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	6917	6298	61652	189845	85509	113827	360010	300977
fixed32_a	taylormath	6956	6922	15729	201253	303803	254862	170733	861728	397566
fixed64	taylormath	13247	13211	20761	206308	835902	399448	396151	2217824	1437792
float	cmath	101477	105844	164146	254705	2025247	344892	535360	1527857	2016053
double	cmath	118830	121306	289300	495248	3026929	614699	843223	2339121	2996070
float	taylormath	101490	105844	163517	255330	2686019	1839398	1434164	8438956	4190184
double	taylormath	118830	121313	288667	494616	5820614	8243824	7006465	34110313	10240876

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.