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(* \chaptertitle{PDFFun}{PDF Functions} *)
open Utility
open Pdf
(* \section{Types} *)
(* \intf Postscript calculator functions. *)
type calculator =
| If of calculator list
| IfElse of calculator list * calculator list
| Bool of bool
| Float of float
| Int of int32
| Abs | Add | Atan | Ceiling | Cos | Cvi | Cvr
| Div | Exp | Floor | Idiv | Ln | Log | Mod
| Mul | Neg | Round | Sin | Sqrt | Sub | Truncate
| And | Bitshift | Eq | Ge | Gt | Le | Lt | Ne | Not
| Or | Xor | Copy | Exch | Pop | Dup | Index | Roll
(* \intf Sampled functions. *)
type sampled =
{size : int list;
order : int;
encode : float list;
decode : float list;
bps : int;
samples : int32 array}
(* \intf Interpolation functions. *)
and interpolation =
{c0 : float list;
c1 : float list;
n : float}
(* \intf Stitching functions. *)
and stitching =
{functions : pdf_fun list;
bounds : float list;
stitch_encode : float list}
(* \intf Collect the above types into a single type. *)
and pdf_fun_kind =
| Interpolation of interpolation
| Stitching of stitching
| Sampled of sampled
| Calculator of calculator list
(* \intf Main type. *)
and pdf_fun =
{func : pdf_fun_kind;
domain : float list;
range : float list option}
(* \section{Printing functions} *)
(* Build a string of a calculator function. For debug only, since could exceed
string length limit. *)
let rec string_of_calculator_inner = function
| If l ->
string_of_calculator l ^ " if"
| IfElse (l, l') ->
string_of_calculator l ^ " " ^ string_of_calculator l' ^ " ifelse"
| Bool true -> "true"
| Bool false -> "false"
| Float f -> string_of_float f
| Int i -> Int32.to_string i
| Abs -> "abs" | Add -> "add" | Atan -> "atan" | Ceiling -> "ceiling"
| Cos -> "cos" | Cvi -> "cvi" | Cvr -> "cvr" | Div -> "div"
| Exp -> "exp" | Floor -> "floor" | Idiv -> "idiv" | Ln -> "ln"
| Log -> "log" | Mod -> "mod" | Mul -> "mul" | Neg -> "neg"
| Round -> "round" | Sin -> "sin" | Sqrt -> "sqrt" | Sub -> "sub"
| Truncate -> "truncate" | And -> "and" | Bitshift -> "bitshift"
| Eq -> "eq" | Ge -> "ge" | Gt -> "gt" | Le -> "le" | Lt -> "lt"
| Ne -> "ne" | Not -> "not" | Or -> "or" | Xor -> "xor"
| Copy -> "copy" | Exch -> "exch" | Pop -> "pop" | Dup -> "dup"
| Index -> "index" | Roll -> "roll"
and string_of_calculator cs =
let ops =
fold_right ( ^ ) (interleave " " (map string_of_calculator_inner cs)) ""
in
"{" ^ ops ^ "}"
(* Print a function out for debug. *)
let rec print_function f =
print_string "Domain...\n";
print_floats f.domain;
begin match f.range with
| None -> print_string "null range\n"
| Some values -> print_floats values
end;
match f.func with
| Sampled s ->
print_string "Sampled\n";
print_string "size: ";
print_ints s.size;
print_string "order: ";
print_int s.order;
print_string "\nencode:\n";
print_floats s.encode;
print_string "decode:\n";
print_floats s.decode;
print_string "original bits per sample..\n";
print_int s.bps;
print_string "\ndata:\n";
print_int32s (Array.to_list s.samples)
| Interpolation i ->
print_string "Interpolation\n";
print_string "C0:\n";
print_floats i.c0;
print_string "C1:\n";
print_floats i.c1;
Printf.printf "n = %f\n" i.n;
| Stitching s ->
print_string "Stitching\n";
iter print_function s.functions;
print_string "Bounds:\n";
print_floats s.bounds;
print_string "Encode:\n";
print_floats s.stitch_encode;
| Calculator c ->
print_string "Calculator:\n";
print_string (string_of_calculator c)
(* \section{Parsing Calculator Functions} *)
let keyword_of_string = function
| "abs" -> Abs | "add" -> Add | "atan" -> Atan | "ceiling" -> Ceiling
| "cos" -> Cos | "cvr" -> Cvr | "div" -> Div | "exp" -> Exp
| "floor" -> Floor | "idiv" -> Idiv | "ln" -> Ln | "log" -> Log
| "mod" -> Mod | "mul" -> Mul | "neg" -> Neg | "round" -> Round
| "sin" -> Sin | "sqrt" -> Sqrt | "sub" -> Sub
| "truncate" -> Truncate | "and" -> And | "bitshift" -> Bitshift
| "eq" -> Eq | "ge" -> Ge | "gt" -> Gt | "le" -> Le | "lt" -> Lt
| "ne" -> Ne | "not" -> Not | "or" -> Or | "xor" -> Xor
| "copy" -> Copy | "exch" -> Exch | "pop" -> Pop
| "dup" -> Dup | "index" -> Index | "roll" -> Roll
| s ->
Printf.eprintf "%s" ("Bad keyword " ^ s); raise (Assert_failure ("", 0, 0))
let parse_calculator s =
let lexemes =
Cgenlex.lex (Pdfio.input_of_bytestream (bytestream_of_string s))
in
let rec strip_outer_braces = function
| Cgenlex.Ident ("{" | "}")::t ->
rev (strip_outer_braces (rev t))
| x -> x
and group_operators = function
| [] -> []
| Cgenlex.Ident "{"::t ->
let ops, rest = cleavewhile (neq (Cgenlex.Ident "}")) t in
ops::group_operators (tl rest)
| h::t -> [h]::group_operators t
and parse = function
| [] -> []
| l::l'::[Cgenlex.Ident "ifelse"]::t ->
IfElse (procss l, procss l')::parse t
| l::[Cgenlex.Ident "if"]::t -> If (procss l)::parse t
| [Cgenlex.Ident "true"]::t -> Bool true::parse t
| [Cgenlex.Ident "false"]::t -> Bool false::parse t
| [Cgenlex.Float f]::t -> Float f::parse t
| [Cgenlex.Int i]::t-> Int (i32ofi i)::parse t (* FIXME: range *)
| [Cgenlex.Ident x]::t -> keyword_of_string x::parse t
| h::_ -> (*i Printf.eprintf (Cgenlex.string_of_tokens h); i*) raise (Failure "Bad lexeme")
and procss lexemes =
try
parse (group_operators (strip_outer_braces lexemes))
with
_ -> raise (Pdf.PDFError "Cannot parse Type 4 function")
in
procss lexemes
(* \section{Parsing functions} *)
let rec parse_function pdf f =
let f = direct pdf f in
let getnum_direct o = getnum (direct pdf o) in
let domain =
match lookup_fail "No /Domain" pdf "/Domain" f with
| Array ns -> map getnum_direct ns
| _ -> raise (PDFError "Bad /Domain")
and range =
match lookup_direct pdf "/Range" f with
| Some (Array ns) -> Some (map getnum_direct ns)
| _ -> None
in
let func =
match lookup_fail "no /FunctionType" pdf "/FunctionType" f with
| Integer 0 ->
let size =
match lookup_fail "no /Size (sampled fun)" pdf "/Size" f with
| Array ns ->
map
(function
| Integer n -> n
| _ -> raise (PDFError "bad /Size (sampled fun)"))
ns
| _ -> raise (PDFError "Bad /Size (sampled fun)")
in
let order =
match lookup_direct pdf "/Order" f with
| Some (Integer n) -> n
| _ -> 1
and encode =
match lookup_direct pdf "/Encode" f with
| Some (Array ns) when length ns = 2 * length size ->
map getnum ns
| _ ->
interleave_lists
(many 0. (length size))
(map (fun x -> float (x - 1)) size)
and decode =
match lookup_direct pdf "/Decode" f with
| Some (Array ns) -> map getnum ns
| _ ->
match range with
| Some r -> r
| None -> raise (PDFError "No /Range")
in
let bitspersample =
match
lookup_fail "no /BitsPerSample" pdf "/BitsPerSample" f
with
| Integer i -> i
| _ -> raise (PDFError "Bad /BitsPerSample")
in
let data =
Pdfcodec.decode_pdfstream pdf f;
let samples =
fold_left ( * ) 1 size * (length decode / 2)
and bitstream =
match f with
| Stream {contents = _, Got data} ->
Pdfio.bitstream_of_input (Pdfio.input_of_bytestream data)
| _ -> raise (Assert_failure ("", 0, 0))
in
let data = Array.make samples 1l in
for i = 0 to Array.length data - 1 do
data.(i) <- Pdfio.getval_32 bitstream bitspersample
done;
data
in
Sampled
{size = size;
order = order;
encode = encode;
decode = decode;
bps = bitspersample;
samples = data}
| Integer 2 ->
let c0 =
match lookup_direct pdf "/C0" f with
| Some (Array ns) -> map getnum_direct ns
| _ -> [0.]
and c1 =
match lookup_direct pdf "/C1" f with
| Some (Array ns) -> map getnum_direct ns
| _ -> [1.]
and n =
getnum (lookup_fail "No /N in Type 2 fun" pdf "/N" f)
in
Interpolation {c0 = c0; c1 = c1; n = n}
| Integer 3 ->
let functions =
match lookup_fail "no /Functions" pdf "/Functions" f with
| Array fs -> fs
| _ -> raise (PDFError "Bad /Functions")
and bounds =
match lookup_fail "no /Bounds" pdf "/Bounds" f with
| Array fs -> fs
| _ -> raise (PDFError "Bad /Bounds")
and encode =
match lookup_fail "no /Encode" pdf "/Encode" f with
| Array fs -> fs
| _ -> raise (PDFError "Bad /Bounds")
in
Stitching
{functions = map (parse_function pdf) functions;
bounds = map getnum_direct bounds;
stitch_encode = map getnum_direct encode}
| Integer 4 ->
(* Read contents of stream, build string, parse. *)
Pdfcodec.decode_pdfstream pdf f;
begin match f with
| Stream {contents = _, Got data} ->
Calculator (parse_calculator (string_of_bytestream data))
| _ -> raise (PDFError "This is not a function")
end
| _ -> raise (PDFError "Unknown function type")
in
{domain = domain; range = range; func = func}
(* \section{Evaluating Sampled Functions} *)
(* \intf Inappropriate inputs have been given to a function. *)
exception BadFunctionEvaluation of string
let interpolate x xmin xmax ymin ymax =
ymin +. ((x -. xmin) *. ((ymax -. ymin) /. (xmax -. xmin)))
(* Evaluate a sampled function. We support only linear interpolation and then
only sensibly for one-dimensional functions. Although results will be
produced for higher dimensions, the results will not be fully accurate. *)
let eval_function_sampled f s clamped_inputs =
(* 1. Encode the input values *)
(*i Printf.printf "eval_function_sampled: length of clamped_inputs %i input is
* %f\n" (length clamped_inputs) (hd clamped_inputs); i*)
let range =
match f.range with
| None -> raise (BadFunctionEvaluation "No Range")
| Some r -> r
in
let d, d' = split (pairs_of_list f.domain)
and e, e' = split (pairs_of_list s.encode)
and dec, dec' = split (pairs_of_list s.decode)
and r, r' = split (pairs_of_list range) in
let encoded_inputs =
map5 interpolate clamped_inputs d d' e e'
in
(* 2. Clip to the size of the table dimension *)
let clamped_encoded_inputs =
map2
(fun i s -> fmin (fmax i 0.) (float s -. 1.))
encoded_inputs
s.size
in
let read_table inputs =
let vals_to_read = length range / 2 in
let size = s.size in
if length size <> length inputs then
raise (BadFunctionEvaluation "Incompatible /Size with inputs");
let pos =
let multipliers =
let inputs' = if length inputs = 1 then [] else ilist 1 (length inputs - 1) in
1::
map
(function x -> fold_left ( * ) 1 (take size x)) inputs'
(* (ilist_fail_null 1 (length inputs - 1)) *)
in
(*i Printf.printf "%i inputs and %i multipliers" (length inputs)
(length multipliers); i*)
fold_left ( + ) 0 (map2 ( * ) inputs multipliers) * vals_to_read
in
Array.to_list (Array.sub s.samples pos vals_to_read)
in
(* 3. Read values from table. For now, just linear iterpolation. *)
let ceilings =
map (fun x -> toint (ceil x)) clamped_encoded_inputs
and floors =
map (fun x -> toint (floor x)) clamped_encoded_inputs
in
let outputs =
let ceiling_results = read_table ceilings
and floor_results = read_table floors in
(*i Printf.printf "length of floors: %i, length of ceilings %i\n"
(List.length ceiling_results) (List.length floor_results); i*)
map2
(fun x y ->
Int32.to_float x /. 2. +. Int32.to_float y /. 2.)
ceiling_results
floor_results
in
(* 4. Decode output values *)
let outputs_decoded =
map5
interpolate
outputs
(many 0. (length outputs))
(many (2. ** float s.bps -. 1.) (length outputs))
dec dec'
in
map3 (fun x r r' -> fmin (fmax x r) r') outputs_decoded r r'
(* \section{Evaluating Calculator Functions} *)
let eval_function_calculator clamped_inputs ops =
let s =
ref (map (fun i -> Float i) (rev clamped_inputs))
and typecheck () =
raise (BadFunctionEvaluation "Type error")
in
let rec getfloat () =
match !s with
| Int i::r -> s := r; i32tof i
| Float f::r -> s := r; f
| _ -> typecheck ()
and getint () =
match !s with
| Int i::r -> s := r; i
| _ -> typecheck ()
and getfloats () =
let x = getfloat () in x, getfloat ()
and getints () =
let x = getint () in x, getint ()
in
let rec eval k =
(*i flprint (string_of_calculator !s);
Printf.printf "Now op %s\n\n" (string_of_calculator [k]); i*)
match k with
| If l ->
begin match !s with
| Bool b::r -> s := r; if b then iter eval l
| _ -> typecheck ()
end
| IfElse (l, l') ->
begin match !s with
| Bool b::r -> s := r; iter eval (if b then l else l')
| _ -> typecheck ()
end
| (Bool _ | Float _ | Int _) as immediate ->
s =| immediate
| Abs ->
begin match !s with
| Float f::r -> s := Float (fabs f)::r
| Int i::r ->
let out =
if i = Int32.min_int
then (Float (i32tof Int32.max_int))
else (Int (Int32.abs i))
in
s := out::r
| _ -> typecheck ()
end
| Add ->
begin match !s with
| Int i::Int i'::r ->
s := Int (i32add i i')::r (*r FIXME: Overflow to float *)
| Int i::Float f::r
| Float f::Int i::r ->
s := Float (i32tof i +. f)::r
| Float f::Float f'::r ->
s := Float (f +. f')::r
| _ -> typecheck ()
end
| Atan ->
let num, den = getfloats () in
let result = atan2 num den in
s := Float result::tl (tl !s)
| Ceiling ->
begin match !s with
| Float f::r -> s := Float (ceil f)::r
| Int _::_ -> ()
| _ -> typecheck ()
end
| Cos ->
let f = getfloat () in
s := Float (cos (rad_of_deg f))::!s
| Cvi ->
begin match !s with
| Int _::r -> ()
| Float f::r -> s := Int (Int32.of_float (floor f))::r
| _ -> typecheck ()
end
| Cvr ->
begin match !s with
| Int i::r -> s := Float (i32tof i)::r
| Float f::r -> ()
| _ -> typecheck ()
end
| Div ->
let n, n' = getfloats () in
s := Float (n /. n')::!s
| Exp ->
let bse, exponent = getfloats () in
s := Float (bse ** exponent)::!s
| Floor ->
begin match !s with
| Int i::r -> ()
| Float f::r -> s := Int (Int32.of_float (floor f))::r
| _ -> typecheck ()
end
| Idiv ->
let i, i' = getints () in
s := Int (i32div i i')::!s
| Ln ->
let f = getfloat () in
s := Float (log f)::!s
| Log ->
let f = getfloat () in
s := Float (log10 f)::!s
| Mod ->
let i, i' = getints () in
s := Int (Int32.rem i i')::!s
| Mul ->
begin match !s with
| Int i::Int i'::r ->
s := Int (i32mul i i')::r (*r FIXME: Overflow to float *)
| Int i::Float f::r
| Float f::Int i::r ->
s := Float (i32tof i *. f)::r
| Float f::Float f'::r ->
s := Float (f *. f')::r
| _ -> typecheck ()
end
| Neg ->
begin match !s with
| Float f::r -> s := Float (~-.f)::r
| Int i::r ->
let out =
if i = Int32.min_int
then (Float (i32tof Int32.max_int))
else (Int (Int32.neg i))
in
s := out::r
| _ -> typecheck ()
end
| Round ->
begin match !s with
| Int _::_ -> ()
| Float f::r -> s := Int (Int32.of_float (round f))::r
| _ -> typecheck ()
end
| Sin ->
let f = getfloat () in
s := Float (sin (rad_of_deg f))::!s
| Sqrt ->
let f = getfloat () in
s := Float (sqrt f)::!s
| Sub ->
begin match !s with
| Int i::Int i'::r ->
s := Int (i32sub i' i)::r (*r \FIXME{Overflow to float} *)
| Int i::Float f::r ->
s := Float (f -. i32tof i)::r
| Float f::Int i::r ->
s := Float (i32tof i -. f)::r
| Float f::Float f'::r ->
s := Float (f' -. f)::r
| _ -> typecheck ()
end
| Truncate ->
begin match !s with
| Int _::_ -> ()
| Float f::r -> s := Int (i32ofi (toint f))::r
| _ -> typecheck ()
end
| And ->
begin match !s with
| Int i::Int i'::r ->
s := Int (Int32.logand i i')::r
| Bool b::Bool b'::r ->
s := Bool (b && b')::r
| _ -> typecheck ()
end
| Bitshift ->
let i = getint () in
let shift = i32toi (getint ()) in
let r =
if i < 0l
then Int32.shift_left i shift
else Int32.shift_right_logical i (abs shift)
in
s := Int r::!s
| Eq ->
begin match !s with
| a::b::r -> s := Bool (a = b)::r
| _ -> typecheck ()
end
| Ge ->
begin match !s with
| a::b::r -> s := Bool (a >= b)::r
| _ -> typecheck ()
end
| Gt ->
begin match !s with
| a::b::r -> s := Bool (a > b)::r
| _ -> typecheck ()
end
| Le ->
begin match !s with
| a::b::r -> s := Bool (a <= b)::r
| _ -> typecheck ()
end
| Lt ->
begin match !s with
| a::b::r -> s := Bool (a < b)::r
| _ -> typecheck ()
end
| Ne ->
begin match !s with
| a::b::r -> s := Bool (a <> b)::r
| _ -> typecheck ()
end
| Not ->
begin match !s with
| Int i::r ->
s := Int (Int32.lognot i)::r
| Bool b::Bool b'::r ->
s := Bool (not b)::r
| _ -> typecheck ()
end
| Or ->
begin match !s with
| Int i::Int i'::r ->
s := Int (Int32.logor i i')::r
| Bool b::Bool b'::r ->
s := Bool (b || b')::r
| _ -> typecheck ()
end
| Xor ->
begin match !s with
| Int i::Int i'::r ->
s := Int (Int32.logxor i i')::r
| Bool b::Bool b'::r ->
s := Bool (b |&| b')::r
| _ -> typecheck ()
end
| Copy ->
begin match !s with
| Int i::r when i >= 0l ->
s := take r (i32toi i) @ r
| _ -> typecheck ()
end
| Exch ->
begin match !s with
| a::b::r -> s := b::a::r
| _ -> typecheck ()
end
| Pop ->
begin match !s with
| a::r -> s := r
| _ -> typecheck ()
end
| Dup ->
begin match !s with
| a::r -> s := a::a::r
| _ -> typecheck ()
end
| Index ->
begin match !s with
| Int i::r when i >= 0l ->
let v = select (i32toi i + 1) r in
s := v::r
| _ -> typecheck ()
end
| Roll ->
let rec rotate j l =
if j = 0 then l else
match l with
| [] -> []
| h::t -> rotate (j - 1) (t @ [h])
and rotate_o j l =
if j = 0 then l else
match rev l with
| [] -> []
| h::t -> rotate (j - 1) ([h] @ rev t)
in
begin match !s with
| Int j::Int n::r when n >= 1l ->
let j = i32toi j and n = i32toi n in
let vals, rest = cleave r n in
let newvals =
if j > 0
then rotate j vals
else rotate_o ~-j vals
in
s := newvals @ rest
| _ -> typecheck ()
end
in
try
iter eval ops;
rev_map
(function
| Float x -> x
| _ -> raise (BadFunctionEvaluation "Type 4"))
!s
with
_ -> raise (BadFunctionEvaluation "Type 4")
(* \section{Evaluating functions} *)
(* \intf Evaluate a function on some inputs. *)
let rec eval_function f inputs =
let rec clampvals vals domain =
let clampval v d d' =
if v < d then d else if v > d' then d' else v
in
match vals, domain with
| [], [] -> []
| v::vs, d::d'::ds ->
clampval v d d'::clampvals vs ds
| _ -> raise (BadFunctionEvaluation "Domain wrong")
in
let clamped_inputs = clampvals inputs f.domain in
let outputs =
match f.func with
| Calculator ops ->
eval_function_calculator clamped_inputs ops
| Sampled s ->
eval_function_sampled f s clamped_inputs
| Interpolation f ->
let interp n c0 c1 i =
try
map2
(fun c0 c1 -> c0 +. i ** n *. (c1 -. c0)) c0 c1
with
Invalid_argument _ ->
raise (BadFunctionEvaluation "Interpolation")
in
flatten (map (interp f.n f.c0 f.c1) clamped_inputs)
| Stitching s ->
(* iPrintf.printf "First input to function: %f\n" (hd
* clamped_inputs); *)
match clamped_inputs, f.domain with
| [i], [d0; d1] ->
let points = [d0] @ s.bounds @ [d1] in
(*i flprint "points: ";
iter (Printf.printf "%f ") points;
flprint "\n";
flprint "s.stitch_encode: ";
iter (Printf.printf "%f ") s.stitch_encode;
flprint "\n"; i*)
let rec extract_subfunction points funs n =
match points, funs with
| p::p'::_, f::_ when i >= p && i < p' -> p, p', f, n
| p::p'::_, f::_ when i = p' -> p, p', f, n
| _::ps, _::fs -> extract_subfunction ps fs (n + 1)
| _ ->
raise (BadFunctionEvaluation "stitching: funs")
in
let d0', d1', f, n = extract_subfunction points s.functions 0 in
(*i Printf.printf "d0 = %f, d1 = %f, n = %i\n" d0' d1' n;
* i*)
let encode =
try
let a, b, c, d =
select (n + 1) points, select (n + 2) points,
select (n * 2 + 1) s.stitch_encode, select (n * 2 + 2) s.stitch_encode
in
(* Printf.printf "a = %f, b %f, c = %f, d = %f\n" a b
* c d; *)
fun x -> interpolate x a b c d
with
Invalid_argument "select" ->
raise (BadFunctionEvaluation "stitching: encode/domain")
in
eval_function f [encode i]
| _ -> raise (BadFunctionEvaluation "stitching: Bad arity")
in
let outputs =
match f.range with
| None -> outputs
| Some range -> clampvals outputs range
in
(* iter (Printf.printf "%f ") outputs;
flprint "\n"; *)
outputs
(*i let eval_function f vals =
Printf.printf "eval_function\n";
iter (Printf.printf "%f ") vals;
flprint "\n";
print_function f;
flprint "\n";
let vals = eval_function f vals in
iter (Printf.printf "%f ") vals;
flprint "\n";
vals i*)
let funtype_of_function f =
match f.func with
| Interpolation _ -> 2
| Stitching _ -> 3
| Sampled _ -> 0
| Calculator _ -> 4
let mkreal x = Pdf.Real x
let entries_of_interpolation i =
["/C0", Pdf.Array (map mkreal i.c0);
"/C1", Pdf.Array (map mkreal i.c1);
"/N", Pdf.Real i.n]
let rec entries_of_stitching pdf i =
(* Add the functions as objects. *)
let numbers =
map (fun f -> Pdf.Indirect (Pdf.addobj pdf (pdfobject_of_function pdf f))) i.functions
in
["/Functions", Pdf.Array numbers;
"/Bounds", Pdf.Array (map mkreal i.bounds);
"/Encode", Pdf.Array (map mkreal i.stitch_encode)]
and extra_entries_of_function pdf f =
match f.func with
| Interpolation i -> entries_of_interpolation i
| Stitching s -> entries_of_stitching pdf s
| Sampled s -> []
| Calculator c -> []
and pdfobject_of_function pdf f =
let domain =
Pdf.Array (map (function x -> Pdf.Real x) f.domain)
and range =
match f.range with
| None -> []
| Some fs -> ["/Range", Pdf.Array (map (function x -> Pdf.Real x) fs)]
in
Pdf.Dictionary
(["/FunctionType", Pdf.Integer (funtype_of_function f); "/Domain", domain]
@ range @ extra_entries_of_function pdf f)
|