#+begin_src elisp tangle: no :results none
(setq pwd (file-name-directory buffer-file-name))
(setq name (file-name-nondirectory (substring buffer-file-name 0 -4)))
(setq lib (concat pwd "lib/"))
(setq testdir (concat pwd "test/"))
(setq mli (concat lib name ".mli"))
(setq ml  (concat lib name ".ml"))
(setq test-ml  (concat testdir name ".ml"))
(org-babel-tangle)
#+end_src 

* Bit string
  :PROPERTIES:
  :ml:       lib/bitstring.ml
  :mli:      lib/bitstring.mli
  :test-ml:  test/bitstring.ml
  :header-args: :noweb yes :comments both
  :END:

  We define here a data type to handle bit strings efficiently.
  When the bit string contains less than 64 bits, it is stored
  internally in a 63-bit integer and uses bitwise instructions.
  When more than 63 bits are required, the =zarith= library is used to
  consider the bit string as a multi-precision integer.

  
** Single-integer implementation

   #+begin_src ocaml :tangle (eval ml)
module One = struct

  let of_int x =
    assert (x > 0); x

  let numbits _ = 63
  let zero = 0
  let is_zero x = x = 0
  let shift_left x i = x lsl i
  let shift_right x i = x lsr i
  let shift_left_one i = 1 lsl i
  let testbit x i = ( (x lsr i) land 1 ) = 1  
  let logor a b = a lor b
  let neg a = - a 
  let logxor a b = a lxor b
  let logand a b = a land b
  let lognot a = lnot a
  let minus_one a = a - 1
  let plus_one a = a + 1

  let popcount = function
    | 0 -> 0
    | r  -> Util.popcnt (Int64.of_int r) 

  let trailing_zeros r = 
    Util.trailz (Int64.of_int r) 

  let hamdist a b =
    a lxor b
    |> popcount 

  let pp ppf s = 
    Format.fprintf ppf "@[@[%a@]@]" (Util.pp_bitstring 64) 
      (Z.of_int s)

end
   #+end_src

** Zarith implementation

   #+begin_src ocaml :tangle (eval ml)
module Many = struct

  let of_z x = x
  let zero = Z.zero
  let is_zero x = x = Z.zero
  let shift_left = Z.shift_left 
  let shift_right = Z.shift_right
  let shift_left_one i = Z.shift_left Z.one i
  let testbit = Z.testbit
  let logor = Z.logor
  let logxor = Z.logxor
  let logand = Z.logand
  let lognot = Z.lognot
  let neg = Z.neg
  let minus_one = Z.pred
  let plus_one = Z.succ
  let trailing_zeros = Z.trailing_zeros
  let hamdist = Z.hamdist
  let numbits i = max (Z.numbits i) 64

  let popcount z = 
    if z = Z.zero then 0 else Z.popcount z

  let pp ppf s = 
    Format.fprintf ppf "@[@[%a@]@]" (Util.pp_bitstring (Z.numbits s))  s

end
   #+end_src

** Type

   #+begin_src ocaml :tangle (eval mli)
type t
   #+end_src

   #+begin_src ocaml :tangle (eval ml)
type t =
  | One of int
  | Many of Z.t
   #+end_src

** Tests header

   #+begin_src ocaml :tangle (eval test-ml)
open Common.Bitstring
let check msg x = Alcotest.(check bool) msg true x
let test_all () =
    let x = 8745687 in
    let one_x  = of_int x in
    let z = Z.shift_left (Z.of_int x) 64 in
    let many_x = of_z z in
   #+end_src
** General implementation
   
*** ~of_int~
    
    Creates a bit string from an ~int~.

    #+begin_src ocaml :tangle (eval mli)
val of_int : int -> t
    #+end_src
   
    #+begin_src ocaml :tangle (eval ml)
let of_int x =
  One (One.of_int x)
    #+end_src

    #+begin_src ocaml :tangle (eval test-ml)
    Alcotest.(check bool) "of_x" true (one_x = (of_int x));
    #+end_src

*** ~of_z~
    
    Creates a bit string from an ~Z.t~ multi-precision integer.

    #+begin_src ocaml :tangle (eval mli)
val of_z : Z.t -> t
    #+end_src
   
    #+begin_src ocaml :tangle (eval ml)
let of_z x =
  if Z.numbits x < 64 then One (Z.to_int x) else Many (Many.of_z x)
    #+end_src

    #+begin_src ocaml :tangle (eval test-ml)
    Alcotest.(check bool) "of_z" true (one_x = (of_z (Z.of_int x)));
    #+end_src

*** ~zero~

    ~zero n~ creates a zero bit string with ~n~ bits.
    
    #+begin_src ocaml :tangle (eval mli)
val zero : int -> t
    #+end_src
   
    #+begin_src ocaml :tangle (eval ml)
let zero = function
| n when n < 64 -> One (One.zero)
| _ -> Many (Many.zero)
    #+end_src

*** ~numbits~

    Returns the number of bits used to represent the bit string.
    
    #+begin_src ocaml :tangle (eval mli)
val numbits : t -> int
    #+end_src
   
    #+begin_src ocaml :tangle (eval ml)
let numbits = function
| One x -> One.numbits x
| Many x -> Many.numbits x
    #+end_src

*** ~is_zero~

    True if all the bits of the bit string are zero.
    
    #+begin_src ocaml :tangle (eval mli)
val is_zero : t -> bool
    #+end_src
   
    #+begin_src ocaml :tangle (eval ml)
let is_zero = function
  | One x -> One.is_zero x
  | Many x -> Many.is_zero x
    #+end_src

*** ~neg~

    Returns the negative of the integer interpretation of the bit string.

    #+begin_example
neg (of_int x) = neg (of_int (-x))
    #+end_example

    #+begin_src ocaml :tangle (eval mli)
val neg : t -> t
    #+end_src

    #+begin_src ocaml :tangle (eval ml)
let neg = function
  | One x -> One (One.neg x)
  | Many x -> Many (Many.neg x)
    #+end_src

*** ~shift_left~

    ~shift_left t n~ returns a new bit strings with all the bits
    shifted ~n~ positions to the left.
    
    #+begin_src ocaml :tangle (eval mli)
val shift_left : t -> int -> t
    #+end_src
    
    #+begin_src ocaml :tangle (eval ml)
let shift_left x i = match x with
| One x -> One (One.shift_left x i)
| Many x -> Many (Many.shift_left x i)
    #+end_src

    #+begin_src ocaml :tangle (eval test-ml)
    Alcotest.(check bool) "shift_left1" true (of_int (x lsl 3) = shift_left one_x 3);
    Alcotest.(check bool) "shift_left2" true (of_z (Z.shift_left z 3) = shift_left many_x 3);
    Alcotest.(check bool) "shift_left3" true (of_z (Z.shift_left z 100) = shift_left many_x 100);
    #+end_src

*** ~shift_right~

    ~shift_right t n~ returns a new bit strings with all the bits
    shifted ~n~ positions to the right.
    
    #+begin_src ocaml :tangle (eval mli)
val shift_right : t -> int -> t
    #+end_src
    
    #+begin_src ocaml :tangle (eval ml)
let shift_right x i = match x with
| One x -> One (One.shift_right x i)
| Many x -> Many (Many.shift_right x i)
    #+end_src

    #+begin_src ocaml :tangle (eval test-ml)
    Alcotest.(check bool) "shift_right1" true (of_int (x lsr 3) = shift_right one_x 3);
    Alcotest.(check bool) "shift_right2" true (of_z (Z.shift_right z 3) = shift_right many_x 3);
    #+end_src

*** ~shift_left_one~

    ~shift_left_one size n~ returns a new bit strings with the ~n~-th
    bit set to one.
    It is equivalent as shifting ~1~ by ~n~ bits to the left.
    ~size~ is the total number of bits of the bit string.
    
    #+begin_src ocaml :tangle (eval mli)
val shift_left_one : int -> int -> t
    #+end_src
    
    #+begin_src ocaml :tangle (eval ml)
let shift_left_one = function
| n when n < 64 -> fun i -> One (One.shift_left_one i)
| _ -> fun i -> Many (Many.shift_left_one i)

    #+end_src

    #+begin_src ocaml :tangle (eval test-ml)
    Alcotest.(check bool) "shift_left_one1" true (of_int (1 lsl 3) = shift_left_one 4 3);
    Alcotest.(check bool) "shift_left_one2" true (of_z (Z.shift_left Z.one 200) = shift_left_one 300 200);
    #+end_src

*** ~testbit~

    ~testbit t n~ is true if the ~n~-th bit of the bit string ~t~ is
    set to ~1~.
    
    #+begin_src ocaml :tangle (eval mli)
val testbit : t -> int -> bool
    #+end_src
   
    #+begin_src ocaml :tangle (eval ml)
let testbit = function
| One x -> One.testbit x
| Many x -> Many.testbit x
    #+end_src

    #+begin_src ocaml :tangle (eval test-ml)
    Alcotest.(check bool) "testbit1" true  (testbit (of_int 8) 3);
    Alcotest.(check bool) "testbit2" false (testbit (of_int 8) 2);
    Alcotest.(check bool) "testbit3" false (testbit (of_int 8) 4);
    Alcotest.(check bool) "testbit4" true  (testbit (of_z (Z.of_int 8)) 3);
    Alcotest.(check bool) "testbit5" false (testbit (of_z (Z.of_int 8)) 2);
    Alcotest.(check bool) "testbit6" false (testbit (of_z (Z.of_int 8)) 4);
    #+end_src

*** ~logor~

    Bitwise logical or.

    #+begin_src ocaml :tangle (eval mli)
val logor : t -> t -> t
    #+end_src

    #+begin_src ocaml :tangle (eval ml)
let logor a b = 
  match a,b with
  | One a, One b -> One (One.logor a b)
  | Many a, Many b -> Many (Many.logor a b)
  | _ -> invalid_arg "Bitstring.logor"
    #+end_src

    #+begin_src ocaml :tangle (eval test-ml)
    Alcotest.(check bool) "logor1" true (of_int (1 lor 2) = logor (of_int 1) (of_int 2));
    Alcotest.(check bool) "logor2" true (of_z (Z.of_int (1 lor 2)) = logor (of_z Z.one) (of_z (Z.of_int 2)));
    #+end_src

*** ~logxor~

    Bitwise logical exclusive or.

    #+begin_src ocaml :tangle (eval mli)
val logxor : t -> t -> t
    #+end_src

    #+begin_src ocaml :tangle (eval ml)
let logxor a b = 
  match a,b with
  | One a, One b -> One (One.logxor a b)
  | Many a, Many b -> Many (Many.logxor a b)
  | _ -> invalid_arg "Bitstring.logxor"

    #+end_src

    #+begin_src ocaml :tangle (eval test-ml)
    Alcotest.(check bool) "logxor1" true (of_int (1 lxor 2) = logxor (of_int 1) (of_int 2));
    Alcotest.(check bool) "logxor2" true (of_z (Z.of_int (1 lxor 2)) = logxor (of_z Z.one) (of_z (Z.of_int 2)));
    #+end_src

*** ~logand~

    Bitwise logical and.

    #+begin_src ocaml :tangle (eval mli)
val logand : t -> t -> t
    #+end_src

    #+begin_src ocaml :tangle (eval ml)
let logand a b = 
  match a,b with
  | One a, One b -> One (One.logand a b)
  | Many a, Many b -> Many (Many.logand a b)
  | _ -> invalid_arg "Bitstring.logand"

    #+end_src

    #+begin_src ocaml :tangle (eval test-ml)
    Alcotest.(check bool) "logand1" true (of_int (1 land 3) = logand (of_int 1) (of_int 3));
    Alcotest.(check bool) "logand2" true (of_z (Z.of_int (1 land 3)) = logand (of_z Z.one) (of_z (Z.of_int 3)));
    #+end_src

*** ~lognot~

    Bitwise logical negation.

    #+begin_src ocaml :tangle (eval mli)
val lognot : t -> t 
    #+end_src

    #+begin_src ocaml :tangle (eval ml)
let lognot = function
| One x -> One (One.lognot x)
| Many x -> Many (Many.lognot x)
    #+end_src

*** ~minus_one~

    Takes the integer representation of the bit string and removes one.

    #+begin_example
minus_one (of_int 10) = of_int 9
    #+end_example

    #+begin_src ocaml :tangle (eval mli)
val minus_one : t -> t
    #+end_src

    #+begin_src ocaml :tangle (eval ml)
let minus_one = function
| One x -> One (One.minus_one x)
| Many x -> Many (Many.minus_one x)
    #+end_src

*** ~plus_one~

    Takes the integer representation of the bit string and adds one.

    #+begin_example
plus_one (of_int 10) = of_int 11
    #+end_example

    #+begin_src ocaml :tangle (eval mli)
val plus_one : t -> t
    #+end_src

    #+begin_src ocaml :tangle (eval ml)
let plus_one = function
| One x -> One (One.plus_one x)
| Many x -> Many (Many.plus_one x)
    #+end_src

*** ~trailing_zeros~

    Returns the number of trailing zeros in the bit string.
    
    #+begin_src ocaml :tangle (eval mli)
val trailing_zeros : t -> int
    #+end_src

    #+begin_src ocaml :tangle (eval ml)
let trailing_zeros = function
| One x -> One.trailing_zeros x
| Many x -> Many.trailing_zeros x
    #+end_src

*** ~hamdist~

    Returns the Hamming distance, i.e. the number of bits differing
    between two bit strings.
    
    #+begin_src ocaml :tangle (eval mli)
val hamdist : t -> t -> int
    #+end_src

    #+begin_src ocaml :tangle (eval ml)
let hamdist a b = match a, b with
| One a, One b -> One.hamdist a b
| Many a, Many b -> Many.hamdist a b
| _ -> invalid_arg "Bitstring.hamdist"
    #+end_src

*** ~popcount~

    Returns the number of bits set to one in the bit string.
    
    #+begin_src ocaml :tangle (eval mli)
val popcount : t -> int
    #+end_src

    #+begin_src ocaml :tangle (eval ml)
let popcount = function
| One x -> One.popcount x
| Many x -> Many.popcount x
    #+end_src
    
*** ~to_list~

    Converts a bit string into a list of integers indicating the
    positions where the bits are set to ~1~. The first value for the
    position is not ~0~ but ~1~.
    
    #+begin_example
Bitstring.to_list (of_int 5);;
- : int list = [1; 3]
    #+end_example

    #+begin_src ocaml :tangle (eval mli)
val to_list : ?accu:(int list) -> t -> int list                                                              
    #+end_src

    #+begin_src ocaml :tangle (eval ml)
let rec to_list ?(accu=[]) = function
  | t when (is_zero t) -> List.rev accu
  | t -> let newlist =
           (trailing_zeros t + 1)::accu
         in
         logand t @@ minus_one t
         |> (to_list [@tailcall]) ~accu:newlist
    #+end_src

   #+begin_src ocaml :tangle (eval test-ml)
    Alcotest.(check bool) "to_list" true ([ 1 ; 3 ; 4 ; 6 ] = (to_list (of_int 45)));
   #+end_src

*** ~permutations~

    ~permutations m n~ generates the list of all possible ~n~-bit
    strings with ~m~ bits set to ~1~.
    Algorithm adapted from [[https://graphics.stanford.edu/~seander/bithacks.html#NextBitPermutation][Bit twiddling hacks]].
    
    #+begin_example
Bitstring.permutations 2 4
|> List.map (fun x -> Format.asprintf "%a" Bitstring.pp x) ;;
- : string list =
["++--------------------------------------------------------------";
 "+-+-------------------------------------------------------------";
 "-++-------------------------------------------------------------";
 "+--+------------------------------------------------------------";
 "-+-+------------------------------------------------------------";
 "--++------------------------------------------------------------"]
    #+end_example
    
    #+begin_src ocaml :tangle (eval mli)
val permutations : int -> int -> t list
    #+end_src

    #+begin_src ocaml :tangle (eval ml)
let permutations m n =                                                                   

  let rec aux k u rest =
    if k=1 then
      List.rev (u :: rest)
    else
      let t   = logor u @@ minus_one u in
      let t'  = plus_one t in
      let not_t = lognot t in
      let neg_not_t = neg not_t in
      let t'' = shift_right (minus_one @@ logand not_t neg_not_t) (trailing_zeros u + 1) in
      (*
      let t'' = shift_right (minus_one (logand (lognot t) t')) (trailing_zeros u + 1) in
      ,*)
      (aux [@tailcall]) (k-1) (logor t' t'') (u :: rest)
  in
  aux (Util.binom n m) (minus_one (shift_left_one n m)) []
    #+end_src

    #+begin_src ocaml :tangle (eval test-ml)
check "permutations"
  (permutations 2 4 = List.map of_int
     [ 3 ; 5 ; 6 ; 9 ; 10 ; 12 ]);
    #+end_src

** Printers

   Printers can print as a string (~pp_string~) or as an integer (~pp_int~).

   #+begin_src ocaml :tangle (eval mli)
val pp : Format.formatter -> t -> unit
   #+end_src

   #+begin_src ocaml :tangle (eval ml)
let pp ppf = function
| One x -> One.pp ppf x
| Many x -> Many.pp ppf x
   #+end_src

** Tests

   #+begin_src ocaml :tangle (eval test-ml)
   ()
   
let tests = [
    "all", `Quick, test_all;
  ]
   #+end_src