QMATH(3) FreeBSD Library Functions Manual 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
Name Description
Q_INI(3) initialise a Q number
Numeric functions which operate on two Q numbers
Name Description
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
Name Description
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
Name Description
Q_QABS(3) absolute value
Q_Q2D(3) double representation
Q_Q2F(3) float representation
Comparison and logic functions
Name Description
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
Name Description
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
Name Description
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
Name Description
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
Name Description
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
Name Description
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
The qmath functions and this manual page were written by Lawrence Stewart
<lstewart@FreeBSD.org> and sponsored by Netflix, Inc.
FreeBSD 13.1-RELEASE-p6 July 4, 2019 FreeBSD 13.1-RELEASE-p6
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