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sir-pinecone/float16.c

Created January 21, 2022 16:13
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float16.c
// Branch-free implementation of half-precision (16 bit) floating point
// Copyright 2006 Mike Acton <macton@gmail.com>
//
// Permission is hereby granted, free of charge, to any person obtaining a
// copy of this software and associated documentation files (the "Software"),
// to deal in the Software without restriction, including without limitation
// the rights to use, copy, modify, merge, publish, distribute, sublicense,
// and/or sell copies of the Software, and to permit persons to whom the
// Software is furnished to do so, subject to the following conditions:
//
// The above copyright notice and this permission notice shall be included
// in all copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
// THE SOFTWARE
//
// Half-precision floating point format
// ------------------------------------
//
// | Field | Last | First | Note
// |----------|------|-------|----------
// | Sign | 15 | 15 |
// | Exponent | 14 | 10 | Bias = 15
// | Mantissa | 9 | 0 |
//
// Compiling
// ---------
//
// Preferred compile flags for GCC:
// -O3 -fstrict-aliasing -std=c99 -pedantic -Wall -Wstrict-aliasing
//
// This file is a C99 source file, intended to be compiled with a C99
// compliant compiler. However, for the moment it remains combatible
// with C++98. Therefore if you are using a compiler that poorly implements
// C standards (e.g. MSVC), it may be compiled as C++. This is not
// guaranteed for future versions.
//
#include "half.h"
// Load immediate
static inline uint32_t _uint32_li( uint32_t a )
{
return (a);
}
// Decrement
static inline uint32_t _uint32_dec( uint32_t a )
{
return (a - 1);
}
// Increment
static inline uint32_t _uint32_inc( uint32_t a )
{
return (a + 1);
}
// Complement
static inline uint32_t _uint32_not( uint32_t a )
{
return (~a);
}
// Negate
static inline uint32_t _uint32_neg( uint32_t a )
{
return (-a);
}
// Extend sign
static inline uint32_t _uint32_ext( uint32_t a )
{
return (((int32_t)a)>>31);
}
// And
static inline uint32_t _uint32_and( uint32_t a, uint32_t b )
{
return (a & b);
}
// Exclusive Or
static inline uint32_t _uint32_xor( uint32_t a, uint32_t b )
{
return (a ^ b);
}
// And with Complement
static inline uint32_t _uint32_andc( uint32_t a, uint32_t b )
{
return (a & ~b);
}
// Or
static inline uint32_t _uint32_or( uint32_t a, uint32_t b )
{
return (a | b);
}
// Shift Right Logical
static inline uint32_t _uint32_srl( uint32_t a, int sa )
{
return (a >> sa);
}
// Shift Left Logical
static inline uint32_t _uint32_sll( uint32_t a, int sa )
{
return (a << sa);
}
// Add
static inline uint32_t _uint32_add( uint32_t a, uint32_t b )
{
return (a + b);
}
// Subtract
static inline uint32_t _uint32_sub( uint32_t a, uint32_t b )
{
return (a - b);
}
// Multiply
static inline uint32_t _uint32_mul( uint32_t a, uint32_t b )
{
return (a * b);
}
// Select on Sign bit
static inline uint32_t _uint32_sels( uint32_t test, uint32_t a, uint32_t b )
{
const uint32_t mask = _uint32_ext( test );
const uint32_t sel_a = _uint32_and( a, mask );
const uint32_t sel_b = _uint32_andc( b, mask );
const uint32_t result = _uint32_or( sel_a, sel_b );
return (result);
}
// Select Bits on mask
static inline uint32_t _uint32_selb( uint32_t mask, uint32_t a, uint32_t b )
{
const uint32_t sel_a = _uint32_and( a, mask );
const uint32_t sel_b = _uint32_andc( b, mask );
const uint32_t result = _uint32_or( sel_a, sel_b );
return (result);
}
// Load Immediate
static inline uint16_t _uint16_li( uint16_t a )
{
return (a);
}
// Extend sign
static inline uint16_t _uint16_ext( uint16_t a )
{
return (((int16_t)a)>>15);
}
// Negate
static inline uint16_t _uint16_neg( uint16_t a )
{
return (-a);
}
// Complement
static inline uint16_t _uint16_not( uint16_t a )
{
return (~a);
}
// Decrement
static inline uint16_t _uint16_dec( uint16_t a )
{
return (a - 1);
}
// Shift Left Logical
static inline uint16_t _uint16_sll( uint16_t a, int sa )
{
return (a << sa);
}
// Shift Right Logical
static inline uint16_t _uint16_srl( uint16_t a, int sa )
{
return (a >> sa);
}
// Add
static inline uint16_t _uint16_add( uint16_t a, uint16_t b )
{
return (a + b);
}
// Subtract
static inline uint16_t _uint16_sub( uint16_t a, uint16_t b )
{
return (a - b);
}
// And
static inline uint16_t _uint16_and( uint16_t a, uint16_t b )
{
return (a & b);
}
// Or
static inline uint16_t _uint16_or( uint16_t a, uint16_t b )
{
return (a | b);
}
// Exclusive Or
static inline uint16_t _uint16_xor( uint16_t a, uint16_t b )
{
return (a ^ b);
}
// And with Complement
static inline uint16_t _uint16_andc( uint16_t a, uint16_t b )
{
return (a & ~b);
}
// And then Shift Right Logical
static inline uint16_t _uint16_andsrl( uint16_t a, uint16_t b, int sa )
{
return ((a & b) >> sa);
}
// Shift Right Logical then Mask
static inline uint16_t _uint16_srlm( uint16_t a, int sa, uint16_t mask )
{
return ((a >> sa) & mask);
}
// Add then Mask
static inline uint16_t _uint16_addm( uint16_t a, uint16_t b, uint16_t mask )
{
return ((a + b) & mask);
}
// Select on Sign bit
static inline uint16_t _uint16_sels( uint16_t test, uint16_t a, uint16_t b )
{
const uint16_t mask = _uint16_ext( test );
const uint16_t sel_a = _uint16_and( a, mask );
const uint16_t sel_b = _uint16_andc( b, mask );
const uint16_t result = _uint16_or( sel_a, sel_b );
return (result);
}
// Count Leading Zeros
static inline uint32_t _uint32_cntlz( uint32_t x )
{
#ifdef __GNUC__
/* NOTE: __builtin_clz is undefined for x == 0 */
/* On PowerPC, this will map to insn: cntlzw */
/* On Pentium, this will map to insn: clz */
uint32_t is_x_nez_msb = _uint32_neg( x );
uint32_t nlz = __builtin_clz( x );
uint32_t result = _uint32_sels( is_x_nez_msb, nlz, 0x00000020 );
return (result);
#else
const uint32_t x0 = _uint32_srl( x, 1 );
const uint32_t x1 = _uint32_or( x, x0 );
const uint32_t x2 = _uint32_srl( x1, 2 );
const uint32_t x3 = _uint32_or( x1, x2 );
const uint32_t x4 = _uint32_srl( x3, 4 );
const uint32_t x5 = _uint32_or( x3, x4 );
const uint32_t x6 = _uint32_srl( x5, 8 );
const uint32_t x7 = _uint32_or( x5, x6 );
const uint32_t x8 = _uint32_srl( x7, 16 );
const uint32_t x9 = _uint32_or( x7, x8 );
const uint32_t xA = _uint32_not( x9 );
const uint32_t xB = _uint32_srl( xA, 1 );
const uint32_t xC = _uint32_and( xB, 0x55555555 );
const uint32_t xD = _uint32_sub( xA, xC );
const uint32_t xE = _uint32_and( xD, 0x33333333 );
const uint32_t xF = _uint32_srl( xD, 2 );
const uint32_t x10 = _uint32_and( xF, 0x33333333 );
const uint32_t x11 = _uint32_add( xE, x10 );
const uint32_t x12 = _uint32_srl( x11, 4 );
const uint32_t x13 = _uint32_add( x11, x12 );
const uint32_t x14 = _uint32_and( x13, 0x0f0f0f0f );
const uint32_t x15 = _uint32_srl( x14, 8 );
const uint32_t x16 = _uint32_add( x14, x15 );
const uint32_t x17 = _uint32_srl( x16, 16 );
const uint32_t x18 = _uint32_add( x16, x17 );
const uint32_t x19 = _uint32_and( x18, 0x0000003f );
return ( x19 );
#endif
}
// Count Leading Zeros
static inline uint16_t _uint16_cntlz( uint16_t x )
{
#ifdef __GNUC__
uint16_t nlz32 = (uint16_t)_uint32_cntlz( (uint32_t)x );
uint32_t nlz = _uint32_sub( nlz32, 16 );
return (nlz);
#else
const uint16_t x0 = _uint16_srl( x, 1 );
const uint16_t x1 = _uint16_or( x, x0 );
const uint16_t x2 = _uint16_srl( x1, 2 );
const uint16_t x3 = _uint16_or( x1, x2 );
const uint16_t x4 = _uint16_srl( x3, 4 );
const uint16_t x5 = _uint16_or( x3, x4 );
const uint16_t x6 = _uint16_srl( x5, 8 );
const uint16_t x7 = _uint16_or( x5, x6 );
const uint16_t x8 = _uint16_not( x7 );
const uint16_t x9 = _uint16_srlm( x8, 1, 0x5555 );
const uint16_t xA = _uint16_sub( x8, x9 );
const uint16_t xB = _uint16_and( xA, 0x3333 );
const uint16_t xC = _uint16_srlm( xA, 2, 0x3333 );
const uint16_t xD = _uint16_add( xB, xC );
const uint16_t xE = _uint16_srl( xD, 4 );
const uint16_t xF = _uint16_addm( xD, xE, 0x0f0f );
const uint16_t x10 = _uint16_srl( xF, 8 );
const uint16_t x11 = _uint16_addm( xF, x10, 0x001f );
return ( x11 );
#endif
}
uint16_t
half_from_float( uint32_t f )
{
const uint32_t one = _uint32_li( 0x00000001 );
const uint32_t f_s_mask = _uint32_li( 0x80000000 );
const uint32_t f_e_mask = _uint32_li( 0x7f800000 );
const uint32_t f_m_mask = _uint32_li( 0x007fffff );
const uint32_t f_m_hidden_bit = _uint32_li( 0x00800000 );
const uint32_t f_m_round_bit = _uint32_li( 0x00001000 );
const uint32_t f_snan_mask = _uint32_li( 0x7fc00000 );
const uint32_t f_e_pos = _uint32_li( 0x00000017 );
const uint32_t h_e_pos = _uint32_li( 0x0000000a );
const uint32_t h_e_mask = _uint32_li( 0x00007c00 );
const uint32_t h_snan_mask = _uint32_li( 0x00007e00 );
const uint32_t h_e_mask_value = _uint32_li( 0x0000001f );
const uint32_t f_h_s_pos_offset = _uint32_li( 0x00000010 );
const uint32_t f_h_bias_offset = _uint32_li( 0x00000070 );
const uint32_t f_h_m_pos_offset = _uint32_li( 0x0000000d );
const uint32_t h_nan_min = _uint32_li( 0x00007c01 );
const uint32_t f_h_e_biased_flag = _uint32_li( 0x0000008f );
const uint32_t f_s = _uint32_and( f, f_s_mask );
const uint32_t f_e = _uint32_and( f, f_e_mask );
const uint16_t h_s = _uint32_srl( f_s, f_h_s_pos_offset );
const uint32_t f_m = _uint32_and( f, f_m_mask );
const uint16_t f_e_amount = _uint32_srl( f_e, f_e_pos );
const uint32_t f_e_half_bias = _uint32_sub( f_e_amount, f_h_bias_offset );
const uint32_t f_snan = _uint32_and( f, f_snan_mask );
const uint32_t f_m_round_mask = _uint32_and( f_m, f_m_round_bit );
const uint32_t f_m_round_offset = _uint32_sll( f_m_round_mask, one );
const uint32_t f_m_rounded = _uint32_add( f_m, f_m_round_offset );
const uint32_t f_m_denorm_sa = _uint32_sub( one, f_e_half_bias );
const uint32_t f_m_with_hidden = _uint32_or( f_m_rounded, f_m_hidden_bit );
const uint32_t f_m_denorm = _uint32_srl( f_m_with_hidden, f_m_denorm_sa );
const uint32_t h_m_denorm = _uint32_srl( f_m_denorm, f_h_m_pos_offset );
const uint32_t f_m_rounded_overflow = _uint32_and( f_m_rounded, f_m_hidden_bit );
const uint32_t m_nan = _uint32_srl( f_m, f_h_m_pos_offset );
const uint32_t h_em_nan = _uint32_or( h_e_mask, m_nan );
const uint32_t h_e_norm_overflow_offset = _uint32_inc( f_e_half_bias );
const uint32_t h_e_norm_overflow = _uint32_sll( h_e_norm_overflow_offset, h_e_pos );
const uint32_t h_e_norm = _uint32_sll( f_e_half_bias, h_e_pos );
const uint32_t h_m_norm = _uint32_srl( f_m_rounded, f_h_m_pos_offset );
const uint32_t h_em_norm = _uint32_or( h_e_norm, h_m_norm );
const uint32_t is_h_ndenorm_msb = _uint32_sub( f_h_bias_offset, f_e_amount );
const uint32_t is_f_e_flagged_msb = _uint32_sub( f_h_e_biased_flag, f_e_half_bias );
const uint32_t is_h_denorm_msb = _uint32_not( is_h_ndenorm_msb );
const uint32_t is_f_m_eqz_msb = _uint32_dec( f_m );
const uint32_t is_h_nan_eqz_msb = _uint32_dec( m_nan );
const uint32_t is_f_inf_msb = _uint32_and( is_f_e_flagged_msb, is_f_m_eqz_msb );
const uint32_t is_f_nan_underflow_msb = _uint32_and( is_f_e_flagged_msb, is_h_nan_eqz_msb );
const uint32_t is_e_overflow_msb = _uint32_sub( h_e_mask_value, f_e_half_bias );
const uint32_t is_h_inf_msb = _uint32_or( is_e_overflow_msb, is_f_inf_msb );
const uint32_t is_f_nsnan_msb = _uint32_sub( f_snan, f_snan_mask );
const uint32_t is_m_norm_overflow_msb = _uint32_neg( f_m_rounded_overflow );
const uint32_t is_f_snan_msb = _uint32_not( is_f_nsnan_msb );
const uint32_t h_em_overflow_result = _uint32_sels( is_m_norm_overflow_msb, h_e_norm_overflow, h_em_norm );
const uint32_t h_em_nan_result = _uint32_sels( is_f_e_flagged_msb, h_em_nan, h_em_overflow_result );
const uint32_t h_em_nan_underflow_result = _uint32_sels( is_f_nan_underflow_msb, h_nan_min, h_em_nan_result );
const uint32_t h_em_inf_result = _uint32_sels( is_h_inf_msb, h_e_mask, h_em_nan_underflow_result );
const uint32_t h_em_denorm_result = _uint32_sels( is_h_denorm_msb, h_m_denorm, h_em_inf_result );
const uint32_t h_em_snan_result = _uint32_sels( is_f_snan_msb, h_snan_mask, h_em_denorm_result );
const uint32_t h_result = _uint32_or( h_s, h_em_snan_result );
return (uint16_t)(h_result);
}
uint32_t
half_to_float( uint16_t h )
{
const uint32_t h_e_mask = _uint32_li( 0x00007c00 );
const uint32_t h_m_mask = _uint32_li( 0x000003ff );
const uint32_t h_s_mask = _uint32_li( 0x00008000 );
const uint32_t h_f_s_pos_offset = _uint32_li( 0x00000010 );
const uint32_t h_f_e_pos_offset = _uint32_li( 0x0000000d );
const uint32_t h_f_bias_offset = _uint32_li( 0x0001c000 );
const uint32_t f_e_mask = _uint32_li( 0x7f800000 );
const uint32_t f_m_mask = _uint32_li( 0x007fffff );
const uint32_t h_f_e_denorm_bias = _uint32_li( 0x0000007e );
const uint32_t h_f_m_denorm_sa_bias = _uint32_li( 0x00000008 );
const uint32_t f_e_pos = _uint32_li( 0x00000017 );
const uint32_t h_e_mask_minus_one = _uint32_li( 0x00007bff );
const uint32_t h_e = _uint32_and( h, h_e_mask );
const uint32_t h_m = _uint32_and( h, h_m_mask );
const uint32_t h_s = _uint32_and( h, h_s_mask );
const uint32_t h_e_f_bias = _uint32_add( h_e, h_f_bias_offset );
const uint32_t h_m_nlz = _uint32_cntlz( h_m );
const uint32_t f_s = _uint32_sll( h_s, h_f_s_pos_offset );
const uint32_t f_e = _uint32_sll( h_e_f_bias, h_f_e_pos_offset );
const uint32_t f_m = _uint32_sll( h_m, h_f_e_pos_offset );
const uint32_t f_em = _uint32_or( f_e, f_m );
const uint32_t h_f_m_sa = _uint32_sub( h_m_nlz, h_f_m_denorm_sa_bias );
const uint32_t f_e_denorm_unpacked = _uint32_sub( h_f_e_denorm_bias, h_f_m_sa );
const uint32_t h_f_m = _uint32_sll( h_m, h_f_m_sa );
const uint32_t f_m_denorm = _uint32_and( h_f_m, f_m_mask );
const uint32_t f_e_denorm = _uint32_sll( f_e_denorm_unpacked, f_e_pos );
const uint32_t f_em_denorm = _uint32_or( f_e_denorm, f_m_denorm );
const uint32_t f_em_nan = _uint32_or( f_e_mask, f_m );
const uint32_t is_e_eqz_msb = _uint32_dec( h_e );
const uint32_t is_m_nez_msb = _uint32_neg( h_m );
const uint32_t is_e_flagged_msb = _uint32_sub( h_e_mask_minus_one, h_e );
const uint32_t is_zero_msb = _uint32_andc( is_e_eqz_msb, is_m_nez_msb );
const uint32_t is_inf_msb = _uint32_andc( is_e_flagged_msb, is_m_nez_msb );
const uint32_t is_denorm_msb = _uint32_and( is_m_nez_msb, is_e_eqz_msb );
const uint32_t is_nan_msb = _uint32_and( is_e_flagged_msb, is_m_nez_msb );
const uint32_t is_zero = _uint32_ext( is_zero_msb );
const uint32_t f_zero_result = _uint32_andc( f_em, is_zero );
const uint32_t f_denorm_result = _uint32_sels( is_denorm_msb, f_em_denorm, f_zero_result );
const uint32_t f_inf_result = _uint32_sels( is_inf_msb, f_e_mask, f_denorm_result );
const uint32_t f_nan_result = _uint32_sels( is_nan_msb, f_em_nan, f_inf_result );
const uint32_t f_result = _uint32_or( f_s, f_nan_result );
return (f_result);
}
// half_add
// --------
//
// (SUM) uint16_t z = half_add( x, y );
// (DIFFERENCE) uint16_t z = half_add( x, -y );
//
// * Difference of ZEROs is always +ZERO
// * Sum round with guard + round + sticky bit (grs)
// * QNaN + <x> = QNaN
// * <x> + +INF = +INF
// * <x> - -INF = -INF
// * INF - INF = SNaN
//
// Will have exactly (0 ulps difference) the same result as:
// (Round up)
//
// union FLOAT_32
// {
// float f32;
// uint32_t u32;
// };
//
// union FLOAT_32 fx = { .u32 = half_to_float( x ) };
// union FLOAT_32 fy = { .u32 = half_to_float( y ) };
// union FLOAT_32 fz = { .f32 = fx.f32 + fy.f32 };
// uint16_t z = float_to_half( fz );
//
uint16_t
half_add( uint16_t x, uint16_t y )
{
const uint16_t one = _uint16_li( 0x0001 );
const uint16_t msb_to_lsb_sa = _uint16_li( 0x000f );
const uint16_t h_s_mask = _uint16_li( 0x8000 );
const uint16_t h_e_mask = _uint16_li( 0x7c00 );
const uint16_t h_m_mask = _uint16_li( 0x03ff );
const uint16_t h_m_msb_mask = _uint16_li( 0x2000 );
const uint16_t h_m_msb_sa = _uint16_li( 0x000d );
const uint16_t h_m_hidden = _uint16_li( 0x0400 );
const uint16_t h_e_pos = _uint16_li( 0x000a );
const uint16_t h_e_bias_minus_one = _uint16_li( 0x000e );
const uint16_t h_m_grs_carry = _uint16_li( 0x4000 );
const uint16_t h_m_grs_carry_pos = _uint16_li( 0x000e );
const uint16_t h_grs_size = _uint16_li( 0x0003 );
const uint16_t h_snan = _uint16_li( 0xfe00 );
const uint16_t h_e_mask_minus_one = _uint16_li( 0x7bff );
const uint16_t h_grs_round_carry = _uint16_sll( one, h_grs_size );
const uint16_t h_grs_round_mask = _uint16_sub( h_grs_round_carry, one );
const uint16_t x_e = _uint16_and( x, h_e_mask );
const uint16_t y_e = _uint16_and( y, h_e_mask );
const uint16_t is_y_e_larger_msb = _uint16_sub( x_e, y_e );
const uint16_t a = _uint16_sels( is_y_e_larger_msb, y, x);
const uint16_t a_s = _uint16_and( a, h_s_mask );
const uint16_t a_e = _uint16_and( a, h_e_mask );
const uint16_t a_m_no_hidden_bit = _uint16_and( a, h_m_mask );
const uint16_t a_em_no_hidden_bit = _uint16_or( a_e, a_m_no_hidden_bit );
const uint16_t b = _uint16_sels( is_y_e_larger_msb, x, y);
const uint16_t b_s = _uint16_and( b, h_s_mask );
const uint16_t b_e = _uint16_and( b, h_e_mask );
const uint16_t b_m_no_hidden_bit = _uint16_and( b, h_m_mask );
const uint16_t b_em_no_hidden_bit = _uint16_or( b_e, b_m_no_hidden_bit );
const uint16_t is_diff_sign_msb = _uint16_xor( a_s, b_s );
const uint16_t is_a_inf_msb = _uint16_sub( h_e_mask_minus_one, a_em_no_hidden_bit );
const uint16_t is_b_inf_msb = _uint16_sub( h_e_mask_minus_one, b_em_no_hidden_bit );
const uint16_t is_undenorm_msb = _uint16_dec( a_e );
const uint16_t is_undenorm = _uint16_ext( is_undenorm_msb );
const uint16_t is_both_inf_msb = _uint16_and( is_a_inf_msb, is_b_inf_msb );
const uint16_t is_invalid_inf_op_msb = _uint16_and( is_both_inf_msb, b_s );
const uint16_t is_a_e_nez_msb = _uint16_neg( a_e );
const uint16_t is_b_e_nez_msb = _uint16_neg( b_e );
const uint16_t is_a_e_nez = _uint16_ext( is_a_e_nez_msb );
const uint16_t is_b_e_nez = _uint16_ext( is_b_e_nez_msb );
const uint16_t a_m_hidden_bit = _uint16_and( is_a_e_nez, h_m_hidden );
const uint16_t b_m_hidden_bit = _uint16_and( is_b_e_nez, h_m_hidden );
const uint16_t a_m_no_grs = _uint16_or( a_m_no_hidden_bit, a_m_hidden_bit );
const uint16_t b_m_no_grs = _uint16_or( b_m_no_hidden_bit, b_m_hidden_bit );
const uint16_t diff_e = _uint16_sub( a_e, b_e );
const uint16_t a_e_unbias = _uint16_sub( a_e, h_e_bias_minus_one );
const uint16_t a_m = _uint16_sll( a_m_no_grs, h_grs_size );
const uint16_t a_e_biased = _uint16_srl( a_e, h_e_pos );
const uint16_t m_sa_unbias = _uint16_srl( a_e_unbias, h_e_pos );
const uint16_t m_sa_default = _uint16_srl( diff_e, h_e_pos );
const uint16_t m_sa_unbias_mask = _uint16_andc( is_a_e_nez_msb, is_b_e_nez_msb );
const uint16_t m_sa = _uint16_sels( m_sa_unbias_mask, m_sa_unbias, m_sa_default );
const uint16_t b_m_no_sticky = _uint16_sll( b_m_no_grs, h_grs_size );
const uint16_t sh_m = _uint16_srl( b_m_no_sticky, m_sa );
const uint16_t sticky_overflow = _uint16_sll( one, m_sa );
const uint16_t sticky_mask = _uint16_dec( sticky_overflow );
const uint16_t sticky_collect = _uint16_and( b_m_no_sticky, sticky_mask );
const uint16_t is_sticky_set_msb = _uint16_neg( sticky_collect );
const uint16_t sticky = _uint16_srl( is_sticky_set_msb, msb_to_lsb_sa);
const uint16_t b_m = _uint16_or( sh_m, sticky );
const uint16_t is_c_m_ab_pos_msb = _uint16_sub( b_m, a_m );
const uint16_t c_inf = _uint16_or( a_s, h_e_mask );
const uint16_t c_m_sum = _uint16_add( a_m, b_m );
const uint16_t c_m_diff_ab = _uint16_sub( a_m, b_m );
const uint16_t c_m_diff_ba = _uint16_sub( b_m, a_m );
const uint16_t c_m_smag_diff = _uint16_sels( is_c_m_ab_pos_msb, c_m_diff_ab, c_m_diff_ba );
const uint16_t c_s_diff = _uint16_sels( is_c_m_ab_pos_msb, a_s, b_s );
const uint16_t c_s = _uint16_sels( is_diff_sign_msb, c_s_diff, a_s );
const uint16_t c_m_smag_diff_nlz = _uint16_cntlz( c_m_smag_diff );
const uint16_t diff_norm_sa = _uint16_sub( c_m_smag_diff_nlz, one );
const uint16_t is_diff_denorm_msb = _uint16_sub( a_e_biased, diff_norm_sa );
const uint16_t is_diff_denorm = _uint16_ext( is_diff_denorm_msb );
const uint16_t is_a_or_b_norm_msb = _uint16_neg( a_e_biased );
const uint16_t diff_denorm_sa = _uint16_dec( a_e_biased );
const uint16_t c_m_diff_denorm = _uint16_sll( c_m_smag_diff, diff_denorm_sa );
const uint16_t c_m_diff_norm = _uint16_sll( c_m_smag_diff, diff_norm_sa );
const uint16_t c_e_diff_norm = _uint16_sub( a_e_biased, diff_norm_sa );
const uint16_t c_m_diff_ab_norm = _uint16_sels( is_diff_denorm_msb, c_m_diff_denorm, c_m_diff_norm );
const uint16_t c_e_diff_ab_norm = _uint16_andc( c_e_diff_norm, is_diff_denorm );
const uint16_t c_m_diff = _uint16_sels( is_a_or_b_norm_msb, c_m_diff_ab_norm, c_m_smag_diff );
const uint16_t c_e_diff = _uint16_sels( is_a_or_b_norm_msb, c_e_diff_ab_norm, a_e_biased );
const uint16_t is_diff_eqz_msb = _uint16_dec( c_m_diff );
const uint16_t is_diff_exactly_zero_msb = _uint16_and( is_diff_sign_msb, is_diff_eqz_msb );
const uint16_t is_diff_exactly_zero = _uint16_ext( is_diff_exactly_zero_msb );
const uint16_t c_m_added = _uint16_sels( is_diff_sign_msb, c_m_diff, c_m_sum );
const uint16_t c_e_added = _uint16_sels( is_diff_sign_msb, c_e_diff, a_e_biased );
const uint16_t c_m_carry = _uint16_and( c_m_added, h_m_grs_carry );
const uint16_t is_c_m_carry_msb = _uint16_neg( c_m_carry );
const uint16_t c_e_hidden_offset = _uint16_andsrl( c_m_added, h_m_grs_carry, h_m_grs_carry_pos );
const uint16_t c_m_sub_hidden = _uint16_srl( c_m_added, one );
const uint16_t c_m_no_hidden = _uint16_sels( is_c_m_carry_msb, c_m_sub_hidden, c_m_added );
const uint16_t c_e_no_hidden = _uint16_add( c_e_added, c_e_hidden_offset );
const uint16_t c_m_no_hidden_msb = _uint16_and( c_m_no_hidden, h_m_msb_mask );
const uint16_t undenorm_m_msb_odd = _uint16_srl( c_m_no_hidden_msb, h_m_msb_sa );
const uint16_t undenorm_fix_e = _uint16_and( is_undenorm, undenorm_m_msb_odd );
const uint16_t c_e_fixed = _uint16_add( c_e_no_hidden, undenorm_fix_e );
const uint16_t c_m_round_amount = _uint16_and( c_m_no_hidden, h_grs_round_mask );
const uint16_t c_m_rounded = _uint16_add( c_m_no_hidden, c_m_round_amount );
const uint16_t c_m_round_overflow = _uint16_andsrl( c_m_rounded, h_m_grs_carry, h_m_grs_carry_pos );
const uint16_t c_e_rounded = _uint16_add( c_e_fixed, c_m_round_overflow );
const uint16_t c_m_no_grs = _uint16_srlm( c_m_rounded, h_grs_size, h_m_mask );
const uint16_t c_e = _uint16_sll( c_e_rounded, h_e_pos );
const uint16_t c_em = _uint16_or( c_e, c_m_no_grs );
const uint16_t c_normal = _uint16_or( c_s, c_em );
const uint16_t c_inf_result = _uint16_sels( is_a_inf_msb, c_inf, c_normal );
const uint16_t c_zero_result = _uint16_andc( c_inf_result, is_diff_exactly_zero );
const uint16_t c_result = _uint16_sels( is_invalid_inf_op_msb, h_snan, c_zero_result );
return (c_result);
}
// half_mul
// --------
//
// May have 0 or 1 ulp difference from the following result:
// (Round to nearest)
// NOTE: Rounding mode differs between conversion and multiply
//
// union FLOAT_32
// {
// float f32;
// uint32_t u32;
// };
//
// union FLOAT_32 fx = { .u32 = half_to_float( x ) };
// union FLOAT_32 fy = { .u32 = half_to_float( y ) };
// union FLOAT_32 fz = { .f32 = fx.f32 * fy.f32 };
// uint16_t z = float_to_half( fz );
//
uint16_t
half_mul( uint16_t x, uint16_t y )
{
const uint32_t one = _uint32_li( 0x00000001 );
const uint32_t h_s_mask = _uint32_li( 0x00008000 );
const uint32_t h_e_mask = _uint32_li( 0x00007c00 );
const uint32_t h_m_mask = _uint32_li( 0x000003ff );
const uint32_t h_m_hidden = _uint32_li( 0x00000400 );
const uint32_t h_e_pos = _uint32_li( 0x0000000a );
const uint32_t h_e_bias = _uint32_li( 0x0000000f );
const uint32_t h_m_bit_count = _uint32_li( 0x0000000a );
const uint32_t h_m_bit_half_count = _uint32_li( 0x00000005 );
const uint32_t h_nan_min = _uint32_li( 0x00007c01 );
const uint32_t h_e_mask_minus_one = _uint32_li( 0x00007bff );
const uint32_t h_snan = _uint32_li( 0x0000fe00 );
const uint32_t m_round_overflow_bit = _uint32_li( 0x00000020 );
const uint32_t m_hidden_bit = _uint32_li( 0x00100000 );
const uint32_t a_s = _uint32_and( x, h_s_mask );
const uint32_t b_s = _uint32_and( y, h_s_mask );
const uint32_t c_s = _uint32_xor( a_s, b_s );
const uint32_t x_e = _uint32_and( x, h_e_mask );
const uint32_t x_e_eqz_msb = _uint32_dec( x_e );
const uint32_t a = _uint32_sels( x_e_eqz_msb, y, x );
const uint32_t b = _uint32_sels( x_e_eqz_msb, x, y );
const uint32_t a_e = _uint32_and( a, h_e_mask );
const uint32_t b_e = _uint32_and( b, h_e_mask );
const uint32_t a_m = _uint32_and( a, h_m_mask );
const uint32_t b_m = _uint32_and( b, h_m_mask );
const uint32_t a_e_amount = _uint32_srl( a_e, h_e_pos );
const uint32_t b_e_amount = _uint32_srl( b_e, h_e_pos );
const uint32_t a_m_with_hidden = _uint32_or( a_m, h_m_hidden );
const uint32_t b_m_with_hidden = _uint32_or( b_m, h_m_hidden );
const uint32_t c_m_normal = _uint32_mul( a_m_with_hidden, b_m_with_hidden );
const uint32_t c_m_denorm_biased = _uint32_mul( a_m_with_hidden, b_m );
const uint32_t c_e_denorm_unbias_e = _uint32_sub( h_e_bias, a_e_amount );
const uint32_t c_m_denorm_round_amount = _uint32_and( c_m_denorm_biased, h_m_mask );
const uint32_t c_m_denorm_rounded = _uint32_add( c_m_denorm_biased, c_m_denorm_round_amount );
const uint32_t c_m_denorm_inplace = _uint32_srl( c_m_denorm_rounded, h_m_bit_count );
const uint32_t c_m_denorm_unbiased = _uint32_srl( c_m_denorm_inplace, c_e_denorm_unbias_e );
const uint32_t c_m_denorm = _uint32_and( c_m_denorm_unbiased, h_m_mask );
const uint32_t c_e_amount_biased = _uint32_add( a_e_amount, b_e_amount );
const uint32_t c_e_amount_unbiased = _uint32_sub( c_e_amount_biased, h_e_bias );
const uint32_t is_c_e_unbiased_underflow = _uint32_ext( c_e_amount_unbiased );
const uint32_t c_e_underflow_half_sa = _uint32_neg( c_e_amount_unbiased );
const uint32_t c_e_underflow_sa = _uint32_sll( c_e_underflow_half_sa, one );
const uint32_t c_m_underflow = _uint32_srl( c_m_normal, c_e_underflow_sa );
const uint32_t c_e_underflow_added = _uint32_andc( c_e_amount_unbiased, is_c_e_unbiased_underflow );
const uint32_t c_m_underflow_added = _uint32_selb( is_c_e_unbiased_underflow, c_m_underflow, c_m_normal );
const uint32_t is_mul_overflow_test = _uint32_and( c_e_underflow_added, m_round_overflow_bit );
const uint32_t is_mul_overflow_msb = _uint32_neg( is_mul_overflow_test );
const uint32_t c_e_norm_radix_corrected = _uint32_inc( c_e_underflow_added );
const uint32_t c_m_norm_radix_corrected = _uint32_srl( c_m_underflow_added, one );
const uint32_t c_m_norm_hidden_bit = _uint32_and( c_m_norm_radix_corrected, m_hidden_bit );
const uint32_t is_c_m_norm_no_hidden_msb = _uint32_dec( c_m_norm_hidden_bit );
const uint32_t c_m_norm_lo = _uint32_srl( c_m_norm_radix_corrected, h_m_bit_half_count );
const uint32_t c_m_norm_lo_nlz = _uint16_cntlz( c_m_norm_lo );
const uint32_t is_c_m_hidden_nunderflow_msb = _uint32_sub( c_m_norm_lo_nlz, c_e_norm_radix_corrected );
const uint32_t is_c_m_hidden_underflow_msb = _uint32_not( is_c_m_hidden_nunderflow_msb );
const uint32_t is_c_m_hidden_underflow = _uint32_ext( is_c_m_hidden_underflow_msb );
const uint32_t c_m_hidden_underflow_normalized_sa = _uint32_srl( c_m_norm_lo_nlz, one );
const uint32_t c_m_hidden_underflow_normalized = _uint32_sll( c_m_norm_radix_corrected, c_m_hidden_underflow_normalized_sa );
const uint32_t c_m_hidden_normalized = _uint32_sll( c_m_norm_radix_corrected, c_m_norm_lo_nlz );
const uint32_t c_e_hidden_normalized = _uint32_sub( c_e_norm_radix_corrected, c_m_norm_lo_nlz );
const uint32_t c_e_hidden = _uint32_andc( c_e_hidden_normalized, is_c_m_hidden_underflow );
const uint32_t c_m_hidden = _uint32_sels( is_c_m_hidden_underflow_msb, c_m_hidden_underflow_normalized, c_m_hidden_normalized );
const uint32_t c_m_normalized = _uint32_sels( is_c_m_norm_no_hidden_msb, c_m_hidden, c_m_norm_radix_corrected );
const uint32_t c_e_normalized = _uint32_sels( is_c_m_norm_no_hidden_msb, c_e_hidden, c_e_norm_radix_corrected );
const uint32_t c_m_norm_round_amount = _uint32_and( c_m_normalized, h_m_mask );
const uint32_t c_m_norm_rounded = _uint32_add( c_m_normalized, c_m_norm_round_amount );
const uint32_t is_round_overflow_test = _uint32_and( c_e_normalized, m_round_overflow_bit );
const uint32_t is_round_overflow_msb = _uint32_neg( is_round_overflow_test );
const uint32_t c_m_norm_inplace = _uint32_srl( c_m_norm_rounded, h_m_bit_count );
const uint32_t c_m = _uint32_and( c_m_norm_inplace, h_m_mask );
const uint32_t c_e_norm_inplace = _uint32_sll( c_e_normalized, h_e_pos );
const uint32_t c_e = _uint32_and( c_e_norm_inplace, h_e_mask );
const uint32_t c_em_nan = _uint32_or( h_e_mask, a_m );
const uint32_t c_nan = _uint32_or( a_s, c_em_nan );
const uint32_t c_denorm = _uint32_or( c_s, c_m_denorm );
const uint32_t c_inf = _uint32_or( c_s, h_e_mask );
const uint32_t c_em_norm = _uint32_or( c_e, c_m );
const uint32_t is_a_e_flagged_msb = _uint32_sub( h_e_mask_minus_one, a_e );
const uint32_t is_b_e_flagged_msb = _uint32_sub( h_e_mask_minus_one, b_e );
const uint32_t is_a_e_eqz_msb = _uint32_dec( a_e );
const uint32_t is_a_m_eqz_msb = _uint32_dec( a_m );
const uint32_t is_b_e_eqz_msb = _uint32_dec( b_e );
const uint32_t is_b_m_eqz_msb = _uint32_dec( b_m );
const uint32_t is_b_eqz_msb = _uint32_and( is_b_e_eqz_msb, is_b_m_eqz_msb );
const uint32_t is_a_eqz_msb = _uint32_and( is_a_e_eqz_msb, is_a_m_eqz_msb );
const uint32_t is_c_nan_via_a_msb = _uint32_andc( is_a_e_flagged_msb, is_b_e_flagged_msb );
const uint32_t is_c_nan_via_b_msb = _uint32_andc( is_b_e_flagged_msb, is_b_m_eqz_msb );
const uint32_t is_c_nan_msb = _uint32_or( is_c_nan_via_a_msb, is_c_nan_via_b_msb );
const uint32_t is_c_denorm_msb = _uint32_andc( is_b_e_eqz_msb, is_a_e_flagged_msb );
const uint32_t is_a_inf_msb = _uint32_and( is_a_e_flagged_msb, is_a_m_eqz_msb );
const uint32_t is_c_snan_msb = _uint32_and( is_a_inf_msb, is_b_eqz_msb );
const uint32_t is_c_nan_min_via_a_msb = _uint32_and( is_a_e_flagged_msb, is_b_eqz_msb );
const uint32_t is_c_nan_min_via_b_msb = _uint32_and( is_b_e_flagged_msb, is_a_eqz_msb );
const uint32_t is_c_nan_min_msb = _uint32_or( is_c_nan_min_via_a_msb, is_c_nan_min_via_b_msb );
const uint32_t is_c_inf_msb = _uint32_or( is_a_e_flagged_msb, is_b_e_flagged_msb );
const uint32_t is_overflow_msb = _uint32_or( is_round_overflow_msb, is_mul_overflow_msb );
const uint32_t c_em_overflow_result = _uint32_sels( is_overflow_msb, h_e_mask, c_em_norm );
const uint32_t c_common_result = _uint32_or( c_s, c_em_overflow_result );
const uint32_t c_zero_result = _uint32_sels( is_b_eqz_msb, c_s, c_common_result );
const uint32_t c_nan_result = _uint32_sels( is_c_nan_msb, c_nan, c_zero_result );
const uint32_t c_nan_min_result = _uint32_sels( is_c_nan_min_msb, h_nan_min, c_nan_result );
const uint32_t c_inf_result = _uint32_sels( is_c_inf_msb, c_inf, c_nan_min_result );
const uint32_t c_denorm_result = _uint32_sels( is_c_denorm_msb, c_denorm, c_inf_result);
const uint32_t c_result = _uint32_sels( is_c_snan_msb, h_snan, c_denorm_result );
return (uint16_t)(c_result);
}
Raw
float16.h
#ifndef HALF_H
#define FLOAT16_H
#include <stdint.h>
uint32_t half_to_float( uint16_t h );
uint16_t half_from_float( uint32_t f );
uint16_t half_add( uint16_t arg0, uint16_t arg1 );
uint16_t half_mul( uint16_t arg0, uint16_t arg1 );
static inline uint16_t
half_sub( uint16_t ha, uint16_t hb )
{
// (a-b) is the same as (a+(-b))
return half_add( ha, hb ^ 0x8000 );
}
#endif /* FLOAT16_H */
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