OpenVDB 11.0.0

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Half.h File Reference
`#include <openvdb/Platform.h>`
`#include <openvdb/version.h>`
`#include <iostream>`
`#include <stdint.h>`
`#include <stdio.h>`
`#include <limits>`

Go to the source code of this file.

## Classes
union | imath_half_uif
| a type for both C-only programs and C++ to use the same utilities More...
class | half
class | numeric_limits< openvdb::math::internal::half >

## Namespaces
namespace | openvdb
namespace | openvdb::v11_0
namespace | openvdb::v11_0::math
namespace | openvdb::v11_0::math::internal
namespace | std

## Macros
#define | IMATH_HALF_NO_LOOKUP_TABLE
#define | VDBB_HALF_DENORM_MIN 5.96046448e-08
| Smallest positive denormalized half.
#define | VDB_HALF_NRM_MIN 6.10351562e-05
| Smallest positive normalized half.
#define | VDB_HALF_MIN 6.10351562e-05f
| Smallest positive normalized half.
#define | VDB_HALF_MAX 65504.0
| Largest positive half.
#define | VDB_HALF_EPSILON 0.00097656
| Smallest positive e for which `half(1.0 + e) != half(1.0)`
#define | VDB_HALF_MANT_DIG 11
| Number of digits in mantissa (significand + hidden leading 1)
#define | VDB_HALF_DIG 3
#define | VDB_HALF_DECIMAL_DIG 5
#define | VDB_HALF_RADIX 2
| Base of the exponent.
#define | VDBB_HALF_DENORM_MIN_EXP -13
#define | VDB_HALF_MAX_EXP 16
#define | VDBB_HALF_DENORM_MIN_10_EXP -4
#define | VDB_HALF_MAX_10_EXP 4

## Typedefs
typedef union openvdb::v11_0::math::internal::imath_half_uif | imath_half_uif_t
| a type for both C-only programs and C++ to use the same utilities
typedef uint16_t | imath_half_bits_t
| a type for both C-only programs and C++ to use the same utilities
typedef imath_half_bits_t | half
| if we're in a C-only context, alias the half bits type to half

## Functions
static float | imath_half_to_float (imath_half_bits_t h)
static imath_half_bits_t | imath_float_to_half (float f)
OPENVDB_API void | printBits (std::ostream &os, half h)
OPENVDB_API void | printBits (std::ostream &os, float f)
OPENVDB_API void | printBits (char c[19], half h)
OPENVDB_API void | printBits (char c[35], float f)
OPENVDB_API std::ostream & | operator<< (std::ostream &os, half h)
| Output h to os, formatted as a float.
OPENVDB_API std::istream & | operator>> (std::istream &is, half &h)
| Input h from is.

## Detailed Description

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:

            s    e-127
    f = (-1)  * 2      * 1.m

If e is 0, and m is not zero, f is a denormalized number:

            s    -126
    f = (-1)  * 2      * 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:

            s    e-15
    h = (-1)  * 2     * 1.m

If e is 0, and m is not zero, h is a denormalized number:

            S    -14
    h = (-1)  * 2     * 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.

## Macro Definition Documentation

## ◆ IMATH_HALF_NO_LOOKUP_TABLE

#define IMATH_HALF_NO_LOOKUP_TABLE

## ◆ VDB_HALF_DECIMAL_DIG

#define VDB_HALF_DECIMAL_DIG 5

Number of base-10 digits that are necessary to uniquely represent all distinct values:

`ceil(HALF_MANT_DIG * log10(2) + 1) => 4.31... -> 5`

## ◆ VDB_HALF_DIG

#define VDB_HALF_DIG 3

Number of base 10 digits that can be represented without change:

`floor( (HALF_MANT_DIG - 1) * log10(2) ) => 3.01... -> 3`

## ◆ VDB_HALF_EPSILON

#define VDB_HALF_EPSILON 0.00097656

Smallest positive e for which `half(1.0 + e) != half(1.0)`

## ◆ VDB_HALF_MANT_DIG

#define VDB_HALF_MANT_DIG 11

Number of digits in mantissa (significand + hidden leading 1)

## ◆ VDB_HALF_MAX

#define VDB_HALF_MAX 65504.0

Largest positive half.

## ◆ VDB_HALF_MAX_10_EXP

#define VDB_HALF_MAX_10_EXP 4

Maximum positive integer such that 10 raised to that power is a normalized half

## ◆ VDB_HALF_MAX_EXP

#define VDB_HALF_MAX_EXP 16

Maximum positive integer such that `HALF_RADIX` raised to the power of one less than that integer is a normalized half

## ◆ VDB_HALF_MIN

#define VDB_HALF_MIN 6.10351562e-05f

Smallest positive normalized half.

## ◆ VDB_HALF_NRM_MIN

#define VDB_HALF_NRM_MIN 6.10351562e-05

Smallest positive normalized half.

## ◆ VDB_HALF_RADIX

#define VDB_HALF_RADIX 2

Base of the exponent.

## ◆ VDBB_HALF_DENORM_MIN

#define VDBB_HALF_DENORM_MIN 5.96046448e-08

Smallest positive denormalized half.

## ◆ VDBB_HALF_DENORM_MIN_10_EXP

#define VDBB_HALF_DENORM_MIN_10_EXP -4

Minimum positive integer such that 10 raised to that power is a normalized half

## ◆ VDBB_HALF_DENORM_MIN_EXP

#define VDBB_HALF_DENORM_MIN_EXP -13

Minimum negative integer such that `HALF_RADIX` raised to the power of one less than that integer is a normalized half