package bap-std

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package bap-std
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Shortcut for bitvectors that represent addresses

include module type of Bitvector with type t = addr and type endian = endian and type comparator_witness = Bitvector.comparator_witness

Bitvector -- an integer with modular arithmentics.

Overview

A numeric value with the 2-complement binary representation. It is good for representing addresses, offsets and other arithmetic values.

Each value is attributed by a bitwidth and signedness. All arithmetic operations over values are done modulo their widths. It is an error to apply arithmetic operation to values with different widths. Default implementations will raise a an exception, however there exists a family of modules that provide arithmetic operations lifted to an Or_error.t monad. It is suggested to use them, if you know what kind of operands you're expecting.

Clarification on size-morphism

Size-monomorphic operations (as opposed to size-polymorphic comparison) doesn't allow to compare two operands with different sizes, and either raise an exception or return Error. If we would have a type safe interface, with type t defined as type 'a t, where 'a stands for size, then size-monomorphic operations will have type 'a t -> 'a t -> _, and size-polymorphic 'a t -> 'b t -> _.

By default, size-polymorphic comparison is used. To understand the ordering relation one can think that a lexical ordering is specified on a tuple (x,n), where x is the number and n is the size. For example, the following sequence is in an ascending order:

0x0:1, 0x0:32, 0x0:64, 0x1:1, 0x1:32, 0xD:4, 0xDEADBEEF:32

.

A size-monomorphic interfaced is exposed in a Mono submodule. So if you want a monomorphic map, then just use Mono.Map module. Note, Mono submodule doesn't provide Table, since we cannot guarantee that all keys in a hash-table have equal size.

Clarification on signs

By default all numbers represented by a bitvector are considered unsigned. This includes comparisons, e.g., of_int (-1) ~width:32 is greater than zero. If you need to perform a signed operation, you can use the signed operator to temporary cast your value to the signed kind. We use word "temporary" to emphasize that, the signedness property won't propagate to the result of the operation, e.g. result of the following expression: Int_exn.(signed x / y) will not be signed. In other words each new value is created unsigned.

If any operand of a binary operation is signed, then a signed version of an operation is used, i.e., the other operand is upcasted to the signed kind.

Remember to use explicit casts, whenever you really need a signed representation. Examples:

let x = of_int ~-6 ~width:8
let y = to_int x          (* y = 250 *)
let z = to_int (signed x) (* z = ~-6 *)
let zero = of_int 0 ~width:8
let p = x < zero          (* p = false *)
let q = signed x < zero   (* p = true *)

Clarification on string representation

As a part of Identifiable interface bitvector provides a pair of complement functions: to_string and of_string, that provides facilities to store bitvector as a human readable string, and to restore it from string. The format of the representation is the following (in EBNF):

repr  = [sign], [base], digit, {digit}, ":", size, [kind]
  sign  = "+" | "-";
base  = "0x" | "0b" | "0o";
size  = dec, {dec};
digit = dec | oct | hex;
dec   = ?decimal digit?;
oct   = ?octal digit?;
hex   = ?hexadecimal digit?;
kind  = u | s

Examples: 0x5D:32s, 0b0101:16u, 5:64, +5:8, +0x5D:16.

If base is omitted base-10 is assumed. If the kind is ommited, then the usigned kind is assumed. The output format is always in a hex representation with a full prefix. .

type t = addr

word is an abbreviation to Bitvector.t

Common Interfaces

A bitvector is a value, first of all, so it supports a common set of a value interface: it can be stored, compared, it can be a key in a dictionary, etc. Moreover, being a number it can be compared with zero and applied to a common set of integer operations.

include Regular.Std.Regular.S with type t := t with type comparator_witness = Bitvector.comparator_witness
include sig ... end
val t_of_sexp : Sexplib.Sexp.t -> t
val sexp_of_t : t -> Sexplib.Sexp.t
val bin_t : t Core_kernel.Std.Bin_prot.Type_class.t
val bin_read_t : t Core_kernel.Std.Bin_prot.Read.reader
val __bin_read_t__ : (int -> t) Core_kernel.Std.Bin_prot.Read.reader
val bin_reader_t : t Core_kernel.Std.Bin_prot.Type_class.reader
val bin_size_t : t Core_kernel.Std.Bin_prot.Size.sizer
val bin_write_t : t Core_kernel.Std.Bin_prot.Write.writer
val bin_writer_t : t Core_kernel.Std.Bin_prot.Type_class.writer
include Regular.Std.Printable.S with type t := t
val to_string : t -> string

to_string x returns a human-readable representation of x

val str : unit -> t -> string

str () t is formatted output function that matches "%a" conversion format specifier in functions, that prints to string, e.g., sprintf, failwithf, errorf and, suprisingly all Lwt printing function, including Lwt_io.printf and logging (or any other function with type ('a,unit,string,...) formatN`. Example:

Or_error.errorf "type %a is not valid for %a"
Type.str ty Exp.str exp
val pps : unit -> t -> string

synonym for str

val ppo : Core_kernel.Std.out_channel -> t -> unit

will print to a standard output_channel, useful for using in printf, fprintf, etc.

val pp_seq : Format.formatter -> t Core_kernel.Std.Sequence.t -> unit

prints a sequence of values of type t

this will include pp function from Core that has type t printer, and can be used in Format.printf family of functions

include Core_kernel.Std.Pretty_printer.S with type t := t
include Core_kernel.Std.Comparable.S_binable with type t := t with type comparator_witness = Bitvector.comparator_witness
val (>=) : t -> t -> bool
val (<=) : t -> t -> bool
val (=) : t -> t -> bool
val (>) : t -> t -> bool
val (<) : t -> t -> bool
val (<>) : t -> t -> bool
val equal : t -> t -> bool
val compare : t -> t -> int
val min : t -> t -> t
val max : t -> t -> t
val ascending : t -> t -> int
val descending : t -> t -> int
val between : t -> low:t -> high:t -> bool
val clamp_exn : t -> min:t -> max:t -> t
val clamp : t -> min:t -> max:t -> t Core_kernel.Or_error.t
module Replace_polymorphic_compare : sig ... end
type comparator_witness = Bitvector.comparator_witness
val validate_lbound : min:t Core_kernel.Maybe_bound.t -> t Core_kernel.Validate.check
val validate_ubound : max:t Core_kernel.Maybe_bound.t -> t Core_kernel.Validate.check
val validate_bound : min:t Core_kernel.Maybe_bound.t -> max:t Core_kernel.Maybe_bound.t -> t Core_kernel.Validate.check
val comparator : (t, comparator_witness) Core_kernel.Comparator.comparator
module Map : sig ... end
module Set : sig ... end
include Core_kernel.Std.Hashable.S_binable with type t := t
val hash : t -> int
val hashable : t Core_kernel.Std.Hashable.Hashtbl.Hashable.t
module Table : sig ... end
module Hash_set : sig ... end
module Hash_queue : sig ... end
include Regular.Std.Data.S with type t := t
type info = string * [ `Ver of string ] * string option

name,Ver v,desc information attached to a particular reader or writer.

val version : string

Data representation version. After any change in data representation the version should be increased.

Serializers that are derived from a data representation must have the same version as a version of the data structure, from which it is derived. This kind of serializers can only read and write data of the same version.

Other serializers can actually read and write data independent on its representation version. A serializer, that can't store data of current version simply shouldn't be added to a set of serializers.

It is assumed, that if a reader and a writer has the same name and version, then whatever was written by the writer should be readable by the reader. The round-trip equality is not required, thus it is acceptable if some information is lost.

It is also possible, that a reader and a writer that has the same name are compatible. In that case it is recommended to use semantic versioning.

val size_in_bytes : ?ver:string -> ?fmt:string -> t -> int

size_in_bytes ?ver ?fmt datum returns the amount of bytes that is needed to represent datum in the given format and version

val of_bytes : ?ver:string -> ?fmt:string -> Regular.Std.bytes -> t

of_bytes ?ver ?fmt bytes deserializes a value from bytes.

val to_bytes : ?ver:string -> ?fmt:string -> t -> Regular.Std.bytes

to_bytes ?ver ?fmt datum serializes a datum to a sequence of bytes.

val blit_to_bytes : ?ver:string -> ?fmt:string -> Regular.Std.bytes -> t -> int -> unit

blit_to_bytes ?ver ?fmt buffer datum offset copies a serialized representation of datum into a buffer, starting from the offset.

val of_bigstring : ?ver:string -> ?fmt:string -> Core_kernel.Std.bigstring -> t

of_bigstring ?ver ?fmt buf deserializes a datum from bigstring

val to_bigstring : ?ver:string -> ?fmt:string -> t -> Core_kernel.Std.bigstring

of_bigstring ?ver ?fmt datum serializes a datum to a sequence of bytes represented as bigstring

val blit_to_bigstring : ?ver:string -> ?fmt:string -> Core_kernel.Std.bigstring -> t -> int -> unit

blit_to_bigstring ?ver ?fmt buffer datum offset copies a serialized representation of datum into a buffer, starting from offset.

module Io : sig ... end

Input/Output functions for the given datum.

module Cache : sig ... end

Data cache.

val add_reader : ?desc:string -> ver:string -> string -> t Regular.Std.reader -> unit

add_reader ?desc ~ver name reader registers a new reader with a provided name, version ver and optional description desc

val add_writer : ?desc:string -> ver:string -> string -> t Regular.Std.writer -> unit

add_writer ?desc ~ver name writer registers a new writer with a provided name, version ver and optional description desc

val available_readers : unit -> info list

available_reader () lists available readers for the data type

val default_reader : unit -> info

default_reader returns information about default reader

val set_default_reader : ?ver:string -> string -> unit

set_default_reader ?ver name sets new default reader. If version is not specifed then the latest available version is used. Raises an exception if a reader with a given name doesn't exist.

val with_reader : ?ver:string -> string -> (unit -> 'a) -> 'a

with_reader ?ver name operation temporary sets a default reader to a reader with a specified name and version. The default reader is restored after operation is finished.

val available_writers : unit -> info list

available_writer () lists available writers for the data type

val default_writer : unit -> info

default_writer returns information about the default writer

val set_default_writer : ?ver:string -> string -> unit

set_default_writer ?ver name sets new default writer. If version is not specifed then the latest available version is used. Raises an exception if a writer with a given name doesn't exist.

val with_writer : ?ver:string -> string -> (unit -> 'a) -> 'a

with_writer ?ver name operation temporary sets a default writer to a writer with a specified name and version. The default writer is restored after operation is finished.

val default_printer : unit -> info option

default_writer optionally returns an information about default printer

val set_default_printer : ?ver:string -> string -> unit

set_default_printer ?ver name sets new default printer. If version is not specifed then the latest available version is used. Raises an exception if a printer with a given name doesn't exist.

val with_printer : ?ver:string -> string -> (unit -> 'a) -> 'a

with_printer ?ver name operation temporary sets a default printer to a printer with a specified name and version. The default printer is restored after operation is finished.

Low level access to serializers

val find_reader : ?ver:string -> string -> t Regular.Std.reader option

find_reader ?ver name lookups a reader with a given name. If version is not specified, then a reader with maximum version is returned.

val find_writer : ?ver:string -> string -> t Regular.Std.writer option

find_writer ?ver name lookups a writer with a given name. If version is not specified, then a writer with maximum version is returned.

Bitvector implements a common set of operations that are expected from integral values.

include Integer.S with type t := t
include Integer.Base with type t := t
val abs : t -> t

abs x absolute value of x

val neg : t -> t

neg x = -x

val add : t -> t -> t

add x y is x + y

val sub : t -> t -> t

sub x y is x - y

val mul : t -> t -> t

mul x y is x * y

val div : t -> t -> t

div x y is x / y

val modulo : t -> t -> t

modulo x y is x mod y

val lnot : t -> t

lnot x is a logical negation of x (1-complement)

logand x y is a conjunction of x and y

val logand : t -> t -> t

logand x y is a conjunction of x and y

val logor : t -> t -> t

logor x y is a disjunction of x and y

val logxor : t -> t -> t

logxor x y is exclusive or between x and y

val lshift : t -> t -> t

lshift x y shift x by y bits left

val rshift : t -> t -> t

rshift x y shift x by y bits to the right

val arshift : t -> t -> t

arshift x y shift x by y bits to the right and fill with the sign bit.

A common set of infix operators

val (~-) : t -> t

~-x = neg x

val (+) : t -> t -> t

x + y = add x y

val (-) : t -> t -> t

x - y = sub x y

val (*) : t -> t -> t

x * y = mul x y

val (/) : t -> t -> t

x / y = div x y

val (mod) : t -> t -> t

x mod y = modulo x y

val (land) : t -> t -> t

x land y = logand x y

val (lor) : t -> t -> t

x lor y = logor x y

val (lxor) : t -> t -> t

lxor x y = logxor x y

val (lsl) : t -> t -> t

x lsl y = lshift x y

val (lsr) : t -> t -> t

x lsr y = rshift x y

val (asr) : t -> t -> t

x asr y = arshift x y

module Mono : Core_kernel.Std.Comparable with type t := t

A comparable interface with size-monomorphic comparison.

type endian = endian =
  1. | LittleEndian
    (*

    least significant byte comes first

    *)
  2. | BigEndian
    (*

    most significant byte comes first

    *)

Specifies the order of bytes in a word.

include sig ... end
val endian_of_sexp : Sexplib.Sexp.t -> endian
val sexp_of_endian : endian -> Sexplib.Sexp.t
val compare_endian : endian -> endian -> int
val bin_endian : endian Core_kernel.Std.Bin_prot.Type_class.t
val bin_read_endian : endian Core_kernel.Std.Bin_prot.Read.reader
val __bin_read_endian__ : (int -> endian) Core_kernel.Std.Bin_prot.Read.reader
val bin_reader_endian : endian Core_kernel.Std.Bin_prot.Type_class.reader
val bin_size_endian : endian Core_kernel.Std.Bin_prot.Size.sizer
val bin_write_endian : endian Core_kernel.Std.Bin_prot.Write.writer
val bin_writer_endian : endian Core_kernel.Std.Bin_prot.Type_class.writer

Constructors

val of_string : string -> t

of_string s parses a bitvector from a string representation defined in section Clarification on string representation.

val of_bool : bool -> t

of_bool x is a bitvector with length 1 and value b0 if x is false and b1 otherwise.

val of_int : width:int -> int -> t

of_int ~width n creates a bitvector of the specified bit-width with the value equal to n. If bits of the n that doesn't fit into width are ignored.

val of_int32 : ?width:int -> int32 -> t

of_int32 ?width n creates a bitvector of the specified bit-width with the value equal to n. If bits of the n that doesn't fit into width are ignored. Parameter width defaults to 32.

val of_int64 : ?width:int -> int64 -> t

of_int32 ?width n creates a bitvector of the specified bit-width with the value equal to n. If bits of the n that doesn't fit into width are ignored. Parameter width defaults to 32.

Some predefined constant constructors

val b0 : t

b0 = of_bool false

val b1 : t

b1 = of_bool true

Helpful shortcuts

val one : int -> t

one width number one with a specified width, is a shortcut for of_int 1 ~width

val zero : int -> t

zero width zero with a specified width, is a shortcut for of_int 0 ~width

val ones : int -> t

ones width is a number with a specified width, and all bits set to 1. It is a shortcut for lnot (zero width)

val of_binary : ?width:int -> endian -> string -> t

of_binary ?width endian num creates a bitvector from a string interpreted as a sequence of bytes in a specified order.

The result is always positive and unsigned. num argument is not shared. width defaults to String.length num

Conversions to OCaml built in integer types

val to_int : t -> int Core_kernel.Std.Or_error.t

to_int x projects x in to OCaml int.

val to_int32 : t -> int32 Core_kernel.Std.Or_error.t

to_int32 x projects x in to int32

val to_int64 : t -> int64 Core_kernel.Std.Or_error.t

to_int64 x projects x in to int64

val to_int_exn : t -> int

to_int_exn x projects x in to OCaml int.

  • since 1.3
val to_int32_exn : t -> int32

to_int32_exn x projects x in to int32

  • since 1.3
val to_int64_exn : t -> int64

to_int64_exn x projects x in to int64

  • since 1.3

printf "%a" pp x prints x into a formatter. This is a default printer, controlled by set_default_printer. Multiple formats are available, see the available_writers for the actual list of formats and a format description. Out of box it defaults to pp_hex_full. Note, the printf function from examples refers to the Format.printf, thus it is assumed that the Format module is open in the scope.

val pp_hex : t Regular.Std.printer

printf "%a" pp_hex x prints x in the hexadecimal format omitting suffixes, and the prefix if it is not necessary. Example,

# printf "%a\n" pp_hex (Word.of_int32 0xDEADBEEFl);;
0xDEADBEEF
# printf "%a\n" pp_hex (Word.of_int32 0x1);;
1
val pp_dec : t Regular.Std.printer

printf "%a" pp_dec x prints x in the decimal format omitting suffixes and prefixes. Example,

# printf "%a\n" pp_dec (Word.of_int32 0xDEADBEEFl);;
3735928559
# printf "%a\n" pp_dec (Word.of_int32 0x1);;
1
  • since 1.3
val pp_oct : t Regular.Std.printer

printf "%a" pp_oct x prints x in the octal format omitting suffixes, and the prefix if it is not necessary. Example,

# printf "%a\n" pp_oct (Word.of_int32 0xDEADBEEFl);;
0o33653337357
# printf "%a\n" pp_oct (Word.of_int32 0x1);;
1
  • since 1.3
val pp_bin : t Regular.Std.printer

printf "%a" pp_bin x prints x in the binary (0 and 1) format omitting suffixes, and the prefix if it is not necessary. Example,

# printf "%a\n" pp_bin (Word.of_int32 0xDEADBEEFl);;
0b11011110101011011011111011101111
# printf "%a\n" pp_bin (Word.of_int32 0x1);;
1
  • since 1.3
val pp_hex_full : t Regular.Std.printer

printf "%a" pp_hex_full x prints x in the hexadecimal format with suffixes, and the prefix if it is necessary. Example,

# printf "%a\n" pp_hex_full (Word.of_int32 0xDEADBEEFl);;
0xDEADBEEF:32u
             # printf "%a\n" pp_hex_full (Word.of_int32 0x1);;
1:32u
val pp_dec_full : t Regular.Std.printer

printf "%a" pp_dec_full x prints x in the decimal format with suffixes and prefixes. Example,

# printf "%a\n" pp_dec_full (Word.of_int32 0xDEADBEEFl);;
3735928559:32u
             # printf "%a\n" pp_dec_full (Word.of_int32 0x1);;
1:32u
  • since 1.3
val pp_oct_full : t Regular.Std.printer

printf "%a" pp_oct_full x prints x in the octal format with suffixes, and the prefix if it is necessary. Example,

# printf "%a\n" pp_oct_full (Word.of_int32 0xDEADBEEFl);;
0o33653337357:32u
                # printf "%a\n" pp_oct_full (Word.of_int32 0x1);;
1:32u
val pp_bin_full : t Regular.Std.printer

printf "%a" pp_bin_full x prints x in the binary (0 and 1) format omitting suffixes, and the prefix if it is necessary. Example,

# printf "%a\n" pp_bin_full (Word.of_int32 0xDEADBEEFl);;
0b11011110101011011011111011101111:32u
                                     # printf "%a\n" pp_bin_full (Word.of_int32 0x1);;
1:32u
val pp_generic : ?case:[ `upper | `lower ] -> ?prefix:[ `auto | `base | `none | `this of string ] -> ?suffix:[ `none | `full | `size ] -> ?format:[ `hex | `dec | `oct | `bin ] -> t Regular.Std.printer

pp_generic ?case ?prefix ?suffix ?format ppf x - a printer to suit all tastes.

Note: this is a generic printer factory that should be used if none of the nine preinstantiated suits you.

  • parameter prefix

    defines whether or not a number is prefixed:

    • `auto (default) - a prefix that corresponds to the chosen format is printed if it is necessary to disambiguate a number from a decimal representation;
    • `base - a corresponding prefix is always printed;
    • `none - the prefix is never printed;
    • `this p - the user specified prefix p is always printed;
  • parameter suffix

    defines how the suffix should be printed:

    • `none (default) - the suffix is never printed;
    • `full - a full suffix that denotes size and signedness is printed, e.g., 0xDE:32s is a signed integer modulo 32.
    • `size - only the modulo is printed, e.g., 0xDE:32s is printed as 0xDE:32
  • parameter format

    defines the textual representation format:

    • hex (default) - hexadecimal
    • dec - decimal
    • oct - octal
    • bin - binary (0 and 1).
  • parameter case

    defines the case of hexadecimal letters

val string_of_value : ?hex:bool -> t -> string

string_of_value ?hex x returns a textual representation of the x value, i.e., ignores size and signedness. If hex is true (default), then it is in the hexadecimal representation, otherwise the decimal representation is used. The returned value is not prefixed. No leading zeros are printed. If a value is signed and negative, then a leading negative sign is printed. Hexadecimal letter literals are printed in a lowercase format.

val signed : t -> t

signed t casts t to a signed type, so that any operations applied on t will be signed.

val unsigned : t -> t

unsigned t casts t to an unsigned type, so that any operations applied to it will interpret t as an unsigned word.

  • since 1.3
val is_zero : t -> bool

is_zero bv is true iff all bits are set to zero.

val is_one : t -> bool

is_ones bv is true if the least significant bit is equal to one

val bitwidth : t -> int

bitwidth bv return a bit-width, i.e., the amount of bits

val extract : ?hi:int -> ?lo:int -> t -> t Core_kernel.Std.Or_error.t

extract bv ~hi ~lo extracts a subvector from bv, starting from bit hi and ending with lo. Bits are enumerated from right to left (from least significant to most), starting from zero. hi maybe greater than size.

hi defaults to width bv - 1 lo defaults to 0.

Example:

extract (of_int 17 ~width:8) ~hi:4 ~lo:3 will result in a two bit vector consisting of the forth and third bits, i.e., equal to a number 2.

lo and hi should be non-negative numbers. lo must be less then a width bv and hi >= lo.

val extract_exn : ?hi:int -> ?lo:int -> t -> t

extract_exn bv ~hi ~lo is the same as extract, but will raise an exception on error.

val concat : t -> t -> t

concat b1 b2 concatenates two bitvectors

val (@.) : t -> t -> t

b1 @. b2 is concat b1 b2

val succ : t -> t

succ n returns next value after n. It is not guaranteed that signed (succ n) > signed n

val pred : t -> t

pred n returns a value preceding n.

val nsucc : t -> int -> t

nsucc m n is Fn.apply_n_times ~n succ m, but more efficient.

val npred : t -> int -> t

npred m n is Fn.apply_n_times ~n pred addr, but more efficient.

val (++) : t -> int -> t

a ++ n is nsucc a n

val (--) : t -> int -> t

a -- n is npred a n

Iteration over bitvector components

val enum_bytes : t -> endian -> t seq

enum_bytes x order returns a sequence of bytes of x in a specified order. Each byte is represented as a bitvector itself.

val enum_chars : t -> endian -> char seq

enum_bytes x order returns bytes of x in a specified order, with bytes represented by char type

val enum_bits : t -> endian -> bool seq

enum_bits x order returns bits of x in a specified order. order defines only the ordering of words in a bitvector, bits will always be in MSB first order. The length of the sequence is always a power of 8.

Comparison with zero

Note, we're not including With_zero interface, since it refers to the `Sign` module, that is available only in core_kernel >= 113.33.00.

val validate_positive : t Core_kernel.Std.Validate.check

validate_positive validates that a value is positive.

val validate_non_negative : t Core_kernel.Std.Validate.check

validate_non_negative validates that a value is non negative.

val validate_negative : t Core_kernel.Std.Validate.check

validate_negative validates that a value is negative.

val validate_non_positive : t Core_kernel.Std.Validate.check

validate_non_positive validates that a value is not positive.

val is_positive : t -> bool

is_positive x is true if x is greater than zero. Always true if x is unsigned.

val is_non_negative : t -> bool

is_non_negative x is true if x is greater than or equal to zero. Tautology if x is unsigned.

val is_negative : t -> bool

is_negative x is true if x is strictly less than zero. It is a contradiction if x is not signed.

val is_non_positive : t -> bool

is_non_positive x is true if x is less than zero. It is a contradiction if x is not signed.

module Int_err : sig ... end
module Int_exn : Integer.S with type t = t

Arithmetic that raises exceptions.

module Unsafe : Integer.S with type t = t

Arithmentic operations that doesn't check the widths.

module Stable : sig ... end

Stable marshaling interface.

module Trie : sig ... end

Prefix trees for bitvectors.

val memref : ?disp:int -> ?index:int -> ?scale:size -> addr -> addr

memref ?disp ?index ?scale base mimics a memory reference syntax in gas assembler, dis(base,index,scale) assembler operation. It returns address at base + index * scale + dis.

  • parameter disp

    stands for displacement and defaults to 0

  • parameter index

    defaults for 0

  • parameter scale

    defaults to `r8

    All operations are taken modulo $2^n$ , where n = bitwidth base.