In This Issue...
New Source for 65802's
I talked to Constantine Geromnimon at Alliance Computers this morning. His company has ordered hundreds of 65802's, and offers them to you at $49.95 each. They expect their next shipment to come in around the middle of January, so now is the time to order. Call them at (718) 672-0684, or write to P. O. Box 408, Corona, NY 11368.
EPROM Programmer
A new EPROM Programmer, called the PROmGRAMER, is out from SCRG (the makers of quikLoader). This one burns anything from 2716's up to 27256's, and retails at $149.50. We'll sell 'em to you for a nice round $140. The software comes on disk, with instructions for loading it into EPROM for the quikLoader card.
Tom Weishaar Writes Again!
If you are among the throng who mourn the passing of Softalk, and particularly of the many informative columns such as DOStalk by Tom Weishaar, you will be as glad as I am that Tom has started publishing his own monthly newsletter.
Called "Open-Apple", you can subscribe for $24. In an unprecedented move toward international goodwill and the wholesome exchange of information, Tom has set the price the same for everyone, everywhere. We promptly sent him a check. If you love your Apple, do likewise. Send to Open-Apple, 10026 Roe, Overland Park, Kansas 66207. If you are cautious, send no money; Tom will bill you with the first issue, and you can cancel if you lose interest.
Someone pointed out last week that this series is getting a little long. Well, we are nearing the end. What we are doing is probably unprecedented in the industry: listing the source code and explaining it for a large commercially valuable software product. It takes time and space to break precedents.
This month's installment completes the normal set of math functions, with sine, cosine, and arc tangent. We even slipped in a simple form of the tangent function. Still to come are the formatted INPUT and PRINT routines.
Some Elementary Info:
Trigonometry is a frightening word. (If it doesn't scare you, skip ahead several paragraphs.) The "-ometry" refers to measurement, but what is a "trigon". Believe it or not, "trigon" is another name for a triangle. Trigon means three sides, and figures with three sides just happen to also have three angles. "Trig" (a nice nickname) is a branch of mathematics dealing with triangles, without which we could not fly to the moon, draw a map, or build bridges. Strangely enough, much of electronics also uses trig funtions ... are electrons triangular?
When I took trig in high school, long before the day of personal calculators, we used trig tables. (These were not articles of furniture made in the local woodshop, but rather long lists of strange numbers printed and bound into books.) The tables contained values for various ratios of the sides of a triangle having one 90-degree angle. Now we use calculators or computers, but obviously the trig tables would not fit in them. Instead, approximation formulas are used.
In high school, we talked about six different ratios: sine, cosine, tangent, cotangent, secant, and cosecant. When it is all boiled down, we really only need the sine; all the rest are derivable from those. The sine function gives a a number for any angle. We frequently need to be able to go from a trig value back to an angle, and the most useful function for that is called the inverse tangent, or arctangent.
Even though I have been talking about triangles, trig functions are even more related to circles. We compute functions of the angle between any two radii, like the hands on an old-fashioned, pre-digital wrist watch. When we start talking about circles, we get into radians vs. degrees.
Just as scientists like logarithms to the base e (rather than 10), they also like trig functions based on angles expressed in radians, rather than degrees. Degrees were invented back in Babylon, I understand, and are nice and clean: 360 make a complete circle. Radians are not clean: 360 degrees is two-times-pi radians. Nevertheless, many physical and electronic formulas simplify when angles are expressed in radians. Consequently, calculators and computer languages usually expect your angles to be expressed in radians. Some allow both options. Applesoft expects radians, and so do my DP18 programs.
We commonly think of an angle as being somewhere between 0 and 360 degrees, or the equivalent range in radians. However, angles can actually be any number, from -infinity to +infinity. The numbers beyond one complete circle are valid, but they don't buy much. If you stand in one place and spin around 1445 degrees (4*360 + 5) you will end up pointing the same direction as if you merely swiveled 5 degrees. Therefore the first step in a sine function calculation involves subtracting out all the multiples of a full circle from the angle.
The arctangent function could return an infinite number of answers, but that is impractical. We will return only the principal value, which is the one closest to 0. All others are that value plus or minus any number of full circles. In DP18 the ATN function may have one or two arguments. If you only have one argument, the result will be an angle between -pi/2 and +pi/2. If you specify two arguments, a value between -pi and +pi will be returned.
The Nitty-Gritty:
Enough of this preliminary stuff, let's get into the code. In the listing which follows, you will find entries for four functions: SIN, COS, TAN, and ATN.
Perhaps the easiest is the TAN function, at lines 2530-2630. Since tan=sin/cos, that is all this code does. We lose a little speed and possibly some precision with this simplistic solution, but the TAN function is relatively rarely called.
Next in difficulty is the COS function, lines 1630-1710. Since cos(-x)=cos(x), we start by making the sign positive (lines 1690-1700. Since cos(x)=sin(x+pi/2), we add pi/2 and fall into the SIN function. Simple, but effective.
The SIN function gets more interesting. For very very small angles, within the precision of 20 digits, sin(x)=x. Lines 1780-1810 check for exponents below -10; all angles smaller than 10^-10 are small enough that sin(x)=x.
Next we take advantage of the fact that sin(-x)=-sin(x), at lines 1820-1860. We remember the sign by shoving it on the stack, and force the sign of x positive.
Lines 1870-1950 get the principal angle. I divide x by twopi, and throw away the integral part. The fractional part that remains is a fraction of a full circle, a value between 0 and .999999...9 (not radians, and not degrees either). Note that if x was extremely large there will be no fractional part, and the remainder will be zero. Some SIN function calculators give an error message when this happens, but I chose to let it ride.
Lines 1960-2000 multiply the circle-fraction by four. This gives a number between 0 and 3.99999...9, which I will refer to later as the "circle fraction times four", or c-f-t-f. The integer part is effectively a quadrant number, and the fractional part a fraction within the quadrant:
|
1 | 0
---------
2 | 3
|
Lines 2010-2030 determine if the angle is in the first (0) quadrant. If so, no folding need be done.
Lines 2040-2070 determine if the angle is in the second (1) quadrant. If so, we skip ahead to apply the fact that sin(pi/2 + x) = sin(pi/2 - x).
Lines 2080-2160 are executed if the angle is in the 3rd or 4th quadrants (integral part is 2 or 3). Here I apply the fact that sin(pi+x)=-sin(x). I pull the saved sign off the stack, complement it, and shove it back on (lines 2090-2110). Then I subtract 2 from the c-f-t-f, yielding a number between 0 and 1.99999...9. We have folded the third and fourth quadrants over the first and second quadrants. Next lines 2170-2190 determine if the result was in the first quadrant or not.
Lines 2200-2240 fold a second quadrant number into the first quadrant, by applying the fact that sin(pi/2+x) = sin(pi/2-x). Subtacting the c-f-t-f from 2 flips us into the first quadrant.
Lines 2260-2270 pull the sign off the stack and make it the sign of the angle. Remember that now the angle is a fraction (between 0 and .99999...9) of a quadrant. After all these folding operations, the angle might again be very very small, so lines 2280-2300 check for that possibility. If so, sin(x)=x, but that is only true when x is in radians. Lines 2490-2520 convert the quadrant-fraction to radians by multiplying by pi/2, and exits.
Lines 2310-2470 handle larger angles by computing x*P/Q, where P and Q are polynomials in x^2. The constants for P and Q are given in lines 1420-1550, and come from the Hart book. [ I should mention here that I wrote those constants with pretty periods separating groups of five digits. This will not assemble in some older versions of the S-C Macro Assembler. If you get a syntax error, just leave out the periods. ]
Turning the Tables:
ATN is hardest to compute. First we have to deal with the two variants of calls, having one or two arguments. While all the previous function programs were called with the argument already in DAC, DP.ATN is called immediately after parsing the ATN-token. Lines 2960-3070 parse and process the following parentheses and whatever is between them.
Lines 2960-2970 require an opening parenthesis. Line 3070 requires the closing parenthesis. In between we expect one expression, or two expressions separated by a comma. If there is only one, we fake a second one (= 1.0).
What are the two arguments? Looking at a cartesian system, with the vector shown below, the arguments are (Y,X). If you call with one argument, it is (Y/X).
By using two separate arguments, rather than just the ratio, we can tell which of the four quadrants the vector was in. DP.ATAN will return a value between -pi and +pi, depending on the two signs. If you specify only the ratio, DP.ATAN will return a value between 0 and +pi depending on the sign.
Lines 3120-3160 save the two signs in bits 6 and 7 of UV.SIGN. Way at the end, lines 4100 and following, UV.SIGN determines the final value. If the sign of the denominator (X-vector) was negative, the composite vector is in the 2nd or 3rd quadrant: computing pi - angle gives a result between pi/2 and pi.
If the numerator (Y-vector) was negative, the composite vector is in the 3rd or 4th quadrant. Flipping the sign gives a result between 0 and -pi.
Lines 3180-3220 check for special cases of Y=0 or X=0. If the first argument (Y-vector) is zero, the angle is 0 or pi depending on the sign of the second argument. If the second argument (X-vector) is zero, the angle is either +pi/2 or -pi/2, depending on the sign of the first argument. What if both arguments are zero? That should produce an error message, but I am overlooking it: I will return an angle of 0 in this case.
If neither argument is zero, some special checks are made to see if the value of the ratio is very small or very large. I check before actually dividing, so the divide routine won't kick out on an overflow error. If the ratio would be greater than 10^20, I return a value of pi/2. This is accurate within the precision of DP18. On the other hand, if the ratio is smaller than 10^-63 I return 0. If neither extreme is true, I go ahead an divide to get the actual ratio. Then I check for an extremely small ratio, in which case atan(x)=x.
If we find our way down to line 3390, the ratio is between 10^-10 and 10^20. That is still too large a range for comfort, so we apply the fact that atan(1/x) = atan(pi/2 - x). If the ratio of Y/X is greater than 1.0, then we take the reciprocal and remember that we did so. This in effect folds the range at pi/4. The resulting argument range is between 10^-10 and 1. The variable N holds either 0 or 2 as a flag: 0 if we were already under 1, 2 if we formed the reciprocal.
The shape of the curve of the arctangent function between 0 and 1 (an angle between 0 and pi/4) is deceptive. It looks nice and easy, but a polynomial over that range with 20 digits of precision is much too long. We can easily reduce the range still further by applying another identity. If the reduced argument is now already below tan(pi/12), fine. If not, calculating (x*sqr(3)-1) / (sqr(3)+x) will bring it into that range. If we have to apply that formula, N will be incremented (making it 1 or 3).
The curve between 0 and tan(pi/12) looks almost like a straight line to the naked eye, but it really is far from straight. It takes a ratio of the form P/xQ where P and Q are polynomials in x^2. The coefficients are given in lines 2650-2770, again from Hart. The ratio is computed in lines 3800-3960.
Lines 3970-4080 start the unfolding process. The variable N is either 0, 1, 2, or 3 by this time. If N is 0, no folding was done. If N is 1, only folding above pi/12 was done. If N is 2, only folding above pi/4 was done. If N is 3, both folds were done. These lines convert the angle back to the correct value, using a table of addends and an optional sign flip:
N unfolding formula
---------------------------------------
0 none
1 pi/6 + x
2 pi/2 - x
3 pi/2 - ( pi/6 + x) = pi/3 - x
That's it! We already discussed the code beyond line 4100, which figures out which quadrant the angle is in.
Any questions?
1000 *SAVE S.DP18 TRIG 1010 *-------------------------------- 1020 AS.CHRGET .EQ $B1 1030 AS.CHRGOT .EQ $B7 1040 AS.CHKCLS .EQ $DEBB 1050 AS.CHKOPN .EQ $DEB8 1060 *-------------------------------- 1070 POLY.1 .EQ $FFFF 1080 POLY.N .EQ $FFFF 1090 DADD .EQ $FFFF 1100 DSUB .EQ $FFFF 1110 DMULT .EQ $FFFF 1120 DDIV .EQ $FFFF 1130 DP.INT .EQ $FFFF 1140 DP.EXP .EQ $FFFF 1150 DP.TRUE .EQ $FFFF 1160 DP.FALSE .EQ $FFFF 1170 MOVE.DAC.ARG .EQ $FFFF 1180 MOVE.YA.ARG.1 .EQ $FFFF 1190 MOVE.YA.DAC.1 .EQ $FFFF 1200 SWAP.ARG.DAC .EQ $FFFF 1210 MOVE.DAC.TEMP1 .EQ $FFFF 1220 MOVE.DAC.TEMP2 .EQ $FFFF 1230 MOVE.DAC.TEMP3 .EQ $FFFF 1240 MOVE.TEMP1.DAC .EQ $FFFF 1250 MOVE.TEMP1.ARG .EQ $FFFF 1260 MOVE.TEMP2.ARG .EQ $FFFF 1270 MOVE.TEMP3.ARG .EQ $FFFF 1280 PUSH.DAC.STACK .EQ $FFFF 1290 POP.STACK.ARG .EQ $FFFF 1300 *-------------------------------- 1310 DAC.EXPONENT .BS 1 1320 DAC.HI .BS 10 1330 DAC.SIGN .BS 1 1340 *-------------------------------- 1350 ARG.EXPONENT .BS 1 1360 ARG.HI .BS 10 1370 ARG.SIGN .BS 1 1380 *-------------------------------- 1390 N .BS 1 1400 UV.SIGN .BS 1 1410 *-------------------------------- 1420 P.SIN .EQ * 1430 P.SIN.N .EQ 6 P6*X^6 + P5*X^5 + ... + P1*X + P0 1440 .HS 3C.50312.63884.64664.12845 P6 1450 .HS BE.82818.08039.29577.39110 P5 1460 .HS 40.62919.63490.93113.55230 P4 1470 .HS C2.25642.44036.60338.57070 P3 1480 .HS 43.53892.64053.57788.76289 P2 1490 .HS C4.49326.67470.47152.36677 P1 1500 .HS 45.12596.16380.91365.41816 P0 1510 *-------------------------------- 1520 Q.SIN .EQ * 1530 Q.SIN.N .EQ 2 X^2 + Q1*X + Q0 1540 .HS 43.15743.43316.33194.13935 Q1 1550 .HS 44.80189.66936.87727.15787 Q0 1560 *-------------------------------- 1570 CON.ONE .HS 41.10000.00000.00000.00000 1580 CON.TWO .HS 41.20000.00000.00000.00000 1590 CON.2PI .HS 41.62831.85307.17958.64769 1600 CON.PI.2 .HS 41.15707.96326.79489.66192 1610 CON.PI .HS 41.31415.92653.58979.32385 1620 CON.1..2PI .HS 40.15915.49430.91895.33577 1/2PI 1630 *-------------------------------- 1640 * COS (DAC) 1650 *-------------------------------- 1660 DP.COS LDA #CON.PI.2 PI/2 1670 LDY /CON.PI.2 1680 JSR MOVE.YA.ARG.1 COS(X) = SIN(X+PI/2) 1690 LDA #0 GET ABS(DAC) TO FORCE 1700 STA DAC.SIGN ...COS(-X)=COS(X) 1710 JSR DADD 1720 *-------------------------------- 1730 * SIN (DAC) 1740 * #3371 1750 *-------------------------------- 1760 DP.SIN 1770 *---IF X VERY SMALL...----------- 1780 LDA DAC.EXPONENT 1790 CMP #$40-10 1800 BCS .1 NOT VERY SMALL 1810 RTS VERY SMALL, SIN(X)=X 1820 *---ADJUST FOR SIGN OF X--------- 1830 .1 LDA DAC.SIGN SIN(-X) = - SIN(X) 1840 PHA ...SO SAVE SIGN OF X 1850 LDA #0 ...AND MAKE X POSITIVE 1860 STA DAC.SIGN 1870 *---X*(1/2PI)-------------------- 1880 LDA #CON.1..2PI 1890 LDY /CON.1..2PI 1900 JSR MOVE.YA.ARG.1 1910 JSR DMULT 1920 *---GET FRACTIONAL PART---------- 1930 JSR MOVE.DAC.ARG 1940 JSR DP.INT 1950 JSR DSUB 1960 *---FOLD QUADRANTS INTO ONE------ 1970 JSR MOVE.DAC.ARG MULTIPLY BY FOUR 1980 JSR DADD BY DOUBLING TWICE 1990 JSR MOVE.DAC.ARG 2000 JSR DADD 0 <= DAC < 4 2010 LDA DAC.EXPONENT IS DAC < 1? 2020 CMP #$41 2030 BCC .4 ...YES, IT IS IN 1ST QUADRANT 2040 *---2ND, 3RD, OR 4TH------------- 2050 LDA DAC.HI 2060 CMP #$20 IS DAC < 2.0? 2070 BCC .3 ...YES, 1ST OR 2ND QUADRANT 2080 *---FOLD 3RD-4TH OVER 1ST-2ND---- 2090 PLA ...NO, FLIP SIGN FOR 2100 EOR #$80 3RD OR 4TH QUADRANTS 2110 PHA 2120 LDA #CON.TWO FOLD 3RD & 4TH OVER 1ST & 2ND 2130 LDY /CON.TWO 2140 JSR MOVE.YA.ARG.1 2150 JSR SWAP.ARG.DAC 2160 JSR DSUB 2170 LDA DAC.EXPONENT 2180 CMP #$41 2190 BCC .4 ...ALREADY IN 1ST 2200 *---FOLD 2ND OVER 1ST------------ 2210 .3 LDA #CON.TWO LET X=2-X 2220 LDY /CON.TWO 2230 JSR MOVE.YA.ARG.1 2240 JSR DSUB 2250 *---ANGLE NOW IN 1ST QUADRANT---- 2260 .4 PLA PUT FINAL SIGN ON X 2270 STA DAC.SIGN 2280 LDA DAC.EXPONENT CHECK FOR VERY SMALL 2290 CMP #$40-9 2300 BCC .5 ...YES, SIN(X)=X*PI/2 2310 JSR MOVE.DAC.ARG PREPARE FOR POLYNOMIALS 2320 JSR MOVE.DAC.TEMP1 X IN TEMP1 2330 JSR DMULT X*X IN TEMP2 2340 JSR MOVE.DAC.TEMP2 2350 LDA #P.SIN 2360 LDY /P.SIN 2370 LDX #P.SIN.N 2380 JSR POLY.N 2390 JSR MOVE.DAC.TEMP3 2400 LDA #Q.SIN 2410 LDY /Q.SIN 2420 LDX #Q.SIN.N 2430 JSR POLY.1 2440 JSR MOVE.TEMP3.ARG 2450 JSR DDIV P/Q 2460 JSR MOVE.TEMP1.ARG XP/Q 2470 JMP DMULT 2480 *-------------------------------- 2490 .5 LDA #CON.PI.2 FOR VERY SMALL X 2500 LDY /CON.PI.2 SIN(2X/PI) = X*PI/2 2510 JSR MOVE.YA.ARG.1 2520 JMP DMULT 2530 *-------------------------------- 2540 * TAN (DAC) = SIN(DAC) / COS(DAC) 2550 *-------------------------------- 2560 DP.TAN JSR PUSH.DAC.STACK SAVE ANGLE 2570 JSR DP.SIN TAN=SIN/COS 2580 JSR POP.STACK.ARG GET ANGLE 2590 JSR PUSH.DAC.STACK SAVE SIN 2600 JSR SWAP.ARG.DAC 2610 JSR DP.COS GET COSINE 2620 JSR POP.STACK.ARG GET SIN 2630 JMP DDIV SIN/COS 2640 *-------------------------------- 2650 P.ATN .EQ * HART # 5505 2660 P.ATN.N .EQ 3 P3*X^3 + P2*X^2 + P1*X + P0 2670 .HS 42.12595.80226.30295.47240 P3 2680 .HS 43.12557.91664.37980.65520 P2 2690 .HS 43.29892.80380.69396.22448 P1 2700 .HS 43.19720.30956.84935.02854 P0 2710 *-------------------------------- 2720 Q.ATN .EQ * 2730 Q.ATN.N .EQ 4 X^4 + Q3X^3 + Q2X^2 + Q1X + Q0 2740 .HS 42.37066.08632.20190.23801 Q3 2750 .HS 43.20769.26817.33604.63361 Q2 2760 .HS 43.36466.24032.97707.76242 Q1 2770 .HS 43.19720.30956.84935.02861 Q0 2780 *-------------------------------- 2790 ATN.TBL.H 2800 .DA /CON.PI.6 2810 .DA /CON.PI.2 2820 .DA /CON.PI.3 2830 ATN.TBL.L 2840 .DA #CON.PI.6 2850 .DA #CON.PI.2 2860 .DA #CON.PI.3 2870 *-------------------------------- 2880 CON.TAN.PI.12 .HS 40.26794.91924.31122.70647 2890 CON.PI.6 .HS 40.52359.87755.98298.87308 2900 CON.PI.3 .HS 41.10471.97551.19659.77462 2910 CON.SQR.3 .HS 41.17320.50807.56887.72935 2920 *-------------------------------- 2930 * ATN FUNCTION 2940 * 1 OR 2 ARGUMENTS 2950 *-------------------------------- 2960 DP.ATN JSR AS.CHRGET 2970 JSR AS.CHKOPN CHECK FOR ( 2980 JSR DP.EXP GET EXPRESSION 2990 JSR PUSH.DAC.STACK 3000 JSR DP.TRUE IN CASE 1 ARGUMENT 3010 JSR AS.CHRGOT 3020 CMP #', TWO-ARG? 3030 BNE .1 NO 3040 JSR AS.CHRGET GOBBLE , 3050 JSR DP.EXP YES,GET OTHER ONE 3060 .1 JSR POP.STACK.ARG GET 1ST ARG BACK 3070 JSR AS.CHKCLS REQUIRE ")" 3080 *-------------------------------- 3090 * ATN (ARG,DAC) ARG/DAC 3100 *-------------------------------- 3110 DP.ATAN 3120 LDA DAC.SIGN SAVE BOTH SIGNS 3130 ASL SIGN OF DENOMINATOR 3140 LDA ARG.SIGN SIGN OF NUMERATOR 3150 ROR BIT 7 = DENOM SIGN 3160 STA UV.SIGN BIT 6 = NUMER SIGN 3170 *---CHECK FOR BOUNDARIES--------- 3180 LDA DAC.EXPONENT CHECK DENOMINATOR 3190 BEQ .1 ...V/0, SO RETURN PI/2 3200 SEC 3210 LDA ARG.EXPONENT 3220 BEQ .12 ...0/U, SO RETURN 0 3230 SBC DAC.EXPONENT 3240 BMI .13 3250 CMP #20 IF >10^20, RETURN PI/2 3260 BCC .11 ...NOT >10^20 3270 .1 LDA #CON.PI.2 V/0 OR OVERFLOW 3280 LDY /CON.PI.2 SO RETURN PI/2 3290 JSR MOVE.YA.DAC.1 3300 JMP DP.ATN.C 3310 .13 CMP #-63 IF <10^-63, RETURN 0 3320 BCS .11 3330 .12 JSR DP.FALSE RETURN 0 3340 .14 JMP DP.ATN.B 3350 .11 JSR DDIV CALCULATE V/U 3360 LDA DAC.EXPONENT 3370 CMP #$40-10 IF X VERY SMALL, ATAN(X)=X 3380 BCC .14 ...VERY SMALL INDEED! 3390 *---FOLD AT PI/4----------------- 3400 LDA #0 GET ABS(X), BECAUSE 3410 STA DAC.SIGN SIGNS ALREADY REMEMBERED 3420 STA N 3430 LDA DAC.EXPONENT IS X<1? 3440 CMP #$41 3450 BCC .3 ...YES, X<1 3460 LDA #CON.ONE FORM RECIPROCAL 3470 LDY /CON.ONE 3480 JSR MOVE.YA.ARG.1 3490 JSR DDIV 1/X 3500 LDA #2 AND REMEMBER WE DID IT 3510 STA N 3520 *---FOLD AT PI/12---------------- 3530 .3 JSR MOVE.DAC.TEMP1 SAVE X 3540 LDA #CON.TAN.PI.12 TAN(PI/12) 3550 LDY /CON.TAN.PI.12 3560 JSR MOVE.YA.ARG.1 3570 JSR DSUB IS X>TAN(PI/12)? 3580 LDA DAC.SIGN 3590 PHA 3600 JSR MOVE.TEMP1.DAC RESTORE X 3610 PLA 3620 BPL .4 ...NO, WE DON'T HAVE TO FOLD 3630 INC N ...YES, SO FORM 3640 LDA #CON.SQR.3 (X*SQR(3)-1) / (SQR(3)+X) 3650 LDY /CON.SQR.3 3660 JSR MOVE.YA.ARG.1 3670 JSR DMULT X*SQR(3) 3680 JSR MOVE.DAC.ARG 3690 JSR DP.TRUE 3700 JSR DSUB X*SQR(3)-1 3710 JSR MOVE.DAC.TEMP2 SAVE IT 3720 JSR MOVE.TEMP1.ARG GET X 3730 LDA #CON.SQR.3 3740 LDY /CON.SQR.3 3750 JSR MOVE.YA.DAC.1 3760 JSR DADD SQR(3)+X 3770 JSR MOVE.TEMP2.ARG 3780 JSR DDIV THE ANSWER 3790 JSR MOVE.DAC.TEMP1 SAVE FOLDED-UP X 3800 *---ATAN(0...PI/12)-------------- 3810 .4 JSR MOVE.DAC.ARG 3820 JSR DMULT X^2 3830 JSR MOVE.DAC.TEMP2 SAVE X^2 3840 LDA #P.ATN 3850 LDY /P.ATN 3860 LDX #P.ATN.N 3870 JSR POLY.N 3880 JSR MOVE.DAC.TEMP3 3890 LDA #Q.ATN 3900 LDY /Q.ATN 3910 LDX #Q.ATN.N 3920 JSR POLY.1 3930 JSR MOVE.TEMP3.ARG GET P 3940 JSR DDIV P/Q 3950 JSR MOVE.TEMP1.ARG GET X 3960 JSR DMULT P(X^2)/Q(X^2)*X 3970 *---UNFOLD FROM PI/12, PI/4------ 3980 LDX N 0, 1, 2, OR 3 3990 BEQ DP.ATN.B ...NO ADDEND 4000 DEX 0, 1, OR 2 4010 BEQ .5 ...NO COMPLEMENT 4020 LDA DAC.SIGN ATAN(1/X)=ATAN(PI/2 - X) 4030 EOR #$80 4040 STA DAC.SIGN 4050 .5 LDA ATN.TBL.L,X GET A(N) 4060 LDY ATN.TBL.H,X 4070 JSR MOVE.YA.ARG.1 4080 JSR DADD X + A(N) 4090 *---UNFOLD INTO QUADRANTS-------- 4100 DP.ATN.B 4110 BIT UV.SIGN TEST SIGN OF DENOMINATOR 4120 BPL DP.ATN.C ...POSITIVE, 1ST OR 4TH 4130 LDA #CON.PI ...NEGATIVE, 2ND OR 3RD 4140 LDY /CON.PI SO DO PI-X 4150 JSR MOVE.YA.ARG.1 4160 JSR DSUB 4170 *-------------------------------- 4180 DP.ATN.C 4190 BIT UV.SIGN TEST SIGN OF NUMERATOR 4200 BVC .6 ...POSITIVE, 1ST OR 2ND 4210 LDA DAC.SIGN ...NEGATIVE, 3RD OR 4TH 4220 EOR #$80 -X 4230 STA DAC.SIGN 4240 .6 RTS 4250 *-------------------------------- |
In recent issues of AAL there have been several articles on the 65C02 and how to get it running in the Apple II+. I too was keen to get a 65C02 working in my machine, and had spent some time trying to get first a 1MHz part and then a 2MHz part to work.
William D. O'Ryan's letter in the June 84 AAL prompted me to try again and I am happy to report that the modification he described does work (replacing the LS257's at B6 and B7 with F257's). I wanted to find exactly why I could not simply substitute a 65C02 for a 6502, and so I spent some time looking at the circuit and specifications, using an oscilloscope to check my results.
The reasons that I eventually came up with are as follows. The Apple II circuit relies on various 'features' of the 6502 so that all the various parts of the Apple will work. The circuit diagram shows that the system timing is derived from o/0; the 6502 actually expects system timing to be derived from o/2. There is a slight delay between these two signals: on a 6502 it is about 50ns and on a 65C02 it is about 30ns. This difference in delays is what causes the problems when fitting a 65C02.
To simplify its circuit design the Apple uses a rather dirty trick when reading data from RAM memory. Normally when the 6502 reads data it expects the data on the bus to be valid 100ns before the end of o/2, and it latches the data into its internal registers when o/2 changes. The setup time allows the data bus to settle into a consistent state before being read. The Apple reduces the setup time to about 45 ns (worst case). This setup time would be ample for the 65C02 were it not for the shift between o/0 and o/2; this shift reduces the setup time to 25ns. A 2MHz 65C02 specifies a MINIMUM 40ns setup time; obviously there is a -15ns tolerance on the setup time, and hence the processor works erratically when timings fall into worst case conditions.
The tolerance is regained by substituting 74F257's for the two 74LS257's at board locations B6 and B7. These two chips multiplex the RAM data and the keyboard data; in doing so they add a delay of 30ns worst case to the data. By substituting F257's, the added delay is reduced to 5 ns; this changes the tolerance on the data setup time from -15ns to +10ns.
The Apple //e must use a slightly modified technique when reading data from RAM which explains why a 65C02 works in it without any modifications. I cannot check this as I do not have a //e circuit description. Anyway, it is probably all inside the MMU chip.
[ The 65816 specifications state a minimum read data setup time of 50ns, 10ns longer than the 65C02. One AAL reader has called us to report that the 65802 works wonderfully well in his old II+, even better than the original 6502. Some of you have wondered where to get the F257's: try Jameco Electronics, 1355 Shoreway Road, Belmont, CA 94002, phone (415) 592-8097. Their ad in Byte, Dec '84, page 349, says they have 74F257's at $1.79 each. (editor) ]
This is a useful book. The kind you want to keep, read, and constantly use as a reference. About 400 pages thick, 6x9, published by Brady Communications at $19.95.
Gary, a lawyer in Vancouver, has been serious about Apples since 1978 (almost as long as me). He's a long-time subscriber to AAL, Call APPLE, and other sources of the in-depth knowledge crammed into his book. He's also a programmer, with serious software on the market such as "Modem Magician". He knows what he's writing about, and writes it well.
A walk through the chapters may be the quickest way to get the measure of the book.
1--condensed history of Apple; intro. to binary, hex, and assembly language.
2--inside the 6502 itself: zero page, stack, registers, status, opcodes, address modes, I/O, interrupts, and the memory layout in the //e.
3--the Apple monitor: the commands explained, plus a table of the most useful subroutines in the monitor ROM.
4--Applesoft: memory map, tokenization, variable storage, integer and real numbers, the CHRGET subroutine, linking to assembly language programs, subroutines in ROM, and more.
5--DOS: internal structure, memory map, page 3 vectors, VTOC, catalog, track/sector lists, RWTS, and a read.sector program. ProDOS: memory map, page 3 vectors, volume bit map, directory, MLI, and a read.block program.
6--character input and the keyboard: RDKEY, 80-column firmware, RDCHAR, reading a line, changing input devices, encoding of keys, auto-repeat, type-ahead, all about RESET.
7--character and graphic output: too much to list here, all the way through double hi-res.
8--memory management: bank switching of ROM and RAM, auxiliary RAM, running co-resident programs.
9--speaker and cassette ports: music and voice.
10--game port: experiments, push button inputs, annunciators, strobe.
11--peripheral slots: I/O memory locations, slot ROM, expansion ROM, scratchpad RAM, auxiliary slot, software protocols.
Many useful and interesting programs are listed in the book. There is an optional diskette available (coupon bound in the book offers it for $20). The diskette also includes a few bonus utility programs for use with DOS 3.3, including RAMDISK and DISK MAP.
Each chapter ends with a bibliography of related books, manuals, and articles. (You'll find lots of references to AAL.)
If you grew along with Apple, as I did, you probably don't really need this book. On the other hand, you will still enjoy it, and probably want it for you collection. If you are relatively new, and having difficulty gathering all the information from past publications and scattered sources, you will want Gary's book too.
As you might suspect, we like the book so well we have decided to stock it. You can get from us for $18 plus shipping (and tax where applicable).
In the October AAL I presented a general memory mover written in 65802 code. I stated that the MVP and MVN instructions took 3 cycles-per-byte during the move. I was wrong.
In looking through small tiny print in the preliminary documentation for the chip, I came across the number "7". Shocked, I wrote a little test program which moved 10000 bytes 1000 times. That means the MVN in my test would move a total of 10,000,000 bytes. With a stop watch I clocked the running time at just under 70 seconds. If it had been 3 cycles-per-byte, the test would have run in 30 seconds.
I don't know how I got that "3" in my head, but the right number is "7". Still considerably faster than 6502, though.
1000 *SAVE S.TIME MVN 1010 .OP 65816 1020 .OR $300 1030 *-------------------------------- 1040 CNTR .EQ 0 AND 1 1050 *-------------------------------- 1060 MVN.TIMER 1070 CLC 65816 MODE 1080 XCE 1090 REP #$30 16-BIT MODE 1100 *-------------------------------- 1110 LDA ##1000 1120 STA CNTR 1130 *-------------------------------- 1140 .1 LDX ##$3000 Source start address 1150 LDY ##$4000 Destination start address 1160 LDA ##9999 # Bytes - 1 1170 MVN 0,0 1180 DEC CNTR 1190 * BNE .1 1200 *-------------------------------- 1210 SEC RETURN TO 6502 MODE 1220 XCE 1230 RTS |
Division by seven is a necessary step for hi-res plotting routines. The quotient is the byte index on a given scan line. The remainder gives the bit position within that byte.
The hi-res code inside the Applesoft ROMs uses a subtraction loop to divide by seven, which can loop up to 36 times at 7 cycles per loop. This is a maximum of over 250 cycles, which is why super-fast hi-res usually uses lookup tables for the quotient and remainder.
I stumbled on a faster way of dividing any value up to 255 by seven. This is not directly usable by standard hi-res, because the x-coordinate can be as large as 279. My trick also does not give the remainder, just the quotient.
Here is the program, along with a test routine which tries every value from 0 to $FF, printing the quotient. The output from the test program is also shown, and you can see that the quotient is correct in every case. Can you explain why it works?
[ Hint: 1/7 = 1/8 + 1/64 + 1/512 + 1/4096 + ... ]
1000 *SAVE S.FUNNY DIVIDE BY SEVEN 1010 *-------------------------------- 1020 BYTE .EQ 0 1030 *-------------------------------- 1040 T LDA #0 1050 STA BYTE 1060 .2 LDX #14 1070 .1 CPX #7 1080 BNE .4 1090 LDA #$A0 1100 JSR $FDED 0000000000000000 0101010101010101 1110 .4 JSR DIVIDE.BY.SEVEN 0202020202020202 0303030303030303 1120 JSR $FDDA 0404040404040404 0505050505050505 1130 INC BYTE 0606060606060606 0707070707070707 1140 BEQ .3 0808080808080808 0909090909090909 1150 DEX 0A0A0A0A0A0A0A0A 0B0B0B0B0B0B0B0B 1160 BNE .1 0C0C0C0C0C0C0C0C 0D0D0D0D0D0D0D0D 1170 JSR $FD8E 0E0E0E0E0E0E0E0E 0F0F0F0F0F0F0F0F 1180 JMP .2 1010101010101010 1111111111111111 1190 .3 RTS 1212121212121212 1313131313131313 1200 *--------------------------- 1414141414141414 1515151515151515 1210 DIVIDE.BY.SEVEN 1616161616161616 1717171717171717 1220 LDA BYTE 1818181818181818 1919191919191919 1230 LSR 1A1A1A1A1A1A1A1A 1B1B1B1B1B1B1B1B 1240 LSR 1C1C1C1C1C1C1C1C 1D1D1D1D1D1D1D1D 1250 LSR 1E1E1E1E1E1E1E1E 1F1F1F1F1F1F1F1F 1260 ADC BYTE 2020202020202020 2121212121212121 1270 ROR 2222222222222222 2323232323232323 1280 LSR 24242424 1290 LSR 1300 ADC BYTE 1310 ROR 1320 LSR 1330 LSR 1340 RTS 1350 *-------------------------------- |
It is possible to divide by 3 or 15 using a program based on the same principle as the divide-by-seven above. Here is the code for those.
1210 DIVIDE.BY.FIFTEEN 1220 LDA BYTE 1230 LSR 1240 LSR 1250 LSR 1260 LSR 1270 ADC BYTE 1280 ROR 1290 LSR 1300 LSR 1310 LSR 1320 ADC BYTE 1330 ROR 1340 LSR 1350 LSR 1360 LSR 1370 RTS | 1210 DIVIDE.BY.THREE 1220 LDA BYTE 1230 LSR 1240 LSR 1250 ADC BYTE 1260 ROR 1270 LSR 1280 ADC BYTE 1290 ROR 1300 LSR 1310 ADC BYTE 1320 ROR 1330 LSR 1340 ADC BYTE 1350 ROR 1360 LSR 1370 RTS |
Using the divide by 15, you could make a divide by ten. First multiply the original number by three (by shifting one bit left and adding), then divide by 15 using the above program, and then by 2 (by shifting one bit right). Since 3X/30 = X/10, there you have it.
Have you ever wondered what would happen if you added, in the 6502 decimal mode,values that were not decimal? I have. I also wondered if any of the results might be useful.
For example, what happens if I add 0 to $0A, in decimal mode? The following little piece of code will tell me:
CLC
SED set decimal mode
LDA #$0A
ADC #0
CLD clear decimal mode
JMP $FDDA monitor print byte routine
Lo! The $0A turns into $10! It makes sense, because of course adding zero does not change anything. But the automatic "decimal adjust" that occurs after the add when the 6502 is in decimal mode detects the "A" nybble, generates a carry to the next nybble, and subtracts $0A.
It turns out the same process turns $0B into $11, $0C into $12, and so on up to $0F into $15.
That is a useful result! That means that I can convert a hex nybble to BCD byte by merely adding zero when in decimal mode!
A little further experimentation will lead to another useful trick. If I add first $90 and then $40, both additions in decimal mode, a value between $00 and $0F will be converted to the ASCII code for the digits 0-9 and letter A-F. Believe it or not!
The first addition, of $90, gives us $90-$9F. The automatic "decimal adjust" does nothing to $90-$99, and carry will be clear afterwards. If the intermediate result was $9A-$9F, the decimal adjust will first generate a nybble carry because the A-F nybble is greater than 9, and reduce that nybble by A. The nybble carry will increment the 9 nybble to A, which gets reduced back to 0 and a byte carry is set. This means we end up with $90-$99 with carry clear or $00-$05 with carry set.
Adding $40 in the next step brings the $90-$99 up to $30-$39 (with carry out of the byte, which we will ignore). The $00-$05 will be brought up to $41-$45, ASCII codes for A-F. Voila!
Useful, but maybe not the best. It turns out that a more traditional approach is only one byte longer and saves a few cycles. With the value $00-$0F in the A-register:
CMP #$0A
BCC .1 0-9
ADC #6 convert A-F to $11-16
.1 ADC #$30
will convert to ASCII.
1000 *SAVE S.HEX TO DEC 1010 T LDX #0 1020 .1 TXA 1030 JSR $FDDA 1040 LDA #"-" 1050 JSR $FDED 1060 TXA 1070 *-------------------------------- 1080 SED 1090 CLC 1110 ADC #0 1120 CLD 1130 *-------------------------------- 1140 JSR $FDDA 1150 LDA #"-" 1160 JSR $FDED 1170 TXA 1180 *-------------------------------- 1190 SED 1200 CLC 1210 ADC #$90 1220 ADC #$40 1230 CLD 1240 *-------------------------------- 1250 JSR $FDDA 1260 LDA #"-" 1270 JSR $FDED 1280 TXA 1290 *-------------------------------- 1300 CMP #10 1310 BCC .2 1320 ADC #6 1330 .2 ADC #$30 1340 *-------------------------------- 1350 JSR $FDDA 1360 *-------------------------------- 1370 JSR $FD8E 1380 INX 1390 CPX #16 1400 BCC .1 1410 RTS |
There is a lot of grumbling going on, or at least so says the media. Supposedly Mac owners are MAD over Apple's $995 price tag for the 512K upgrade kit. And the fact that new buyers get a lower system price makes them even madder.
If it's true, then I guess the computer "for the rest of us" has found a market with a real-estate or Detroit mentality. Haven't they noticed that prices on virtually all electronic items go down every year? (I always say, "If houses and cars had gone the way electronics has over the last 30 years, we would now be able to buy a 3-bedroom home for two dollars and a nice car for 50 cents. Of course they would both fit on the head of a pin....")
I remember when I bought my Apple, with two rows of 4K RAM chips totalling 8K bytes. Adding another row of 4K chips would have cost me about $50. The price at that time for one set of 8 16K chips was $520. Through a special arrangement at Mostek, members of our local club were able to get them for $150. So to raise my Apple from 8K to 48K cost me $450. Retail price would have been $1560, plus tax.
Looking back even further, I found a letter from a Raymond Hoobler to the editor of the Journal of Dentistry, from October 1976. Ray owned an Apple 1, which was populated with 1K RAM chips. He was VERY happy with Apple's promise of an upgrade kit consisting of 4K RAM chips for ONLY $500!
It will not be too long before the price of 256K RAMs drops. Then we can start grumbling about the price of 4-megabyte upgrade kits. Or, we could rejoice at the blessings of ever improving technology, mass marketing, and understanding wives.
Look on page A23 in the Apple Supplement in the back of the December 1984 issue of Byte for an excellent article for the hi-res graphics buff: "Preshift-Table Graphics on Your Apple", by Bill Budge, Gregg Williams, and Rob Moore.
The article presents another of Bill Budge's secrets for fast animation using block graphics. If you want to move a block a few dots left or right, it is time-consuming to shift the 7-bits-in-8 dot images. Older techniques stored pre-shifted sets for each image that might be moved. The neater method described in this article stores a 14x256 byte table of all possible shifts of all possible bytes, and uses a fast lookup technique. I am not going to repeat all that here ... get the article.
The article also included some sample programs that used two other tables: a 192 entry address table for the addresses of each hi-res line, and a 280 entry table for the quotient and remainder of each horizontal position. Both of these tables were originally generated by Applesoft programs, and BSAVEd. The example program BLOADed them.
It dawned on me that a machine language program to generate those two tables would take less than half a page of code and be considerably faster than BLOADing pre-generated tables. Furthermore, once the tables were generated, the half-page of code could be overlaid with other programs or data. In a commercial product, this could cut down the boot time significantly.
First I wrote a program to generate the 192 addresses. This was almost a hand-compilation of the Applesoft program in the Byte article, but not quite. (I wrote the comments in near- Basic, as you can see.)
Then I merged into that program the stuff to generate the first 192 quotients and remainders. This is the horizontal dot position divided by 7 (7 dots per byte) to give the byte position on the line and the bit position in that byte.
After the 192 trips through that code, I added a loop to generate the rest of the Q/R pairs, from dot position 192 up to 279.
I timed the program by running it 250 times. All 250 took roughly 3 seconds, which means building the tables once takes about 12 milliseconds. Compare that to loading them from disk, which would take at least a half second.
I haven't tried it yet, but I think the preshift tables which were the meat of the Byte article could also be generated by a machine language program much quicker than BLOADing the same. And since the program only needs to be used once, during initialization, it too could be burned after using.
1000 *SAVE S.MAKE HIRES ADDRS 1010 *-------------------------------- 1020 I .EQ 0 1030 JL .EQ 1 1040 JH .EQ 2 1050 K .EQ 3 1060 Q .EQ 4 1070 R .EQ 5 1080 *-------------------------------- 1090 ADDRL .EQ $900 1100 ADDRH .EQ $9C0 1110 QUO.1 .EQ $A80 1120 QUO.2 .EQ QUO.1+192 1130 REM.1 .EQ QUO.1+280 1140 REM.2 .EQ REM.1+192 1150 *-------------------------------- 1160 BUILD LDX #0 FOR X = 0 TO 191 STEP 1 1170 STX I FOR I = 0 TO $50 STEP $28 1180 STX JL FOR J = 0 TO $0380 STEP $0080 1190 STX JH 1200 STX K FOR K = 0 TO $1C STEP $04 1210 STX Q QUOTIENT = 0 1220 STX R REMAINDER = 0 1230 *---BUILD NEXT HI-RES ADDR------- 1240 .1 LDA I 1250 ORA JL 1260 STA ADDRL,X 1270 LDA #$20 1280 ORA JH 1290 ORA K 1300 STA ADDRH,X 1310 *---SAVE NEXT Q/R PAIR----------- 1320 LDA Q 1330 STA QUO.1,X 1340 LDA R 1350 STA REM.1,X 1360 *---NEXT K----------------------- 1370 CLC 1380 LDA K 1390 ADC #4 1400 STA K 1410 EOR #$20 1420 BNE .2 1430 *---NEXT J----------------------- 1440 STA K 1450 LDA JL 1460 EOR #$80 1470 STA JL 1480 BNE .2 1490 INC JH 1500 LDA JH 1510 EOR #4 1520 BNE .2 1530 *---NEXT I----------------------- 1540 STA JH 1550 CLC 1560 LDA I 1570 ADC #$28 1580 STA I 1590 *---BUMP Q/R PAIR---------------- 1600 .2 INC R R COUNTS 0...6 1610 LDA R 1620 EOR #7 IF R=7, MAKE 0 AND BUMP Q 1630 BNE .3 ...NOT 7 YET 1640 STA R ...R=7, SO MAKE IT 0 1650 INC Q AND BUMP Q 1660 *---NEXT X----------------------- 1670 .3 INX 1680 CPX #192 1690 BCC .1 1700 *---NOW FINISH Q/R PAIRS--------- 1710 *---BETWEEN 192 AND 279---------- 1720 LDX #0 FOR X = 0 TO 280-192-1 1730 .4 LDA Q 1740 STA QUO.2,X 1750 LDA R 1760 STA REM.2,X 1770 *---BUMP Q/R PAIR AS BEFORE------ 1780 INC R 1790 LDA R 1800 EOR #7 1810 BNE .5 1820 STA R 1830 INC Q 1840 *---NEXT X----------------------- 1850 .5 INX 1860 CPX #280-192 1870 BCC .4 1880 RTS 1890 *-------------------------------- 1900 .LIST OFF |
John Blankenship has put together an Applesoft enhancement package, at a mouth-watering price. (See his ad elsewhere in this issue for his $20 introductory offer.) He sent me a review copy, so I tried it out.
BBASIC is a large chunk of machine language code that sits between HIMEM and the DOS file buffers. It also sits between you and Applesoft, hiding itself behind a facade of new editing and listing features. BBASIC takes control even in direct mode, giving you an EDIT command, structured listings, and the ability to skip out of long catalogs.
In pure BBASIC, line numbers are used only as line numbers, not as destinations for GOTOs or GOSUBs. A built-in RENUM command soon convinces you to live this way and like it. In place of line-number branches, you use alphabetic "names" for subroutines, and WHEN-ELSE-ENDWHEN for logic flow. John has also added WHILE-ENDWHILE, REPEAT-UNTIL, CASE, and other structured looping and branching words.
During execution, a special COMPILE verb creates a table of "names" used in your program. This speeds up execution.
Hires Text generation is built-in, along with some extensions to the hires graphics. Musical tone generation with control over pitch, duration, and timbre is also included. You also get SORT, SEARCH, and PRINT USING.
I am just scratching the surface. I didn't like every feature, but there is plenty left over. Worth a lot more than $20.
By the way, if John's name sounds familiar, it may be because he is the author of "The Apple House", a book on controlling your home published by Prentice-Hall. John also is a Professor at DeVry Institute.
I have recently resolved a compatibility problem between two desirable sets of DOS 3.3 patches: the RAMdisk of the 192K Neptune extended memory card, and the DOS Dater that comes with Applied Engineering's Timemaster II. It seems they both want to put patches into the same "unused" spaces inside DOS.
After examining the two patches carefully, I found out which parts of the patches were overlapping. Being unable to find a truly unused area inside DOS, I used the technique on page 7.3 of "Beneath Apple DOS" of placing routines in the "safe" area between DOS and its buffers. This seems to work fine. [ Until you try to run some other program that does the same thing, like PLE... (editor) ]
The file DATER.OBJ0 contains the DOS.DATER patch that I use. I noticed that the patch could be placed anywhere, since there are no internal references. Using an Applesoft program (part of my HELLO), I move the DOS buffers down far enough to fit this code in, and then BLOAD the patches.
100 PRINT CHR$(4)"BRUN AUTO NEPTUNE"
110 PRINT "PSEUDO DISK INSTALLED"
120 POKE 40192,128 : REM Lower the buffers
130 PRINT CHR$(4)"MAXFILES 3"
140 PRINT "BUFFERS MOVED"
150 PRINT CHR$(4)"BLOAD DATER.OBJ0,A$9CD0"
160 POKE 45571,15 :REM Patch file name length
170 POKE 42883,14
180 POKE 44085,208 :REM Hook DOS to the DATER code
190 POKE 44086,156
200 PRINT "DOS DATER INSTALLED"
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