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Fuchsia Fixed-Point Library (FFL)

Introduction

FFL is a C++ template library for fixed-point arithmetic. The library is primarily intended to support the Zircon kernel scheduler however, it is sufficiently general to be useful wherever fixed-point computations are needed.

FFL is motivated by the following requirements:

• Availability: Fixed-point support is not yet ratified in the standard library. We need a solution today.
• Dependency: Many alternatives require additional dependencies. We prefer to depend only on the standard library.
• Rounding: Many alternatives, including the proposal for the standard library, have poor or ill-defined rounding behavior. We need well-defined rounding with reasonable general-purpose stability, such as convergent rounding.

General Usage

The main user-facing type in FFL is the value template type ffl::Fixed<typename Integer, size_t FractionalBits>. This template accepts an integer type for the underlying value and the number of bit to use to represent the fractional component. Naturally, the range of the integer component of the fixed-point value is defined by the difference between the number of bits of the underlying integer type and the number of bits reserved for the fractional part.

ffl::Fixed behaves similarly to plain integers. The type supports most of the same arithmetic operators: addition, subtraction, negation, multiplication, and division, as well as all of the comparison operators.

#include <ffl/fixed.h>

using ffl::Fixed;

Fixed<int32_t, 31> UnitaryRatio(Fixed<int32_t, 0> a, Fixed<int32_t, 0> b) {
if (a > b)
return b / a;
else
return a / b;
}

Fixed<uint8_t, 0> Blend(Fixed<uint8_t, 0> color0, Fixed<uint8_t, 0> color1, Fixed<uint8_t, 7> alpha) {
return alpha * color0 + (Fixed<uint8_t, 7>{1} - alpha) * color1;
}

Expressions and Managing Precision

FFL supports arithmetic and comparisons with mixed precisions. These operations may consider both the source and destination precisions at compile time to select an appropriate strategy for intermediate computations.

In order to consider the destination precision, the evaluation of an expression involving ffl::Fixed values is deferred until the expression is assigned to a ffl::Fixed variable. To facilitate this deferred evaluation, the arithmetic operators return instances of the template type ffl::Expression, which captures the arithmetic operation and the arguments involved. The arguments may be instances of ffl::Fixed, plain integers, or other instances of ffl::Expression returned by other operators and utility functions. With this approach, compound expressions result in expression trees that follow the C++ order of operations.

In many cases the use of expression trees is transparent to the user, as in the following example:

#include <ffl/fixed.h>

using ffl::Fixed;

struct Point2d {
Fixed<int16_t, 0> x;
Fixed<int16_t, 0> y;
};

Point2d LinearInterpolate(Point2d p0, Point2d p1, Fixed<int32_t, 16> t) {
return {p0.x + t * (p1.x - p0.x), p0.y + t * (p1.y - p0.y)};
}

In this example the arithmetic expressions and assignments happen together and there is no need to consider the intermediate Expression objects.

In some cases it is useful to be aware of the intermediate Expression objects. The following example uses intermediate expressions to make the overall computation more readable:

#include <ffl/fixed.h>

using ffl::Fixed;

// TODO

Mixed Precision

FFL uses the following rules when performing mixed precision arithmetic:

• Addition and subtraction convert to the least precision between the two operands before computing an intermediate result.
• Multiplication produces an intermediate result with precision resolution sufficient to hold the sum of both the fractional and integral bit depths of the operands.
• Division produces an intermediate result with the resolution of the target format.

Comparisons convert to the least precision between the two operands before performing the comparison. Care must be taken when comparing a fixed-point value with an plain integer: plain integers are promoted to a fixed-point value with a zero bit fractional component. This means that comparisons with plain integers round the fixed-point value to a whole integer value before comparison. Sometimes this is the desired outcome however, it can lead to unexpected results if you are unaware of this behavior.

Consider the following example that presents two functions to determine whether given fixed-point value is zero:

#include <ffl/fixed.h>

using ffl::Fixed;
using ffl::FromRatio;

template <typename Integer, size_t FractionalBits>
constexpr bool IsZero1(Fixed<Intefer, FractionalBits> value) {
return value == 0;
}

template <typename Integer, size_t FractionalBits>
constexpr bool IsZero2(Fixed<Intefer, FractionalBits> value) {
return value == Fixed<Integer, FractionalBits>{0};
}

constexpr Fixed<int, 1> kOneHalf = FromRatio(1, 2);

// Round-half-to-even rounds one-half down to zero.
static_assert(IsZero1(kOneHalf) == true);

// Fixed-to-fixed comparison of one-half is not equal to zero.
static_assert(IsZero2(kOneHalf) == false);

In this example IsZero1 compares the fractional value kOneHalf with plain integer zero. This results in kOneHalf being rounded to a value with zero fractional bits, due to the promotion of the literal 0 to a fixed-point value. The convergent rounding policy rounds kOneHalf towards zero because it is the nearest even integer.

In contrast, IsZero2 explicitly converts the literal 0 to a fixed-point value with the same precision as the argument. Because both arguments of the comparison have the same precision, the values are directly compared without rounding.

Both types of comparison are valid. Which one to use depends on the what the situation requires. Keep in mind that comparisons always convert to the least precision when comparing with plain integers and the right choice will be clear.

Intermediate Values and Saturation

Saturation is one important difference between fixed-point arithmetic in FFL and regular integer arithmetic. Regular integer arithmetic over or underflows when the result exceeds the range of the integral type. In contrast, FFL uses intermediate values with sufficient range for the computation. When an intermediate value is finally assigned to a fixed-point variable the value is clamped to precision of the destination type.

Coercing Resolution

Some arithmetic operations take the target resolution into account when computing intermediate values (only division at the time of this writing). The target resolution may be influenced using the ToResolution<FractionalBits>() utility function. This utility functions by inserting a resolution node into the expression tree at the point of invocation; deeper nodes that consider target resolution will consider the resolution given by this node instead of the final resolution.

#include <ffl/fixed.h>

using ffl::Fixed;
using ffl::Round;
using ffl::ToResolution;

constexpr int32_t Divide(int32_t numerator, int32_t denominator) {
const Fixed<int32_t, 0> fixed_numerator{numerator};
const Fixed<int32_t, 0> fixed_denominator{denominator};

// Perform division with 2bit fractional resolution for optimum convergent
// rounding of the quotient.
return Round<int32_t>(ToResolution<2>(fixed_numerator / fixed_denominator));
}