.\" .\" Copyright (c) 2018 Netflix, Inc. .\" All rights reserved. .\" .\" Redistribution and use in source and binary forms, with or without .\" modification, are permitted provided that the following conditions .\" are met: .\" 1. Redistributions of source code must retain the above copyright .\" notice, this list of conditions, and the following disclaimer, .\" without modification, immediately at the beginning of the file. .\" 2. The name of the author may not be used to endorse or promote products .\" derived from this software without specific prior written permission. .\" .\" THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND .\" ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE .\" IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE .\" ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE FOR .\" ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL .\" DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS .\" OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) .\" HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT .\" LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY .\" OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF .\" SUCH DAMAGE. .\" .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.