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Closest to e (Posted on 2012-04-18) Difficulty: 3 of 5
This is in continuation of Closest to pi.

(A) As we are aware, the approximate value of e with some decimals is 2.7182818284590...

Write an expression using all digits (0-9) once each, to achieve the closest value to e. The digits 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0 must be arranged strictly in this order. You can only use the five symbols: parentheses , addition, subtraction, multiplication and division symbols that is, (), +, -, *, /

Concatenating two or more digits is allowed. Each symbol can occur at most once in the given expression. Only standard order of operations is followed that is, multiplication does not take precedence over division, addition does not take precedence over multiplication, and so on.

For example : (123+456)/7890 is a valid expression. However, expressions like 1+234/567 + 8/9 -0 or, 1+3*2456/78-90 are not allowed.

(B) Write an expression closest to e, if all the other conditions in (A) remain unaltered, but the restriction of each symbol occurring at most one is withdrawn. For example, expressions like 1+234/567 + 8/9 -0 is allowed. However, expressions like: 1+3*2456/78-90 are not allowed.

(C) What are the respective answers to (A) and (B) if a sixth symbol, that is, the decimal point is allowed?

No Solution Yet Submitted by K Sengupta    
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Solution computer solution (so far) for part C, subpart B Comment 6 of 6 |

The below version of the program for close to e, which has been running for over two days and is still running although having been compiled in QB64, has so far already produced a much better approximation to e than the version without decimal points.

The new evaluation is 1/((2+3*.4)/(5/6+7/.890)) ~= 2.71828183520599.

DECLARE SUB addOn ()
DEFDBL A-Z
DIM SHARED stack(11)
DIM SHARED upTo, stackLevel
DIM SHARED h$, best, declevel

best = 9999
stack(1) = 1: stackLevel = 1: upTo = 1: declevel = 0
h$ = "1"

OPEN "close2ep.txt" FOR OUTPUT AS #2

addOn

CLOSE 2

SUB addOn
nextOne = upTo + 1

SELECT CASE nextOne
    CASE 10
        nextOne = 0
    CASE 1
        ' this is the end
        ' evaluate
        IF stackLevel = 1 THEN
            diff = ABS(stack(1) - 2.718281828#)
            IF diff <= best THEN
                best = diff
                PRINT h$, stack(1)
                PRINT #2, h$, stack(1)
            END IF
            EXIT SUB
        ELSE
            nextOne = 999
        END IF
END SELECT

'first try concatenation
IF nextOne < 999 THEN
    lc$ = RIGHT$(h$, 1)
    ix = INSTR("0123456789", lc$)
    IF ix THEN
        saveDecLevel = declevel
        h$ = h$ + LTRIM$(STR$(nextOne))
        IF declevel = 0 THEN
            stack(stackLevel) = 10 * stack(stackLevel) + nextOne
        ELSE
            declevel = declevel + 1
            stack(stackLevel) = stack(stackLevel) + nextOne / 10 ^ declevel
        END IF
        utSave = upTo
        upTo = nextOne
        addOn
        declevel = saveDecLevel
        upTo = utSave
        IF declevel = 0 THEN
            stack(stackLevel) = stack(stackLevel) \ 10
        ELSE
            stack(stackLevel) = stack(stackLevel) - nextOne / 10 ^ declevel
        END IF
        h$ = LEFT$(h$, LEN(h$) - 1)

        IF declevel = 0 THEN
            declevel = 1

            h$ = h$ + "." + LTRIM$(STR$(nextOne))
            stack(stackLevel) = stack(stackLevel) + nextOne / 10 ^ declevel
            h$ = LEFT$(h$, LEN(h$) - 2)
            utSave = upTo
            upTo = nextOne
            addOn
            declevel = saveDecLevel
            upTo = utSave
            stack(stackLevel) = stack(stackLevel) - nextOne / 10
        END IF
    END IF

END IF

'try comma
IF nextOne < 999 THEN
    saveDecLevel = declevel
    stackLevel = stackLevel + 1
    stack(stackLevel) = nextOne
    h$ = h$ + "," + LTRIM$(STR$(nextOne))
    utSave = upTo
    upTo = nextOne
    nextOneSave = nextOne
    declevel = 0
    addOn
    nextOne = nextOneSave
    upTo = utSave
    stackLevel = stackLevel - 1
    h$ = LEFT$(h$, LEN(h$) - 2)

    stackLevel = stackLevel + 1
    declevel = 1
    stack(stackLevel) = nextOne / 10
    h$ = h$ + ",." + LTRIM$(STR$(nextOne))
    utSave = upTo
    upTo = nextOne
    nextOneSave = nextOne
    addOn
    nextOne = nextOneSave
    upTo = utSave
    stackLevel = stackLevel - 1
    h$ = LEFT$(h$, LEN(h$) - 3)
    declevel = saveDecLevel

END IF

GOSUB tryDiadic

EXIT SUB

tryDiadic:
IF stackLevel >= 2 THEN
    s1 = stack(stackLevel - 1)
    s2 = stack(stackLevel)
    stackLevel = stackLevel - 1

    stack(stackLevel) = s1 + s2
    h$ = h$ + "+"
    saveDecLevel = declevel: declevel = 0: addOn: declevel = saveDecLevel
    h$ = LEFT$(h$, LEN(h$) - 1)

    stack(stackLevel) = s1
    stackLevel = stackLevel + 1
    stack(stackLevel) = s2

    s1 = stack(stackLevel - 1)
    s2 = stack(stackLevel)
    stackLevel = stackLevel - 1

    stack(stackLevel) = s1 - s2
    h$ = h$ + "-"
    saveDecLevel = declevel: declevel = 0: addOn: declevel = saveDecLevel
    h$ = LEFT$(h$, LEN(h$) - 1)

    stack(stackLevel) = s1
    stackLevel = stackLevel + 1
    stack(stackLevel) = s2

    s1 = stack(stackLevel - 1)
    s2 = stack(stackLevel)
    stackLevel = stackLevel - 1

    stack(stackLevel) = s1 * s2
    h$ = h$ + "*"
    saveDecLevel = declevel: declevel = 0: addOn: declevel = saveDecLevel
    h$ = LEFT$(h$, LEN(h$) - 1)

    stack(stackLevel) = s1
    stackLevel = stackLevel + 1
    stack(stackLevel) = s2

    s1 = stack(stackLevel - 1)
    s2 = stack(stackLevel)

    IF s2 <> 0 THEN
        stackLevel = stackLevel - 1

        stack(stackLevel) = s1 / s2
        h$ = h$ + "/"
        saveDecLevel = declevel: declevel = 0: addOn: declevel = saveDecLevel
        h$ = LEFT$(h$, LEN(h$) - 1)

        stack(stackLevel) = s1
        stackLevel = stackLevel + 1
        stack(stackLevel) = s2
    END IF

END IF
RETURN


END SUB

The RPN output that was translated at the top of this comment is:

1,2,3,.4*+,5,6/,7,.890/+//          2.718281835205992

 

 


  Posted by Charlie on 2012-04-21 01:11:00
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