Q:

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 losing, but that's OK for my application.
The IEEE 16-bit format would be great.

A:

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. 

A:

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.

A:

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);

