Jensen's Device
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This task is an exercise in call by name.
You are encouraged to solve this task according to the task description, using any language you may know.
Jensen's Device is a computer programming technique devised by Danish computer scientist Jørn Jensen after studying the ALGOL 60 Report.
The following program was proposed to illustrate the technique. It computes the 100th harmonic number:
begin
integer i;
real procedure sum (i, lo, hi, term);
value lo, hi;
integer i, lo, hi;
real term;
comment term is passed by-name, and so is i;
begin
real temp;
temp := 0;
for i := lo step 1 until hi do
temp := temp + term;
sum := temp
end;
comment note the correspondence between the mathematical notation and the call to sum;
print (sum (i, 1, 100, 1/i))
end
The above exploits call by name to produce the correct answer (5.187...). It depends on the assumption that an expression passed as an actual parameter to a procedure would be re-evaluated in the caller's context every time the corresponding formal parameter's value was required. If the last parameter to sum had been passed by value, and assuming the initial value of i were 1, the result would have been 100 × 1/1 = 100.
Moreover, the first parameter to sum, representing the "bound" variable of the summation, must also be passed by name (or at least by reference), otherwise changes to it (made within sum) would not be visible in the caller's context when computing each of the values to be added. (On the other hand, the global variable does not have to use the same identifier, in this case i, as the formal parameter.)
Donald Knuth later proposed the Man or Boy Test as a more rigorous exercise.
11l
<lang 11l>F sum(&i, lo, hi, term)
V temp = 0.0
i = lo
L i <= hi
temp += term()
i++
R temp
F main()
Int i print(sum(&i, 1, 100, () -> 1 / @i))
main()</lang>
- Output:
5.18738
Ada
<lang ada>with Ada.Text_IO; use Ada.Text_IO;
procedure Jensen_Device is
function Sum
( I : not null access Float;
Lo, Hi : Float;
F : access function return Float
) return Float is
Temp : Float := 0.0;
begin
I.all := Lo;
while I.all <= Hi loop
Temp := Temp + F.all;
I.all := I.all + 1.0;
end loop;
return Temp;
end Sum;
I : aliased Float;
function Inv_I return Float is
begin
return 1.0 / I;
end Inv_I;
begin
Put_Line (Float'Image (Sum (I'Access, 1.0, 100.0, Inv_I'Access)));
end Jensen_Device;</lang>
5.18738E+00
ALGOL 60
Honor given where honor is due. In Algol 60, 'call by name' is the default argument evaluation.
begin
integer i;
real procedure sum (i, lo, hi, term);
value lo, hi;
integer i, lo, hi;
real term;
comment term is passed by-name, and so is i;
begin
real temp;
temp := 0;
for i := lo step 1 until hi do
temp := temp + term;
sum := temp
end;
comment note the correspondence between the mathematical notation and the call to sum;
print (sum (i, 1, 100, 1/i))
end
ALGOL 68
<lang algol68>BEGIN
INT i;
PROC sum = (REF INT i, INT lo, hi, PROC REAL term)REAL:
COMMENT term is passed by-name, and so is i COMMENT
BEGIN
REAL temp := 0;
i := lo;
WHILE i <= hi DO # ALGOL 68 has a "for" loop but it creates a distinct #
temp +:= term; # variable which would not be shared with the passed "i" #
i +:= 1 # Here the actual passed "i" is incremented. #
OD;
temp
END;
COMMENT note the correspondence between the mathematical notation and the call to sum COMMENT
print (sum (i, 1, 100, REAL: 1/i))
END</lang> Output: +5.18737751763962e +0
AppleScript
<lang AppleScript>set i to 0
on jsum(i, lo, hi, term)
set {temp, i's contents} to {0, lo}
repeat while i's contents ≤ hi
set {temp, i's contents} to {temp + (term's f(i)), (i's contents) + 1}
end repeat
return temp
end jsum
script term_func
on f(i)
return 1 / i
end f
end script
return jsum(a reference to i, 1, 100, term_func)</lang> Output: 5.18737751764
ARM Assembly
<lang ARM Assembly>
/* ARM assembly Raspberry PI */ /* program jensen.s */ /* compil as with option -mcpu=<processor> -mfpu=vfpv4 -mfloat-abi=hard */ /* link with gcc */
/* Constantes */ .equ EXIT, 1 @ Linux syscall /* Initialized data */ .data
szFormat: .asciz "Result = %.8f \n" .align 4
/* UnInitialized data */ .bss
/* code section */ .text .global main main:
mov r0,#1 @ first indice mov r1,#100 @ last indice adr r2,funcdiv @ address function bl funcSum vcvt.f64.f32 d1, s0 @ conversion double float for print by C ldr r0,iAdrszFormat @ display format vmov r2,r3,d1 @ parameter function printf for float double bl printf @ display float double
100: @ standard end of the program
mov r0, #0 @ return code mov r7, #EXIT @ request to exit program svc 0 @ perform system call
iAdrszFormat: .int szFormat /******************************************************************/ /* function sum */ /******************************************************************/ /* r0 contains begin */ /* r1 contains end */ /* r2 contains address function */
/* r0 return result */ funcSum:
push {r0,r3,lr} @ save registers
mov r3,r0
mov r0,#0 @ init r0
vmov s3,r0 @ and s3
vcvt.f32.s32 s3, s3 @ convert in float single précision (32bits)
1: @ begin loop
mov r0,r3 @ loop indice -> parameter function blx r2 @ call function address in r2 vadd.f32 s3,s0 @ addition float add r3,#1 @ increment indice cmp r3,r1 @ end ? ble 1b @ no loop vmov s0,s3 @ return float result in s0
100:
pop {r0,r3,lr} @ restaur registers
bx lr @ return
/******************************************************************/ /* compute 1/r0 */ /******************************************************************/ /* r0 contains the value */ /* r0 return result */ funcdiv:
push {r1,lr} @ save registers
vpush {s1} @ save float registers
cmp r0,#0 @ division by zero -> end
beq 100f
ldr r1,fUn @ load float constant 1.0
vmov s0,r1 @ in float register s3
vmov s1,r0 @
vcvt.f32.s32 s1, s1 @conversion in float single précision (32 bits)
vdiv.f32 s0,s0,s1 @ division 1/r0
@ and return result in s0
100:
vpop {s1} @ restaur float registers
pop {r1,lr} @ restaur registers
bx lr @ return
fUn: .float 1
</lang>
BASIC256
<lang BASIC256>subroutine Evaluation() lo = 1 : hi = 100 : temp = 0 for i = lo to hi temp += (1/i) ##r(i) next i print temp end subroutine
call Evaluation() end</lang>
- Output:
5.18737751764
BBC BASIC
<lang bbcbasic> PRINT FNsum(j, 1, 100, FNreciprocal)
END
DEF FNsum(RETURN i, lo, hi, RETURN func)
LOCAL temp
FOR i = lo TO hi
temp += FN(^func)
NEXT
= temp
DEF FNreciprocal = 1/i</lang>
Output:
5.18737752
Bracmat
<lang bracmat>( ( sum
= I lo hi Term temp
. !arg:((=?I),?lo,?hi,(=?Term))
& 0:?temp
& !lo:?!I
& whl
' ( !!I:~>!hi
& !temp+!Term:?temp
& 1+!!I:?!I
)
& !temp
)
& sum$((=i),1,100,(=!i^-1)) );</lang> Output:
14466636279520351160221518043104131447711/2788815009188499086581352357412492142272
C
<lang c>#include <stdio.h>
int i; double sum(int *i, int lo, int hi, double (*term)()) {
double temp = 0;
for (*i = lo; *i <= hi; (*i)++)
temp += term();
return temp;
}
double term_func() { return 1.0 / i; }
int main () {
printf("%f\n", sum(&i, 1, 100, term_func));
return 0;
}</lang> Output: 5.18738
Alternatively, C's macros provide a closer imitation of ALGOL's call-by-name semantics: <lang c>#include <stdio.h>
int i;
- define sum(i, lo_byname, hi_byname, term) \
({ \
int lo = lo_byname; \
int hi = hi_byname; \
\
double temp = 0; \
for (i = lo; i <= hi; ++i) \
temp += term; \
temp; \
})
int main () {
printf("%f\n", sum(i, 1, 100, 1.0 / i));
return 0;
}</lang> Output: 5.187378
C#
Can be simulated via lambda expressions: <lang csharp>using System;
class JensensDevice {
public static double Sum(ref int i, int lo, int hi, Func<double> term)
{
double temp = 0.0;
for (i = lo; i <= hi; i++)
{
temp += term();
}
return temp;
}
static void Main()
{
int i = 0;
Console.WriteLine(Sum(ref i, 1, 100, () => 1.0 / i));
}
}</lang>
C++
<lang cpp>
- include <iostream>
- define SUM(i,lo,hi,term)\
[&](const int _lo,const int _hi){\
decltype(+(term)) sum{};\
for (i = _lo; i <= _hi; ++i) sum += (term);\
return sum;\
}((lo),(hi))
int i; double sum(int &i, int lo, int hi, double (*term)()) {
double temp = 0;
for (i = lo; i <= hi; i++)
temp += term();
return temp;
} double term_func() { return 1.0 / i; }
int main () {
std::cout << sum(i, 1, 100, term_func) << std::endl; std::cout << SUM(i,1,100,1.0/i) << "\n"; return 0;
}</lang> Output: 5.18738 5.18738
Clipper
With hindsight Algol60 provided this feature in a way that is terrible for program maintenance, because the calling code looks innocuous. <lang Clipper>// Jensen's device in Clipper (or Harbour) // A fairly direct translation of the Algol 60 // John M Skelton 11-Feb-2012
function main() local i ? transform(sum(@i, 1, 100, {|| 1 / i}), "##.###############")
// @ is the quite rarely used pass by ref, {|| ...} is a
// code block (an anonymous function, here without arguments)
// The @i makes it clear that something unusual is occurring;
// a called function which modifies a parameter is commonly
// poor design!
return 0
function sum(i, lo, hi, bFunc) local temp := 0 for i = lo to hi
temp += eval(bFunc)
next i return temp </lang>
Common Lisp
Common Lisp does not have call-by-name for functions; however, it can be directly simulated by a macro wrapping selected parameters in lambdas.
<lang lisp>(declaim (inline %sum))
(defun %sum (lo hi func)
(loop for i from lo to hi sum (funcall func i)))
(defmacro sum (i lo hi term)
`(%sum ,lo ,hi (lambda (,i) ,term)))</lang>
<lang lisp>CL-USER> (sum i 1 100 (/ 1 i)) 14466636279520351160221518043104131447711/2788815009188499086581352357412492142272 CL-USER> (float (sum i 1 100 (/ 1 i))) 5.1873775</lang>
D
There are better ways to do this in D, but this is closer to the original Algol version: <lang d>double sum(ref int i, in int lo, in int hi, lazy double term) pure @safe /*nothrow @nogc*/ {
double result = 0.0;
for (i = lo; i <= hi; i++)
result += term();
return result;
}
void main() {
import std.stdio;
int i; sum(i, 1, 100, 1.0/i).writeln;
}</lang>
- Output:
5.18738
DWScript
Must use a "while" loop, as "for" loop variables are restricted to local variable for code clarity, and this indeed a case where any kind of extra clarity helps. <lang delphi>function sum(var i : Integer; lo, hi : Integer; lazy term : Float) : Float; begin
i:=lo;
while i<=hi do begin
Result += term;
Inc(i);
end;
end;
var i : Integer;
PrintLn(sum(i, 1, 100, 1.0/i));</lang> Output: 5.187...
E
In E, the distinct mutable locations behind assignable variables can be reified as Slot objects. The E language allows a variable name (noun) to be bound to a particular slot, and the slot of an already-bound noun to be extracted, using the & operator.
(The definition of the outer i has been moved down to emphasize that it is unrelated to the i inside of sum.)
<lang e>pragma.enable("one-method-object") # "def _.get" is experimental shorthand def sum(&i, lo, hi, &term) { # bind i and term to passed slots
var temp := 0
i := lo
while (i <= hi) { # E has numeric-range iteration but it creates a distinct
temp += term # variable which would not be shared with the passed i
i += 1
}
return temp
} {
var i := null
sum(&i, 1, 100, def _.get() { return 1/i })
}</lang>
1/i is not a noun, so there is no slot associated with it; so we use def _.get() { return 1/i } to define a slot object which does the computation when it is read as a slot.
The value returned by the above program (expression) is 5.187377517639621.
This emulation of the original call-by-name is of course unidiomatic; a natural version of the same computation would be:
<lang e>def sum(lo, hi, f) {
var temp := 0
for i in lo..hi { temp += f(i) }
return temp
} sum(1, 100, fn i { 1/i })</lang>
Elixir
<lang elixir>defmodule JensenDevice do
def task, do: sum( 1, 100, fn i -> 1 / i end ) defp sum( i, high, _term ) when i > high, do: 0 defp sum( i, high, term ) do temp = term.( i ) temp + sum( i + 1, high, term ) end
end
IO.puts JensenDevice.task</lang>
- Output:
5.1873775176396215
Erlang
No call by name, no macros, so I use a fun(ction). Actually, the the macro part is a lie. Somebody else, that knows how, could do a parse transform.
<lang Erlang> -module( jensens_device ).
-export( [task/0] ).
task() ->
sum( 1, 100, fun (I) -> 1 / I end ).
sum( I, High, _Term ) when I > High -> 0; sum( I, High, Term ) ->
Temp = Term( I ), Temp + sum( I + 1, High, Term ).
</lang>
- Output:
4> jensens_device:task(). 5.1873775176396215
F#
<lang fsharp> printfn "%.14f" (List.fold(fun n g->n+1.0/g) 0.0 [1.0..100.0]);; </lang>
- Output:
5.18737751763962
Factor
Similar to the Java and Kotlin examples: <lang factor>: sum ( lo hi term -- x ) [ [a,b] ] dip map-sum ; inline
1 100 [ recip ] sum .</lang>
This version is a bit closer to the original, as it increments i in the caller's namespace.
<lang factor>SYMBOL: i
- sum ( i lo hi term -- x )
[ [a,b] ] dip pick [ inc ] curry compose map-sum nip ; inline
i 1 100 [ recip ] sum .</lang>
- Output:
5+522561233577855727314756256041670736351/2788815009188499086581352357412492142272
Forth
This version passes i on the stack:
<lang forth>: sum 0 s>f 1+ swap ?do i over execute f+ loop drop ;
- noname s>f 1 s>f fswap f/ ; 1 100 sum f.</lang>
Output: 5.18737751763962
The following version passes i and 1/i as execution tokens and is thus closer to the original, but less idiomatic:
<lang forth>fvariable ii \ i is a Forth word that we need
- sum ( xt1 lo hi xt2 -- r )
0e swap 1+ rot ?do ( addr xt r1 ) i s>f over execute f! dup execute f+ loop 2drop ;
' ii 1 100 :noname 1e ii f@ f/ ; sum f.</lang>
Fortran
Fortran does not offer call-by-name in the manner of the Algol language. It passes parameters by reference (i.e. by passing the storage address) and alternatively uses copy-in, copy-out to give the same effect, approximately, as by reference. If a parameter is an arithmetic expression, it will be evaluated and its value stored in a temporary storage area, whose address will be passed to the routine. This evaluation is done once only for each call, thus vitiating the repeated re-evaluation required by Jensen's device every time within the routine that the parameter is accessed. So, this will not work<lang Fortran> FUNCTION SUM(I,LO,HI,TERM)
SUM = 0
DO I = LO,HI
SUM = SUM + TERM
END DO
END FUNCTION SUM
WRITE (6,*) SUM(I,1,100,1.0/I)
END</lang>
Here, type declarations have been omitted to save space because they won't help - until there appears a "BY NAME" or some such phrasing. Although variable I in the calling routine will have its value adjusted as the DO-loop in SUM proceeds (the parameter being passed by reference), this won't affect the evaluation of 1.0/I, which will be performed once using whatever value is in the caller's variable (it is uninitialised, indeed, undeclared also and so by default an integer) then the function is invoked with the address of the location containing that result. The function will make many references to that result, obtaining the same value each time. The fact that the caller's I will be changed each time doesn't matter.
Fortran does offer a facility to pass a function as a parameter using the EXTERNAL declaration, as follows - SUM is a F90 library function, so a name change to SUMJ: <lang Fortran> FUNCTION SUMJ(I,LO,HI,TERM) !Attempt to follow Jensen's Device...
INTEGER I !Being by reference is workable.
INTEGER LO,HI !Just as any other parameters.
EXTERNAL TERM !Thus, not a variable, but a function.
SUMJ = 0
DO I = LO,HI !The specified span.
SUMJ = SUMJ + TERM(I) !Number and type of parameters now apparent.
END DO !TERM will be evaluated afresh, each time.
END FUNCTION SUMJ !So, almost there.
FUNCTION THIS(I) !A function of an integer.
INTEGER I
THIS = 1.0/I !Convert to floating-point.
END !Since 1/i will mostly give zero.
PROGRAM JENSEN !Aspiration.
EXTERNAL THIS !Thus, not a variable, but a function.
INTEGER I !But this is a variable, not a function.
WRITE (6,*) SUMJ(I,1,100,THIS) !No statement as to the parameters of THIS.
END</lang>
The result of this is 5.187378, however it does not follow the formalism of Jensen's Device. The invocation statement SUMJ(I,1,100,THIS) does not contain the form of the function but only its name, and the function itself is defined separately. This means that the convenience of different functions via the likes of SUM(I,1,100,1.0/I**2) is unavailable, a separately-defined function with its own name must be defined for each such function. Further, the SUM routine must invoke TERM(I) itself, explicitly supplying the appropriate parameter. And the fact that variable I is a parameter to SUM is an irrelevance, and might as well be omitted from SUMJ.
Incidentally, a subroutine such as TEST(A,B) invoked as TEST(X,X) enables the discovery of copy-in, copy-out parameter passing. Within the routine, modify the value of A and look to see if B suddenly has a new value also.
FreeBASIC
<lang freebasic>Sub Evaluation
Dim As Integer i, lo = 1, hi = 100
Dim As Double temp = 0
For i = lo To hi
temp += (1/i) r(i)
Next i
Print temp
End Sub
Evaluation Sleep</lang>
- Output:
5.187377517639621
Go
<lang go>package main
import "fmt"
var i int
func sum(i *int, lo, hi int, term func() float64) float64 {
temp := 0.0
for *i = lo; *i <= hi; (*i)++ {
temp += term()
}
return temp
}
func main() {
fmt.Printf("%f\n", sum(&i, 1, 100, func() float64 { return 1.0 / float64(i) }))
}</lang>
- Output:
5.187378
Groovy
Solution: <lang groovy>def sum = { i, lo, hi, term ->
(lo..hi).sum { i.value = it; term() }
} def obj = [:] println (sum(obj, 1, 100, { 1 / obj.value }))</lang>
Output:
5.1873775176
Haskell
<lang haskell>import Control.Monad.ST import Data.STRef
sum_ :: STRef s Double -> Double -> Double -> ST s Double -> ST s Double sum_ ref_i lo hi term = sum <$> mapM ((>> term) . writeSTRef ref_i) [lo .. hi]
foo :: Double foo =
runST $
do i <- newSTRef undefined -- initial value doesn't matter
sum_ i 1 100 $ recip <$> readSTRef i
main :: IO () main = print foo</lang>
- Output:
5.187377517639621
Huginn
<lang huginn>harmonic_sum( i, lo, hi, term ) {
temp = 0.0;
i *= 0.0;
i += lo;
while ( i <= hi ) {
temp += term();
i += 1.0;
}
return ( temp );
}
main() {
i = 0.0;
print( "{}\n".format( harmonic_sum( i, 1.0, 100.0, @[i](){ 1.0 / i; } ) ) );
}</lang>
- Output:
5.18737751764
Icon and Unicon
Traditional call by name and reference are not features of Icon/Unicon. Procedures parameters are passed by value (immutable types) and reference (mutable types). However, a similar effect may be accomplished by means of co-expressions. The example below was selected for cleanliness of calling.
<lang Icon>record mutable(value) # record wrapper to provide mutable access to immutable types
procedure main()
A := mutable() write( sum(A, 1, 100, create 1.0/A.value) )
end
procedure sum(A, lo, hi, term)
temp := 0
every A.value := lo to hi do
temp +:= @^term
return temp
end</lang>
Refreshing the co-expression above is more expensive to process but to avoid it requires unary alternation in the call. <lang Icon> write( sum(A, 1, 100, create |1.0/A.value) ) ...
temp +:= @term</lang>
Alternately, we can use a programmer defined control operator (PDCO) approach that passes every argument as a co-expression. Again the refresh co-expression/unary iteration trade-off can be made. The call is cleaner looking but the procedure code is less clear. Additionally all the parameters are passed as individual co-expressions. <lang Icon> write( sum{A.value, 1, 100, 1.0/A.value} ) ... procedure sum(X) ...
every @X[1] := @X[2] to @X[3] do
temp +:= @^X[4]</lang>
J
Solution: <lang j>jensen=: monad define
'name lo hi expression'=. y temp=. 0 for_n. lo+i.1+hi-lo do. (name)=. n temp=. temp + ".expression end.
)</lang> Example: <lang j> jensen 'i';1;100;'1%i'
5.18738</lang>
Note, however, that in J it is reasonably likely that the expression (or an obvious variation on the expression) can deal with the looping itself. And in typical use this often simplifies to entering the expression and data directly on the command line.
And another obvious variation here would be turning the expression into a named entity (if it has some lasting usefulness).
Java
This is Java 8.
<lang java>import java.util.function.*; import java.util.stream.*;
public class Jensen {
static double sum(int lo, int hi, IntToDoubleFunction f) {
return IntStream.rangeClosed(lo, hi).mapToDouble(f).sum();
}
public static void main(String args[]) {
System.out.println(sum(1, 100, (i -> 1.0/i)));
}
} </lang> The program prints '5.187377517639621'.
Java 7 is more verbose, but under the hood does essentially the same thing:
<lang java>public class Jensen2 {
interface IntToDoubleFunction {
double apply(int n);
}
static double sum(int lo, int hi, IntToDoubleFunction f) {
double res = 0;
for (int i = lo; i <= hi; i++)
res += f.apply(i);
return res;
}
public static void main(String args[]) {
System.out.println(
sum(1, 100,
new IntToDoubleFunction() {
public double apply(int i) { return 1.0/i;}
}));
}
} </lang>
JavaScript
Uses an object o instead of integer pointer i, as the C example does.
<lang javascript>var obj;
function sum(o, lo, hi, term) {
var tmp = 0; for (o.val = lo; o.val <= hi; o.val++) tmp += term(); return tmp;
}
obj = {val: 0}; alert(sum(obj, 1, 100, function() {return 1 / obj.val}));</lang> The alert shows us '5.187377517639621'.
Joy
<lang Joy>100 [0] [[1.0 swap /] dip +] primrec.</lang> Joy does not have named parameters. Neither i nor 1/i are visible in the program.
jq
The technique used in the Javascript example can also be used in jq, but in jq it is more idiomatic to use "." to refer to the current term. For example, using sum/3 defined below, we can write: sum(1; 100; 1/.) to perform the task. <lang jq>def sum(lo; hi; term):
reduce range(lo; hi+1) as $i (0; . + ($i|term));
- The task:
sum(1;100;1/.)</lang>
- Output:
$ jq -n -f jensen.jq 5.187377517639621
Julia
<lang julia>macro sum(i, loname, hiname, term)
return quote
lo = $loname
hi = $hiname
tmp = 0.0
for i in lo:hi
tmp += $term
end
return tmp
end
end
i = 0 @sum(i, 1, 100, 1.0 / i)</lang>
Kotlin
<lang scala>fun sum(lo: Int, hi: Int, f: (Int) -> Double) = (lo..hi).sumByDouble(f)
fun main(args: Array<String>) = println(sum(1, 100, { 1.0 / it }))</lang>
Lua
<lang Lua> function sum(var, a, b, str)
local ret = 0
for i = a, b do
ret = ret + setfenv(loadstring("return "..str), {[var] = i})()
end
return ret
end print(sum("i", 1, 100, "1/i")) </lang>
M2000 Interpreter
The definition of the lazy function has two statements. First statement is a Module with one argument, the actual name of Jensen`s_Device, which make the function to get the same scope as module Jensen`s_Device, and the second statement is =1/i which return the expression.
<lang M2000 Interpreter> Module Jensen`s_Device {
Def double i
Report Lazy$(1/i) ' display the definition of the lazy function
Function Sum (&i, lo, hi, &f()) {
def double temp
For i= lo to hi {
temp+=f()
}
=temp
}
Print Sum(&i, 1, 100, Lazy$(1/i))==5.1873775176392 ' true
Print i=101 ' true
} Jensen`s_Device </lang>
Using Decimal for better accuracy. change &i to &any to show that: when any change, change i, so f() use this i. <lang M2000 Interpreter> Module Jensen`s_Device {
Def decimal i
Function Sum (&any, lo, hi, &f()) {
def decimal temp
For any= lo to hi {
temp+=f()
}
=temp
}
Print Sum(&i, 1, 100, Lazy$(1/i))=5.1873775176396202608051176755@ ' true
Print i=101 ' true
} Jensen`s_Device </lang>
Many other examples use single float. So this is one for single. <lang M2000 Interpreter> Module Jensen`s_Device {
Def single i
Function Sum (&any, lo, hi, &f()) {
def single temp
For any= lo to hi {
temp+=f()
}
=temp
}
Print Sum(&i, 1, 100, Lazy$(1/i))=5.187378~ ' true
Print i=101 ' true
} Jensen`s_Device </lang>
M4
<lang M4>define(`for',
`ifelse($#,0,``$0, `ifelse(eval($2<=$3),1, `pushdef(`$1',$2)$4`'popdef(`$1')$0(`$1',incr($2),$3,`$4')')')')
define(`sum',
`pushdef(`temp',0)`'for(`$1',$2,$3,
`define(`temp',eval(temp+$4))')`'temp`'popdef(`temp')')
sum(`i',1,100,`1000/i')</lang>
Output:
5142
Mathematica / Wolfram Language
<lang Mathematica>sum[term_, i_, lo_, hi_] := Block[{temp = 0},
Do[temp = temp + term, {i, lo, hi}];
temp];
SetAttributes[sum, HoldFirst];</lang>
Output:
In[2]:= sum[1/i, i, 1, 100] Out[2]= 14466636279520351160221518043104131447711/2788815009188499086581352357412492142272 In[3]:=N[sum[1/i, i, 1, 100]] Out[3]:=5.18738
Maxima
<lang maxima>mysum(e, v, lo, hi) := block([s: 0], for i from lo thru hi do s: s + subst(v=i, e), s)$
mysum(1/n, n, 1, 10); 7381/2520
/* compare with builtin sum */ sum(1/n, n, 1, 10); 7381/2520
/* what if n is assigned a value ? */ n: 200$
/* still works */ mysum(1/n, n, 1, 10); 7381/2520</lang>
NetRexx
<lang netrexx> import COM.ibm.netrexx.process.
class JensensDevice
properties static
interpreter=NetRexxA
exp=Rexx ""
termMethod=Method
method main(x=String[]) static
say sum('i',1,100,'1/i')
method sum(i,lo,hi,term) static SIGNALS IOException,NoSuchMethodException,IllegalAccessException,InvocationTargetException
sum=0
loop iv=lo to hi
sum=sum+termeval(i,iv,term)
end
return sum
method termeval(i,iv,e) static returns Rexx SIGNALS IOException,NoSuchMethodException,IllegalAccessException,InvocationTargetException
if e\=exp then interpreter=null
exp=e
if interpreter=null then do
termpgm='method term('i'=Rexx) static returns rexx;return' e
fw=FileWriter("termpgm.nrx")
fw.write(termpgm,0,termpgm.length)
fw.close
interpreter=NetRexxA()
interpreter.parse([String 'termpgm.nrx'],[String 'nocrossref'])
termClass=interpreter.getClassObject(null,'termpgm')
classes=[interpreter.getClassObject('netrexx.lang', 'Rexx', 0)]
termMethod=termClass.getMethod('term', classes)
end
return Rexx termMethod.invoke(null,[iv])
</lang>
Nim
<lang nim>var i: int
proc harmonicSum(i: var int, lo, hi, term): float =
i = lo while i <= hi: result += term() inc i
echo harmonicSum(i, 1, 100, proc: float = 1.0 / float(i))</lang> Output:
5.1873775176396206e+00
Objeck
<lang objeck> bundle Default {
class Jensens {
i : static : Int;
function : Sum(lo : Int, hi : Int, term : () ~ Float) ~ Float {
temp := 0.0;
for(i := lo; i <= hi; i += 1;) {
temp += term();
};
return temp; }
function : term() ~ Float {
return 1.0 / i;
}
function : Main(args : String[]) ~ Nil {
Sum(1, 100, term() ~ Float)->PrintLine();
}
}
} </lang>
Output: 5.18738
OCaml
<lang ocaml>let i = ref 42 (* initial value doesn't matter *)
let sum' i lo hi term =
let result = ref 0. in
i := lo;
while !i <= hi do
result := !result +. term ();
incr i
done;
!result
let () =
Printf.printf "%f\n" (sum' i 1 100 (fun () -> 1. /. float !i))</lang>
Output: 5.187378
Oforth
<lang Oforth>: mysum(lo, hi, term) | i | 0 lo hi for: i [ i term perform + ] ;</lang>
- Output:
mysum(1, 100, #inv) println 5.18737751763962 mysum(1, 100, #[ sq inv ]) println 1.63498390018489
Oz
Translation using mutable references and an anonymous function: <lang oz>declare
fun {Sum I Lo Hi Term}
Temp = {NewCell 0.0}
in
I := Lo
for while:@I =< Hi do
Temp := @Temp + {Term}
I := @I + 1
end
@Temp
end
I = {NewCell unit}
in
{Show {Sum I 1 100 fun {$} 1.0 / {Int.toFloat @I} end}}</lang>
Idiomatic code: <lang oz>declare
fun {Sum Lo Hi F}
{FoldL {Map {List.number Lo Hi 1} F} Number.'+' 0.0}
end
in
{Show {Sum 1 100 fun {$ I} 1.0/{Int.toFloat I} end}}</lang>
PARI/GP
GP does not have pass-by-reference semantics for user-generated functions, though some predefined functions do. PARI programming allows this, though such a solution would essentially be identical to the C solution above.
Pascal
<lang pascal>{$MODE objFPC} type
tTerm = function(i: integer):real;
function term(i:integer):real; Begin
term := 1/i;
end;
function sum(var i: LongInt;
lo,hi: integer;
term:tTerm):real;
Begin
result := 0; i := lo; while i<=hi do begin result := result+term(i); inc(i); end;
end;
var
i : LongInt;
Begin
writeln(sum(i,1,100,@term));
end. </lang> Out
5.1873775176396206E+000
Perl
<lang perl>my $i; sub sum {
my ($i, $lo, $hi, $term) = @_;
my $temp = 0;
for ($$i = $lo; $$i <= $hi; $$i++) {
$temp += $term->();
}
return $temp;
}
print sum(\$i, 1, 100, sub { 1 / $i }), "\n";</lang> Output: 5.18737751763962
Or you can take advantage of the fact that elements of the @_ are aliases of the original: <lang perl>my $i; sub sum {
my (undef, $lo, $hi, $term) = @_;
my $temp = 0;
for ($_[0] = $lo; $_[0] <= $hi; $_[0]++) {
$temp += $term->();
}
return $temp;
}
print sum($i, 1, 100, sub { 1 / $i }), "\n";</lang> Output: 5.18737751763962
Phix
Not really as asked for (implicit assumption replaced with explicit parameter) but this gives the required result.
I could also have done what C and PHP are doing, though in Phix I'd have to explicitly assign the static var within the loop.
I wholeheartedly agree with the comment on the Clipper example.
<lang Phix>function sumr(integer lo, hi, rid)
atom res = 0
for i=lo to hi do
res += call_func(rid,{i})
end for
return res
end function
function reciprocal(atom i) return 1/i end function
?sumr(1, 100, routine_id("reciprocal"))</lang>
- Output:
5.187377518
PHP
<lang php>$i; function sum (&$i, $lo, $hi, $term) {
$temp = 0;
for ($i = $lo; $i <= $hi; $i++) {
$temp += $term();
}
return $temp;
}
echo sum($i, 1, 100, create_function(, 'global $i; return 1 / $i;')), "\n"; //Output: 5.18737751764 (5.1873775176396)
function sum ($lo,$hi) {
$temp = 0;
for ($i = $lo; $i <= $hi; $i++)
{
$temp += (1 / $i);
}
return $temp;
} echo sum(1,100);
//Output: 5.1873775176396 </lang>
PicoLisp
<lang PicoLisp>(scl 6)
(de jensen (I Lo Hi Term)
(let Temp 0
(set I Lo)
(while (>= Hi (val I))
(inc 'Temp (Term))
(inc I) )
Temp ) )
(let I (box) # Create indirect reference
(format
(jensen I 1 100 '(() (*/ 1.0 (val I))))
*Scl ) )</lang>
Output:
-> "5.187383"
PureBasic
<lang PureBasic>Prototype.d func()
Global i
Procedure.d Sum(*i.Integer, lo, hi, *term.func)
Protected Temp.d For i=lo To hi temp + *term() Next ProcedureReturn Temp
EndProcedure
Procedure.d term_func()
ProcedureReturn 1/i
EndProcedure
Answer.d = Sum(@i, 1, 100, @term_func())</lang>
Python
<lang python>class Ref(object):
def __init__(self, value=None):
self.value = value
def harmonic_sum(i, lo, hi, term):
# term is passed by-name, and so is i
temp = 0
i.value = lo
while i.value <= hi: # Python "for" loop creates a distinct which
temp += term() # would not be shared with the passed "i"
i.value += 1 # Here the actual passed "i" is incremented.
return temp
i = Ref()
- note the correspondence between the mathematical notation and the
- call to sum it's almost as good as sum(1/i for i in range(1,101))
print harmonic_sum(i, 1, 100, lambda: 1.0/i.value)</lang>
or
<lang python> def harmonic_sum(i, lo, hi, term):
return sum(term() for i[0] in range(lo, hi + 1))
i = [0] print(harmonic_sum(i, 1, 100, lambda: 1.0 / i[0])) </lang>
or
<lang python> def harmonic_sum(i, lo, hi, term):
return sum(eval(term) for i[0] in range(lo, hi + 1))
i = [0] print(harmonic_sum(i, 1, 100, "1.0 / i[0]")) </lang>
Output: 5.18737751764
R
R uses a call by need evaluation strategy where function inputs are evaluated on demand and then cached; functions can bypass the normal argument evaluation by using functions substitute and match.call to access the parse tree of the as-yet-unevaluated arguments, and using parent.frame to access the scope of the caller. There are some proposed conventions to do this in a way that is less confusing to the user of a function; however, ignoring conventions we can come disturbingly close to the ALGOL call-by-name semantics.
<lang R>sum <- function(var, lo, hi, term)
eval(substitute({
.temp <- 0;
for (var in lo:hi) {
.temp <- .temp + term
}
.temp
}, as.list(match.call()[-1])),
enclos=parent.frame())
sum(i, 1, 100, 1/i) #prints 5.187378
- and because of enclos=parent.frame(), the term can involve variables in the caller's scope:
x <- -1 sum(i, 1, 100, i^x) #5.187378</lang>
Racket
Racket happens to have an Algol 60-language, so Jensen's Device can be written just as Jørn Jensen did at Regnecentralen.
<lang racket>
- lang algol60
begin
integer i;
real procedure sum (i, lo, hi, term);
value lo, hi;
integer i, lo, hi;
real term;
comment term is passed by-name, and so is i;
begin
real temp;
temp := 0;
for i := lo step 1 until hi do
temp := temp + term;
sum := temp
end;
comment note the correspondence between the mathematical notation and the call to sum;
printnln (sum (i, 1, 100, 1/i))
end </lang>
But of course you can also use the more boring popular alternative of first class functions:
<lang racket>
- lang racket/base
(define (sum lo hi f)
(for/sum ([i (in-range lo (add1 hi))]) (f i)))
(sum 1 100 (λ(i) (/ 1.0 i))) </lang>
Raku
(formerly Perl 6)
Rather than playing tricks like Perl 5 does, the declarations of the formal parameters are quite straightforward in Raku: <lang perl6>sub sum($i is rw, $lo, $hi, &term) {
my $temp = 0;
loop ($i = $lo; $i <= $hi; $i++) {
$temp += term;
}
return $temp;
}
my $i; say sum $i, 1, 100, { 1 / $i };</lang> Note that the C-style "for" loop is pronounced "loop" in Raku, and is the only loop statement that actually requires parens.
Rascal
<lang rascal>public num Jenssen(int lo, int hi, num (int i) term){ temp = 0; while (lo <= hi){ temp += term(lo); lo += 1;} return temp; }</lang>
With as output:
<lang rascal>rascal>Jenssen(1, 100, num(int i){return 1.0/i;}) num: 5.18737751763962026080511767565825315790897212670845165317653395662</lang>
REXX
<lang rexx>/*REXX program demonstrates Jensen's device (via call subroutine, and args by name). */ parse arg d . /*obtain optional argument from the CL.*/ if d== | d=="," then d= 100 /*Not specified? Then use the default.*/ numeric digits d /*use D decimal digits (9 is default)*/ say 'using ' d " decimal digits:" /*display what's being used for digits.*/ say say sum( i, 1, 100, "1/i" ) /*invoke SUM (100th harmonic number).*/ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ sum: procedure; parse arg j,start,finish,exp; $= 0
interpret 'do' j "=" start 'to' finish "; $=$+" exp '; end'
/*comment ──── ═ ─── ═════ ──── ══════ ────────── ═══ ───────── */ /*comment lit var lit var lit var literal var literal */
return $</lang>
- output when using the default input:
using 100 decimal digits: 5.187377517639620260805117675658253157908972126708451653176533956587219557532550496605687768923120415
- output when using the input: 1000
(Shown at three-quarter size and with 200 characters per line.)
using 1000 decimal digits: 5.187377517639620260805117675658253157908972126708451653176533956587219557532550496605687768923120413552951372900080959485764334902003859251284547479399606488677719356437701034351417501628003612133813 93634033610397170258150385609229760925775852490242015786454123413833660918987060275907253504512582948807527866739590394714709377905509971663909084580816222756304901297019081913723833776150679344482592 19985786828216280140988475651174867766685160764730429716983310052063466701008405663630740646670436720827975050329078640945579952223172461998152578702106818073281191723171032278163615245743308956980821 10786794204451169328900410057940565163334352244388766863157323818250401277131246550164879348955299573048040410736739783727083287179928615106959660501145265658411572959372901925824344377263363761945330 17905075097606740175205276891748232922334187250177881689092871712673549165589217457070884105311065936887252732260150280756519586504475363590572034459636088593436136141078274322996362525543164325745468 2
Ring
<lang ring>
- Project : Jensen's Device
decimals(14) i = 100 see sum(i,1,100,"1/n") + nl
func sum(i,lo,hi,term)
temp = 0
for n = lo to hi step 1
eval("num = " + term)
temp = temp + num
next
return temp
</lang> Output:
5.18737751763962
Ruby
Here, setting the variable and evaluating the term are truly executed in the "outer" context: <lang ruby>def sum(var, lo, hi, term, context)
sum = 0.0
lo.upto(hi) do |n|
sum += eval "#{var} = #{n}; #{term}", context
end
sum
end p sum "i", 1, 100, "1.0 / i", binding # => 5.18737751763962</lang>
But here is the Ruby way to do it: <lang ruby>def sum2(lo, hi)
lo.upto(hi).inject(0.0) {|sum, n| sum += yield n}
end p sum2(1, 100) {|i| 1.0/i} # => 5.18737751763962</lang>
Even more concise: (requires ruby >= 2.4) <lang ruby> def sum lo, hi, &term
(lo..hi).sum(&term)
end p sum(1,100){|i| 1.0/i} # => 5.187377517639621
- or using Rational:
p sum(1,100){|i| Rational(1,i)} # => 14466636279520351160221518043104131447711 / 2788815009188499086581352357412492142272 </lang>
Scala
Actually, the i parameter needs to be passed by reference, as done in so many
examples here, so that changes made to it reflect on the parameter that was passed. Scala
supports passing parameters by name, but not by reference, which means it can't change the
value of any parameter passed. The code below gets around that by creating a mutable integer
class, which is effectively the same as passing by reference.
<lang scala>class MyInt { var i: Int = _ } val i = new MyInt def sum(i: MyInt, lo: Int, hi: Int, term: => Double) = {
var temp = 0.0
i.i = lo
while(i.i <= hi) {
temp = temp + term
i.i += 1
}
temp
} sum(i, 1, 100, 1.0 / i.i)</lang>
Result:
res2: Double = 5.187377517639621
Scheme
Scheme procedures do not support call-by-name. Scheme macros, however, do:
<lang scheme> (define-syntax sum
(syntax-rules ()
((sum var low high . body)
(let loop ((var low)
(result 0))
(if (> var high)
result
(loop (+ var 1)
(+ result . body)))))))
</lang>
(exact->inexact (sum i 1 100 (/ 1 i))) 5.18737751763962
Seed7
Seed7 supports call-by-name with function parameters:
<lang seed7> $ include "seed7_05.s7i";
include "float.s7i";
var integer: i is 0;
const func float: sum (inout integer: i, in integer: lo, in integer: hi,
ref func float: term) is func
result
var float: sum is 0.0
begin
for i range lo to hi do
sum +:= term;
end for;
end func;
const proc: main is func
begin writeln(sum(i, 1, 100, 1.0/flt(i)) digits 6); end func;
</lang>
Output:
5.187378
Sidef
<lang ruby>var i; func sum (i, lo, hi, term) {
var temp = 0;
for (*i = lo; *i <= hi; (*i)++) {
temp += term.run;
};
return temp;
}; say sum(\i, 1, 100, { 1 / i });</lang>
- Output:
5.18737751763962026080511767565825315790899
Simula
Compare with Algol 60, in Simula 67 'call by name' is specified with name. It is a true 'call by name' evaluation not a 'procedure parameter' emulation.<lang simula>comment Jensen's Device; begin
integer i;
real procedure sum (i, lo, hi, term);
name i, term;
value lo, hi;
integer i, lo, hi;
real term;
comment term is passed by-name, and so is i;
begin
integer j;
real temp;
temp := 0;
for j := lo step 1 until hi do
begin
i := j;
temp := temp + term
end;
sum := temp
end;
comment note the correspondence between the mathematical notation and the call to sum;
outreal (sum (i, 1, 100, 1/i), 7, 14)
end</lang>
- Output:
5.187378&+000
Standard ML
<lang sml>val i = ref 42 (* initial value doesn't matter *)
fun sum' (i, lo, hi, term) = let
val result = ref 0.0
in
i := lo; while !i <= hi do ( result := !result + term (); i := !i + 1 ); !result
end
val () =
print (Real.toString (sum' (i, 1, 100, fn () => 1.0 / real (!i))) ^ "\n")</lang>
Output: 5.18737751764
Swift
<lang swift>var i = 42 // initial value doesn't matter
func sum(inout i: Int, lo: Int, hi: Int, @autoclosure term: () -> Double) -> Double {
var result = 0.0
for i = lo; i <= hi; i++ {
result += term()
}
return result
}
println(sum(&i, 1, 100, 1 / Double(i)))</lang>
(Prior to Swift 1.2, replace @autoclosure term: () -> Double with term: @autoclosure () -> Double.)
- Output:
5.187378
Tcl
Here, we set the value of the passed variable in the caller's frame. We then evaluate the passed term there too. <lang tcl>proc sum {var lo hi term} {
upvar 1 $var x
set sum 0.0
for {set x $lo} {$x < $hi} {incr x} {
set sum [expr {$sum + [uplevel 1 [list expr $term]]}]
}
return $sum
} puts [sum i 1 100 {1.0/$i}] ;# 5.177377517639621</lang> However, the solution is expressed more simply like this <lang tcl>proc sum2 {lo hi lambda} {
set sum 0.0
for {set n $lo} {$n < $hi} {incr n} {
set sum [expr {$sum + [apply $lambda $n]}]
}
return $sum
} puts [sum2 1 100 {i {expr {1.0/$i}}}] ;# 5.177377517639621</lang>
VBA
<lang vb> Private Function sum(i As String, ByVal lo As Integer, ByVal hi As Integer, term As String) As Double
Dim temp As Double
For k = lo To hi
temp = temp + Evaluate(Replace(term, i, k))
Next k
sum = temp
End Function Sub Jensen_Device()
Debug.Print sum("i", 1, 100, "1/i")
Debug.Print sum("i", 1, 100, "i*i")
Debug.Print sum("j", 1, 100, "sin(j)")
End Sub </lang>
- Output:
5,18737751763962 338350 -0,12717101366042
Wren
As Wren doesn't support call by name, call by reference nor pointers we need to 'box' the global numeric variable 'i' and use a function for 'term' to simulate Jensen's device. This works because all user defined types are reference types and functions can capture external variables. <lang ecmascript>class Box {
construct new(v) { _v = v }
v { _v }
v=(value) { _v = value }
}
var i = Box.new(0) // any initial value will do here
var sum = Fn.new { |i, lo, hi, term|
var temp = 0
i.v = lo
while (i.v <= hi) {
temp = temp + term.call()
i.v = i.v + 1
}
return temp
}
var s = sum.call(i, 1, 100, Fn.new { 1/i.v }) System.print(s)</lang>
- Output:
5.1873775176396
Yabasic
<lang Yabasic>sub Evaluation()
lo = 1 : hi = 100 : temp = 0
for i = lo to hi
temp = temp + (1/i) //r(i)
next i
print temp
end sub
Evaluation()</lang>
- Output:
5.18738
zkl
zkl doesn't support call by name/address but does have reference objects. Using an explicit call to term: <lang zkl>fcn sum(ri, lo,hi, term){
temp:=0.0; ri.set(lo);
do{ temp+=term(ri); } while(ri.inc()<hi); // inc return previous value
return(temp);
} sum(Ref(0), 1,100, fcn(ri){ 1.0/ri.value }).println();</lang> Using function application/deferred(lazy) objects, we can make the function call implicit (addition forces evaluation of the LHS): <lang zkl>fcn sum2(ri, lo,hi, term){
temp:=0.0; ri.set(lo);
do{ temp=term + temp; } while(ri.inc()<hi); // inc return previous value
return(temp);
} ri:=Ref(0); sum2(ri, 1,100, 'wrap(){ 1.0/ri.value }).println();</lang> In this case, we can call sum or sum2 and it does the same thing (the ri parameter will be ignored).
Of course, as others have pointed out, this can be expressed very simply: <lang zkl>fcn sum3(lo,hi, term){ [lo..hi].reduce('wrap(sum,i){ sum + term(i) },0.0) } sum3(1,100, fcn(i){ 1.0/i }).println();</lang>
- Output:
5.187378 5.187378 5.187378
ZX Spectrum Basic
<lang zxbasic>10 DEF FN r(x)=1/x 20 LET f$="FN r(i)" 30 LET lo=1: LET hi=100 40 GO SUB 1000 50 PRINT temp 60 STOP 1000 REM Evaluation 1010 LET temp=0 1020 FOR i=lo TO hi 1030 LET temp=temp+VAL f$ 1040 NEXT i 1050 RETURN </lang>
- Output:
5.1873775