by GUY COUSINEAU
(The following article is another in the series Exploring
SmartBASIC, by Guy Cousineau and was provided by Ron Mitchell,
president of AUFG, via ANN. It originally appeared in the Jul/Aug
In several of the previous articles on SMARTBASIC, I have
mentioned variable commands. Although most of them are mathematical
functions, a few are not; notably FRE and USR. Variable commands
are those commands which pass a parameter within brackets: eg
INT(123.45). The parameter is evaluated by the function in order
to determine the result.
Presumably to save on interpretation and parsing code, the
designers of SMARTBASIC adopted a complicated technique, which
dynamically relocates these variable commands based on the LOMEM
setting. Each of the variables is defined as an array and the
array simply points back to the execution routine for each function.
When you are playing around with memory and accidently write
where you should not, the variable commands are invariably the
first ones to suffer. When they start misbehaving, the best thing
to do is reboot.
In this article, we will briefly cover the use of arithmetic
and algebraic functions. I will not attempt to describe the calculation
method, for even if I understood it completely, it would take
several pages to explain. The purpose of this article is to remind
you that these functions are there and clarify their use as required.
INT does just what you might expect; it extracts the integer
value of a real variable. Since it truncates rather than round
off, statistical calculations will be more precise if you use
INT(x+.5). You will also find that certain numbers truncate in
a strange fashion. I have never noted the exact numbers, but
the floating point has difficulty handling numbers like .001.
For this reason, I often use INT(x+.50001). This helps to avoid
those nasty INTEGER values which wind up being 37.9999997.
ABS takes the absolute value of a number by removing its sign
thus ABS(-12) will yield 12. You can use ABS to make a number
negative with something like -1*ABS(x).
SGN will report on the sign of a variable. SGN will return 0
if the variable is 0, -1 for negative values, and +1 for positive
values. SGN can be used in conjunction with ON GOTO in the following
999 REM make decision on sign of x
1000 sx=SGN(x)+2: REM make result 1,2,3
1010 ON sx GOTO 2000,3000,4000
2000 REM handle negative
3000 REM handle zero
4000 REM handle positive
LOG takes the natural LOG (base e) of a number. If you are
curious, the value of e is 2.718281828....You can come close
to this value by asking SMARTBASIC for the LOG(10). If you want
to take use 10 LOGS, just divide the LOG value by LOG(10):
999 REM subroutine to take base 10 log of x
EXP is the complementary function which raises a to the power
of the argument. Thus EXP(2) is the same as e^2. This function
is redundant for powers of 10 since you can use 10^2 or 1.0E+2.
It would be tedious, however to write 2.718281828^2.
SQR extracts the square root. Thus the Pythagorean theorem
would be calculated by hyp = SQR(s1^2 + s2^2). The square root
can also be expressed with hyp = (s1^2 + s2^2) ^ (1/2), but SQR
is more convenient.
Before discussing the TRIG functions, a bit about RADIANS.
Computers insist on working with radians rather than degrees.
If you remember your high school trigonometry, there are pi
radians in 180 degrees or about 57.3 degrees per radian. pi
(despite what textbooks might say) has the value 3.141592657
and you can define RAD=180/3.141592657 to use as a conversion
from degrees to radians. This will become clearer in a moment.
SIN takes the sine of the specified radian. If you would
rather work in degrees, use something like SIN(45/RAD) to evaluate
the sine of 45 degrees.
COS takes the cosine of the specified radian. Again, if you
remember your high school math, COS(x)=SIN(90-x). Thus COS(45)
should be the same as SIN(45). Define RAD as outlined above and
print SIN(45/RAD) and COS(45/RAD).
If you change the value of pi in the equation
to a different value like 3.141592655, you will see that the
values are not the same. Thus the value given above is the CORRECT
one for working with ADAMs floating point accumulator.
TAN takes the tangent and ATN takes the arc-tangent; the latter
function is difficult to calculate manually.
All other TRIG functions can be evaluated using the 4 functions
above; you just have to remember how its done. I must admit
I have forgotten.
INT executes at 10672(29B0).
It verifies that the number is in floating point format. It then
checks if the number is less than 1 in which case 0 is returned.
It then juggles the number around to drop the decimal portion.
ABS executes at 2276(08E4).
It simply resets the sign bit in the floating point accumulator.
This is why ABS cannot be used with INTEGER variables.
SGN executes at 2285(08ED). It starts by checking the value
of the exponent in the FPA. If zero it simply returns which yields
a zero value. It then sets the value in the FPA to +1 or -1 depending
on the original sign of the number.
LOG executes at 3604(0E14).
It checks for zero and negative values and then calls a power
series calculator to do a recursive calculation.
EXP executes at 3816(0EEB).
It uses a power series calculator to approximate the required
SQR executes at 3678(0E5E).
It does the calculation the way we learned it when we learned
about logs. It takes the LOG of the value, divides it by 2 and
recalculates the exponent. This is like doing e^(log(x)/2).
SIN executes at 3954(0F72).
This is the workhorse which uses a power series calculator to
approximate the required value.
COS executes at 3946(0F6A).
It calculates the value from the formula COS(x)=SIN(x+pi/2),
in radians, of course.
TAN executes at 3912(OF48).
It calculates the value from the formula TAN=SIN/COS.
ATN executes at 4180(1054). It also uses a power series calculator.
The following is a list of routines used by the MATH functions.
They are very complex and should likely not be fiddled with.
The addresses are included for information purposes only.
4156-4179 POWER SERIES CALCULATOR
It takes a value from a table pointed to by HL. Copies FPA1 to
FPA2, multiplies the first value at (HL) by FPA1 and multiplies
the original result by the next table value. It is called by
the controlling routine as long as there are values in the table.
4255-4269 ADD PI FRACTIONS
Depending on the entry point, these routines add or subtract
pi/2 or pi/4 to the current value in the floating point accumulator.
4270-4355 POWER SERIES CALCULATOR
This one basically multiplies the 2 numbers in the FPAs,
then multiplies each component by values from the power table
and finally adds the 2 halves together.
This is a set of values used by the various math routines when
calling the power series calculators. Some are in floating point
and others in integer format. You will find values corresponding
to LOG(2) 1/LOG(2) pi/2 2/pi pi/4 1/11 1/9 1/7 1/5 1/3 but NOT
pi or e.
That is the extent of my coverage of MATH functions.
Next time, random numbers...
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