QMATH(3)

NAME
qmath - fixed-point math library based on the “Q” number format

SYNOPSIS
#include <sys/qmath.h>

DESCRIPTION
The qmath data types and APIs support fixed-point math based on the “Q” number format. The APIs have been built around the following data types: s8q_t, u8q_t, s16q_t, u16q_t, s32q_t, u32q_t, s64q_t, and u64q_t, which are referred to generically in the earlier API definitions as QTYPE. The ITYPE refers to the stdint(7) integer types. NTYPE is used to refer to any numeric type and is therefore a superset of QTYPE and ITYPE.

This scheme can represent Q numbers with [2, 4, 6, 8, 16, 32, 48] bits of precision after the binary radix point, depending on the rpshft argument to Q_INI(). The number of bits available for the integral component is not explicitly specified, and implicitly consumes the remaining available bits of the chosen Q data type.

Operations on Q numbers maintain the precision of their arguments. The fractional component is truncated to fit into the destination, with no rounding. None of the operations is affected by the floating-point environment.

For more details, see the IMPLEMENTATION DETAILS below.

LIST OF FUNCTIONS

  Functions which create/initialise a Q number
Q_INI(3) initialise a Q number

  Numeric functions which operate on two Q numbers
Q_QADDQ(3) addition Q_QDIVQ(3) division Q_QMULQ(3) multiplication Q_QSUBQ(3) subtraction Q_NORMPREC(3) normalisation Q_QMAXQ(3) maximum function Q_QMINQ(3) minimum function Q_QCLONEQ(3) identical copy Q_QCPYVALQ(3) representational copy

  Numeric functions which apply integers to a Q number
Q_QADDI(3) addition Q_QDIVI(3) division Q_QMULI(3) multiplication Q_QSUBI(3) subtraction Q_QFRACI(3) fraction Q_QCPYVALI(3) overwrite

  Numeric functions which operate on a single Q number
Q_QABS(3) absolute value Q_Q2D(3) double representation Q_Q2F(3) float representation

  Comparison and logic functions
Q_SIGNED(3) determine sign Q_LTZ(3) less than zero Q_PRECEQ(3) compare bits Q_QLTQ(3) less than Q_QLEQ(3) less or equal Q_QGTQ(3) greater than Q_QGEQ(3) greater or equal Q_QEQ(3) equal Q_QNEQ(3) not equal Q_OFLOW(3) would overflow Q_RELPREC(3) relative precision

  Functions which manipulate the control/sign data bits
Q_SIGNSHFT(3) sign bit position Q_SSIGN(3) sign bit Q_CRAWMASK(3) control bitmask Q_SRAWMASK(3) sign bitmask Q_GCRAW(3) raw control bits Q_GCVAL(3) value of control bits Q_SCVAL(3) set control bits

  Functions which manipulate the combined integer/fractional data bits
Q_IFRAWMASK(3) integer/fractional bitmask Q_IFVALIMASK(3) value of integer bits Q_IFVALFMASK(3) value of fractional bits Q_GIFRAW(3) raw integer/fractional bits Q_GIFABSVAL(3) absolute value of fractional bits Q_GIFVAL(3) real value of fractional bits Q_SIFVAL(3) set integer/fractional bits Q_SIFVALS(3) set separate integer/fractional values

  Functions which manipulate the integer data bits
Q_IRAWMASK(3) integer bitmask Q_GIRAW(3) raw integer bits Q_GIABSVAL(3) absolute value of integer bits Q_GIVAL(3) real value of integer bits Q_SIVAL(3) set integer bits

  Functions which manipulate the fractional data bits
Q_FRAWMASK(3) fractional bitmask Q_GFRAW(3) raw fractional bits Q_GFABSVAL(3) absolute value of fractional bits Q_GFVAL(3) real value of fractional bits Q_SFVAL(3) set fractional bits

  Miscellaneous functions/variables
Q_NCBITS(3) number of reserved control bits Q_BT(3) C data type Q_TC(3) casted data type Q_NTBITS(3) number of total bits Q_NFCBITS(3) number of control-encoded fractional bits Q_MAXNFBITS(3) number of maximum fractional bits Q_NFBITS(3) number of effective fractional bits Q_NIBITS(3) number of integer bits Q_RPSHFT(3) bit position of radix point Q_ABS(3) absolute value Q_MAXSTRLEN(3) number of characters to render string Q_TOSTR(3) render string Q_SHL(3) left-shifted value Q_SHR(3) right-shifted value Q_DEBUG(3) render debugging information Q_DFV2BFV(3) convert decimal fractional value

IMPLEMENTATION DETAILS
The qmath data types and APIs support fixed-point math based on the “Q” number format. This implementation uses the Q notation Qm.n, where m specifies the number of bits for integral data (excluding the sign bit for signed types), and n specifies the number of bits for fractional data.

The APIs have been built around the following q_t derived data types:

    typedef int8_t		s8q_t;
    typedef uint8_t		u8q_t;
    typedef int16_t		s16q_t;
    typedef uint16_t	u16q_t;
    typedef int32_t		s32q_t;
    typedef uint32_t	u32q_t;
    typedef int64_t		s64q_t;
    typedef uint64_t	u64q_t;

These types are referred to generically in the earlier API definitions as QTYPE, while ITYPE refers to the stdint(7) integer types the Q data types are derived from. NTYPE is used to refer to any numeric type and is therefore a superset of QTYPE and ITYPE.

The 3 least significant bits (LSBs) of all q_t data types are reserved for embedded control data:

-   bits 1-2 specify the binary radix point shift index operand, with 00,01,10,11 == 1,2,3,4.

-   bit 3 specifies the radix point shift index operand multiplier as 2 (0) or 16 (1).

This scheme can therefore represent Q numbers with [2,4,6,8,16,32,48,64] bits of precision after the binary radix point. The number of bits available for the integral component is not explicitly specified, and implicitly consumes the remaining available bits of the chosen Q data type.

Additionally, the most significant bit (MSB) of signed Q types stores the sign bit, with bit value 0 representing a positive number and bit value 1 representing a negative number. Negative numbers are stored as absolute values with the sign bit set, rather than the more typical two's complement representation. This avoids having to bit shift negative numbers, which can result in undefined behaviour from some compilers.

This binary representation used for Q numbers therefore comprises a set of distinct data bit types and associated bit counts. Data bit types/labels, listed in LSB to MSB order, are: control ‘C’, fractional ‘F’, integer ‘I’ and sign ‘S’. The following example illustrates the binary representation of a Q20.8 number represented using a s32q_t variable:

    M                                                             L
    S                                                             S
    B                                                             B
    3 3 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1
    1 0 9 8 7 6 5 4 3 2 1 0 9 8 7 6 5 4 3 2 1 0 9 8 7 6 5 4 3 2 1 0
    S I I I I I I I I I I I I I I I I I I I I F F F F F F F F C C C

Important bit counts are: total, control, control-encoded fractional, maximum fractional, effective fractional and integer bits.

The count of total bits is derived from the size of the q_t data type. For example, a s32q_t has 32 total bits.

The count of control-encoded fractional bits is derived from calculating the number of fractional bits per the control bit encoding scheme. For example, the control bits binary value of 101 encodes a fractional bit count of 2 x 16 = 32 fractional bits.

The count of maximum fractional bits is derived from the difference between the counts of total bits and control/sign bits. For example, a s32q_t has a maximum of 32 - 3 - 1 = 28 fractional bits.

The count of effective fractional bits is derived from the minimum of the control-encoded fractional bits and the maximum fractional bits. For example, a s32q_t with 32 control-encoded fractional bits is effectively limited to 28 fractional bits.

The count of integer bits is derived from the difference between the counts of total bits and all other non-integer data bits (the sum of control, fractional and sign bits.) For example, a s32q_t with 8 effective fractional bits has 32 - 3 - 8 - 1 = 20 integer bits. The count of integer bits can be zero if all available numeric data bits have been reserved for fractional data, e.g., when the number of control-encoded fractional bits is greater than or equal to the underlying Q data type's maximum fractional bits.

EXAMPLES

  Calculating area of a circle with r=4.2 and rpshft=16

    u64q_t a, pi, r;
    char buf[32]
    Q_INI(&a, 0, 0, 16);
    Q_INI(&pi, 3, 14159, 16);
    Q_INI(&r, 4, 2, 16);
    Q_QCLONEQ(&a, r);
    Q_QMULQ(&a, r);
    Q_QMULQ(&a, pi);
    Q_TOSTR(a, -1, 10, buf, sizeof(buf));
    printf("%s\n", buf);

  Debugging
Declare a Q20.8 s32q_t number s32, initialise it with the fixed-point value for 5/3, and render a debugging representation of the variable (including its full precision decimal C-string representation), to the console:

    s32q_t s32;
    Q_INI(&s32, 0, 0, 8);
    Q_QFRACI(&s32, 5, 3);
    char buf[Q_MAXSTRLEN(s32, 10)];
    Q_TOSTR(s32, -1, 10, buf, sizeof(buf));
    printf(Q_DEBUG(s32, "", "\n\ttostr=%s\n\n", 0), buf);

The above code outputs the following to the console:

    "s32"@0x7fffffffe7d4
    	type=s32q_t, Qm.n=Q20.8, rpshft=11, imin=0xfff00001, \
    imax=0xfffff
    	qraw=0x00000d53
    	imask=0x7ffff800, fmask=0x000007f8, cmask=0x00000007, \
    ifmask=0x7ffffff8
    	iraw=0x00000800, iabsval=0x1, ival=0x1
    	fraw=0x00000550, fabsval=0xaa, fval=0xaa
    	tostr=1.664

Note: The "\" present in the rendered output above indicates a manual line break inserted to keep the man page within 80 columns and is not part of the actual output.

SEE ALSO
errno(2), math(3), Q_FRAWMASK(3), Q_IFRAWMASK(3), Q_INI(3), Q_IRAWMASK(3), Q_QABS(3), Q_QADDI(3), Q_QADDQ(3), Q_SIGNED(3), Q_SIGNSHFT(3), stdint(7)

HISTORY
The qmath functions first appeared in FreeBSD 13.0.

AUTHORS
-nosplit The qmath functions and this manual page were written by Lawrence Stewart <lstewart@FreeBSD.org> and sponsored by Netflix, Inc.