08-14-2022, 04:08 AM
Calculating Pi using Leibniz formula with string math. 40 iterations are used for this demo. It runs a bit slow at 200 and the limit&& variable should be increased as iterations increase. What surprised me is that it seems to take thousands of iterations to get anywhere close to pi at 3.14159. Oh well, math isn't perfect, but it beats fishing.
Pete
Code: (Select All)
DIM SHARED betatest%: betatest% = -1
WIDTH 160, 42
_SCREENMOVE 0, 0
limit&& = 150
j = -1
FOR i = 1 TO 40
j = j + 2
IF oldd$ = "" THEN
d$ = "1": oldd$ = "1": oldn$ = "1": n$ = "1"
ELSE
d$ = LTRIM$(STR$(j))
' 2nd denominator * 1st numerator.
a$ = d$: b$ = oldn$: op$ = "*"
CALL string_math(a$, op$, b$, x$, limit&&)
m1$ = x$
' 1st denominator * 2nd numerator.
a$ = oldd$: b$ = n$
CALL string_math(a$, op$, b$, x$, limit&&)
m2$ = x$
' Get common denominator
a$ = d$: b$ = oldd$
CALL string_math(a$, op$, b$, x$, limit&&)
d$ = x$
a$ = m1$: b$ = m2$: IF i / 2 = i \ 2 THEN op$ = "-" ELSE op$ = "+"
CALL string_math(a$, op$, b$, x$, limit&&)
'CALL greatest_common_factor(x$, d$, limit&&)
REM PRINT "oldn$ = "; oldn$; " oldd$ = "; oldd$, "n$ = "; x$; " d$ = "; d$
PRINT "n$ = "; x$; " d$ = "; d$;: COLOR 14, 0: PRINT j: COLOR 7, 0
oldn$ = x$: oldd$ = d$
END IF
'''SLEEP
NEXT
n$ = x$
a$ = x$: b$ = d$: op$ = "/"
CALL string_math(a$, op$, b$, x$, limit&&)
a$ = x$: b$ = "4": op$ = "*"
CALL string_math(a$, op$, b$, x$, limit&&)
PRINT: PRINT "pi = "; x$
END
SUB string_math (stringmatha$, operator$, stringmathb$, runningtotal$, 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:
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: EXIT DO
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$)
gl% = 1
END SELECT
EXIT DO
LOOP
RETURN
END SUB
SUB greatest_common_factor (gfca$, gfcb$, limit&&)
hold_gfca$ = gfca$
hold_gfcb$ = gfcb$
numerator$ = gfca$: denominator$ = gfcb$
' Make both numbers positive.
IF MID$(gfca$, 1, 1) = "-" THEN gfca$ = MID$(gfca$, 2)
IF MID$(gfcb$, 1, 1) = "-" THEN gfcb$ = MID$(gfcb$, 2)
IF gfca$ < gfcb$ THEN SWAP gfca$, gfcb$
i% = 0
DO
i% = i% + 1
stringmatha$ = gfca$: stringmathb$ = gfcb$
operator$ = "/": CALL string_math(stringmatha$, operator$, stringmathb$, runningtotal$, limit&&)
IF INSTR(runningtotal$, ".") THEN runningtotal$ = MID$(runningtotal$, 1, INSTR(runningtotal$, ".") - 1)
stringmatha$ = runningtotal$: stringmathb$ = gfcb$
operator$ = "*": CALL string_math(stringmatha$, operator$, stringmathb$, runningtotal$, limit&&)
stringmatha$ = gfca$: stringmathb$ = runningtotal$
operator$ = "-": CALL string_math(stringmatha$, operator$, stringmathb$, runningtotal$, limit&&)
SWAP gfca$, gfcb$: gfcb$ = runningtotal$
IF runningtotal$ = "0" THEN EXIT DO
IF i% > 9 THEN gfca$ = hold_gfcb$: EXIT SUB
LOOP
stringmatha$ = numerator$: stringmathb$ = gfca$
IF betatest% THEN PRINT "GFC = "; gfca$
operator$ = "/": CALL string_math(stringmatha$, operator$, stringmathb$, runningtotal$, limit&&)
numerator$ = runningtotal$
stringmatha$ = denominator$: stringmathb$ = gfca$
operator$ = "/": CALL string_math(stringmatha$, operator$, stringmathb$, runningtotal$, limit&&)
denominator$ = runningtotal$
IF betatest% THEN PRINT "Fraction: "; hold_gfca$; " / "; hold_gfcb$, numerator$; " / "; denominator$
gfca$ = numerator$: gfcb$ = denominator$
END SUBPete
Shoot first and shoot people who ask questions, later.