32-bit to 16-bit Floating Point Conversion

I need a cross-platform library/algorithm that will convert between 32-bit and 16-bit floating point numbers. I don't need to perform math with the 16-bit numbers; I just need to decrease the size of the 32-bit floats so they can be sent over the network. I am working in C++.

I understand how much precision I would be lo开发者_如何学Gosing, but that's OK for my application.

The IEEE 16-bit format would be great.

---

Complete conversion from single precision to half precision. This is a direct copy from my SSE version, so it's branch-less. It makes use of the fact that -true == ~0 to preform branchless selections (GCC converts if statements into an unholy mess of conditional jumps, while Clang just converts them to conditional moves.)

Update (2019-11-04): reworked to support single and double precision values with fully correct rounding. I also put a corresponding if statement above each branchless select as a comment for clarity. All incoming NaNs are converted to the base quiet NaN for speed and sanity, as there is no way to reliably convert an embedded NaN message between formats.

#include <cstdint> // uint32_t, uint64_t, etc.
#include <cstring> // memcpy
#include <climits> // CHAR_BIT
#include <limits>  // numeric_limits
#include <utility> // is_integral_v, is_floating_point_v, forward

namespace std
{
  template< typename T , typename U >
  T bit_cast( U&& u ) {
    static_assert( sizeof( T ) == sizeof( U ) );
    union { T t; }; // prevent construction
    std::memcpy( &t, &u, sizeof( t ) );
    return t;
  }
} // namespace std

template< typename T > struct native_float_bits;
template<> struct native_float_bits< float >{ using type = std::uint32_t; };
template<> struct native_float_bits< double >{ using type = std::uint64_t; };
template< typename T > using native_float_bits_t = typename native_float_bits< T >::type;

static_assert( sizeof( float ) == sizeof( native_float_bits_t< float > ) );
static_assert( sizeof( double ) == sizeof( native_float_bits_t< double > ) );

template< typename T, int SIG_BITS, int EXP_BITS >
struct raw_float_type_info {
  using raw_type = T;

  static constexpr int sig_bits = SIG_BITS;
  static constexpr int exp_bits = EXP_BITS;
  static constexpr int bits = sig_bits + exp_bits + 1;

  static_assert( std::is_integral_v< raw_type > );
  static_assert( sig_bits >= 0 );
  static_assert( exp_bits >= 0 );
  static_assert( bits <= sizeof( raw_type ) * CHAR_BIT );

  static constexpr int exp_max = ( 1 << exp_bits ) - 1;
  static constexpr int exp_bias = exp_max >> 1;

  static constexpr raw_type sign = raw_type( 1 ) << ( bits - 1 );
  static constexpr raw_type inf = raw_type( exp_max ) << sig_bits;
  static constexpr raw_type qnan = inf | ( inf >> 1 );

  static constexpr auto abs( raw_type v ) { return raw_type( v & ( sign - 1 ) ); }
  static constexpr bool is_nan( raw_type v ) { return abs( v ) > inf; }
  static constexpr bool is_inf( raw_type v ) { return abs( v ) == inf; }
  static constexpr bool is_zero( raw_type v ) { return abs( v ) == 0; }
};
using raw_flt16_type_info = raw_float_type_info< std::uint16_t, 10, 5 >;
using raw_flt32_type_info = raw_float_type_info< std::uint32_t, 23, 8 >;
using raw_flt64_type_info = raw_float_type_info< std::uint64_t, 52, 11 >;
//using raw_flt128_type_info = raw_float_type_info< uint128_t, 112, 15 >;

template< typename T, int SIG_BITS = std::numeric_limits< T >::digits - 1,
  int EXP_BITS = sizeof( T ) * CHAR_BIT - SIG_BITS - 1 >
struct float_type_info 
: raw_float_type_info< native_float_bits_t< T >, SIG_BITS, EXP_BITS > {
  using flt_type = T;
  static_assert( std::is_floating_point_v< flt_type > );
};

template< typename E >
struct raw_float_encoder
{
  using enc = E;
  using enc_type = typename enc::raw_type;

  template< bool DO_ROUNDING, typename F >
  static auto encode( F value )
  {
    using flt = float_type_info< F >;
    using raw_type = typename flt::raw_type;
    static constexpr auto sig_diff = flt::sig_bits - enc::sig_bits;
    static constexpr auto bit_diff = flt::bits - enc::bits;
    static constexpr auto do_rounding = DO_ROUNDING && sig_diff > 0;
    static constexpr auto bias_mul = raw_type( enc::exp_bias ) << flt::sig_bits;
    if constexpr( !do_rounding ) { // fix exp bias
      // when not rounding, fix exp first to avoid mixing float and binary ops
      value *= std::bit_cast< F >( bias_mul );
    }
    auto bits = std::bit_cast< raw_type >( value );
    auto sign = bits & flt::sign; // save sign
    bits ^= sign; // clear sign
    auto is_nan = flt::inf < bits; // compare before rounding!!
    if constexpr( do_rounding ) {
      static constexpr auto min_norm = raw_type( flt::exp_bias - enc::exp_bias + 1 ) << flt::sig_bits;
      static constexpr auto sub_rnd = enc::exp_bias < sig_diff
        ? raw_type( 1 ) << ( flt::sig_bits - 1 + enc::exp_bias - sig_diff )
        : raw_type( enc::exp_bias - sig_diff ) << flt::sig_bits;
      static constexpr auto sub_mul = raw_type( flt::exp_bias + sig_diff ) << flt::sig_bits;
      bool is_sub = bits < min_norm;
      auto norm = std::bit_cast< F >( bits );
      auto subn = norm;
      subn *= std::bit_cast< F >( sub_rnd ); // round subnormals
      subn *= std::bit_cast< F >( sub_mul ); // correct subnormal exp
      norm *= std::bit_cast< F >( bias_mul ); // fix exp bias
      bits = std::bit_cast< raw_type >( norm );
      bits += ( bits >> sig_diff ) & 1; // add tie breaking bias
      bits += ( raw_type( 1 ) << ( sig_diff - 1 ) ) - 1; // round up to half
      //if( is_sub ) bits = std::bit_cast< raw_type >( subn );
      bits ^= -is_sub & ( std::bit_cast< raw_type >( subn ) ^ bits );
    }
    bits >>= sig_diff; // truncate
    //if( enc::inf < bits ) bits = enc::inf; // fix overflow
    bits ^= -( enc::inf < bits ) & ( enc::inf ^ bits );
    //if( is_nan ) bits = enc::qnan;
    bits ^= -is_nan & ( enc::qnan ^ bits );
    bits |= sign >> bit_diff; // restore sign
    return enc_type( bits );
  }

  template< typename F >
  static F decode( enc_type value )
  {
    using flt = float_type_info< F >;
    using raw_type = typename flt::raw_type;
    static constexpr auto sig_diff = flt::sig_bits - enc::sig_bits;
    static constexpr auto bit_diff = flt::bits - enc::bits;
    static constexpr auto bias_mul = raw_type( 2 * flt::exp_bias - enc::exp_bias ) << flt::sig_bits;
    raw_type bits = value;
    auto sign = bits & enc::sign; // save sign
    bits ^= sign; // clear sign
    auto is_norm = bits < enc::inf;
    bits = ( sign << bit_diff ) | ( bits << sig_diff );
    auto val = std::bit_cast< F >( bits ) * std::bit_cast< F >( bias_mul );
    bits = std::bit_cast< raw_type >( val );
    //if( !is_norm ) bits |= flt::inf;
    bits |= -!is_norm & flt::inf;
    return std::bit_cast< F >( bits );
  }
};

using flt16_encoder = raw_float_encoder< raw_flt16_type_info >;

template< typename F >
auto quick_encode_flt16( F && value )
{ return flt16_encoder::encode< false >( std::forward< F >( value ) ); }

template< typename F >
auto encode_flt16( F && value )
{ return flt16_encoder::encode< true >( std::forward< F >( value ) ); }

template< typename F = float, typename X >
auto decode_flt16( X && value )
{ return flt16_encoder::decode< F >( std::forward< X >( value ) ); }

Of course full IEEE support isn't always needed. If your values don't require logarithmic resolution approaching zero, then linearizing them to a fixed point format is much faster, as was already mentioned.

---

Half to float:
float f = ((h&0x8000)<<16) | (((h&0x7c00)+0x1C000)<<13) | ((h&0x03FF)<<13);

Float to half:
uint32_t x = *((uint32_t*)&f);
uint16_t h = ((x>>16)&0x8000)|((((x&0x7f800000)-0x38000000)>>13)&0x7c00)|((x>>13)&0x03ff);

---

std::frexp extracts the significand and exponent from normal floats or doubles -- then you need to decide what to do with exponents that are too large to fit in a half-precision float (saturate...?), adjust accordingly, and put the half-precision number together. This article has C source code to show you how to perform the conversion.

---

Given your needs (-1000, 1000), perhaps it would be better to use a fixed-point representation.

//change to 20000 to SHORT_MAX if you don't mind whole numbers
//being turned into fractional ones
const int compact_range = 20000;

short compactFloat(double input) {
    return round(input * compact_range / 1000);
}
double expandToFloat(short input) {
    return ((double)input) * 1000 / compact_range;
}

This will give you accuracy to the nearest 0.05. If you change 20000 to SHORT_MAX you'll get a bit more accuracy but some whole numbers will end up as decimals on the other end.

---

Why so over-complicated? My implementation does not need any additional library, complies with the IEEE-754 FP16 format, manages both normalized and denormalized numbers, is branch-less, takes about 40-ish clock cycles for a back and forth conversion and ditches NaN or Inf for an extended range. That's the magical power of bit operations.

typedef unsigned short ushort;
typedef unsigned int uint;

uint as_uint(const float x) {
    return *(uint*)&x;
}
float as_float(const uint x) {
    return *(float*)&x;
}

float half_to_float(const ushort x) { // IEEE-754 16-bit floating-point format (without infinity): 1-5-10, exp-15, +-131008.0, +-6.1035156E-5, +-5.9604645E-8, 3.311 digits
    const uint e = (x&0x7C00)>>10; // exponent
    const uint m = (x&0x03FF)<<13; // mantissa
    const uint v = as_uint((float)m)>>23; // evil log2 bit hack to count leading zeros in denormalized format
    return as_float((x&0x8000)<<16 | (e!=0)*((e+112)<<23|m) | ((e==0)&(m!=0))*((v-37)<<23|((m<<(150-v))&0x007FE000))); // sign : normalized : denormalized
}
ushort float_to_half(const float x) { // IEEE-754 16-bit floating-point format (without infinity): 1-5-10, exp-15, +-131008.0, +-6.1035156E-5, +-5.9604645E-8, 3.311 digits
    const uint b = as_uint(x)+0x00001000; // round-to-nearest-even: add last bit after truncated mantissa
    const uint e = (b&0x7F800000)>>23; // exponent
    const uint m = b&0x007FFFFF; // mantissa; in line below: 0x007FF000 = 0x00800000-0x00001000 = decimal indicator flag - initial rounding
    return (b&0x80000000)>>16 | (e>112)*((((e-112)<<10)&0x7C00)|m>>13) | ((e<113)&(e>101))*((((0x007FF000+m)>>(125-e))+1)>>1) | (e>143)*0x7FFF; // sign : normalized : denormalized : saturate
}

Example for how to use it and to check that the conversion is correct:

#include <iostream>

void print_bits(const ushort x) {
    for(int i=15; i>=0; i--) {
        cout << ((x>>i)&1);
        if(i==15||i==10) cout << " ";
        if(i==10) cout << "      ";
    }
    cout << endl;
}
void print_bits(const float x) {
    uint b = *(uint*)&x;
    for(int i=31; i>=0; i--) {
        cout << ((b>>i)&1);
        if(i==31||i==23) cout << " ";
        if(i==23) cout << "   ";
    }
    cout << endl;
}

int main() {
    const float x = 1.0f;
    const ushort x_compressed = float_to_half(x);
    const float x_decompressed = half_to_float(x_compressed);
    print_bits(x);
    print_bits(x_compressed);
    print_bits(x_decompressed);
    return 0;
}

Output:

0 01111111    00000000000000000000000
0 01111       0000000000
0 01111111    00000000000000000000000

I have published an adapted version of this FP32<->FP16 conversion algorithm in this paper with detailed description on how the bit manipulation magic works. In this paper I also provide several ultra-fast conversion algorithms for different 16-bit Posit formats.

---

If you're sending a stream of information across, you could probably do better than this, especially if everything is in a consistent range, as your application seems to have.

Send a small header, that just consists of a float32 minimum and maximum, then you can send across your information as a 16 bit interpolation value between the two. As you also say that precision isn't much of an issue, you could even send 8bits at a time.

Your value would be something like, at reconstruction time:

float t = _t / numeric_limits<unsigned short>::max();  // With casting, naturally ;)
float val = h.min + t * (h.max - h.min);

Hope that helps.

-Tom

---

This question is already a bit old, but for the sake of completeness, you might also take a look at this paper for half-to-float and float-to-half conversion.

They use a branchless table-driven approach with relatively small look-up tables. It is completely IEEE-conformant and even beats Phernost's IEEE-conformant branchless conversion routines in performance (at least on my machine). But of course his code is much better suited to SSE and is not that prone to memory latency effects.

---

This conversion for 16-to-32-bit floating point is quite fast for cases where you do not have to account for infinities or NaNs, and can accept denormals-as-zero (DAZ). I.e. it is suitable for performance-sensitive calculations, but you should beware of division by zero if you expect to encounter denormals.

Note that this is most suitable for x86 or other platforms that have conditional moves or "set if" equivalents.

1. Strip the sign bit off the input
2. Align the most significant bit of the mantissa to the 22nd bit
3. Adjust the exponent bias
4. Set bits to all-zero if the input exponent is zero
5. Re-insert sign bit

The reverse applies for single-to-half-precision, with some additions.

void float32(float* __restrict out, const uint16_t in) {
    uint32_t t1;
    uint32_t t2;
    uint32_t t3;

    t1 = in & 0x7fff;                       // Non-sign bits
    t2 = in & 0x8000;                       // Sign bit
    t3 = in & 0x7c00;                       // Exponent

    t1 <<= 13;                              // Align mantissa on MSB
    t2 <<= 16;                              // Shift sign bit into position

    t1 += 0x38000000;                       // Adjust bias

    t1 = (t3 == 0 ? 0 : t1);                // Denormals-as-zero

    t1 |= t2;                               // Re-insert sign bit

    *((uint32_t*)out) = t1;
};

void float16(uint16_t* __restrict out, const float in) {
    uint32_t inu = *((uint32_t*)&in);
    uint32_t t1;
    uint32_t t2;
    uint32_t t3;

    t1 = inu & 0x7fffffff;                 // Non-sign bits
    t2 = inu & 0x80000000;                 // Sign bit
    t3 = inu & 0x7f800000;                 // Exponent

    t1 >>= 13;                             // Align mantissa on MSB
    t2 >>= 16;                             // Shift sign bit into position

    t1 -= 0x1c000;                         // Adjust bias

    t1 = (t3 > 0x38800000) ? 0 : t1;       // Flush-to-zero
    t1 = (t3 < 0x8e000000) ? 0x7bff : t1;  // Clamp-to-max
    t1 = (t3 == 0 ? 0 : t1);               // Denormals-as-zero

    t1 |= t2;                              // Re-insert sign bit

    *((uint16_t*)out) = t1;
};

Note that you can change the constant 0x7bff to 0x7c00 for it to overflow to infinity.

See GitHub for source code.

---

Most of the approaches described in the other answers here either do not round correctly on conversion from float to half, throw away subnormals which is a problem since 2**-14 becomes your smallest non-zero number, or do unfortunate things with Inf / NaN. Inf is also a problem because the largest finite number in half is a bit less than 2^16. OpenEXR was unnecessarily slow and complicated, last I looked at it. A fast correct approach will use the FPU to do the conversion, either as a direct instruction, or using the FPU rounding hardware to make the right thing happen. Any half to float conversion should be no slower than a 2^16 element lookup table.

The following are hard to beat:

On OS X / iOS, you can use vImageConvert_PlanarFtoPlanar16F and vImageConvert_Planar16FtoPlanarF. See Accelerate.framework.

Intel ivybridge added SSE instructions for this. See f16cintrin.h. Similar instructions were added to the ARM ISA for Neon. See vcvt_f32_f16 and vcvt_f16_f32 in arm_neon.h. On iOS you will need to use the arm64 or armv7s arch to get access to them.

---

This code converts a 32-bit floating point number to 16-bits and back.

#include <x86intrin.h>
#include <iostream>

int main()
{
    float f32;
    unsigned short f16;
    f32 = 3.14159265358979323846;
    f16 = _cvtss_sh(f32, 0);
    std::cout << f32 << std::endl;
    f32 = _cvtsh_ss(f16);
    std::cout << f32 << std::endl;
    return 0;
}

I tested with the Intel icpc 16.0.2:

$ icpc a.cpp

g++ 7.3.0:

$ g++ -march=native a.cpp

and clang++ 6.0.0:

$ clang++ -march=native a.cpp

It prints:

$ ./a.out
3.14159
3.14062

Documentation about these intrinsics is available at:

https://software.intel.com/en-us/node/524287

https://clang.llvm.org/doxygen/f16cintrin_8h.html

---

The question is old and has already been answered, but I figured it would be worth mentioning an open source C++ library that can create 16bit IEEE compliant half precision floats and has a class that acts pretty much identically to the built in float type, but with 16 bits instead of 32. It is the "half" class of the OpenEXR library. The code is under a permissive BSD style license. I don't believe it has any dependencies outside of the standard library.

---

I had this same exact problem, and found this link very helpful. Just import the file "ieeehalfprecision.c" into your project and use it like this :

float myFloat = 1.24;
uint16_t resultInHalf;
singles2halfp(&resultInHalf, &myFloat, 1); // it accepts a series of floats, so use 1 to input 1 float

// an example to revert the half float back
float resultInSingle;
halfp2singles(&resultInSingle, &resultInHalf, 1);

I also change some code (See the comment by the author (James Tursa) in the link) :

#define INT16_TYPE int16_t 
#define UINT16_TYPE uint16_t 
#define INT32_TYPE int32_t 
#define UINT32_TYPE uint32_t

---

I have found an implementation of conversion from half-float to single-float format and back with using of AVX2. There are much more faster than software implementation of these algorithms. I hope it will be useful.

32-bit float to 16-bit float conversion:

#include <immintrin.h"

inline void Float32ToFloat16(const float * src, uint16_t * dst)
{
    _mm_storeu_si128((__m128i*)dst, _mm256_cvtps_ph(_mm256_loadu_ps(src), 0));
}

void Float32ToFloat16(const float * src, size_t size, uint16_t * dst)
{
    assert(size >= 8);

    size_t fullAlignedSize = size&~(32-1);
    size_t partialAlignedSize = size&~(8-1);

    size_t i = 0;
    for (; i < fullAlignedSize; i += 32)
    {
        Float32ToFloat16(src + i + 0, dst + i + 0);
        Float32ToFloat16(src + i + 8, dst + i + 8);
        Float32ToFloat16(src + i + 16, dst + i + 16);
        Float32ToFloat16(src + i + 24, dst + i + 24);
    }
    for (; i < partialAlignedSize; i += 8)
        Float32ToFloat16(src + i, dst + i);
    if(partialAlignedSize != size)
        Float32ToFloat16(src + size - 8, dst + size - 8);
}

16-bit float to 32-bit float conversion:

#include <immintrin.h"

inline void Float16ToFloat32(const uint16_t * src, float * dst)
{
    _mm256_storeu_ps(dst, _mm256_cvtph_ps(_mm_loadu_si128((__m128i*)src)));
}

void Float16ToFloat32(const uint16_t * src, size_t size, float * dst)
{
    assert(size >= 8);

    size_t fullAlignedSize = size&~(32-1);
    size_t partialAlignedSize = size&~(8-1);

    size_t i = 0;
    for (; i < fullAlignedSize; i += 32)
    {
        Float16ToFloat32<align>(src + i + 0, dst + i + 0);
        Float16ToFloat32<align>(src + i + 8, dst + i + 8);
        Float16ToFloat32<align>(src + i + 16, dst + i + 16);
        Float16ToFloat32<align>(src + i + 24, dst + i + 24);
    }
    for (; i < partialAlignedSize; i += 8)
        Float16ToFloat32<align>(src + i, dst + i);
    if (partialAlignedSize != size)
        Float16ToFloat32<false>(src + size - 8, dst + size - 8);
}

---

Thanks Code for decimal to single precision

We actually can try to edit the same code to half precision , however it is not possible with gcc C compiler , so do the following

sudo apt install clang

Then try the following code

// A C code to convert Decimal value to IEEE 16-bit floating point Half precision

#include <stdio.h>

void printBinary(int n, int i)
{
 

    int k;
    for (k = i - 1; k >= 0; k--) {
 
        if ((n >> k) & 1)
            printf("1");
        else
            printf("0");
    }
}
 
typedef union {
    
    __fp16 f;
    struct
    {
        unsigned int mantissa : 10;
        unsigned int exponent : 5;
        unsigned int sign : 1;
 
    } raw;
} myfloat;
 

// Driver Code
int main()
{
    myfloat var;
    var.f = 11;
    printf("%d | ", var.raw.sign);
    printBinary(var.raw.exponent, 5);
    printf(" | ");
    printBinary(var.raw.mantissa, 10);
    printf("\n");
    return 0;
}

Compile the code in your terminal

clang code_name.c -o code_name
./code_name

Here

__fp16

is a 2 byte float data type supported in clang C compiler