What's the best way to do fixed-point math? [closed]
Solution 1:
You can try my fixed point class (Latest available @ https://github.com/eteran/cpp-utilities)
// From: https://github.com/eteran/cpp-utilities/edit/master/Fixed.h
// See also: http://stackoverflow.com/questions/79677/whats-the-best-way-to-do-fixed-point-math
/*
* The MIT License (MIT)
*
* Copyright (c) 2015 Evan Teran
*
* 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.
*/
#ifndef FIXED_H_
#define FIXED_H_
#include <ostream>
#include <exception>
#include <cstddef> // for size_t
#include <cstdint>
#include <type_traits>
#include <boost/operators.hpp>
namespace numeric {
template <size_t I, size_t F>
class Fixed;
namespace detail {
// helper templates to make magic with types :)
// these allow us to determine resonable types from
// a desired size, they also let us infer the next largest type
// from a type which is nice for the division op
template <size_t T>
struct type_from_size {
static const bool is_specialized = false;
typedef void value_type;
};
#if defined(__GNUC__) && defined(__x86_64__)
template <>
struct type_from_size<128> {
static const bool is_specialized = true;
static const size_t size = 128;
typedef __int128 value_type;
typedef unsigned __int128 unsigned_type;
typedef __int128 signed_type;
typedef type_from_size<256> next_size;
};
#endif
template <>
struct type_from_size<64> {
static const bool is_specialized = true;
static const size_t size = 64;
typedef int64_t value_type;
typedef uint64_t unsigned_type;
typedef int64_t signed_type;
typedef type_from_size<128> next_size;
};
template <>
struct type_from_size<32> {
static const bool is_specialized = true;
static const size_t size = 32;
typedef int32_t value_type;
typedef uint32_t unsigned_type;
typedef int32_t signed_type;
typedef type_from_size<64> next_size;
};
template <>
struct type_from_size<16> {
static const bool is_specialized = true;
static const size_t size = 16;
typedef int16_t value_type;
typedef uint16_t unsigned_type;
typedef int16_t signed_type;
typedef type_from_size<32> next_size;
};
template <>
struct type_from_size<8> {
static const bool is_specialized = true;
static const size_t size = 8;
typedef int8_t value_type;
typedef uint8_t unsigned_type;
typedef int8_t signed_type;
typedef type_from_size<16> next_size;
};
// this is to assist in adding support for non-native base
// types (for adding big-int support), this should be fine
// unless your bit-int class doesn't nicely support casting
template <class B, class N>
B next_to_base(const N& rhs) {
return static_cast<B>(rhs);
}
struct divide_by_zero : std::exception {
};
template <size_t I, size_t F>
Fixed<I,F> divide(const Fixed<I,F> &numerator, const Fixed<I,F> &denominator, Fixed<I,F> &remainder, typename std::enable_if<type_from_size<I+F>::next_size::is_specialized>::type* = 0) {
typedef typename Fixed<I,F>::next_type next_type;
typedef typename Fixed<I,F>::base_type base_type;
static const size_t fractional_bits = Fixed<I,F>::fractional_bits;
next_type t(numerator.to_raw());
t <<= fractional_bits;
Fixed<I,F> quotient;
quotient = Fixed<I,F>::from_base(next_to_base<base_type>(t / denominator.to_raw()));
remainder = Fixed<I,F>::from_base(next_to_base<base_type>(t % denominator.to_raw()));
return quotient;
}
template <size_t I, size_t F>
Fixed<I,F> divide(Fixed<I,F> numerator, Fixed<I,F> denominator, Fixed<I,F> &remainder, typename std::enable_if<!type_from_size<I+F>::next_size::is_specialized>::type* = 0) {
// NOTE(eteran): division is broken for large types :-(
// especially when dealing with negative quantities
typedef typename Fixed<I,F>::base_type base_type;
typedef typename Fixed<I,F>::unsigned_type unsigned_type;
static const int bits = Fixed<I,F>::total_bits;
if(denominator == 0) {
throw divide_by_zero();
} else {
int sign = 0;
Fixed<I,F> quotient;
if(numerator < 0) {
sign ^= 1;
numerator = -numerator;
}
if(denominator < 0) {
sign ^= 1;
denominator = -denominator;
}
base_type n = numerator.to_raw();
base_type d = denominator.to_raw();
base_type x = 1;
base_type answer = 0;
// egyptian division algorithm
while((n >= d) && (((d >> (bits - 1)) & 1) == 0)) {
x <<= 1;
d <<= 1;
}
while(x != 0) {
if(n >= d) {
n -= d;
answer += x;
}
x >>= 1;
d >>= 1;
}
unsigned_type l1 = n;
unsigned_type l2 = denominator.to_raw();
// calculate the lower bits (needs to be unsigned)
// unfortunately for many fractions this overflows the type still :-/
const unsigned_type lo = (static_cast<unsigned_type>(n) << F) / denominator.to_raw();
quotient = Fixed<I,F>::from_base((answer << F) | lo);
remainder = n;
if(sign) {
quotient = -quotient;
}
return quotient;
}
}
// this is the usual implementation of multiplication
template <size_t I, size_t F>
void multiply(const Fixed<I,F> &lhs, const Fixed<I,F> &rhs, Fixed<I,F> &result, typename std::enable_if<type_from_size<I+F>::next_size::is_specialized>::type* = 0) {
typedef typename Fixed<I,F>::next_type next_type;
typedef typename Fixed<I,F>::base_type base_type;
static const size_t fractional_bits = Fixed<I,F>::fractional_bits;
next_type t(static_cast<next_type>(lhs.to_raw()) * static_cast<next_type>(rhs.to_raw()));
t >>= fractional_bits;
result = Fixed<I,F>::from_base(next_to_base<base_type>(t));
}
// this is the fall back version we use when we don't have a next size
// it is slightly slower, but is more robust since it doesn't
// require and upgraded type
template <size_t I, size_t F>
void multiply(const Fixed<I,F> &lhs, const Fixed<I,F> &rhs, Fixed<I,F> &result, typename std::enable_if<!type_from_size<I+F>::next_size::is_specialized>::type* = 0) {
typedef typename Fixed<I,F>::base_type base_type;
static const size_t fractional_bits = Fixed<I,F>::fractional_bits;
static const size_t integer_mask = Fixed<I,F>::integer_mask;
static const size_t fractional_mask = Fixed<I,F>::fractional_mask;
// more costly but doesn't need a larger type
const base_type a_hi = (lhs.to_raw() & integer_mask) >> fractional_bits;
const base_type b_hi = (rhs.to_raw() & integer_mask) >> fractional_bits;
const base_type a_lo = (lhs.to_raw() & fractional_mask);
const base_type b_lo = (rhs.to_raw() & fractional_mask);
const base_type x1 = a_hi * b_hi;
const base_type x2 = a_hi * b_lo;
const base_type x3 = a_lo * b_hi;
const base_type x4 = a_lo * b_lo;
result = Fixed<I,F>::from_base((x1 << fractional_bits) + (x3 + x2) + (x4 >> fractional_bits));
}
}
/*
* inheriting from boost::operators enables us to be a drop in replacement for base types
* without having to specify all the different versions of operators manually
*/
template <size_t I, size_t F>
class Fixed : boost::operators<Fixed<I,F>> {
static_assert(detail::type_from_size<I + F>::is_specialized, "invalid combination of sizes");
public:
static const size_t fractional_bits = F;
static const size_t integer_bits = I;
static const size_t total_bits = I + F;
typedef detail::type_from_size<total_bits> base_type_info;
typedef typename base_type_info::value_type base_type;
typedef typename base_type_info::next_size::value_type next_type;
typedef typename base_type_info::unsigned_type unsigned_type;
public:
static const size_t base_size = base_type_info::size;
static const base_type fractional_mask = ~((~base_type(0)) << fractional_bits);
static const base_type integer_mask = ~fractional_mask;
public:
static const base_type one = base_type(1) << fractional_bits;
public: // constructors
Fixed() : data_(0) {
}
Fixed(long n) : data_(base_type(n) << fractional_bits) {
// TODO(eteran): assert in range!
}
Fixed(unsigned long n) : data_(base_type(n) << fractional_bits) {
// TODO(eteran): assert in range!
}
Fixed(int n) : data_(base_type(n) << fractional_bits) {
// TODO(eteran): assert in range!
}
Fixed(unsigned int n) : data_(base_type(n) << fractional_bits) {
// TODO(eteran): assert in range!
}
Fixed(float n) : data_(static_cast<base_type>(n * one)) {
// TODO(eteran): assert in range!
}
Fixed(double n) : data_(static_cast<base_type>(n * one)) {
// TODO(eteran): assert in range!
}
Fixed(const Fixed &o) : data_(o.data_) {
}
Fixed& operator=(const Fixed &o) {
data_ = o.data_;
return *this;
}
private:
// this makes it simpler to create a fixed point object from
// a native type without scaling
// use "Fixed::from_base" in order to perform this.
struct NoScale {};
Fixed(base_type n, const NoScale &) : data_(n) {
}
public:
static Fixed from_base(base_type n) {
return Fixed(n, NoScale());
}
public: // comparison operators
bool operator==(const Fixed &o) const {
return data_ == o.data_;
}
bool operator<(const Fixed &o) const {
return data_ < o.data_;
}
public: // unary operators
bool operator!() const {
return !data_;
}
Fixed operator~() const {
Fixed t(*this);
t.data_ = ~t.data_;
return t;
}
Fixed operator-() const {
Fixed t(*this);
t.data_ = -t.data_;
return t;
}
Fixed operator+() const {
return *this;
}
Fixed& operator++() {
data_ += one;
return *this;
}
Fixed& operator--() {
data_ -= one;
return *this;
}
public: // basic math operators
Fixed& operator+=(const Fixed &n) {
data_ += n.data_;
return *this;
}
Fixed& operator-=(const Fixed &n) {
data_ -= n.data_;
return *this;
}
Fixed& operator&=(const Fixed &n) {
data_ &= n.data_;
return *this;
}
Fixed& operator|=(const Fixed &n) {
data_ |= n.data_;
return *this;
}
Fixed& operator^=(const Fixed &n) {
data_ ^= n.data_;
return *this;
}
Fixed& operator*=(const Fixed &n) {
detail::multiply(*this, n, *this);
return *this;
}
Fixed& operator/=(const Fixed &n) {
Fixed temp;
*this = detail::divide(*this, n, temp);
return *this;
}
Fixed& operator>>=(const Fixed &n) {
data_ >>= n.to_int();
return *this;
}
Fixed& operator<<=(const Fixed &n) {
data_ <<= n.to_int();
return *this;
}
public: // conversion to basic types
int to_int() const {
return (data_ & integer_mask) >> fractional_bits;
}
unsigned int to_uint() const {
return (data_ & integer_mask) >> fractional_bits;
}
float to_float() const {
return static_cast<float>(data_) / Fixed::one;
}
double to_double() const {
return static_cast<double>(data_) / Fixed::one;
}
base_type to_raw() const {
return data_;
}
public:
void swap(Fixed &rhs) {
using std::swap;
swap(data_, rhs.data_);
}
public:
base_type data_;
};
// if we have the same fractional portion, but differing integer portions, we trivially upgrade the smaller type
template <size_t I1, size_t I2, size_t F>
typename std::conditional<I1 >= I2, Fixed<I1,F>, Fixed<I2,F>>::type operator+(const Fixed<I1,F> &lhs, const Fixed<I2,F> &rhs) {
typedef typename std::conditional<
I1 >= I2,
Fixed<I1,F>,
Fixed<I2,F>
>::type T;
const T l = T::from_base(lhs.to_raw());
const T r = T::from_base(rhs.to_raw());
return l + r;
}
template <size_t I1, size_t I2, size_t F>
typename std::conditional<I1 >= I2, Fixed<I1,F>, Fixed<I2,F>>::type operator-(const Fixed<I1,F> &lhs, const Fixed<I2,F> &rhs) {
typedef typename std::conditional<
I1 >= I2,
Fixed<I1,F>,
Fixed<I2,F>
>::type T;
const T l = T::from_base(lhs.to_raw());
const T r = T::from_base(rhs.to_raw());
return l - r;
}
template <size_t I1, size_t I2, size_t F>
typename std::conditional<I1 >= I2, Fixed<I1,F>, Fixed<I2,F>>::type operator*(const Fixed<I1,F> &lhs, const Fixed<I2,F> &rhs) {
typedef typename std::conditional<
I1 >= I2,
Fixed<I1,F>,
Fixed<I2,F>
>::type T;
const T l = T::from_base(lhs.to_raw());
const T r = T::from_base(rhs.to_raw());
return l * r;
}
template <size_t I1, size_t I2, size_t F>
typename std::conditional<I1 >= I2, Fixed<I1,F>, Fixed<I2,F>>::type operator/(const Fixed<I1,F> &lhs, const Fixed<I2,F> &rhs) {
typedef typename std::conditional<
I1 >= I2,
Fixed<I1,F>,
Fixed<I2,F>
>::type T;
const T l = T::from_base(lhs.to_raw());
const T r = T::from_base(rhs.to_raw());
return l / r;
}
template <size_t I, size_t F>
std::ostream &operator<<(std::ostream &os, const Fixed<I,F> &f) {
os << f.to_double();
return os;
}
template <size_t I, size_t F>
const size_t Fixed<I,F>::fractional_bits;
template <size_t I, size_t F>
const size_t Fixed<I,F>::integer_bits;
template <size_t I, size_t F>
const size_t Fixed<I,F>::total_bits;
}
#endif
It is designed to be a near drop in replacement for floats/doubles and has a choose-able precision. It does make use of boost to add all the necessary math operator overloads, so you will need that as well (I believe for this it is just a header dependency, not a library dependency).
BTW, common usage could be something like this:
using namespace numeric;
typedef Fixed<16, 16> fixed;
fixed f;
The only real rule is that the number have to add up to a native size of your system such as 8, 16, 32, 64.
Solution 2:
In modern C++ implementations, there will be no performance penalty for using simple and lean abstractions, such as concrete classes. Fixed-point computation is precisely the place where using a properly engineered class will save you from lots of bugs.
Therefore, you should write a FixedPoint8 class. Test and debug it thoroughly. If you have to convince yourself of its performance as compared to using plain integers, measure it.
It will save you from many a trouble by moving the complexity of fixed-point calculation to a single place.
If you like, you can further increase the utility of your class by making it a template and replacing the old FixedPoint8 with, say, typedef FixedPoint<short, 8> FixedPoint8; But on your target architecture this is not probably necessary, so avoid the complexity of templates at first.
There is probably a good fixed point class somewhere in the internet - I'd start looking from the Boost libraries.
Solution 3:
Does your floating point code actually make use of the decimal point? If so:
First you have to read Randy Yates's paper on Intro to Fixed Point Math: http://www.digitalsignallabs.com/fp.pdf
Then you need to do "profiling" on your floating point code to figure out the appropriate range of fixed-point values required at "critical" points in your code, e.g. U(5,3) = 5 bits to the left, 3 bits to the right, unsigned.
At this point, you can apply the arithmetic rules in the paper mentioned above; the rules specify how to interpret the bits which result from arithmetic operations. You can write macros or functions to perform the operations.
It's handy to keep the floating point version around, in order to compare the floating point vs fixed point results.
Solution 4:
I wouldn't use floating point at all on a CPU without special hardware for handling it. My advice is to treat ALL numbers as integers scaled to a specific factor. For example, all monetary values are in cents as integers rather than dollars as floats. For example, 0.72 is represented as the integer 72.
Addition and subtraction are then a very simple integer operation such as (0.72 + 1 becomes 72 + 100 becomes 172 becomes 1.72).
Multiplication is slightly more complex as it needs an integer multiply followed by a scale back such as (0.72 * 2 becomes 72 * 200 becomes 14400 becomes 144 (scaleback) becomes 1.44).
That may require special functions for performing more complex math (sine, cosine, etc) but even those can be sped up by using lookup tables. Example: since you're using fixed-2 representation, there's only 100 values in the range (0.0,1] (0-99) and sin/cos repeat outside this range so you only need a 100-integer lookup table.
Cheers, Pax.