Fixed-Point Q16.16 library

Table of Contents

• Datatype limits
• Fixed-point functions
  • Conversion functions
  • Basic arithmetic
  • Exponential and transcendental functions
  • Trigonometric functions and helpers
• Example: Solution of the quadratic equation

qFP16 is a compact fixed-point number library intended for use in embedded systems. It includes a variety of transcendental functions and essential operators, carefully chosen for optimal performance.

The format is a signed Q16.16, which is good enough for most purposes.

Datatype limits

The maximum representable value is 32767.999985. The minimum value is -32768.0

The minimum value is also used to represent fp16.overflow for overflow detection, so for some operations it cannot be determined whether it overflowed or the result was the smallest possible value. In practice, this does not matter much.

The smallest unit (machine precision) of the datatype is 1/65536=0.000015259.

Fixed-point functions

All the provided functions operate on 32-bit numbers, qFP16_t, which have 16-bit integer part and 16-bit fractional part.

Conversion functions

Conversion from integers and floating-point values. These conversions retain the numeric value and perform rounding where necessary.

• qFP16_IntToFP() Simply multiplies a by qFP16.one=65536
• qFP16_FPToInt() Divides by qFP16_t and rounds to nearest integer.
• qFP16_FloatToFP() Multiplies by qFP16_t and rounds to nearest value.
• qFP16_FPToFloat() Divides by qFP16.one. Rounding is according to the current floating-point mode
• qFP16_DoubleToFP() Multiplies by qFP16 and rounds to nearest value.
• qFP16_FPToDouble() Divides by qFP16.one. All qFP16_t values fit into a double, so no rounding happens.
• qFP16_FPToA() Converts from qFP16_t to string.
• qFP16_AToFP() Converts from string to qFP16_t.

Basic arithmetic

These functions perform rounding and detect overflows. When overflow is detected, they return qfp16.overflow as a marker value.

• qFP16_Add() Addition
• qFP16_Sub() Subtraction
• qFP16_Mul() Multiplication
• qFP16_Div() Division
• qFP16_Mod() Modulo

Exponential and transcendental functions

Roots, exponents & similar.

• qFP16_Sqrt() Square root. Performs rounding and is accurate to qFP16 limits.
• qFP16_Exp() Exponential function using power series approximation. Accuracy depends on range, worst case +-40 absolute for negative inputs and +-0.003% for positive inputs. Average error is +-1 for neg and +-0.0003% for pos.
• qFP16_Log() Natural logarithm using Newton approximation and qFP16_Exp(). Worst case error +-3 absolute, average error less than 1 unit.
• qFP16_Log2() Logarithm base 2.
• qFP16_IPow() Computes the integer-power of a qFP16_t number
• qFP16_Pow() Modulo

Trigonometric functions and helpers

• qFP16_Sin() Sine for angle in radians
• qFP16_Cos() Cosine for angle in radians
• qFP16_Tan() Tangent for angle in radians
• qFP16_Asin() Inverse of sine, output in radians
• qFP16_Acos() Inverse of cosine, output in radians
• qFP16_Atan() Inverse of tangent, output in radians
• qFP16_Atan2() Arc tangent in radians x,y
• qFP16_Sinh() Hyperbolic sine
• qFP16_Cosh() Hyperbolic cosine
• qFP16_Tanh() Hyperbolic tangent
• qFP16_RadToDeg() Converts angle units from radians to degrees.
• qFP16_DegToRad() Converts angle units from degrees to radians
• qFP16_WrapToPi() Wrap the fixed-point angle in radians to [−pi pi]
• qFP16_WrapTo180() Wrap the fixed-point angle in degrees to [−180 180]

Example: Solution of the quadratic equation

This draft example computes one solution of the quadratic equation by using the fixed point format. Equation is given by:

\( x = \frac{ -b + \sqrt{ b^{2} - 4ac} }{ 2a } \)

#include <stdio.h>
#include <stdlib.h>
#include "qfp16.h"
 
int main( int argc, char *argv[] ) 
{
    qFP16_t a = qFP16_Constant( 1.5f );
    qFP16_t b = qFP16_Constant( 5.2f );
    qFP16_t c = qFP16_Constant( 4.0f );
    qFP16_t tmp;
    char ans[ 10 ];
    
    tmp = qFP16_Mul( qFP16_IntToFP( 4 ), qFP16_Mul( a, c ) );
    tmp = qFP16_Add( -b, qFP16_Sqrt( qFP16_Sub( qFP16_Mul( b, b ), tmp  ) ) );
    tmp = qFP16_Div( tmp, qFP16_Mul( qFP16_IntToFP( 2 ), a ) );
    printf( " result = %s \r\n" , qFP16_FPToA( tmp, ans, 4 ) );
    return EXIT_SUCCESS;
}