Library for fixed point operations with math module based on (mostly) taylor series.

Template dedicated for RISC-V 32 IM.

Usage

Files

All library files can be found in src/lib/. Just copy the files and include them.
Additionally taylormath need to be linked with its cpp file.
The library works with C++:

• std-c++23
• std-c++20
• std-c++17
• std-c++11 if you are not going to use constexpr functionalities (they works but not everywhere)

Usage in code

The template has the following prototype:

template<typename T, typename TC=typename make_fast_int<T>::type, unsigned frac_bits=sizeof(T)*4-1> class fixedpoint;

• T - integer type for calculations.
• TC - integer type for multiplication and division. By default it is fast type of the T (on 64-bits systems it is usually int64 / uint64)
• frac_bits - number - where to put the fraction point.

There are predefinied types:

• Standard types (the fastest TC with at least T size)

  • signed

    • fixed8 - signed 8 bit T, fast 8 bit type for TC, 3 fraction bits
    • fixed16 - signed 16 bit T, fast 16 bit type for TC, 7 fraction bits
    • fixed32 - signed 32 bit T, fast 32 bit type for TC, 15 fraction bits
    • fixed64 - signed 64 bit T, fast 64 bit type for TC, 31 fraction bits

  • unsigned

    • ufixed8 - unsigned 8 bit T, fast 8 bit type for TC, 3 fraction bits
    • ufixed16 - unsigned 16 bit T, fast 16 bit type for TC, 7 fraction bits
    • ufixed32 - unsigned 32 bit T, fast 32 bit type for TC, 15 fraction bits
    • ufixed64 - unsigned 64 bit T, fast 64 bit type for TC, 31 fraction bits

• Accurate types (TC two times larger than T - except [u]int64)

  • signed

    • fixed8_a - signed 8 bit T, fast 16 bit type for TC, 3 fraction bits
    • fixed16_a - signed 16 bit T, fast 32 bit type for TC, 7 fraction bits
    • fixed32_a - signed 32 bit T, fast 64 bit type for TC, 15 fraction bits
    • fixed64_a - signed 64 bit T, fast 64 bit type for TC, 31 fraction bits

  • unsigned

    • ufixed8_a - unsigned 8 bit T, fast 16 bit type for TC, 3 fraction bits
    • ufixed16_a - unsigned 16 bit T, fast 32 bit type for TC, 7 fraction bits
    • ufixed32_a - unsigned 32 bit T, fast 64 bit type for TC, 15 fraction bits
    • ufixed64_a - unsigned 64 bit T, fast 64 bit type for TC, 31 fraction bits

• Simple types (same size of T and TC)

  • signed

    • fixed8_s - signed 8 bit T, 8 bit type for TC, 3 fraction bits
    • fixed16_s - signed 16 bit T, 16 bit type for TC, 7 fraction bits
    • fixed32_s - signed 32 bit T, 32 bit type for TC, 15 fraction bits
    • fixed64_s - signed 64 bit T, 64 bit type for TC, 31 fraction bits

  • unsigned

    • ufixed8_s - unsigned 8 bit T, 8 bit type for TC, 3 fraction bits
    • ufixed16_s - unsigned 16 bit T, 16 bit type for TC, 7 fraction bits
    • ufixed32_s - unsigned 32 bit T, 32 bit type for TC, 15 fraction bits
    • ufixed64_s - unsigned 64 bit T, 64 bit type for TC, 31 fraction bits

Problem with operators

Due to possible inconsistency of result type on calculations of various fixedpoints both operands should be the same type. But it is not something that you must remember because the compiler will remind you.
The solution is to use casting, f.e.:

fixed32 x = 7.35;
fixed64 y = 21.37;
std::cout << x + (fixed32)y << std::endl;
std::cout << (fixed64)x + y << std::endl;

Conversions from IEEE754

All conversions from floats works by multiplying float / double variable by some constant and then casting it to an integer type. It could be faster by using one of the following functions:

• from_ieee754
• from_float
• from_double

all of them might be faster or slower - it depends on the target platform. Results of these functions could be also incorrect so always check results on the new target. Although some assertions were made to avoid errors and thus in case of detection of an unsupported float/double format, these functions will use "standard" multiplication.

Test results

Speed comparision (microseconds per 100000 operations on ESP32C3@160MHz)

STARTING TESTS

type	library	addition	subtraction	multiplication	division	sin	sqrt	asin	log	exp
fixed32_s	taylormath	6982	6916	6925	53469	190834	78826	112980	342369	303379
fixed32_a	taylormath	6956	6923	15722	201881	305426	261547	171984	878552	396626
fixed64	taylormath	13249	13211	20760	205678	836790	384513	392332	2186356	1434525
float	cmath	101480	105843	164144	254701	2025215	344885	535346	1527835	2016006
double	cmath	118830	121311	289294	495233	3026886	614698	843206	2339086	2996029
float	taylormath	101485	105844	163513	255336	2685984	1839360	1433765	8430123	4188721
double	taylormath	118835	121303	288670	494603	5820529	8243711	7005986	34129127	10240732

Taylormath calculates until increasing accuracy is not possible, so the more accurate type, the more time is needed to calculate a function result.

Taylormath accuracy

Differences with double and cmath as reference.