::sqrt(bfloat16(4.0f))), 2.0f);
    279   VERIFY_IS_APPROX(static_cast<float>(sqrt(bfloat16(4.0f))), 2.0f);
    280 
    281   VERIFY_IS_APPROX(static_cast<float>(numext::pow(bfloat16(0.0f), bfloat16(1.0f))), 0.0f);
    282   VERIFY_IS_APPROX(static_cast<float>(pow(bfloat16(0.0f), bfloat16(1.0f))), 0.0f);
    283   VERIFY_IS_APPROX(static_cast<float>(numext::pow(bfloat16(2.0f), bfloat16(2.0f))), 4.0f);
    284   VERIFY_IS_APPROX(static_cast<float>(pow(bfloat16(2.0f), bfloat16(2.0f))), 4.0f);
    285 
    286   VERIFY_IS_EQUAL(static_cast<float>(numext::exp(bfloat16(0.0f))), 1.0f);
    287   VERIFY_IS_EQUAL(static_cast<float>(exp(bfloat16(0.0f))), 1.0f);
    288   VERIFY_IS_APPROX(static_cast<float>(numext::exp(bfloat16(EIGEN_PI))), 20.f + static_cast<float>(EIGEN_PI));
    289   VERIFY_IS_APPROX(static_cast<float>(exp(bfloat16(EIGEN_PI))), 20.f + static_cast<float>(EIGEN_PI));
    290 
    291   VERIFY_IS_EQUAL(static_cast<float>(numext::expm1(bfloat16(0.0f))), 0.0f);
    292   VERIFY_IS_EQUAL(static_cast<float>(expm1(bfloat16(0.0f))), 0.0f);
    293   VERIFY_IS_APPROX(static_cast<float>(numext::expm1(bfloat16(2.0f))), 6.375f);
    294   VERIFY_IS_APPROX(static_cast<float>(expm1(bfloat16(2.0f))), 6.375f);
    295 
    296   VERIFY_IS_EQUAL(static_cast<float>(numext::log(bfloat16(1.0f))), 0.0f);
    297   VERIFY_IS_EQUAL(static_cast<float>(log(bfloat16(1.0f))), 0.0f);
    298   VERIFY_IS_APPROX(static_cast<float>(numext::log(bfloat16(10.0f))), 2.296875f);
    299   VERIFY_IS_APPROX(static_cast<float>(log(bfloat16(10.0f))), 2.296875f);
    300 
    301   VERIFY_IS_EQUAL(static_cast<float>(numext::log1p(bfloat16(0.0f))), 0.0f);
    302   VERIFY_IS_EQUAL(static_cast<float>(log1p(bfloat16(0.0f))), 0.0f);
    303   VERIFY_IS_APPROX(static_cast<float>(numext::log1p(bfloat16(10.0f))), 2.390625f);
    304   VERIFY_IS_APPROX(static_cast<float>(log1p(bfloat16(10.0f))), 2.390625f);
    305 }
    306 
    307 void test_trigonometric_functions()
    308 {
    309   VERIFY_IS_APPROX(numext::cos(bfloat16(0.0f)), bfloat16(cosf(0.0f)));
    310   VERIFY_IS_APPROX(cos(bfloat16(0.0f)), bfloat16(cosf(0.0f)));
    311   VERIFY_IS_APPROX(numext::cos(bfloat16(EIGEN_PI)), bfloat16(cosf(EIGEN_PI)));
    312   // VERIFY_IS_APPROX(numext::cos(bfloat16(EIGEN_PI/2)), bfloat16(cosf(EIGEN_PI/2)));
    313   // VERIFY_IS_APPROX(numext::cos(bfloat16(3*EIGEN_PI/2)), bfloat16(cosf(3*EIGEN_PI/2)));
    314   VERIFY_IS_APPROX(numext::cos(bfloat16(3.5f)), bfloat16(cosf(3.5f)));
    315 
    316   VERIFY_IS_APPROX(numext::sin(bfloat16(0.0f)), bfloat16(sinf(0.0f)));
    317   VERIFY_IS_APPROX(sin(bfloat16(0.0f)), bfloat16(sinf(0.0f)));
    318   // VERIFY_IS_APPROX(numext::sin(bfloat16(EIGEN_PI)), bfloat16(sinf(EIGEN_PI)));
    319   VERIFY_IS_APPROX(numext::sin(bfloat16(EIGEN_PI/2)), bfloat16(sinf(EIGEN_PI/2)));
    320   VERIFY_IS_APPROX(numext::sin(bfloat16(3*EIGEN_PI/2)), bfloat16(sinf(3*EIGEN_PI/2)));
    321   VERIFY_IS_APPROX(numext::sin(bfloat16(3.5f)), bfloat16(sinf(3.5f)));
    322 
    323   VERIFY_IS_APPROX(numext::tan(bfloat16(0.0f)), bfloat16(tanf(0.0f)));
    324   VERIFY_IS_APPROX(tan(bfloat16(0.0f)), bfloat16(tanf(0.0f)));
    325   // VERIFY_IS_APPROX(numext::tan(bfloat16(EIGEN_PI)), bfloat16(tanf(EIGEN_PI)));
    326   // VERIFY_IS_APPROX(numext::tan(bfloat16(EIGEN_PI/2)), bfloat16(tanf(EIGEN_PI/2)));
    327   // VERIFY_IS_APPROX(numext::tan(bfloat16(3*EIGEN_PI/2)), bfloat16(tanf(3*EIGEN_PI/2)));
    328   VERIFY_IS_APPROX(numext::tan(bfloat16(3.5f)), bfloat16(tanf(3.5f)));
    329 }
    330 
    331 void test_array()
    332 {
    333   typedef Array<bfloat16,1,Dynamic> ArrayXh;
    334   Index size = internal::random<Index>(1,10);
    335   Index i = internal::random<Index>(0,size-1);
    336   ArrayXh a1 = ArrayXh::Random(size), a2 = ArrayXh::Random(size);
    337   VERIFY_IS_APPROX( a1+a1, bfloat16(2)*a1 );
    338   VERIFY( (a1.abs() >= bfloat16(0)).all() );
    339   VERIFY_IS_APPROX( (a1*a1).sqrt(), a1.abs() );
    340 
    341   VERIFY( ((a1.min)(a2) <= (a1.max)(a2)).all() );
    342   a1(i) = bfloat16(-10.);
    343   VERIFY_IS_EQUAL( a1.minCoeff(), bfloat16(-10.) );
    344   a1(i) = bfloat16(10.);
    345   VERIFY_IS_EQUAL( a1.maxCoeff(), bfloat16(10.) );
    346 
    347   std::stringstream ss;
    348   ss << a1;
    349 }
    350 
    351 void test_product()
    352 {
    353   typedef Matrix<bfloat16,Dynamic,Dynamic> MatrixXh;
    354   Index rows  = internal::random<Index>(1,EIGEN_TEST_MAX_SIZE);
    355   Index cols  = internal::random<Index>(1,EIGEN_TEST_MAX_SIZE);
    356   Index depth = internal::random<Index>(1,EIGEN_TEST_MAX_SIZE);
    357   MatrixXh Ah = MatrixXh::Random(rows,depth);
    358   MatrixXh Bh = MatrixXh::Random(depth,cols);
    359   MatrixXh Ch = MatrixXh::Random(rows,cols);
    360   MatrixXf Af = Ah.cast<float>();
    361   MatrixXf Bf = Bh.cast<float>();
    362   MatrixXf Cf = Ch.cast<float>();
    363   VERIFY_IS_APPROX(Ch.noalias()+=Ah*Bh, (Cf.noalias()+=Af*Bf).cast<bfloat16>());
    364 }
    365 
    366 EIGEN_DECLARE_TEST(bfloat16_float)
    367 {
    368   CALL_SUBTEST(test_numtraits());
    369   for(int i = 0; i < g_repeat; i++) {
    370     CALL_SUBTEST(test_conversion());
    371     CALL_SUBTEST(test_arithmetic());
    372     CALL_SUBTEST(test_comparison());
    373     CALL_SUBTEST(test_basic_functions());
    374     CALL_SUBTEST(test_trigonometric_functions());
    375     CALL_SUBTEST(test_array());
    376     CALL_SUBTEST(test_product());
    377   }
    378 }