Floating Point Representation¶
The half type is a 16-bit floating number, compatible with the IEEE 754-2008 binary16 type.
Representation of a 32-bit float¶
We assume that a float, f, is an IEEE 754 single-precision
floating point number, whose bits are arranged as follows:
31 (msb)
|
| 30 23
| | |
| | | 22 0 (lsb)
| | | | |
X XXXXXXXX XXXXXXXXXXXXXXXXXXXXXXX
s e m
s is the sign-bit, e is the exponent and m is the significand.
If e is between 1 and 254, f is a normalized number:
f = (-1)s * 2e-127 * 1.m
If e is 0, and m is not zero, f is a denormalized number:
f = (-1)s * 2-126 * 0.m
If e and m are both zero, f is zero:
f = 0.0
If e is 255, f is an “infinity” or “not a number” (NAN),
depending on whether m is zero or not.
Examples:
0 00000000 00000000000000000000000 = 0.0
0 01111110 00000000000000000000000 = 0.5
0 01111111 00000000000000000000000 = 1.0
0 10000000 00000000000000000000000 = 2.0
0 10000000 10000000000000000000000 = 3.0
1 10000101 11110000010000000000000 = -124.0625
0 11111111 00000000000000000000000 = +infinity
1 11111111 00000000000000000000000 = -infinity
0 11111111 10000000000000000000000 = NAN
1 11111111 11111111111111111111111 = NAN
Representation of a 16-bit half¶
Here is the bit-layout for a half number, h:
15 (msb)
|
| 14 10
| | |
| | | 9 0 (lsb)
| | | | |
X XXXXX XXXXXXXXXX
s e m
s is the sign-bit, e is the exponent and m is the significand.
If e is between 1 and 30, h is a normalized number:
h = (-1)s * 2e-15 * 1.m
If e is 0, and m is not zero, h is a denormalized number:
h = (-1) s * 2-14 * 0.m
If e and m are both zero, h is zero:
h = 0.0
If e is 31, h is an “infinity” or “not a number” (NAN),
depending on whether m is zero or not.
Examples:
0 00000 0000000000 = 0.0
0 01110 0000000000 = 0.5
0 01111 0000000000 = 1.0
0 10000 0000000000 = 2.0
0 10000 1000000000 = 3.0
1 10101 1111000001 = -124.0625
0 11111 0000000000 = +infinity
1 11111 0000000000 = -infinity
0 11111 1000000000 = NAN
1 11111 1111111111 = NAN
Conversion via Lookup Table¶
Converting from half to float is performed by default using a
lookup table. There are only 65,536 different half numbers; each
of these numbers has been converted and stored in a table pointed
to by the imath_half_to_float_table pointer.
Prior to Imath v3.1, conversion from float to half was accomplished with the help of an exponent look table, but this is now replaced with explicit bit shifting.
Conversion via Hardware¶
For Imath v3.1, the conversion routines have been extended to use F16C SSE instructions whenever present and enabled by compiler flags.
Conversion via Bit-Shifting¶
If F16C SSE instructions are not available, conversion can be accomplished by a bit-shifting algorithm. For half-to-float conversion, this is generally slower than the lookup table, but it may be preferable when memory limits preclude storing of the 65,536-entry lookup table.
The lookup table symbol is included in the compilation even if
IMATH_HALF_USE_LOOKUP_TABLE is false, because application code
using the exported half.h may choose to enable the use of the table.
An implementation can eliminate the table from compilation by
defining the IMATH_HALF_NO_LOOKUP_TABLE preprocessor symbol.
Simply add:
#define IMATH_HALF_NO_LOOKUP_TABLE
before including half.h, or define the symbol on the compile
command line.
Furthermore, an implementation wishing to receive FE_OVERFLOW
and FE_UNDERFLOW floating point exceptions when converting
float to half by the bit-shift algorithm can define the
preprocessor symbol IMATH_HALF_ENABLE_FP_EXCEPTIONS prior to
including half.h:
#define IMATH_HALF_ENABLE_FP_EXCEPTIONS
Conversion Performance Comparison¶
Testing on a Core i9, the timings are approximately:
half to float:
table: 0.71 ns / call
no table: 1.06 ns / call
f16c: 0.45 ns / call
float-to-half:
original: 5.2 ns / call
no exp table + opt: 1.27 ns / call
f16c: 0.45 ns / call
Note: the timing above depends on the distribution of the floats in question.
