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I'm adding SQR to my new faster string math routines...
#11
Between string math and pi, you've created some real rabbit-hole posts.  I've been opening WAY too many browser tabs lately, and am having trouble deciding to close them.  I've become a tab hoarder.


Anyway, I'll just leave a couple of those tabs here:
https://en.wikipedia.org/wiki/Arbitrary-...arithmetic
https://en.wikipedia.org/wiki/Division_algorithm


My own (very) early experiments with high-precision math pretty much stayed in the double-precision range, using elementary textbook methods.  I never tried anything fancy.  Of course, those were the pre-internet days and my local & school libraries were somewhat limited....
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#12
(08-18-2022, 05:20 PM)JRace Wrote: Between string math and pi, you've created some real rabbit-hole posts.  I've been opening WAY too many browser tabs lately, and am having trouble deciding to close them.  I've become a tab hoarder.


Anyway, I'll just leave a couple of those tabs here:
https://en.wikipedia.org/wiki/Arbitrary-...arithmetic
https://en.wikipedia.org/wiki/Division_algorithm


My own (very) early experiments with high-precision math pretty much stayed in the double-precision range, using elementary textbook methods.  I never tried anything fancy.  Of course, those were the pre-internet days and my local & school libraries were somewhat limited....

I usually have 30+ tabs open. I love tabs. Rabbit holes, not so much. That is a current worry I have over this square root process, because I need to marry it to my numerator / denominator method, used in my calculator build.

Did you see the discrepancy I posted in regard to QB64 SQR and string math for SQR(78)?

Anyway, here's the latest, greatest Square Root installment. This one use LINE INPUT and string validation, so we can start to put in very large numbers, provided the limit&& variable is increased in the code. Un remark the betatest% variable to see the square root calculation process...

Thanks for the additional links. These are fun reads!

Code: (Select All)
$CHECKING:OFF
WIDTH 160, 42
_SCREENMOVE 0, 0
DO
    limit&& = 32
    'betatest% = -1
    IF betatest% THEN limit&& = 16

    DO
        INPUT "Number: "; x$: PRINT

        IF x$ = "" THEN SYSTEM

        IF LEFT$(x$, 1) = "-" THEN
            PRINT "Negatives not allowed. Redo..": _DELAY 2: PRINT
        ELSE
            validate_string x$
            IF INSTR(x$, "invalid") = 0 THEN EXIT DO
            PRINT "Sorry, "; x$: _DELAY 1: PRINT
        END IF
    LOOP

    x# = VAL(x$) ' Needed for QB64 SQR() comparison only.
    oldy$ = ""

    IF INSTR(x$, ".") THEN
        decx$ = MID$(x$, 1, INSTR(x$, ".") - 1)
        x$ = MID$(x$, 1, INSTR(x$, ".") - 1) + MID$(x$, INSTR(x$, ".") + 1)
        IF LEN(x$) = 1 THEN x$ = x$ + "0"
    ELSE
        decx$ = x$
    END IF

    j&& = LEN(decx$)

    ' VAL() okay, one character eval.
    IF VAL(RIGHT$(LTRIM$(STR$(j&&)), 1)) / 2 = VAL(RIGHT$(LTRIM$(STR$(j&&)), 1)) \ 2 THEN
        i&& = 1 ' Even number length.
    ELSE
        i&& = 0 ' Odd number length.
    END IF

    DO
        stringmatha$ = z$: stringmathb$ = k$
        string_math z$, "-", k$, runningtotal$, terminating_decimal%, limit&&
        z$ = runningtotal$ + (MID$(x$, i&&, 2))
        IF LEFT$(z$, 1) = "0" THEN z$ = MID$(z$, 2) ' Remove leading zeros

        oldy$ = ""
        FOR j&& = 1 TO 10
            IF i&& > 1 THEN
                string_math sqrt$, "*", "2", y$, terminating_decimal%, limit&&
                y$ = y$ + LTRIM$(STR$(j&&))
            ELSE
                y$ = LTRIM$(STR$(j&&))
            END IF

            string_math y$, "*", LTRIM$(STR$(j&&)), runningtotal$, terminating_decimal%, limit&&

            string_compare runningtotal$, z$, gl%
            IF gl% > -1 THEN
                IF gl% = 0 THEN
                    h% = 0: oldy$ = y$ ' Perfect square division.
                ELSE
                    h% = 1
                END IF

                string_math oldy$, "*", LTRIM$(STR$(j&& - h%)), runningtotal$, terminating_decimal%, limit&&
                IF STRING$(LEN(z$), "0") = z$ AND runningtotal$ = "0" AND i&& >= LEN(decx$) THEN EXIT DO

                IF dp&& = 0 THEN ' Limited to && size unless converted to string.
                    IF i&& >= LEN(decx$) THEN
                        dp&& = INT(LEN(decx$) / 2 + .5)
                        IF dp&& = 0 THEN dp&& = -1
                    END IF
                END IF

                IF betatest% THEN PRINT "Sqrt "; sqrt$; " * 2 = ";: COLOR 2, 0: PRINT LTRIM$(STR$(VAL(sqrt$) * 2));: COLOR 7, 0: PRINT LTRIM$(STR$(j&& - h%)); " * "; LTRIM$(STR$(j&& - h%)); " ="; VAL(oldy$) * (j&& - h%)
                sqrt$ = sqrt$ + LTRIM$(STR$(j&& - h%))

                string_math oldy$, "*", LTRIM$(STR$(j&& - h%)), runningtotal$, terminating_decimal%, limit&&
                k$ = runningtotal$

                IF betatest% THEN PRINT "Remainder "; z$; " minus "; k$; " = ";
                EXIT FOR
            END IF
            oldy$ = y$
        NEXT

        IF betatest% THEN
            string_math stringmatha$, "-", stringmathb$, runningtotal$, terminating_decimal%, limit&&
            PRINT runningtotal$; " sqrt = "; sqrt$
        END IF

        i&& = i&& + 2
        IF LEN(z$) >= limit&& THEN EXIT DO
        x$ = x$ + "00"
    LOOP

    PRINT

    IF dp&& THEN
        sqrt$ = MID$(sqrt$, 0, dp&& + 1) + "." + MID$(sqrt$, dp&& + 1)
    END IF

    _CLIPBOARD$ = sqrt$
    PRINT "QB64 SQR:"; SQR(x#)
    PRINT "Pete SQR: "; sqrt$: _DELAY 1: CLEAR
    PRINT
LOOP

SUB string_math (stringmatha$, operator$, stringmathb$, runningtotal$, terminating_decimal%, limit&&)
    SELECT CASE operator$
        CASE "+", "-"
            GOTO string_add_subtract
        CASE "*"
            GOTO string_multiply
        CASE "/"
            GOTO string_divide
        CASE ELSE
            PRINT "Error, no operator selected. operator$ = "; operator$
    END SELECT

    string_divide:
    terminating_decimal% = 0: divsign% = 0: divremainder& = 0: divremainder$ = "": divplace& = 0: divplace2& = 0: quotient$ = "": divcarry& = 0
    operationdivision% = -1
    divbuffer& = LEN(stringmathb$) - LEN(stringmatha$)
    IF divbuffer& < 0 THEN divbuffer& = 0
    d2dividend$ = stringmatha$
    d1divisor$ = stringmathb$
    IF LEFT$(d1divisor$, 1) = "0" AND LEN(d1divisor$) = 1 THEN PRINT "Division by zero not allowed.": divsign% = 0: operationdivision% = 0: EXIT SUB
    IF LEFT$(d1divisor$, 1) = "-" THEN divsign% = -1: d1divisor$ = MID$(d1divisor$, 2)
    IF LEFT$(d2dividend$, 1) = "-" THEN
        IF divsign% THEN
            divsign% = 0
        ELSE
            divsign% = -1
        END IF
        d2dividend$ = MID$(d2dividend$, 2)
    END IF
    IF INSTR(d1divisor$, ".") <> 0 THEN
        DO UNTIL RIGHT$(d1divisor$, 1) <> "0"
            d1divisor$ = MID$(d1divisor$, 1, LEN(d1divisor$) - 1) ' Strip off trailing zeros
        LOOP
        divplace& = LEN(d1divisor$) - INSTR(d1divisor$, ".")
        d1divisor$ = MID$(d1divisor$, 1, INSTR(d1divisor$, ".") - 1) + MID$(d1divisor$, INSTR(d1divisor$, ".") + 1) ' Strip off decimal point.
        DO UNTIL LEFT$(d1divisor$, 1) <> "0"
            d1divisor$ = MID$(d1divisor$, 2) ' Strip off leading zeros for divisors smaller than .1
        LOOP
    END IF

    IF INSTR(d2dividend$, ".") <> 0 THEN
        d2dividend$ = d2dividend$ + STRING$(divplace& - LEN(d2dividend$) - INSTR(d2dividend$, "."), "0") ' Add any zeros based on the length of dividend at decimal - length of divisor at decimal. If less than zero, nothing added.
        divplace2& = INSTR(d2dividend$, ".")
        DO UNTIL RIGHT$(d2dividend$, 1) <> "0"
            d2dividend$ = MID$(d2dividend$, 1, LEN(d2dividend$) - 1) ' Strip off trailing zeros
        LOOP
        d2dividend$ = MID$(d2dividend$, 1, INSTR(d2dividend$, ".") - 1) + MID$(d2dividend$, INSTR(d2dividend$, ".") + 1) ' Strip off decimal point.
    ELSE
        d2dividend$ = d2dividend$ + STRING$(divplace&, "0") ' Add any zeros based on the length of dividend at decimal - length of divisor at decimal. If less than zero, nothing added.
        divplace& = 0
    END IF
    DO
        DO
            divremainder& = divremainder& + 1: divremainder$ = divremainder$ + MID$(d2dividend$, divremainder&, 1)
            IF MID$(d2dividend$, divremainder&, 1) = "" THEN
                IF divremainder$ = STRING$(LEN(divremainder$), "0") AND LEN(quotient$) > LEN(d2dividend$) THEN
                    divflag% = -1
                    terminating_decimal% = -1
                    EXIT DO
                END IF
                divcarry& = divcarry& + 1
                IF divcarry& = 1 THEN divplace3& = divremainder& - 1
                IF divcarry& > limit&& + 1 + divbuffer& THEN
                    divflag% = -2: EXIT DO
                END IF
                divremainder$ = divremainder$ + "0" ' No more digits to bring down.
            END IF
            IF LEN(divremainder$) > LEN(d1divisor$) OR LEN(divremainder$) = LEN(d1divisor$) AND divremainder$ >= d1divisor$ THEN EXIT DO
            quotient$ = quotient$ + "0"
        LOOP
        IF divflag% THEN divflag% = 0: EXIT DO
        FOR div_i% = 9 TO 1 STEP -1
            stringmatha$ = LTRIM$(STR$(div_i%)): stringmathb$ = d1divisor$
            m_product$ = "": GOSUB string_multiply
            tempcutd$ = divremainder$ ' divremainder$ can be 00 or other leading zero values.
            DO
                IF LEN(tempcutd$) = 1 THEN EXIT DO
                IF LEFT$(tempcutd$, 1) = "0" THEN
                    tempcutd$ = MID$(tempcutd$, 2)
                ELSE
                    EXIT DO
                END IF
            LOOP
            IF LEN(tempcutd$) > LEN(m_product$) OR LEN(tempcutd$) = LEN(m_product$) AND m_product$ <= tempcutd$ THEN EXIT FOR
        NEXT
        quotient$ = quotient$ + LTRIM$(STR$(div_i%))
        stringmatha$ = LTRIM$(STR$(div_i%)): stringmathb$ = d1divisor$
        m_product$ = "": GOSUB string_multiply
        operator$ = "-"
        stringmatha$ = divremainder$
        stringmathb$ = m_product$
        GOSUB string_add_subtract
        divremainder$ = stringmatha$
        operator$ = "/"
    LOOP
    IF divplace& = 0 AND divplace2& = 0 THEN divplace& = divplace3&
    IF divplace2& THEN divplace& = divplace& + divplace2& - 1
    IF quotient$ = "" THEN divplace& = 0 ' dividend is zero.
    IF divplace& OR divplace2& THEN
        quotient$ = MID$(quotient$, 1, divplace&) + "." + MID$(quotient$, divplace& + 1)
        DO UNTIL RIGHT$(quotient$, 1) <> "0"
            quotient$ = MID$(quotient$, 1, LEN(quotient$) - 1) ' Strip off trailing zeros
        LOOP
        IF RIGHT$(quotient$, 1) = "." THEN quotient$ = MID$(quotient$, 1, LEN(quotient$) - 1) ' Strip off abandoned decimal.
    END IF
    DO UNTIL LEFT$(quotient$, 1) <> "0"
        quotient$ = MID$(quotient$, 2) ' Strip off leading zeros
    LOOP
    IF quotient$ = "" THEN quotient$ = "0": divsign% = 0
    operationdivision% = 0
    stringmathb$ = quotient$: quotient$ = ""

    IF stringmathb$ = "overflow" THEN divsign% = 0: operationdivision% = 0: EXIT SUB

    runningtotal$ = stringmathb$: stringmathb$ = ""
    IF divsign% THEN runningtotal$ = "-" + runningtotal$

    IF stringmathround$ <> "" THEN runningtotal$ = runningtotal$ + stringmathround$
    operationdivision% = 0
    EXIT SUB

    string_multiply:
    m_decimal_places& = 0: m_product$ = ""
    fac1$ = stringmatha$: fac2$ = stringmathb$ ' Make numbers whole numbers and remove any - sign.
    IF LEFT$(fac1$, 1) = "-" THEN fac1$ = MID$(fac1$, 2): m_sign% = -1
    IF LEFT$(fac2$, 1) = "-" THEN fac2$ = MID$(fac2$, 2): IF m_sign% THEN m_sign% = 0 ELSE m_sign% = -1
    IF INSTR(fac1$, ".") <> 0 THEN m_decimal_places& = LEN(fac1$) - INSTR(fac1$, "."): fac1$ = MID$(fac1$, 1, INSTR(fac1$, ".") - 1) + MID$(fac1$, INSTR(fac1$, ".") + 1)
    IF INSTR(fac2$, ".") <> 0 THEN m_decimal_places& = m_decimal_places& + LEN(fac2$) - INSTR(fac2$, "."): fac2$ = MID$(fac2$, 1, INSTR(fac2$, ".") - 1) + MID$(fac2$, INSTR(fac2$, ".") + 1)
    FOR m_i& = LEN(fac2$) TO 1 STEP -1 ' Multiply each charater top and bottom.
        m_k& = m_l&
        m_x2$ = MID$(fac2$, m_i&, 1)
        FOR m_j& = LEN(fac1$) TO 1 STEP -1
            m_x1$ = MID$(fac1$, m_j&, 1)
            IF m_product$ <> "" THEN
                m_add$ = LTRIM$(STR$(VAL(m_x1$) * VAL(m_x2$))) + STRING$(m_k&, "0")
                m_t& = 0: m_xproduct$ = "": m_carry% = 0
                DO ' Add multiplied characters together.
                    m_x3$ = MID$(m_add$, LEN(m_add$) - m_t&, 1)
                    m_x4$ = MID$(m_product$, LEN(m_product$) - m_t&, 1)
                    IF m_x3$ = "" AND m_x4$ = "" THEN
                        IF m_carry% THEN m_xproduct$ = "1" + m_xproduct$
                        EXIT DO
                    END IF
                    m_g% = VAL(m_x3$) + VAL(m_x4$) + m_carry%
                    IF m_g% >= 10 THEN m_g% = m_g% - 10: m_carry% = 1 ELSE m_carry% = 0
                    m_xproduct$ = LTRIM$(STR$(m_g%)) + m_xproduct$
                    m_t& = m_t& + 1
                LOOP
                m_product$ = m_xproduct$: m_xproduct$ = ""
            ELSE
                m_product$ = LTRIM$(STR$(VAL(m_x1$) * VAL(m_x2$))) + STRING$(m_k&, "0") ' First loop makes variable here.
            END IF
            m_k& = m_k& + 1 ' Adds trailing zeros multiplication
        NEXT
        m_l& = m_l& + 1 ' Used to reset value for m_k& adding one trailing zer for each loop.
    NEXT
    fac1$ = "": fac2$ = "": m_l& = 0: m_k& = 0: m_t& = 0
    IF m_decimal_places& > LEN(m_product$) THEN m_product$ = STRING$(m_decimal_places& - LEN(m_product$), "0") + m_product$ ' Add any leading zeros to a decimal. Ex: .02 * .01 is factored as 002. It needs one leading zero before adding the decimal point, .0002.
    IF m_decimal_places& AND m_product$ <> "0" THEN ' Replace any decimal point.
        m_product$ = MID$(m_product$, 1, LEN(m_product$) - m_decimal_places&) + "." + MID$(m_product$, LEN(m_product$) - m_decimal_places& + 1)
    END IF
    DO UNTIL LEFT$(m_product$, 1) <> "0" ' Remove leading zeros.
        m_product$ = MID$(m_product$, 2)
    LOOP
    IF m_decimal_places& THEN
        DO UNTIL RIGHT$(m_product$, 1) <> "0" ' Remove trailing zeros in a decimal sum.
            m_product$ = MID$(m_product$, 1, LEN(m_product$) - 1)
        LOOP
    END IF
    IF m_product$ = "" THEN m_product$ = "0": m_sign% = 0
    IF RIGHT$(m_product$, 1) = "." THEN m_product$ = MID$(m_product$, 1, LEN(m_product$) - 1) ' Remove decimal from the end of an integer total.
    IF operationdivision% THEN m_sign% = 0: RETURN
    stringmathb$ = m_product$: m_product$ = ""

    IF stringmathb$ = "overflow" THEN EXIT SUB

    runningtotal$ = stringmathb$: stringmathb$ = ""
    IF m_sign% THEN runningtotal$ = "-" + runningtotal$: m_sign% = 0
    EXIT SUB

    string_add_subtract:
    IF INSTR(stringmatha$, ".") <> 0 THEN ' Evaluate sum for decimal fraction.
        sumplace& = LEN(stringmatha$) - INSTR(stringmatha$, ".")
        stringmatha$ = MID$(stringmatha$, 1, INSTR(stringmatha$, ".") - 1) + MID$(stringmatha$, INSTR(stringmatha$, ".") + 1) ' Strip out decimal
    END IF
    IF INSTR(stringmathb$, ".") <> 0 THEN ' Evaluate number for decimal fraction.
        numplace& = LEN(stringmathb$) - INSTR(stringmathb$, ".")
        stringmathb$ = MID$(stringmathb$, 1, INSTR(stringmathb$, ".") - 1) + MID$(stringmathb$, INSTR(stringmathb$, ".") + 1) ' Strip out decimal
    END IF
    IF sumplace& > numplace& THEN addsubplace& = sumplace& ELSE addsubplace& = numplace&
    IF sumplace& > addsubplace& THEN
        stringmatha$ = stringmatha$ + STRING$(sumplace& - addsubplace&, "0")
    ELSEIF addsubplace& > sumplace& THEN
        stringmatha$ = stringmatha$ + STRING$(addsubplace& - sumplace&, "0")
    END IF
    IF numplace& > addsubplace& THEN
        stringmathb$ = stringmathb$ + STRING$(numplace& - addsubplace&, "0")
    ELSEIF addsubplace& > numplace& THEN
        stringmathb$ = stringmathb$ + STRING$(addsubplace& - numplace&, "0")
    END IF ' END Decimal evaluations.

    IF LEFT$(stringmatha$, 1) = "-" THEN sign_input$ = "-" ELSE sign_input$ = "+"
    IF LEFT$(stringmathb$, 1) = "-" THEN sign_total$ = "-" ELSE sign_total$ = "+"

    addsubsign% = 0
    SELECT CASE sign_input$ + operator$ + sign_total$
        CASE "+++", "+--"
            operator$ = "+"
            IF LEFT$(stringmathb$, 1) = "-" THEN stringmathb$ = MID$(stringmathb$, 2)
        CASE "++-", "+-+"
            operator$ = "-"
            IF LEFT$(stringmathb$, 1) = "-" THEN stringmathb$ = MID$(stringmathb$, 2)
            GOSUB string_comp
            IF gl% < 0 THEN SWAP stringmatha$, stringmathb$: addsubsign% = -1
        CASE "---", "-++"
            operator$ = "-"
            IF LEFT$(stringmatha$, 1) = "-" THEN stringmatha$ = MID$(stringmatha$, 2)
            IF LEFT$(stringmathb$, 1) = "-" THEN stringmathb$ = MID$(stringmathb$, 2)
            GOSUB string_comp
            IF gl% < 0 THEN SWAP stringmatha$, stringmathb$: addsubsign% = -1
        CASE "--+", "-+-"
            operator$ = "+"
            IF LEFT$(stringmatha$, 1) = "-" THEN stringmatha$ = MID$(stringmatha$, 2)
            IF LEFT$(stringmathb$, 1) = "-" THEN stringmathb$ = MID$(stringmathb$, 2)
            addsubsign% = -1
    END SELECT

    IF LEN(stringmatha$) > LEN(stringmathb$) THEN
        stringmathb$ = STRING$(LEN(stringmatha$) - LEN(stringmathb$), "0") + stringmathb$
    ELSEIF LEN(stringmatha$) < LEN(stringmathb$) THEN
        stringmatha$ = STRING$(LEN(stringmathb$) - LEN(stringmatha$), "0") + stringmatha$
    END IF
    addsubx1$ = ""

    SELECT CASE operator$
        CASE "+", "="
            FOR addsubii& = LEN(stringmatha$) TO 1 STEP -1
                addsubx1% = VAL(MID$(stringmatha$, addsubii&, 1)) + VAL(MID$(stringmathb$, addsubii&, 1)) + addsubcarry%
                IF addsubx1% > 9 THEN addsubx1% = addsubx1% - 10: addsubcarry% = 1 ELSE addsubcarry% = 0
                addsubx1$ = LTRIM$(STR$(addsubx1%)) + addsubx1$
            NEXT
            IF addsubcarry% THEN addsubx1$ = "1" + addsubx1$: addsubcarry% = 0
            GOSUB replace_decimal
        CASE "-"
            FOR addsubii& = LEN(stringmatha$) TO 1 STEP -1
                addsubx1% = VAL(MID$(stringmatha$, addsubii&, 1)) - VAL(MID$(stringmathb$, addsubii&, 1)) + addsubcarry%
                IF addsubx1% < 0 THEN addsubx1% = addsubx1% + 10: addsubcarry% = -1 ELSE addsubcarry% = 0
                addsubx1$ = LTRIM$(STR$(addsubx1%)) + addsubx1$
            NEXT
            IF addsubx1$ <> "" AND addsubx1$ <> STRING$(LEN(addsubx1$), "0") THEN GOSUB replace_decimal
            DO UNTIL LEFT$(addsubx1$, 1) <> "0" ' Remove leading zeros.
                addsubx1$ = MID$(addsubx1$, 2)
            LOOP
            IF addsubx1$ = "" THEN
                addsubx1$ = "0": addsubsign% = 0
            ELSE
                IF addsubcarry% THEN addsubx1$ = "-" + addsubx1$: addsubcarry% = 0
            END IF
    END SELECT

    IF addsubsign% THEN
        IF LEFT$(addsubx1$, 1) = "-" THEN addsubx1$ = MID$(addsubx1$, 2) ELSE addsubx1$ = "-" + addsubx1$
    END IF
    stringmatha$ = addsubx1$: addsubx1$ = ""
    IF operationdivision% THEN RETURN
    stringmathb$ = stringmatha$: stringmatha$ = ""
    IF LEFT$(stringmathb$, 1) = "-" THEN
        stringmathb$ = MID$(stringmathb$, 2)
        n2sign$ = "-"
    ELSE
        n2sign$ = ""
    END IF

    IF stringmathb$ = "overflow" THEN n2sign$ = "": EXIT SUB

    runningtotal$ = n2sign$ + stringmathb$: n2sign$ = ""
    EXIT SUB

    replace_decimal:
    IF addsubplace& THEN
        addsubx1$ = STRING$(addsubplace& - LEN(addsubx1$), "0") + addsubx1$
        addsubx1$ = MID$(addsubx1$, 1, LEN(addsubx1$) - addsubplace&) + "." + MID$(addsubx1$, LEN(addsubx1$) - addsubplace& + 1)
        DO UNTIL RIGHT$(addsubx1$, 1) <> "0" ' Remove trailing zeros in a decimal sum.
            addsubx1$ = MID$(addsubx1$, 1, LEN(addsubx1$) - 1)
            addsubplace& = addsubplace& - 1
        LOOP
        IF RIGHT$(addsubx1$, 1) = "." THEN addsubx1$ = MID$(addsubx1$, 1, LEN(addsubx1$) - 1) ' Number is now an integer.
    END IF
    RETURN

    string_comp:
    DO
        REM Remove trailing zeros after a decimal point.
        IF INSTR(acomp$, ".") THEN
            DO UNTIL RIGHT$(acomp$, 1) <> "0" AND RIGHT$(acomp$, 1) <> "." AND RIGHT$(acomp$, 1) <> "-"
                acomp$ = MID$(acomp$, 1, LEN(acomp$) - 1)
            LOOP
        END IF
        IF INSTR(bcomp$, ".") THEN
            DO UNTIL RIGHT$(bcomp$, 1) <> "0" AND RIGHT$(bcomp$, 1) <> "." AND RIGHT$(bcomp$, 1) <> "-"
                bcomp$ = MID$(bcomp$, 1, LEN(bcomp$) - 1)
            LOOP
        END IF

        IF MID$(acomp$, 1, 2) = "-0" OR acomp$ = "" OR acomp$ = "-" THEN acomp$ = "0"
        IF MID$(bcomp$, 1, 2) = "-0" OR bcomp$ = "" OR bcomp$ = "-" THEN bcomp$ = "0"

        ' A - and +
        IF LEFT$(acomp$, 1) = "-" THEN j% = -1
        IF LEFT$(bcomp$, 1) = "-" THEN k% = -1
        IF k% = 0 AND j% THEN gl% = -1: EXIT DO
        IF j% = 0 AND k% THEN gl% = 1: EXIT DO

        ' A decimal and non-decimal.
        j% = INSTR(acomp$, ".")
        k% = INSTR(bcomp$, ".")
        IF j% = 0 AND k% THEN
            IF acomp$ = "0" THEN gl% = -1 ELSE gl% = 1
            EXIT DO
        END IF
        IF k% = 0 AND j% THEN
            IF bcomp$ = "0" THEN gl% = 1 ELSE gl% = -1
            EXIT DO
        END IF

        ' Both decimals.
        IF j% THEN
            IF acomp$ > bcomp$ THEN
                gl% = 1
            ELSEIF acomp$ = bcomp$ THEN gl% = 0
            ELSEIF acomp$ < bcomp$ THEN gl% = -1
            END IF
            EXIT DO
        END IF

        ' Both positive or both negative whole numbers.
        SELECT CASE LEN(acomp$)
            CASE IS < LEN(bcomp$)
                gl% = -1
            CASE IS = LEN(bcomp$)
                IF acomp$ = bcomp$ THEN
                    gl% = 0
                ELSEIF acomp$ > bcomp$ THEN gl% = 1
                ELSEIF acomp$ < bcomp$ THEN gl% = -1
                END IF
            CASE IS > LEN(bcomp$)
                gl% = 1
        END SELECT
        EXIT DO
    LOOP
    RETURN
END SUB

SUB string_compare (compa$, compb$, gl%)
    DO
        REM Remove trailing zeros after a decimal point.
        IF INSTR(compa$, ".") THEN
            DO UNTIL RIGHT$(compa$, 1) <> "0" AND RIGHT$(compa$, 1) <> "." AND RIGHT$(compa$, 1) <> "-"
                compa$ = MID$(compa$, 1, LEN(compa$) - 1)
            LOOP
        END IF
        IF INSTR(compb$, ".") THEN
            DO UNTIL RIGHT$(compb$, 1) <> "0" AND RIGHT$(compb$, 1) <> "." AND RIGHT$(compb$, 1) <> "-"
                compb$ = MID$(compb$, 1, LEN(compb$) - 1)
            LOOP
        END IF

        IF MID$(compa$, 1, 2) = "-0" OR compa$ = "" OR compa$ = "-" THEN compa$ = "0"
        IF MID$(compb$, 1, 2) = "-0" OR compb$ = "" OR compb$ = "-" THEN compb$ = "0"

        ' A - and +
        IF LEFT$(compa$, 1) = "-" THEN j% = -1
        IF LEFT$(compb$, 1) = "-" THEN k% = -1
        IF k% = 0 AND j% THEN gl% = -1: EXIT DO
        IF j% = 0 AND k% THEN gl% = 1: EXIT DO

        ' A decimal and non-decimal.
        j% = INSTR(compa$, ".")
        k% = INSTR(compb$, ".")
        IF j% = 0 AND k% THEN
            IF compa$ = "0" THEN gl% = -1 ELSE gl% = 1
            EXIT DO
        END IF
        IF k% = 0 AND j% THEN
            IF compb$ = "0" THEN gl% = 1 ELSE gl% = -1
            EXIT DO
        END IF

        ' Both decimals.
        IF j% THEN
            IF compa$ > compb$ THEN
                gl% = 1
            ELSEIF compa$ = compb$ THEN gl% = 0
            ELSEIF compa$ < compb$ THEN gl% = -1
            END IF
            EXIT DO
        END IF

        ' Both positive or both negative whole numbers.
        SELECT CASE LEN(compa$)
            CASE IS < LEN(compb$)
                gl% = -1
            CASE IS = LEN(compb$)
                IF compa$ = compb$ THEN
                    gl% = 0
                ELSEIF compa$ > compb$ THEN gl% = 1
                ELSEIF compa$ < compb$ THEN gl% = -1
                END IF
            CASE IS > LEN(compb$)
                gl% = 1
        END SELECT
        EXIT DO
    LOOP
END SUB

SUB validate_string (stringmathb$)
    DO ' Faux loop.
        vsn_negcnt& = 0: vsn_poscnt& = 0: vsn_depresent& = 0: decimalcnt& = 0: vsn_numberpresent& = 0: vsn_zerospresent& = 0
        IF LEFT$(stringmathb$, 1) = "-" THEN stringmathb$ = MID$(stringmathb$, 2): sm_sign$ = "-" ELSE sm_sign$ = ""
        IF LEFT$(stringmathb$, 1) = "+" THEN IF sm_sign$ <> "-" THEN stringmathb$ = MID$(stringmathb$, 2) ELSE stringmathb$ = "invalid number": EXIT DO
        IF INSTR(UCASE$(stringmathb$), "D") OR INSTR(UCASE$(stringmathb$), "E") THEN ' Evaluate for Scientific Notation.
            FOR sm_i& = 1 TO LEN(stringmathb$)
                validatenum$ = MID$(UCASE$(stringmathb$), sm_i&, 1)
                SELECT CASE validatenum$
                    CASE "+"
                        IF vsn_depresent& THEN vsn_poscnt& = vsn_poscnt& + 1 ELSE stringmathb$ = "invalid number": EXIT DO
                    CASE "-"
                        IF vsn_depresent& THEN vsn_negcnt& = vsn_negcnt& + 1 ELSE stringmathb$ = "invalid number": EXIT DO
                    CASE "0" TO "9"
                        vsn_numberpresent& = -1
                    CASE "D", "E"
                        vsn_depresent& = vsn_depresent& + 1
                        IF decimalcnt& = 0 AND sm_i& <> 2 OR vsn_depresent& > 1 OR vsn_numberpresent& = 0 OR vsn_negcnt& > 1 OR vsn_poscnt& > 1 OR vsn_negcnt& = 1 AND vsn_poscnt& >= 1 THEN vsn_numberpresent& = 0: EXIT FOR
                        vsn_numberpresent& = 0
                        MID$(stringmathb$, sm_i&, 1) = "e" ' Standardize
                    CASE "."
                        decimalcnt& = decimalcnt& + 1
                        IF sm_i& <> 2 THEN vsn_numberpresent& = 0: EXIT FOR
                    CASE ELSE
                        vsn_numberpresent& = 0: EXIT FOR
                END SELECT
            NEXT
            IF decimalcnt& = 0 THEN stringmathb$ = MID$(stringmathb$, 1, 1) + "." + MID$(stringmathb$, 2) ' Standardize "."
            IF vsn_numberpresent& = 0 OR vsn_negcnt& = 1 AND vsn_poscnt& = 1 OR decimalcnt& > 1 OR INSTR(stringmathb$, ".") <> 2 THEN stringmathb$ = "invalid number": EXIT DO
            vsn_depresent& = INSTR(stringmathb$, "e")
            sm_x$ = MID$(stringmathb$, vsn_depresent& + 1, 1) ' Standardize exponent "+" these two lines.
            IF sm_x$ <> "+" AND sm_x$ <> "-" THEN stringmathb$ = MID$(stringmathb$, 1, vsn_depresent&) + "+" + MID$(stringmathb$, vsn_depresent& + 1)
            IF MID$(stringmathb$, vsn_depresent& + 2, 1) = "0" THEN
                IF MID$(stringmathb$, vsn_depresent& + 3, 1) <> "" THEN stringmathb$ = "invalid number": EXIT DO ' No leading zeros allowed in exponent notation.
            END IF
            jjed& = INSTR(stringmathb$, "e") ' Get position of notation.
            valexpside$ = MID$(stringmathb$, jjed&) ' These two lines break up into number and notation
            stringmathb$ = MID$(stringmathb$, 1, jjed& - 1) ' stringmathb$ is +- single digit whole number, decimal point and decimal number. valexpside$ is notation, sign and exponent.
            DO UNTIL RIGHT$(stringmathb$, 1) <> "0" ' Remove any trailing zeros for number. Example 1.0d3 or 1.0000d3, etc.
                stringmathb$ = MID$(stringmathb$, 1, LEN(stringmathb$) - 1)
            LOOP
            IF VAL(MID$(stringmathb$, 1, INSTR(stringmathb$, ".") - 1)) = 0 THEN
                IF RIGHT$(stringmathb$, 1) = "." THEN
                    stringmathb$ = "0.e+0" ' Handles all types of zero entries.
                ELSE
                    stringmathb$ = "invalid number": EXIT DO
                END IF
                EXIT DO
            END IF
            stringmathb$ = sm_sign$ + stringmathb$ + valexpside$
            EXIT DO
        ELSE
            FOR sm_i& = 1 TO LEN(stringmathb$)
                validatenum$ = MID$(stringmathb$, sm_i&, 1)
                SELECT CASE validatenum$
                    CASE "."
                        decimalcnt& = decimalcnt& + 1
                    CASE "0"
                        vsn_zerospresent& = -1
                    CASE "1" TO "9"
                        vsn_numberpresent& = -1
                    CASE "$"
                    CASE ELSE
                        stringmathb$ = "invalid number": EXIT DO
                END SELECT
            NEXT
            IF decimalcnt& > 1 OR vsn_negcnt& > 1 OR vsn_poscnt& > 1 OR vsn_negcnt& >= 1 AND vsn_poscnt& >= 1 THEN
                stringmathb$ = "invalid number": EXIT DO
            END IF

            REM IF INSTR(stringmathb$, "$") THEN GOSUB currency_validate
            REM IF INSTR(stringmathb$, ",") THEN
            REM    GOSUB comma_validation
            REM IF stringmathb$ = "invalid number" THEN exit do
            REM GOSUB comma_removal
            REM END IF

            IF RIGHT$(stringmathb$, 1) = "." THEN stringmathb$ = MID$(stringmathb$, 1, LEN(stringmathb$) - 1)
            DO UNTIL LEFT$(stringmathb$, 1) <> "0" ' Strip off any leading zeros.
                stringmathb$ = MID$(stringmathb$, 2)
            LOOP
            stringmathb$ = sm_sign$ + stringmathb$
            IF INSTR(stringmathb$, ".") THEN
                DO UNTIL RIGHT$(stringmathb$, 1) <> "0" ' Strip off any trailing zeros in a decimal.
                    stringmathb$ = MID$(stringmathb$, 1, LEN(stringmathb$) - 1)
                LOOP
            END IF
            IF RIGHT$(stringmathb$, 1) = "." THEN stringmathb$ = MID$(stringmathb$, 1, LEN(stringmathb$) - 1)
            IF vsn_numberpresent& = 0 THEN
                IF vsn_zerospresent& THEN
                    stringmathb$ = "0"
                ELSE
                    stringmathb$ = "invalid number"
                END IF
            END IF
        END IF
        EXIT DO
    LOOP
END SUB


Pete
Reply
#13
Re sqr(78):
Wolfram Alpha says it's 8.8317608663278468547640427269592539641746394809314178262102029725...
which matches your string math all the way out.

The QB64 value just seems to have an limit-of-precision error due to where calculation stopped before the result was converted to decimal.
QBasic agrees with QB64, while GWBasic sez 8.831760406494141, so GWB is only calculating at single-precision.

(Edited to add: TurboBasic (DOS), FreeBasic, TDM GCC, and Tiny C also agree with QB64 so yeah, it's a double-precision issue.)
Reply
#14
Okay math fans... How about the square root of 2 to 5,000 places in under a minute...

Code: (Select All)
WIDTH 160, 42
_SCREENMOVE 0, 0
DO
    limit&& = 5000 ' Set to something smaller like 128 for instant results.
    'betatest% = -1
    IF betatest% THEN limit&& = 16

    DO
        INPUT "Number: "; x$: PRINT

        IF x$ = "" THEN SYSTEM

        IF LEFT$(x$, 1) = "-" THEN
            PRINT "Negatives not allowed. Redo..": _DELAY 2: PRINT
        ELSE
            validate_string x$
            IF INSTR(x$, "invalid") = 0 THEN EXIT DO
            PRINT "Sorry, "; x$: _DELAY 1: PRINT
        END IF
    LOOP

    x# = VAL(x$) ' Needed for QB64 SQR() comparison only.
    oldy$ = ""

    IF INSTR(x$, ".") THEN
        decx$ = MID$(x$, 1, INSTR(x$, ".") - 1)
        x$ = MID$(x$, 1, INSTR(x$, ".") - 1) + MID$(x$, INSTR(x$, ".") + 1)
        IF LEN(x$) = 1 THEN x$ = x$ + "0"
    ELSE
        decx$ = x$
    END IF

    j&& = LEN(decx$)

    ' VAL() okay, one character eval.
    IF VAL(RIGHT$(LTRIM$(STR$(j&&)), 1)) / 2 = VAL(RIGHT$(LTRIM$(STR$(j&&)), 1)) \ 2 THEN
        i&& = 1 ' Even number length.
    ELSE
        i&& = 0 ' Odd number length.
    END IF

    DO
        stringmatha$ = z$: stringmathb$ = k$
        string_math z$, "-", k$, runningtotal$, terminating_decimal%, limit&&
        z$ = runningtotal$ + (MID$(x$, i&&, 2))
        IF LEFT$(z$, 1) = "0" THEN z$ = MID$(z$, 2) ' Remove leading zeros

        oldy$ = ""
        FOR j&& = 1 TO 10
            IF i&& > 1 THEN
                string_math sqrt$, "*", "2", y$, terminating_decimal%, limit&&
                y$ = y$ + LTRIM$(STR$(j&&))
            ELSE
                y$ = LTRIM$(STR$(j&&))
            END IF

            string_math y$, "*", LTRIM$(STR$(j&&)), runningtotal$, terminating_decimal%, limit&&

            string_compare runningtotal$, z$, gl%
            IF gl% > -1 THEN
                IF gl% = 0 THEN
                    h% = 0: oldy$ = y$ ' Perfect square division.
                ELSE
                    h% = 1
                END IF
                string_math oldy$, "*", LTRIM$(STR$(j&& - h%)), runningtotal$, terminating_decimal%, limit&&
                IF STRING$(LEN(z$), "0") = z$ AND runningtotal$ = "0" AND i&& >= LEN(decx$) THEN EXIT DO

                IF dp&& = 0 THEN ' Limited to && size unless converted to string.
                    IF i&& >= LEN(decx$) THEN
                        dp&& = INT(LEN(decx$) / 2 + .5)
                        IF dp&& = 0 THEN dp&& = -1
                    END IF
                END IF

                IF betatest% THEN PRINT "Sqrt "; sqrt$; " * 2 = ";: COLOR 2, 0: PRINT LTRIM$(STR$(VAL(sqrt$) * 2));: COLOR 7, 0: PRINT LTRIM$(STR$(j&& - h%)); " * "; LTRIM$(STR$(j&& - h%)); " ="; VAL(oldy$) * (j&& - h%)
                sqrt$ = sqrt$ + LTRIM$(STR$(j&& - h%))

                string_math oldy$, "*", LTRIM$(STR$(j&& - h%)), runningtotal$, terminating_decimal%, limit&&
                k$ = runningtotal$

                IF betatest% THEN PRINT "Remainder "; z$; " minus "; k$; " = ";
                EXIT FOR
            END IF
            oldy$ = y$
        NEXT

        IF betatest% THEN
            string_math stringmatha$, "-", stringmathb$, runningtotal$, terminating_decimal%, limit&&
            PRINT runningtotal$; " sqrt = "; sqrt$
        END IF

        i&& = i&& + 2
        IF LEN(z$) >= limit&& THEN EXIT DO
        x$ = x$ + "00"
    LOOP

    PRINT

    IF dp&& THEN
        sqrt$ = MID$(sqrt$, 0, dp&& + 1) + "." + MID$(sqrt$, dp&& + 1)
    END IF

    _CLIPBOARD$ = sqrt$
    PRINT "QB64 SQR:"; SQR(x#)
    PRINT "Pete SQR: "; sqrt$: _DELAY 1: CLEAR
    PRINT
LOOP

SUB string_math (stringmatha$, operator$, stringmathb$, runningtotal$, terminating_decimal%, limit&&)
    DIM AS _INTEGER64 a, b, c, aa, bb, cc, s, ss
    a1$ = stringmatha$: b1$ = stringmathb$

    SELECT CASE operator$
        CASE "+", "-"
            GOTO string_add_subtract_new
        CASE "*"
            GOTO string_multiply_new
        CASE "/"
            GOTO string_divide
        CASE ELSE
            PRINT "Error, no operator selected. operator$ = "; operator$
    END SELECT

    string_divide:
    terminating_decimal% = 0: divsign% = 0: divremainder& = 0: divremainder$ = "": divplace& = 0: divplace2& = 0: quotient$ = "": divcarry& = 0
    operationdivision% = -1
    divbuffer& = LEN(stringmathb$) - LEN(stringmatha$)
    IF divbuffer& < 0 THEN divbuffer& = 0
    d2dividend$ = stringmatha$
    d1divisor$ = stringmathb$
    IF LEFT$(d1divisor$, 1) = "0" AND LEN(d1divisor$) = 1 THEN PRINT "Division by zero not allowed.": divsign% = 0: operationdivision% = 0: EXIT SUB
    IF LEFT$(d1divisor$, 1) = "-" THEN divsign% = -1: d1divisor$ = MID$(d1divisor$, 2)
    IF LEFT$(d2dividend$, 1) = "-" THEN
        IF divsign% THEN
            divsign% = 0
        ELSE
            divsign% = -1
        END IF
        d2dividend$ = MID$(d2dividend$, 2)
    END IF
    IF INSTR(d1divisor$, ".") <> 0 THEN
        DO UNTIL RIGHT$(d1divisor$, 1) <> "0"
            d1divisor$ = MID$(d1divisor$, 1, LEN(d1divisor$) - 1) ' Strip off trailing zeros
        LOOP
        divplace& = LEN(d1divisor$) - INSTR(d1divisor$, ".")
        d1divisor$ = MID$(d1divisor$, 1, INSTR(d1divisor$, ".") - 1) + MID$(d1divisor$, INSTR(d1divisor$, ".") + 1) ' Strip off decimal point.
        DO UNTIL LEFT$(d1divisor$, 1) <> "0"
            d1divisor$ = MID$(d1divisor$, 2) ' Strip off leading zeros for divisors smaller than .1
        LOOP
    END IF

    IF INSTR(d2dividend$, ".") <> 0 THEN
        d2dividend$ = d2dividend$ + STRING$(divplace& - LEN(d2dividend$) - INSTR(d2dividend$, "."), "0") ' Add any zeros based on the length of dividend at decimal - length of divisor at decimal. If less than zero, nothing added.
        divplace2& = INSTR(d2dividend$, ".")
        DO UNTIL RIGHT$(d2dividend$, 1) <> "0"
            d2dividend$ = MID$(d2dividend$, 1, LEN(d2dividend$) - 1) ' Strip off trailing zeros
        LOOP
        d2dividend$ = MID$(d2dividend$, 1, INSTR(d2dividend$, ".") - 1) + MID$(d2dividend$, INSTR(d2dividend$, ".") + 1) ' Strip off decimal point.
    ELSE
        d2dividend$ = d2dividend$ + STRING$(divplace&, "0") ' Add any zeros based on the length of dividend at decimal - length of divisor at decimal. If less than zero, nothing added.
        divplace& = 0
    END IF
    DO
        DO
            divremainder& = divremainder& + 1: divremainder$ = divremainder$ + MID$(d2dividend$, divremainder&, 1)
            IF MID$(d2dividend$, divremainder&, 1) = "" THEN
                IF divremainder$ = STRING$(LEN(divremainder$), "0") AND LEN(quotient$) > LEN(d2dividend$) THEN
                    divflag% = -1
                    terminating_decimal% = -1
                    EXIT DO
                END IF
                divcarry& = divcarry& + 1
                IF divcarry& = 1 THEN divplace3& = divremainder& - 1
                IF divcarry& > limit&& + 1 + divbuffer& THEN
                    divflag% = -2: EXIT DO
                END IF
                divremainder$ = divremainder$ + "0" ' No more digits to bring down.
            END IF
            IF LEN(divremainder$) > LEN(d1divisor$) OR LEN(divremainder$) = LEN(d1divisor$) AND divremainder$ >= d1divisor$ THEN EXIT DO
            quotient$ = quotient$ + "0"
        LOOP
        IF divflag% THEN divflag% = 0: EXIT DO
        FOR div_i% = 9 TO 1 STEP -1
            stringmatha$ = LTRIM$(STR$(div_i%)): stringmathb$ = d1divisor$
            m_product$ = "": GOSUB string_multiply_new
            tempcutd$ = divremainder$ ' divremainder$ can be 00 or other leading zero values.
            DO
                IF LEN(tempcutd$) = 1 THEN EXIT DO
                IF LEFT$(tempcutd$, 1) = "0" THEN
                    tempcutd$ = MID$(tempcutd$, 2)
                ELSE
                    EXIT DO
                END IF
            LOOP
            IF LEN(tempcutd$) > LEN(m_product$) OR LEN(tempcutd$) = LEN(m_product$) AND m_product$ <= tempcutd$ THEN EXIT FOR
        NEXT
        quotient$ = quotient$ + LTRIM$(STR$(div_i%))
        stringmatha$ = LTRIM$(STR$(div_i%)): stringmathb$ = d1divisor$
        m_product$ = "": GOSUB string_multiply_new
        operator$ = "-"
        stringmatha$ = divremainder$
        stringmathb$ = m_product$
        GOSUB string_add_subtract_new
        divremainder$ = stringmatha$
        operator$ = "/"
    LOOP
    IF divplace& = 0 AND divplace2& = 0 THEN divplace& = divplace3&
    IF divplace2& THEN divplace& = divplace& + divplace2& - 1
    IF quotient$ = "" THEN divplace& = 0 ' dividend is zero.
    IF divplace& OR divplace2& THEN
        quotient$ = MID$(quotient$, 1, divplace&) + "." + MID$(quotient$, divplace& + 1)
        DO UNTIL RIGHT$(quotient$, 1) <> "0"
            quotient$ = MID$(quotient$, 1, LEN(quotient$) - 1) ' Strip off trailing zeros
        LOOP
        IF RIGHT$(quotient$, 1) = "." THEN quotient$ = MID$(quotient$, 1, LEN(quotient$) - 1) ' Strip off abandoned decimal.
    END IF
    DO UNTIL LEFT$(quotient$, 1) <> "0"
        quotient$ = MID$(quotient$, 2) ' Strip off leading zeros
    LOOP
    IF quotient$ = "" THEN quotient$ = "0": divsign% = 0
    operationdivision% = 0
    stringmathb$ = quotient$: quotient$ = ""

    IF stringmathb$ = "overflow" THEN divsign% = 0: operationdivision% = 0: EXIT SUB

    runningtotal$ = stringmathb$: stringmathb$ = ""
    IF divsign% THEN runningtotal$ = "-" + runningtotal$

    IF stringmathround$ <> "" THEN runningtotal$ = runningtotal$ + stringmathround$
    operationdivision% = 0
    EXIT SUB

    '------------------------------------------------------------------------
    string_add_subtract_new:
    s = 18

    a$ = stringmatha$: b$ = stringmathb$: op$ = operator$

    IF op$ = "-" THEN
        IF LEFT$(b$, 1) = "-" THEN b$ = MID$(b$, 2) ELSE b$ = "-" + b$
    END IF

    IF INSTR(a$, ".") <> 0 OR INSTR(b$, ".") <> 0 THEN
        decimal% = -1
        IF INSTR(a$, ".") <> 0 THEN
            dec_a&& = LEN(MID$(a$, INSTR(a$, ".") + 1))
            a$ = MID$(a$, 1, INSTR(a$, ".") - 1) + MID$(a$, INSTR(a$, ".") + 1)
        END IF
        IF INSTR(b$, ".") <> 0 THEN
            dec_b&& = LEN(MID$(b$, INSTR(b$, ".") + 1))
            b$ = MID$(b$, 1, INSTR(b$, ".") - 1) + MID$(b$, INSTR(b$, ".") + 1)
        END IF
        ' Line up decimal places by inserting trailing zeros.
        IF dec_b&& > dec_a&& THEN
            j&& = dec_b&&
            a$ = a$ + STRING$(dec_b&& - dec_a&&, "0")
        ELSE
            j&& = dec_a&&
            b$ = b$ + STRING$(dec_a&& - dec_b&&, "0")
        END IF
    END IF

    IF LEFT$(a$, 1) = "-" OR LEFT$(b$, 1) = "-" THEN
        IF LEFT$(a$, 1) = "-" AND LEFT$(b$, 1) = "-" THEN
            sign$ = "--": a$ = MID$(a$, 2): b$ = MID$(b$, 2)
        ELSE
            IF LEFT$(a$, 1) = "-" THEN a$ = MID$(a$, 2): sign_a$ = "-"
            IF LEFT$(b$, 1) = "-" THEN b$ = MID$(b$, 2): sign_b$ = "-"

            IF LEFT$(a1$, 1) = "-" THEN a1_x$ = MID$(a1$, 2) ELSE a1_x$ = a1$
            IF LEFT$(b1$, 1) = "-" THEN b1_x$ = MID$(b1$, 2) ELSE b1_x$ = b1$

            string_compare a1_x$, b1_x$, gl%

            IF gl% < 0 THEN
                IF LEN(sign_b$) THEN sign$ = "-": SWAP a$, b$
            ELSE
                IF LEN(sign_a$) THEN sign$ = "-": SWAP sign_a$, sign_b$
            END IF
        END IF
    END IF

    s = 18: z$ = ""

    DO
        i&& = i&& + s
        x1$ = MID$(a$, LEN(a$) - i&& + 1, s)
        x2$ = MID$(b$, LEN(b$) - i&& + 1, s)
        a = VAL(sign_a$ + x1$) + VAL(sign_b$ + x2$) + c
        IF x1$ + x2$ = "" AND c = 0 THEN EXIT DO
        c = 0
        IF a > VAL(STRING$(s, "9")) THEN a = a - 10 ^ s: c = 1
        IF a < 0 THEN a = a + 10 ^ s: c = -1
        tmp$ = LTRIM$(STR$(a))
        z$ = STRING$(LEN(x1$) - LEN(tmp$), "0") + tmp$ + z$
    LOOP

    IF decimal% THEN
        z$ = MID$(z$, 1, LEN(z$) - j&&) + "." + MID$(z$, LEN(z$) - j&& + 1)
    END IF

    ' Remove any leading zeros.
    DO
        IF LEFT$(z$, 1) = "0" THEN z$ = MID$(z$, 2) ELSE EXIT DO
    LOOP

    IF z$ = "" OR z$ = "0" THEN z$ = "0" ELSE z$ = LEFT$(sign$, 1) + z$

    runningtotal$ = z$ '*'
    sign$ = "": sign_a$ = "": sign_b$ = "": i&& = 0: j&& = 0: decimal% = 0: c = 0
    EXIT SUB

    '------------------------------------------------------------------------
    string_multiply_new:
    z$ = "": sign$ = "": mult&& = 0: h&& = 0: i&& = 0: j&& = 0: c = 0: decimal% = 0
    zz$ = "": ii&& = 0: jj&& = 0
    s = 8: ss = 18

    a$ = stringmatha$: b$ = stringmathb$

    IF INSTR(a$, "-") <> 0 OR INSTR(b$, "-") <> 0 THEN
        IF INSTR(a$, "-") <> 0 AND INSTR(b$, "-") <> 0 THEN
            a$ = MID$(a$, 2): b$ = MID$(b$, 2)
        ELSE
            IF INSTR(a$, "-") <> 0 THEN a$ = MID$(a$, 2) ELSE b$ = MID$(b$, 2)
            sign$ = "-"
        END IF
    END IF

    IF INSTR(a$, ".") <> 0 OR INSTR(b$, ".") <> 0 THEN
        decimal% = -1
        IF INSTR(a$, ".") <> 0 THEN
            dec_a&& = LEN(MID$(a$, INSTR(a$, ".") + 1))
            a$ = MID$(a$, 1, INSTR(a$, ".") - 1) + MID$(a$, INSTR(a$, ".") + 1)
        END IF
        IF INSTR(b$, ".") <> 0 THEN
            dec_b&& = LEN(MID$(b$, INSTR(b$, ".") + 1))
            b$ = MID$(b$, 1, INSTR(b$, ".") - 1) + MID$(b$, INSTR(b$, ".") + 1)
        END IF
    END IF

    IF LEN(a$) < LEN(b$) THEN SWAP a$, b$

    DO
        h&& = h&& + s: i&& = 0
        x2$ = MID$(b$, LEN(b$) - h&& + 1, s)
        WHILE -1
            i&& = i&& + s
            x1$ = MID$(a$, LEN(a$) - i&& + 1, s)
            a = VAL(sign_a$ + x1$) * VAL(sign_b$ + x2$) + c
            IF betatest% THEN PRINT "x1$ = "; x1$;: LOCATE , 20: PRINT "x2$ = "; x2$;: LOCATE , 35: PRINT VAL(x1$) * VAL(x2$) + c;: LOCATE , 55: PRINT "c = "; c;: LOCATE , 75: PRINT "val = "; a,
            c = 0
            tmp$ = LTRIM$(STR$(a))
            IF LEN(tmp$) > s THEN c = VAL(MID$(tmp$, 1, LEN(tmp$) - s)): tmp$ = MID$(tmp$, LEN(tmp$) - s + 1)
            z$ = STRING$(LEN(x1$) - LEN(tmp$), "0") + tmp$ + z$
            IF betatest% THEN LOCATE , 100: PRINT a;: LOCATE , 120: PRINT z$
            IF i&& >= LEN(a$) AND c = 0 THEN EXIT WHILE
        WEND

        jj&& = jj&& + 1

        IF jj&& > 1 THEN
            ii&& = 0: cc = 0
            aa$ = holdaa$
            bb$ = z$ + STRING$((jj&& - 1) * s, "0")
            IF betatest% THEN PRINT "aa$ "; aa$; " + bb$ "; z$;: COLOR 14, 0: PRINT STRING$(jj&& - 1, "0"); " = ";: COLOR 7, 0: SLEEP
            DO
                ii&& = ii&& + ss
                xx1$ = MID$(aa$, LEN(aa$) - ii&& + 1, ss)
                xx2$ = MID$(bb$, LEN(bb$) - ii&& + 1, ss)
                aa = VAL(xx1$) + VAL(xx2$) + cc
                IF xx1$ + xx2$ = "" AND cc = 0 THEN EXIT DO ' Prevents leading zeros.
                cc = 0
                IF aa > VAL(STRING$(ss, "9")) THEN aa = aa - 10 ^ ss: cc = 1
                tmp$ = LTRIM$(STR$(aa))
                zz$ = STRING$(LEN(xx1$) - LEN(tmp$), "0") + tmp$ + zz$
            LOOP

            DO WHILE LEFT$(zz$, 1) = "0"
                IF LEFT$(zz$, 1) = "0" THEN zz$ = MID$(zz$, 2)
            LOOP
            IF zz$ = "" THEN zz$ = "0"

            holdaa$ = zz$
            IF betatest% THEN COLOR 2, 0: PRINT holdaa$: COLOR 7, 0
        ELSE
            holdaa$ = z$ + STRING$(jj&& - 1, "0")
        END IF

        z$ = "": zz$ = ""

    LOOP UNTIL h&& >= LEN(b$)

    z$ = holdaa$

    IF decimal% THEN
        DO UNTIL LEN(z$) >= dec_a&& + dec_b&&
            z$ = "0" + z$
        LOOP

        z$ = MID$(z$, 0, LEN(z$) - (dec_a&& + dec_b&& - 1)) + "." + MID$(z$, LEN(z$) - (dec_a&& + dec_b&&) + 1)

        DO UNTIL RIGHT$(z$, 1) <> "0" AND RIGHT$(z$, 1) <> "."
            z$ = MID$(z$, 1, LEN(z$) - 1)
        LOOP
    END IF

    IF z$ = "" OR z$ = "0" THEN z$ = "0": ELSE z$ = sign$ + z$

    decimal% = 0: sign$ = ""

    runningtotal$ = z$

    EXIT SUB

    replace_decimal:
    IF addsubplace& THEN
        addsubx1$ = STRING$(addsubplace& - LEN(addsubx1$), "0") + addsubx1$
        addsubx1$ = MID$(addsubx1$, 1, LEN(addsubx1$) - addsubplace&) + "." + MID$(addsubx1$, LEN(addsubx1$) - addsubplace& + 1)
        DO UNTIL RIGHT$(addsubx1$, 1) <> "0" ' Remove trailing zeros in a decimal sum.
            addsubx1$ = MID$(addsubx1$, 1, LEN(addsubx1$) - 1)
            addsubplace& = addsubplace& - 1
        LOOP
        IF RIGHT$(addsubx1$, 1) = "." THEN addsubx1$ = MID$(addsubx1$, 1, LEN(addsubx1$) - 1) ' Number is now an integer.
    END IF
    RETURN

    string_comp:
    DO
        ' Remove trailing zeros after a decimal point.
        IF INSTR(acomp$, ".") THEN
            DO UNTIL RIGHT$(acomp$, 1) <> "0" AND RIGHT$(acomp$, 1) <> "." AND RIGHT$(acomp$, 1) <> "-"
                acomp$ = MID$(acomp$, 1, LEN(acomp$) - 1)
            LOOP
        END IF
        IF INSTR(bcomp$, ".") THEN
            DO UNTIL RIGHT$(bcomp$, 1) <> "0" AND RIGHT$(bcomp$, 1) <> "." AND RIGHT$(bcomp$, 1) <> "-"
                bcomp$ = MID$(bcomp$, 1, LEN(bcomp$) - 1)
            LOOP
        END IF

        IF MID$(acomp$, 1, 2) = "-0" OR acomp$ = "" OR acomp$ = "-" THEN acomp$ = "0"
        IF MID$(bcomp$, 1, 2) = "-0" OR bcomp$ = "" OR bcomp$ = "-" THEN bcomp$ = "0"

        ' A - and +
        IF LEFT$(acomp$, 1) = "-" THEN j% = -1
        IF LEFT$(bcomp$, 1) = "-" THEN k% = -1
        IF k% = 0 AND j% THEN gl% = -1: EXIT DO
        IF j% = 0 AND k% THEN gl% = 1: EXIT DO

        ' A decimal and non-decimal.
        j% = INSTR(acomp$, ".")
        k% = INSTR(bcomp$, ".")
        IF j% = 0 AND k% THEN
            IF acomp$ = "0" THEN gl% = -1 ELSE gl% = 1
            EXIT DO
        END IF
        IF k% = 0 AND j% THEN
            IF bcomp$ = "0" THEN gl% = 1 ELSE gl% = -1
            EXIT DO
        END IF

        ' Both decimals.
        IF j% THEN
            IF acomp$ > bcomp$ THEN
                gl% = 1
            ELSEIF acomp$ = bcomp$ THEN gl% = 0
            ELSEIF acomp$ < bcomp$ THEN gl% = -1
            END IF
            EXIT DO
        END IF

        ' Both positive or both negative whole numbers.
        SELECT CASE LEN(acomp$)
            CASE IS < LEN(bcomp$)
                gl% = -1
            CASE IS = LEN(bcomp$)
                IF acomp$ = bcomp$ THEN
                    gl% = 0
                ELSEIF acomp$ > bcomp$ THEN gl% = 1
                ELSEIF acomp$ < bcomp$ THEN gl% = -1
                END IF
            CASE IS > LEN(bcomp$)
                gl% = 1
        END SELECT
        EXIT DO
    LOOP
    RETURN
END SUB

SUB string_compare (compa$, compb$, gl%)
    DO
        ' Remove trailing zeros after a decimal point.
        IF INSTR(compa$, ".") THEN
            DO UNTIL RIGHT$(compa$, 1) <> "0" AND RIGHT$(compa$, 1) <> "." AND RIGHT$(compa$, 1) <> "-"
                compa$ = MID$(compa$, 1, LEN(compa$) - 1)
            LOOP
        END IF
        IF INSTR(compb$, ".") THEN
            DO UNTIL RIGHT$(compb$, 1) <> "0" AND RIGHT$(compb$, 1) <> "." AND RIGHT$(compb$, 1) <> "-"
                compb$ = MID$(compb$, 1, LEN(compb$) - 1)
            LOOP
        END IF

        IF MID$(compa$, 1, 2) = "-0" OR compa$ = "" OR compa$ = "-" THEN compa$ = "0"
        IF MID$(compb$, 1, 2) = "-0" OR compb$ = "" OR compb$ = "-" THEN compb$ = "0"

        ' A - and +
        IF LEFT$(compa$, 1) = "-" THEN j% = -1
        IF LEFT$(compb$, 1) = "-" THEN k% = -1
        IF k% = 0 AND j% THEN gl% = -1: PRINT "1*": EXIT DO
        IF j% = 0 AND k% THEN gl% = 1: PRINT "2*": EXIT DO

        ' A decimal and non-decimal.
        j% = INSTR(compa$, ".")
        k% = INSTR(compb$, ".")

        IF j% = 0 AND k% THEN
            IF compa$ = "0" THEN gl% = -1: PRINT "4*" ELSE gl% = 1: PRINT "5*"
            EXIT DO
        END IF
        IF k% = 0 AND j% THEN
            IF compb$ = "0" THEN gl% = 1: PRINT "6*" ELSE gl% = -1: PRINT "7*"
            EXIT DO
        END IF

        ' Both decimals.
        IF j% THEN
            IF compa$ > compb$ THEN
                gl% = 1: PRINT "8*"
            ELSEIF compa$ = compb$ THEN gl% = 0: PRINT "9*"
            ELSEIF compa$ < compb$ THEN gl% = -1: PRINT "10*"
            END IF
            EXIT DO
        END IF

        ' Both positive or both negative whole numbers.
        SELECT CASE LEN(compa$)
            CASE IS < LEN(compb$)
                gl% = -1
            CASE IS = LEN(compb$)
                IF compa$ = compb$ THEN
                    gl% = 0
                ELSEIF compa$ > compb$ THEN gl% = 1
                ELSEIF compa$ < compb$ THEN gl% = -1
                END IF
            CASE IS > LEN(compb$)
                gl% = 1
        END SELECT
        EXIT DO
    LOOP
END SUB

SUB validate_string (stringmathb$)
    DO ' Faux loop.
        vsn_negcnt& = 0: vsn_poscnt& = 0: vsn_depresent& = 0: decimalcnt& = 0: vsn_numberpresent& = 0: vsn_zerospresent& = 0
        IF LEFT$(stringmathb$, 1) = "-" THEN stringmathb$ = MID$(stringmathb$, 2): sm_sign$ = "-" ELSE sm_sign$ = ""
        IF LEFT$(stringmathb$, 1) = "+" THEN IF sm_sign$ <> "-" THEN stringmathb$ = MID$(stringmathb$, 2) ELSE stringmathb$ = "invalid number": EXIT DO
        IF INSTR(UCASE$(stringmathb$), "D") OR INSTR(UCASE$(stringmathb$), "E") THEN ' Evaluate for Scientific Notation.
            FOR sm_i& = 1 TO LEN(stringmathb$)
                validatenum$ = MID$(UCASE$(stringmathb$), sm_i&, 1)
                SELECT CASE validatenum$
                    CASE "+"
                        IF vsn_depresent& THEN vsn_poscnt& = vsn_poscnt& + 1 ELSE stringmathb$ = "invalid number": EXIT DO
                    CASE "-"
                        IF vsn_depresent& THEN vsn_negcnt& = vsn_negcnt& + 1 ELSE stringmathb$ = "invalid number": EXIT DO
                    CASE "0" TO "9"
                        vsn_numberpresent& = -1
                    CASE "D", "E"
                        vsn_depresent& = vsn_depresent& + 1
                        IF decimalcnt& = 0 AND sm_i& <> 2 OR vsn_depresent& > 1 OR vsn_numberpresent& = 0 OR vsn_negcnt& > 1 OR vsn_poscnt& > 1 OR vsn_negcnt& = 1 AND vsn_poscnt& >= 1 THEN vsn_numberpresent& = 0: EXIT FOR
                        vsn_numberpresent& = 0
                        MID$(stringmathb$, sm_i&, 1) = "e" ' Standardize
                    CASE "."
                        decimalcnt& = decimalcnt& + 1
                        IF sm_i& <> 2 THEN vsn_numberpresent& = 0: EXIT FOR
                    CASE ELSE
                        vsn_numberpresent& = 0: EXIT FOR
                END SELECT
            NEXT
            IF decimalcnt& = 0 THEN stringmathb$ = MID$(stringmathb$, 1, 1) + "." + MID$(stringmathb$, 2) ' Standardize "."
            IF vsn_numberpresent& = 0 OR vsn_negcnt& = 1 AND vsn_poscnt& = 1 OR decimalcnt& > 1 OR INSTR(stringmathb$, ".") <> 2 THEN stringmathb$ = "invalid number": EXIT DO
            vsn_depresent& = INSTR(stringmathb$, "e")
            sm_x$ = MID$(stringmathb$, vsn_depresent& + 1, 1) ' Standardize exponent "+" these two lines.
            IF sm_x$ <> "+" AND sm_x$ <> "-" THEN stringmathb$ = MID$(stringmathb$, 1, vsn_depresent&) + "+" + MID$(stringmathb$, vsn_depresent& + 1)
            IF MID$(stringmathb$, vsn_depresent& + 2, 1) = "0" THEN
                IF MID$(stringmathb$, vsn_depresent& + 3, 1) <> "" THEN stringmathb$ = "invalid number": EXIT DO ' No leading zeros allowed in exponent notation.
            END IF
            jjed& = INSTR(stringmathb$, "e") ' Get position of notation.
            valexpside$ = MID$(stringmathb$, jjed&) ' These two lines break up into number and notation
            stringmathb$ = MID$(stringmathb$, 1, jjed& - 1) ' stringmathb$ is +- single digit whole number, decimal point and decimal number. valexpside$ is notation, sign and exponent.
            DO UNTIL RIGHT$(stringmathb$, 1) <> "0" ' Remove any trailing zeros for number. Example 1.0d3 or 1.0000d3, etc.
                stringmathb$ = MID$(stringmathb$, 1, LEN(stringmathb$) - 1)
            LOOP
            IF VAL(MID$(stringmathb$, 1, INSTR(stringmathb$, ".") - 1)) = 0 THEN
                IF RIGHT$(stringmathb$, 1) = "." THEN
                    stringmathb$ = "0.e+0" ' Handles all types of zero entries.
                ELSE
                    stringmathb$ = "invalid number": EXIT DO
                END IF
                EXIT DO
            END IF
            stringmathb$ = sm_sign$ + stringmathb$ + valexpside$
            EXIT DO
        ELSE
            FOR sm_i& = 1 TO LEN(stringmathb$)
                validatenum$ = MID$(stringmathb$, sm_i&, 1)
                SELECT CASE validatenum$
                    CASE "."
                        decimalcnt& = decimalcnt& + 1
                    CASE "0"
                        vsn_zerospresent& = -1
                    CASE "1" TO "9"
                        vsn_numberpresent& = -1
                    CASE "$"
                    CASE ELSE
                        stringmathb$ = "invalid number": EXIT DO
                END SELECT
            NEXT
            IF decimalcnt& > 1 OR vsn_negcnt& > 1 OR vsn_poscnt& > 1 OR vsn_negcnt& >= 1 AND vsn_poscnt& >= 1 THEN
                stringmathb$ = "invalid number": EXIT DO
            END IF

            REM IF INSTR(stringmathb$, "$") THEN GOSUB currency_validate
            REM IF INSTR(stringmathb$, ",") THEN
            REM GOSUB comma_validation
            REM IF stringmathb$ = "invalid number" THEN exit do
            REM GOSUB comma_removal
            REM END IF

            IF RIGHT$(stringmathb$, 1) = "." THEN stringmathb$ = MID$(stringmathb$, 1, LEN(stringmathb$) - 1)
            DO UNTIL LEFT$(stringmathb$, 1) <> "0" ' Strip off any leading zeros.
                stringmathb$ = MID$(stringmathb$, 2)
            LOOP
            stringmathb$ = sm_sign$ + stringmathb$
            IF INSTR(stringmathb$, ".") THEN
                DO UNTIL RIGHT$(stringmathb$, 1) <> "0" ' Strip off any trailing zeros in a decimal.
                    stringmathb$ = MID$(stringmathb$, 1, LEN(stringmathb$) - 1)
                LOOP
            END IF
            IF RIGHT$(stringmathb$, 1) = "." THEN stringmathb$ = MID$(stringmathb$, 1, LEN(stringmathb$) - 1)
            IF vsn_numberpresent& = 0 THEN
                IF vsn_zerospresent& THEN
                    stringmathb$ = "0"
                ELSE
                    stringmathb$ = "invalid number"
                END IF
            END IF
        END IF
        EXIT DO
    LOOP
END SUB
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