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.Dd July 4, 2019
.Dt QMATH 3
.Os
.Sh NAME
.Nm qmath
.Nd fixed-point math library based on the
.Dq Q
number format
.Sh SYNOPSIS
.In sys/qmath.h
.Sh DESCRIPTION
The
.Nm
data types and APIs support fixed-point math based on the
.Dq Q
number format.
The APIs have been built around the following data types:
.Vt s8q_t ,
.Vt u8q_t ,
.Vt s16q_t ,
.Vt u16q_t ,
.Vt s32q_t ,
.Vt u32q_t ,
.Vt s64q_t ,
and
.Vt u64q_t ,
which are referred to generically in the earlier API definitions as
.Fa QTYPE .
The
.Fa ITYPE
refers to the
.Xr stdint 7
integer types.
.Fa NTYPE
is used to refer to any numeric type and is therefore a superset of
.Fa QTYPE
and
.Fa ITYPE .
.Pp
This scheme can represent Q numbers with
.Bq 2, 4, 6, 8, 16, 32, 48
bits of precision after the binary radix point,
depending on the
.Fa rpshft
argument to
.Fn 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.
.Pp
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.
.Pp
For more details, see the
.Sx IMPLEMENTATION DETAILS
below.
.Sh LIST OF FUNCTIONS
.de Cl
.Bl -column "isgreaterequal" "bessel function of the second kind of the order 0"
.Em "Name	Description"
..
.Ss Functions which create/initialise a Q number
.Cl
.Xr Q_INI 3	initialise a Q number
.El
.Ss Numeric functions which operate on two Q numbers
.Cl
.Xr Q_QADDQ 3	addition
.Xr Q_QDIVQ 3	division
.Xr Q_QMULQ 3	multiplication
.Xr Q_QSUBQ 3	subtraction
.Xr Q_NORMPREC 3	normalisation
.Xr Q_QMAXQ 3	maximum function
.Xr Q_QMINQ 3	minimum function
.Xr Q_QCLONEQ 3	identical copy
.Xr Q_QCPYVALQ 3	representational copy
.El
.Ss Numeric functions which apply integers to a Q number
.Cl
.Xr Q_QADDI 3	addition
.Xr Q_QDIVI 3	division
.Xr Q_QMULI 3	multiplication
.Xr Q_QSUBI 3	subtraction
.Xr Q_QFRACI 3	fraction
.Xr Q_QCPYVALI 3	overwrite
.El
.Ss Numeric functions which operate on a single Q number
.Cl
.Xr Q_QABS 3	absolute value
.Xr Q_Q2D 3	double representation
.Xr Q_Q2F 3	float representation
.El
.Ss Comparison and logic functions
.Cl
.Xr Q_SIGNED 3	determine sign
.Xr Q_LTZ 3	less than zero
.Xr Q_PRECEQ 3	compare bits
.Xr Q_QLTQ 3	less than
.Xr Q_QLEQ 3	less or equal
.Xr Q_QGTQ 3	greater than
.Xr Q_QGEQ 3	greater or equal
.Xr Q_QEQ 3	equal
.Xr Q_QNEQ 3	not equal
.Xr Q_OFLOW 3	would overflow
.Xr Q_RELPREC 3	relative precision
.El
.Ss Functions which manipulate the control/sign data bits
.Cl
.Xr Q_SIGNSHFT 3	sign bit position
.Xr Q_SSIGN 3	sign bit
.Xr Q_CRAWMASK 3	control bitmask
.Xr Q_SRAWMASK 3	sign bitmask
.Xr Q_GCRAW 3	raw control bits
.Xr Q_GCVAL 3	value of control bits
.Xr Q_SCVAL 3	set control bits
.El
.Ss Functions which manipulate the combined integer/fractional data bits
.Cl
.Xr Q_IFRAWMASK 3	integer/fractional bitmask
.Xr Q_IFVALIMASK 3	value of integer bits
.Xr Q_IFVALFMASK 3	value of fractional bits
.Xr Q_GIFRAW 3	raw integer/fractional bits
.Xr Q_GIFABSVAL 3	absolute value of fractional bits
.Xr Q_GIFVAL 3	real value of fractional bits
.Xr Q_SIFVAL 3	set integer/fractional bits
.Xr Q_SIFVALS 3	set separate integer/fractional values
.El
.Ss Functions which manipulate the integer data bits
.Cl
.Xr Q_IRAWMASK 3	integer bitmask
.Xr Q_GIRAW 3	raw integer bits
.Xr Q_GIABSVAL 3	absolute value of integer bits
.Xr Q_GIVAL 3	real value of integer bits
.Xr Q_SIVAL 3	set integer bits
.El
.Ss Functions which manipulate the fractional data bits
.Cl
.Xr Q_FRAWMASK 3	fractional bitmask
.Xr Q_GFRAW 3	raw fractional bits
.Xr Q_GFABSVAL 3	absolute value of fractional bits
.Xr Q_GFVAL 3	real value of fractional bits
.Xr Q_SFVAL 3	set fractional bits
.El
.Ss Miscellaneous functions/variables
.Cl
.Xr Q_NCBITS 3	number of reserved control bits
.Xr Q_BT 3	C data type
.Xr Q_TC 3	casted data type
.Xr Q_NTBITS 3	number of total bits
.Xr Q_NFCBITS 3	number of control-encoded fractional bits
.Xr Q_MAXNFBITS 3	number of maximum fractional bits
.Xr Q_NFBITS 3	number of effective fractional bits
.Xr Q_NIBITS 3	number of integer bits
.Xr Q_RPSHFT 3	bit position of radix point
.Xr Q_ABS 3	absolute value
.Xr Q_MAXSTRLEN 3	number of characters to render string
.Xr Q_TOSTR 3	render string
.Xr Q_SHL 3	left-shifted value
.Xr Q_SHR 3	right-shifted value
.Xr Q_DEBUG 3	render debugging information
.Xr Q_DFV2BFV 3	convert decimal fractional value
.El
.Sh IMPLEMENTATION DETAILS
The
.Nm
data types and APIs support fixed-point math based on the
.Dq Q
number format.
This implementation uses the Q notation
.Em Qm.n ,
where
.Em m
specifies the number of bits for integral data
.Pq excluding the sign bit for signed types ,
and
.Em n
specifies the number of bits for fractional data.
.Pp
The APIs have been built around the following q_t derived data types:
.Bd -literal -offset indent
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;
.Ed
.Pp
These types are referred to generically in the earlier API definitions as
.Fa QTYPE ,
while
.Fa ITYPE
refers to the
.Xr stdint 7
integer types the Q data types are derived from.
.Fa NTYPE
is used to refer to any numeric type and is therefore a superset of
.Fa QTYPE
and
.Fa ITYPE .
.Pp
The 3 least significant bits
.Pq LSBs
of all q_t data types are reserved for embedded control data:
.Bl -dash
.It
bits 1-2 specify the binary radix point shift index operand, with 00,01,10,11 ==
1,2,3,4.
.It
bit 3 specifies the radix point shift index operand multiplier as 2
.Pq 0
or 16
.Pq 1 .
.El
.Pp
This scheme can therefore represent Q numbers with
.Bq 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.
.Pp
Additionally, the most significant bit
.Pq 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.
.Pp
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
.Sq C ,
fractional
.Sq F ,
integer
.Sq I
and sign
.Sq S .
The following example illustrates the binary representation of a Q20.8 number
represented using a s32q_t variable:
.Bd -literal -offset indent
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
.Ed
.Pp
Important bit counts are: total, control, control-encoded fractional, maximum
fractional, effective fractional and integer bits.
.Pp
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.
.Pp
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.
.Pp
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.
.Pp
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.
.Pp
The count of integer bits is derived from the difference between the counts of
total bits and all other non-integer data bits
.Pq 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.
.Sh EXAMPLES
.Ss Calculating area of a circle with r=4.2 and rpshft=16
.Bd -literal -offset indent
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);
.Ed
.Ss Debugging
Declare a Q20.8 s32q_t number
.Fa s32 ,
initialise it with the fixed-point value for 5/3, and render a debugging
representation of the variable
.Pq including its full precision decimal C-string representation ,
to the console:
.Bd -literal -offset indent
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);
.Ed
.Pp
The above code outputs the following to the console:
.Bd -literal -offset indent
"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
.Ed
.Pp
Note: The
.Qq \e
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.
.Sh SEE ALSO
.Xr errno 2 ,
.Xr math 3 ,
.Xr Q_FRAWMASK 3 ,
.Xr Q_IFRAWMASK 3 ,
.Xr Q_INI 3 ,
.Xr Q_IRAWMASK 3 ,
.Xr Q_QABS 3 ,
.Xr Q_QADDI 3 ,
.Xr Q_QADDQ 3 ,
.Xr Q_SIGNED 3 ,
.Xr Q_SIGNSHFT 3 ,
.Xr stdint 7
.Sh HISTORY
The
.Nm
functions first appeared in
.Fx 13.0 .
.Sh AUTHORS
.An -nosplit
The
.Nm
functions and this manual page were written by
.An Lawrence Stewart Aq Mt lstewart@FreeBSD.org
and sponsored by Netflix, Inc.