Jensen's Device

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This task is an exercise in call by name.

Jensen's Device is a computer programming technique devised by Danish computer scientist 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 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, otherwise it would not be possible to compute 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.

ALGOL 68

BEGIN
   INT i;
   PROC sum  = (REF INT ref i, INT lo, hi, PROC REAL term)REAL:
      COMMENT term is passed by-name, and so is i COMMENT
   BEGIN
      REAL temp;
      temp := 0;
      FOR i FROM lo BY 1 TO hi DO
         ref i := i;
         temp := temp + term
      OD;
      # sum := # 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

Output: +5.18737751763962e +0

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.)

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 })
}

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:

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 })

Haskell

Translation of: Python

import Control.Monad
import Control.Monad.ST
import Data.STRef

sum' ref_i lo hi term =
  return sum `ap`
         mapM (\i -> writeSTRef ref_i i >> term) [lo..hi]

foo = runST $ do
        i <- newSTRef 0 -- initial value doesn't matter
        sum' i 1 100 $ return recip `ap` readSTRef i

main = print foo

Output: 5.187377517639621

Python

<python>class Ref(object):

 def __init__(self, value=None):
   self.value=value

i = Ref() def sum(ref_i, lo, hi, term):

  # term is passed by-name, and so is i #
  temp = 0;
  for i in range(lo,hi+1):
     ref_i.value = i;
     temp = temp + term()
  return  temp

note the correspondence between the mathematical notation and the call to sum #

print (sum (i, 1, 100, lambda: 1.0/i.value))</python> Output: 5.18737751764