| Volume 5 -- Issue 2 | November 1984 |
In This Issue...
Apple II Troubleshooting Guide
We have just received a new book from Howard Sams: Apple II+/IIe Troubleshooting & Repair Guide, by Robert C. Brenner. At a glance, it looks like quite a good introduction to the Apple hardware and its potential problems. The first chapter is Basic Troubleshooting, followed by three chapters on Description, Operations, and Specific Troubleshooting for the II Plus, three more similar chapters on the //e, and two chapters on Preventive Maintenance and Advanced Trouble- shooting. Here's a quote from the Introduction:
This book is a detailed troubleshooting and repair document. It is not a treatise on basic computer theory or a discussion of chip operation, registers, busses, and logic gates. It is an all "meat and potatoes" manual to enable the computer user to repair his or her own machine in those 95 percent of circumstances where knowledge and a good reference are enough to find and repair a failure.
List price of the Troubleshooting & Repair Guide is $19.95. Our price will be $18 + shipping.
Apple //e Reference Manual Source
We have located a mail- or phone-order source for the Apple manuals! A reader in New York City phoned to let us know that the McGraw-Hill Bookstore there carries the Apple publications. Apparently the bookstore is also a computer store and an Apple Dealer. The address is McGraw-Hill Bookstore, 1221 Sixth Ave., New York, NY 10020. The phone number is (212) 512-4100.
All material herein is copyrighted by S-C SOFTWARE CORPORATION, all rights reserved. (Apple is a registered trademark of Apple Computer, Inc.)
| 18-Digit Arithmetic, Part 7 | Bob Sander-Cederlof |
Last month we began the implementation of math functions, so it seems appropriate to continue in the same direction. This month we will reveal the LOG and EXP functions.
As always, I turned to "Computer Approximations" for some good algorithms. I mentioned this book last month, and several of you have tried to find copies.
Thanks to Trey Johnson, of Monolith Inc. in San Antonio, for the following information: John Wiley & Sons stopped publishing the book "Computer Approximations" in 1977. They sold the rights to Krieger Publishing Co., and it is now being published under the same title. Trey was quoted a price of $22.50 + shipping. Krieger's address is P. O. Box 9542, Melbourne, FL 32901; phone is (305) 724-9542.
"Computer Approximations" is the only book I have found which lists all the actual coefficients needed to produce good approximations for the whole variety of standard functions. Pages 189-339 are packed solid with nothing by numbers. For example, there are ten pages of numbers for the EXP function alone, providing over 100 different approximation formulas for the EXP function. The chapter covering EXP describes the math behind the approximations. You pick an algorithm according to the precision you need, the number base you are using (2, 10, or whatever), the tradeoff between speed and size, and the range of arguments you will be using. Each algorithm in the book has a number, and I indicate that number in the comments to the programs which follow.
Almost all of the approximations involve these steps:
SIFT: Check the argument for legal range and
easy arguments.
FOLD: Reduce the range of the argument.
POLY: Use a polynomial or a ratio of polynomials
to approximate the function in the reduced
range.
UNFOLD: Expand the result by the reverse of the
processes used to reduce the range.
When we first learned about logarithms in high school, we used tables in books. One set of tables converted normal numbers to logs, and the other converted logs back to normal numbers. The LOG function takes the place of the first set of tables, and the EXP function replaces the second. By the way, those high school logarithms were base 10 logs. The log of a number is the power to which you would have to raise 10 to equal the number. For example, the log base 10 of 1000 is 3; of the square root of 10 is .5.
Scientists prefer base "e" logs. "e" is an irrational number (as is pi) approximately equal to 2.71828182845904523536. Did the original scientists have 2.718281828... fingers? Maybe, if they had to chop firewood (logs?)! Anyway, EXP and LOG in Applesoft work with base e. LOG tells you to what power you would raise e to equal the argument, and EXP raises e to the power of the argument.
One great application of LOG and EXP is to raise any number to any power. Applesoft (as well as DP18) has an exponentiation operator "^" for this purpose, but the code inside does it by calling on EXP and LOG. Here are some mathematical symbols to indicate how it is done:
let z = x^y
then log z = log (x^y)
log z = y log x
exp (log z) = exp (y log x)
x^y = exp (y log x)
Here is the code for the exponentiation operator in DP18:
*-------------------------------------
* EXPONENTIATION: X ^ Y
* (DAC) = Y
* (ARG) = X
*-------------------------------------
DP.POWER
JSR MOVE.DAC.TEMP3 SAVE DAC (POWER) IN TEMP3
JSR SWAP.ARG.DAC
JSR DP.LOG10 GET LOG X
JSR MOVE.TEMP3.ARG GET Y IN ARG
JSR DMULT Y LOG X
JMP DP.EXP10 X ^ Y
Notice I used base 10 log and exp? That is because DP18 is basically decimal. In a binary floating point scheme such as is internal to Applesoft, base 2 log and exp would probably be used. After all, floating point notation is a kind of half-log half-normal notation.
Which leads to the topic of converting from one logarithmic base to another. If my internal subroutines work in base 10, how do I get LOG and EXP to base e? Some more math is due:
suppose e^x = 10^y
then log10 (e^x) = log10 (10^y)
x log10(e) = y log10(10)
x log10(e) = y
Log10(e) is a constant, approximately 0.43429448190325182765. So if I want to know what EXP(3) is, I can first get 3*log10(e) = 1.302..., and 10^1.302... = 20.0855...
EXP Function
Lines 1640-1660 of the program check for a zero argument, which is an easy case: e^0 = 1. Lines 1670-1700 multiply the argument by log10(e), so that EXP10 can be used.
Lines 1730-1740 again sift out the easy case of 10^0, in case DP.EXP10 was called directly.
Lines 1750-1790 begin the folding process. We can cut the range in half by folding all negative arguments on top to the positive range: EXP(-x) = 1/EXP(x).
Lines 1810,1820 further sift, by eliminating arguments larger than 99. If the exponent of the argument is $43 or more, then the argument is 100 or more. Arguments that large are too large. (Indeed, any argument above 63 is too large.) The Applesoft ROM routine for OVERFLOW ERROR will let you know you tried it.
The arguments we have left will be in the range 0 < x < 100. We can further subdivide the range by separating the integer and fractional parts of the argument. Remember that 10^(x+y) = (10^x)*(10^y)? For illustration, suppose the argument is 3.75. Then 10^3.75 = 10^3 * 10^.75 = 5623.4132.... Lines 1830-2100 perform the separation. The variable INTPWR will get the integer part, which may range from 0 to 99. The corresponding digits are zeroed in DAC, and the resulting fraction is re-normalized. If the fractional part is zero, then the log of the fractional part is 1; lines 2080-2100 sift out this special case. This section could be accomplished by using previously covered subroutines, such as DP.INT to get the integer part, and DSUB to get the fractional part. However, that would take considerably longer for only a slight savings in space.
The active part of the argument has now been reduced to the range 0<x<1. The next adjustment will cut that in half. If the argument x<.5, this adjustment will be skipped. Lines 2120-2160 perform the test, and line 2170 saves the result of the test on the stack. We need the result later when we are unfolding. If x >= .5, then lines 2190-2210 subtract .5 from it. If x = .5, then the result after subtraction will be zero. In this case, the correct answer is a known constant, the square root of 10. Lines 2230-2270 load up that value and skip over the POLY part on down to the UNFOLDing. If not exactly .5, we now have a folded argument in the range 0<x<.5, with a flag on the stack indicating whether or not we subtracted .5 to get there. Later, if we DID subtract .5, we will multiply the result of POLY by the square root of 10 to unfold the answer.
We could have arbitrarily subtracted .5, changing the range from 0<x<1 to -.5<x<.5, with the same result. This would have saved the trouble of determining which side of .5 we were on, and of later deciding whether or not to multiply by SQR(10). However, it would also take longer for those cases already under .5, so I decided against it.
The POLY part is lines 2280-2520. This is a ratio of two polynomials, both 8th degree. However, because of derivational and computational reasons, it is actually written and calculated in a different form:
Q(x^2) + xP(x^2)
POLY(x) = ----------------
Q(x^2) - xP(x^2)
Lines 2290-2320 save x and compute x^2. Lines 2330-2380 call on POLY.N (covered last month) to compute the P polynomial, and then multiply the result by x. The constants are given in lines 1440-1490. So that you see the form, I will give it here with the coefficients rounded off:
xP = 31x^7 + 4562x^5 + 134331x^3 + 760254x
Lines 2400-2430 compute the Q-polynomial, by calling POLY.1 (also covered last month). POLY.1 is used when the coefficient of the highest degreed term is 1. We get, approximately,
Q = x^8 + 477x^6 + 29732x^4 + 408437x^2 + 660349
Lines 2440-2520 form the numerator and denominator and divide, giving us a very nice approximation to the function for the folded argument.
Lines 2530-2590 begin the unfolding process, by multiplying by SQR(10) if we previously folded .5<x<1 down to 0<x<.5.
Lines 2600-2660 take care of the integral portion of the original argument, by adding it to the EXPONENT of the result so far. This is equivalent to multiplying by the integral power of ten, but much faster. Isn't base ten nice?
The final adjustment is to take the reciprocal if the original argument was negative, done in lines 2670-2730.
LOG Function
The LOG function is the inverse of the EXP function. Now if we could just run the 6502 backwards....
Log base e is related to log base 10 the same way the exp functions were:
loge x = loge(10) * log10 (x)
Lines 2990-3040 call on the LOG10 subroutine and then multiply the result by the log base e of 10.
The LOG10 routine begins by sifting out the objectionable argument values, at lines 3100-3130. The argument MUST be positive, and MUST NOT be zero. Negative or zero arguments send you to Applesoft's ILLEGAL QUANTITY ERROR.
Lines 3140-3170 separate the exponent from the mantissa of the argument. The exponent represents the power of 10 multiplier, so as an integer it can just be added to the logarithm of the mantissa viewed as a fraction. The exponent is saved in INTPWR, to be processed later. Stuffing $40 in its place in DAC makes the range now .1<=x<1.
Lines 3180-3210 multiply the fraction by SQR(10), which changes the range to
1
------- <= x < SQR(10)
SQR(10)
This can be compensated for later by subtracting .5 from the logarithm of the folded argument.
Lines 3220 further thrash the argument by forming an intermediate argument z = (x-1)/(x+1). This value z will be in the range -.52 < z < +.52, which is a nice symmetrical value to run through a ratio of polynomials. I get lost in the math that motivates this step.
The POLY part is again a ratio of two polynomials. Lines 3330-3440 calculate the numerator, which is approximately
-15z^11 + 301z^9 - 1726z^7 + 4060z^5 - 4192z^3 + 1576z
The denominator, formed in lines 3450-3500, is approximately
z^12-68z^10+764z^8-3200z^6+6122z^4-5432z^2+1815
Dividing at line 3510 gives the logarithm of the value x. To unfold, we need to subtract .5, handled by lines 3860-3920. We also need to add as an integer the power of ten we saved in INTPWR. The latter is trickier, because we must convert a biased binary integer to a signed decimal floating point value.
Lines 3530-3600 un-bias INTPWR. If the exponent happens to be exactly $40, which in un-biased terms is 0, the rest of this step can be skipped (because the log of 10^0 is zero, adding nothing). If not, it is time to build a DP18 value in ARG. Line 3570 saves the sign in ARG.SIGN.
Lines 3610-3620 pre-clear ARG.HI, which is where we will be putting the one or two digits of INTPWR. Line 3630 assumes it will be a one-digit value, and lines 3640-3650 test that assumption. If it is one digit, lines 3730-3780 will shift the digit to the left nybble and store it in ARG.HI. If two digits, lines 3660 will divide by ten to get the high digit as quotient and low digit as remainder. Then lines 3730-3780 will merge the two digits into ARG.HI.
Lines 3790-3840 complete the construction of ARG by storing the exponent and clearing the remaining mantissa bytes. Line 3850 adds the value to the results of the POLY step, lines 3870-3920 subtract .5, and the answer is ready.
1000 *SAVE S.DP18 FUNC LOG 1010 *-------------------------------- 1020 AS.OVRFLW .EQ $E8D5 1030 AS.ILLERR .EQ $E199 1040 *-------------------------------- 1050 POLY.1 .EQ $FFFF 1060 POLY.N .EQ $FFFF 1070 DADD .EQ $FFFF 1080 DSUB .EQ $FFFF 1090 DMULT .EQ $FFFF 1100 DDIV .EQ $FFFF 1110 DP.TRUE .EQ $FFFF 1120 MOVE.YA.ARG.1 .EQ $FFFF 1130 MOVE.YA.DAC.1 .EQ $FFFF 1140 SWAP.DAC.ARG .EQ $FFFF 1150 MOVE.TEMP1.ARG .EQ $FFFF 1160 MOVE.TEMP2.ARG .EQ $FFFF 1170 MOVE.TEMP3.ARG .EQ $FFFF 1180 MOVE.DAC.ARG .EQ $FFFF 1190 MOVE.TEMP3.DAC .EQ $FFFF 1200 MOVE.DAC.TEMP1 .EQ $FFFF 1210 MOVE.DAC.TEMP2 .EQ $FFFF 1220 MOVE.DAC.TEMP3 .EQ $FFFF 1230 NORMALIZE.DAC .EQ $FFFF 1240 *-------------------------------- 1250 DAC.EXPONENT .BS 1 1260 DAC.HI .BS 10 1270 DAC.SIGN .BS 1 1280 *-------------------------------- 1290 ARG.EXPONENT .BS 1 1300 ARG.HI .BS 10 1310 ARG.SIGN .BS 1 1320 *-------------------------------- 1330 SIGN .BS 1 1340 INTPWR .BS 1 1350 *-------------------------------- 1360 CON.ONE .HS 41.10000.00000.00000.00000 1370 CON.1HALF .HS 40.50000.00000.00000.00000 1380 CON.SQR10 .HS 41.31622.77660.16837.93320 1390 *-------------------------------- 1400 * EXP (DAC) E^DAC 1410 * OR 10^DAC 1420 * #1446 IN HART, ET AL 1430 *-------------------------------- 1440 P.EXP .EQ * 1450 P.EXP.N .EQ 3 1460 .HS 42.31341.17940.19730.48777 1470 .HS 44.45618.28316.94656.35848 1480 .HS 46.13433.11347.35855.59034 1490 .HS 46.76025.44794.41265.39434 1500 Q.EXP .EQ * 1510 Q.EXP.N .EQ 4 1520 .HS 43.47705.44030.08207.98775 1530 .HS 45.29732.60655.85996.83303 1540 .HS 46.40843.69796.67736.28236 1550 .HS 46.66034.86505.27141.54491 1560 *-------------------------------- 1570 CON.LOGE .HS 40.43429.44819.03251.82765 1580 *-------------------------------- 1590 DP.EXP.NULL 1600 JMP DP.TRUE E^0 = 10^0 = 1.0 1610 DP.EXP.OVERFLOW 1620 JMP AS.OVRFLW 1630 *-------------------------------- 1640 DP.EXPE 1650 LDA DAC.EXPONENT 1660 BEQ DP.EXP.NULL 1670 LDA #CON.LOGE 1680 LDY /CON.LOGE 1690 JSR MOVE.YA.ARG.1 1700 JSR DMULT CHANGE TO 10^X 1710 *-------------------------------- 1720 DP.EXP10 1730 LDX DAC.EXPONENT 10^0 = 1 1740 BEQ DP.EXP.NULL 1750 *---HANDLE NEGATIVE POWERS------- 1760 LDA DAC.SIGN SAVE FOR 1/EXP IF NEGATIVE 1770 STA SIGN 1780 LDA #0 GET ABS(X) 1790 STA DAC.SIGN 1800 *---SPLIT INTEGER & FRACTION----- 1810 CPX #$43 THREE OR MORE INTEGER DIGITS? 1820 BCS DP.EXP.OVERFLOW YES, OVERFLOW 1830 LDA #0 ...ALL FRACTIONAL 1840 STA INTPWR 1850 CPX #$41 1860 BCC .3 ...NO INTEGRAL PART 1870 LDA DAC.HI ...1 OR 2 DIGITS 1880 LSR 1890 LSR 1900 LSR 1910 LSR 1920 STA INTPWR 1930 LDA DAC.HI 1940 AND #$0F 1950 STA DAC.HI 1960 CPX #$41 ONE OR TWO DIGITS? 1970 BEQ .2 ...ONE DIGIT INTEGER 1980 LDA INTPWR DIGIT*10 1990 ASL 2000 ASL 2010 ADC INTPWR 2020 ASL 2030 ADC DAC.HI 2040 STA INTPWR 2050 LDX #0 2060 STX DAC.HI 2070 .2 JSR NORMALIZE.DAC ADJUST REMAINING FRACTION 2080 BNE .3 FRACTION NOT 0 2090 JSR DP.TRUE 10^0 = 1 2100 JMP .7 2110 *---ADJUST FRACTION SO < .5------ 2120 .3 LDA DAC.EXPONENT 2130 CMP #$40 2140 BCC .4 2150 LDA DAC.HI 2160 CMP #$50 2170 .4 PHP REMEMBER... 2180 BCC .5 ...ALREADY < .5 2190 SBC #$50 2200 STA DAC.HI 2210 JSR NORMALIZE.DAC 2220 BNE .5 ...REST OF FRACTION NOT 0 2230 PLA POP SAVED STATUS 2240 LDA #CON.SQR10 2250 LDY /CON.SQR10 2260 JSR MOVE.YA.DAC.1 2270 JMP .7 2280 *---COMPUTE 10^.XXXX------------- 2290 .5 JSR MOVE.DAC.TEMP1 SAVE X 2300 JSR MOVE.DAC.ARG 2310 JSR DMULT GET X^2 2320 JSR MOVE.DAC.TEMP2 SAVE X^2 2330 LDA #P.EXP COMPUTE P(X^2) 2340 LDY /P.EXP 2350 LDX #P.EXP.N 2360 JSR POLY.N 2370 JSR MOVE.TEMP1.ARG COMPUTE XP(X^2) 2380 JSR DMULT 2390 JSR MOVE.DAC.TEMP3 SAVE XP(X^2) 2400 LDA #Q.EXP COMPUTE Q(X^2) 2410 LDY /Q.EXP 2420 LDX #Q.EXP.N 2430 JSR POLY.1 2440 JSR MOVE.DAC.TEMP2 SAVE Q(X^2) 2450 JSR MOVE.TEMP3.ARG NUMERATOR = Q+XP 2460 JSR DADD Q(X^2)+XP(X^2) 2470 JSR MOVE.DAC.TEMP1 SAVE UMERATOR 2480 JSR MOVE.TEMP2.ARG DENOMINATOR = Q-XP 2490 JSR MOVE.TEMP3.DAC 2500 JSR DSUB Q(X^2)-XP(X^2) 2510 JSR MOVE.TEMP1.ARG 10^.XXX = N/D 2520 JSR DDIV 2530 *---ADJUST BY SQR(10)------------ 2540 PLP SEE IF ADJUSTMENT NEEDED 2550 BCC .7 ...NO 2560 LDA #CON.SQR10 2570 LDY /CON.SQR10 2580 JSR MOVE.YA.ARG.1 2590 JSR DMULT 2600 *---ADD INTEGRAL POWER----------- 2610 .7 CLC 2620 LDA DAC.EXPONENT 2630 ADC INTPWR 2640 BPL .8 ...NO OVERFLOW 2650 JMP DP.EXP.OVERFLOW 2660 .8 STA DAC.EXPONENT 2670 *---ADJUST FOR SIGN-------------- 2680 LDA SIGN GET ORIGINAL SIGN 2690 BPL .9 POSITIVE, WE ARE DONE 2700 LDA #CON.ONE NEGATIVE, FORM RECIPROCAL 2710 LDY /CON.ONE 2720 JSR MOVE.YA.ARG.1 2730 JSR DDIV 2740 .9 RTS 2750 *-------------------------------- 2760 * LN (DAC) LOG E (DAC) 2770 * OR LOG 10 (DAC) 2780 * #2330 IN HART, ET AL 2790 *-------------------------------- 2800 P.LOG .EQ * 2810 P.LOG.N .EQ 5 2820 .HS C2.14933.41871.23101.49868 2830 .HS 43.30132.34734.14748.46138 2840 .HS C4.17255.36265.00653.03387 2850 .HS 44.40598.33123.94476.21513 2860 .HS C4.41923.45602.07081.07911 2870 .HS 44.15764.33484.51127.69255 2880 Q.LOG .EQ * 2890 Q.LOG.N .EQ 6 2900 .HS C2.67696.41190.46224.52758 2910 .HS 43.76357.00230.09155.79877 2920 .HS C4.32000.87986.36664.12225 2930 .HS 44.61216.00041.77468.78069 2940 .HS C4.54315.94950.92575.25735 2950 .HS 44.18149.36120.76616.30282 2960 *-------------------------------- 2970 CON.LN10 .HS 41.23025.85092.99404.56840 2980 *-------------------------------- 2990 DP.LOGE 3000 JSR DP.LOG10 3010 LDA #CON.LN10 CONVERT LOG10 TO LN 3020 LDY /CON.LN10 3030 JSR MOVE.YA.ARG.1 3040 JMP DMULT 3050 *-------------------------------- 3060 DP.LOG.ERR 3070 JMP AS.ILLERR 3080 *-------------------------------- 3090 DP.LOG10 3100 LDA DAC.SIGN CHECK RANGE 3110 BMI DP.LOG.ERR ...NEGATIVE 3120 LDA DAC.EXPONENT 3130 BEQ DP.LOG.ERR ...ZERO 3140 STA INTPWR SAVE POWER OF 10 3150 *---ADJUST RANGE----------------- 3160 LDA #$40 MAKE FRACTION .1 TO .9999 3170 STA DAC.EXPONENT 3180 LDA #CON.SQR10 1/SQR(10) ... SQR(10) 3190 LDY /CON.SQR10 3200 JSR MOVE.YA.ARG.1 3210 JSR DMULT 3220 *---FORM (X-1)/(X+1)------------- 3230 JSR MOVE.DAC.TEMP1 3240 JSR MOVE.DAC.ARG 3250 JSR DP.TRUE GET 1 IN DAC 3260 JSR DSUB X-1 3270 JSR MOVE.DAC.TEMP2 SAVE IT 3280 JSR DP.TRUE GET 1 IN DAC 3290 JSR MOVE.TEMP1.ARG 3300 JSR DADD X+1 3310 JSR MOVE.TEMP2.ARG 3320 JSR DDIV X-1/X+1 3330 *---NUMERATOR = Z*P(Z^2)--------- 3340 JSR MOVE.DAC.TEMP1 SAVE IT 3350 JSR MOVE.DAC.ARG 3360 JSR DMULT Z^2 3370 JSR MOVE.DAC.TEMP2 SAVE Z^2 3380 LDA #P.LOG 3390 LDY /P.LOG 3400 LDX #P.LOG.N 3410 JSR POLY.N 3420 JSR MOVE.TEMP1.ARG 3430 JSR DMULT Z*P(Z^2) 3440 JSR MOVE.DAC.TEMP1 3450 *---DENOMINATOR = Q(Z^2)--------- 3460 LDA #Q.LOG 3470 LDY /Q.LOG 3480 LDX #Q.LOG.N 3490 JSR POLY.1 3500 JSR MOVE.TEMP1.ARG 3510 JSR DDIV Z*P(Z^2)/Q(Z^2) 3520 *---ADD INTEGER POWER------------ 3530 SEC 3540 LDA INTPWR GET POWER OF 10 3550 SBC #$40 3560 BEQ .5 ...0, NO NEED TO ADD ANYTHING 3570 STA ARG.SIGN 3580 BCS .1 ...1 TO 63 3590 EOR #$FF MAKE IT POSITIVE 3600 ADC #1 3610 .1 LDY #0 3620 STY ARG.HI 3630 LDX #$41 3640 CMP #10 3650 BCC .3 1...9 3660 INX 10...63 3670 .2 STA ARG.HI STORE REMAINDER 3680 SBC #10 3690 INY INC. QUOTIENT 3700 BCS .2 ...TRY ANOTHER SUBTRACTION 3710 DEY CORRECT QUOTIENT 3720 TYA GET QUOTIENT 3730 .3 ASL LEFT JUSTIFY 3740 ASL 3750 ASL 3760 ASL 3770 ORA ARG.HI MERGE WITH NEXT DIGIT 3780 STA ARG.HI 3790 STX ARG.EXPONENT $41 OR $42 3800 LDX #9 CLEAR REST OF ARG 3810 LDA #0 3820 .4 STA ARG.HI,X 3830 DEX 3840 BNE .4 3850 JSR DADD 3860 *---SUBTRACT 0.5----------------- 3870 .5 LDA #CON.1HALF 3880 LDY /CON.1HALF 3890 JSR MOVE.YA.ARG.1 3900 LDA #$FF 3910 STA ARG.SIGN 3920 JMP DADD 3930 *-------------------------------- |
| S-C Macro Assembler Version 2.0 | Bill Morgan |
We are now accepting orders for the upgrade to S-C Macro Assembler Version 2.0. Here is a summary of the new features:
Registered owners of S-C Macro Assembler will be able to purchase the upgrade to Version 2.0 for only $20.00. Just send us a check or charge card number, and you will be among the first to have the new version.
| Convert Two Decimal Digits to Binary | Bob Sander-Cederlof |
I have recently been running into more and more uses for the decimal mode in the 6502. In the decimal mode, each byte contains a value from 0 to 99, with the ten's digit in the left nybble and the units digit in the right nybble.
The 6502 has built-in capability to add and subtract values in this format, with automatic carry when a nybble exceeds 9. If you have been following my series on 18-digit arithmetic, you have seen a lot of examples of its use.
A frequent problem that arises is conversion between the decimal form and the binary form of a number. I suppose I have written ten million different programs to do this kind of conversion, on at least a thousand different kinds of computers! (Ever notice that my exaggerations are always in decimal?)
For a small (byte-size) example, suppose a byte contains two decimal digits ($00-$99) and you want to convert it to binary ($00-$63). The first step is to separate the two digits into two different variables. The the ten's digit should be multiplied by ten, and the unit's digit added.
Lines 1390-1510 in the listing perform these steps, but there are a few tricks. Lines 1410-1420 strip out the unit's digit and save it in LOW, and lines 1440-1450 save the high digit in HIGH. Notice that I did not shift the high digit down, so it is really the ten's digit times 16 (call it "tens*16").
Lines 1460-1500 multiply the tens*16 by 10/16. Then line 1500 adds the unit's digit.
The program in lines 1010-1190 is a test driver, which calls the DEC.HEX.2 routine 100 times with successive values in the A-register between $00 and $99. DEC.HEX.2 returns with the converted value ($00-$63 in the A-register, and the test driver prints out the value. If everything is okay, the hexadecimal numbers from $00 through $63 will be displayed.
DEC.HEX.2 as written takes 18 bytes plus two variables in page zero. If the variables are not in page zero, the program will take an additional four bytes.
A faster program which takes only a few more bytes, and does not use any variables in RAM other than the stack, is shown in lines 1200-1340. Lines 1220-1260 convert the ten's digit into an index 0-9 in the X-register. Line 1270 retrieves the original number from the stack. Lines 1290-1300 add a value from the table, indexed by the ten's digit, giving a total which is the converted number.
The values in the table consist of one byte each, having selected so that they subtract out the hexadecimal value of the ten's digit and add back the value of that digit-times-ten in binary. For example, if the original number was $58 (meaning decimal 58 in BCD storage format), we will add the value $E2 (which is 50-$50). $58+$E2 = $3A, which is the correct hexadecimal conversion.
I recently worked on a consulting project which included a lot of mixed decimal and hexadecimal calculations. The project was implemented on a 6511 chip, which has only 192 bytes of RAM. That is total, including the stack! We also had 4096 bytes of EPROM. The system operates in a real-time mode with relatively high-speed interrupts occurring. With these constraints, every routine had to be written to use the minimum amount of RAM and to be as fast as possible. A few extra bytes of code would be all right, because 4096 bytes of EPROM was more than enough. In situations like this, programs like the one in lines 1200-1300 come in real handy.
1000 *SAVE S.QUICK DEC-HEX 1010 *-------------------------------- 1020 T LDA #0 1030 STA 0 1040 .1 LDA 0 1050 JSR DEC.HEX.2 1060 JSR $FDDA 1070 LDA #" " 1080 JSR $FDED 1090 JSR $FDED 1100 SED 1110 CLC 1120 LDA 0 1130 ADC #1 1140 STA 0 1150 CLD 1160 CMP #0 1170 BNE .1 1180 RTS 1190 *-------------------------------- 1200 DEC.HEX 1210 PHA SAVE BYTE 1220 LSR 1230 LSR 1240 LSR 1250 LSR 1260 TAX HI NYBBLE TO X 1270 PLA GET ORIG BYTE 1280 CLC 1290 ADC TBL,X 1300 RTS 1310 *-------------------------------- 1320 TBL .DA #0-0,#10-$10,#20-$20,#30-$30 1330 .DA #40-$40,#50-$50,#60-$60 1340 .DA #70-$70,#80-$80,#90-$90 1350 *-------------------------------- 1360 LOW .EQ 1 1370 HIGH .EQ 2 1380 *-------------------------------- 1390 DEC.HEX.2 1400 PHA 1410 AND #$0F SAVE LOW NYBBLE 1420 STA LOW 1430 PLA 1440 AND #$F0 GET HIGH NYBBLE 1450 STA HIGH 1460 LSR /2 1470 LSR /4 1480 ADC HIGH /4*5 1490 LSR /8*5 = *10/16 1500 ADC LOW + LOW NYBBLE 1510 RTS 1520 *-------------------------------- |
| A Whole Megabyte for your Apple //e | Bob Sander-Cederlof |
Both Applied Engineering and Saturn have announced 1 Mbyte cards for the //e. Saturn's, I understand, plugs into any slot 1-7; this of course makes it a little non-standard compared to other //e memory expanders when it comes to software access.
The new board from Applied Engineering, called RAM WORKS, fits in the //e auxiliary slot. You get 80 column text and double hi-res, with anywhere from 64K to 1 Megabyte of expansion RAM in 64K or 256K increments. You can buy RAM WORKS already expanded, or expand it yourself later. Prices: 64K = $179, 128K = $249, 256K = $449, 512K = $799, and 1Meg = $1499. The first 512K fits one a normal size card, about 6 inches long. The second 512K come in a piggy-back card which attaches to the main card. Another option, an RGB video generator ($129), attaches to the front of the memory card.
The megabyte is divided into 16 chapters of 64K each. You select which one is active by storing a value from $00 to $0F in a register at $C073. Then the normal //e maze of soft switches lets you access the active chapter the same way you would access Apple's standard 64K card.
RAM WORKS has some new design ideas, for which patents are pending, including a power saving circuit and a video refresh circuit. The latter eliminates the annoying screen flicker that normally occurs when you switch chapters with older expansion cards.
Low cost software options available with RAM WORKS include disk emulation for DOS and ProDOS, and workspace expansion for Appleworks. Standard ProDOS will turn Apple's RAM card into a half-size RAMdisk, but with RAM WORKS you get a full megabyte!
If you like the idea of souping up your //e, one of these boards plus a new 65802 processor may be just the ticket!
| 65816 News | Bill Morgan |
Did you see the Infoworld article a few weeks ago (November 5 issue) about the 65816? That story mentioned a plug-in board for the Apple II containing a 65816 processor and extra RAM. Well, I spoke today with Larry Hittel of Com Log, producers of that board, and it does sound very interesting.
Com Log intended their board, the Apple16, to be a developers' tool, rather than a consumer item, or an Apple hot-rod device. They were therefore a little surprised and overwhelmed by the response to the Infoworld story: When I talked to Larry they had exactly one board in stock, and it was waiting for purchase order paperwork from Apple Computer. They are a month or two away from full production quantities.
The Apple16 board uses DMA (Direct Memory Access) to take control of the Apple, shutting down the 6502 and taking over the address bus. They have found that the DMA does not function properly in Apples earlier than Revision 4, due to problems with the bus driver chips on the motherboard.
The 65816 chips are designed to operate at 8 MHz and are currently testing out at 2-4 MHz, but, in order to maintain compatibility with the Apple, the Com Log processor is clocked at 1 MHz.
To the '816, the 64K of Apple memory, both RAM and ROM, is bank 0. Bank 1 echoes the Apple from 0-DFFF, but contains space for new EPROM at E000-FFFF. Banks 2 and 3 are reserved for more new EPROM. Banks 4-7 are the on-board RAM, consisting of one set of either 64K or 256K chips. Banks 8-255 are available on an expansion connector, intended for a future separate memory board. There is abort logic to provide an interrupt on access to non-existent memory.
Com Log is selling the boards now with no EPROMs. They are working on an operating system and an Applesoft interpreter, but those are still some time away. No price has been set for the firmware yet.
The current price of the Apple16 board is $395 with no RAM, $450 with 64K, and $795 with 256K. They are not expecting to have them available in production quantities until January or later, by which time the prices might change. Contact Com Log Corporation at 11056 N. 23rd Dr., Suite 104, Phoenix, AZ 85029. Phone (602) 248-0769.
That Infoworld story quoted an Apple spokesman as saying that the 65816 was to be used in an earlier project that had been shelved. That project is being dusted off and revived, now that the 65816 chips are really coming through. We've been hearing of it as the Apple //x. According to an article in the November 19 issue of Infoworld about an interview with Woz, the //x is still not a fixed design and will not be ready for market until 1986. There's always something new to look forward to!
| New DP18 Square Root Subroutine | Bob Sander-Cederlof |
Even after bending over backwards to be certain I had the best possible SQR implementation in the October AAL, I still found some ways to improve it. Last night I found some more information in a book called "Software Manual for the Elementary Functions", by William Cody and William Waite, Prentice-Hall, 1980.
They pointed out that in general an extra Newton iteration took less time than a complex method of getting an initial approximation which would be accurate enough to avoid one iteration. In other words, using a cubic polynomial like I did in October is just not worth it. Not worth the time, and not worth the space.
They further pointed out that it is best to compute the last Newton iteration in a slightly different fashion, to avoid shifting out the last significant digit. The normal iteration computes (x/y + y)*.5. Re-arrangement to y+(x/y-y)*.5 is better. Since it takes an extra step, it should only be used the last time.
To see the difference, consider the example below. I have used a precision of just 3 digits (instead of 18 or 20)to simplify the illustration:
let x=.253, and y=.5
then x/y=.506
x/y+y=1.00 (truncating to 3 places)
(x/y+y)*.5 = .500, which is wrong
x/y-y=.006
(x/y-y)*.5=.003
y+(x/y-y)*.5 = .503, which is correct.
My new SQR version uses a much faster method for getting the first approximation. The first two digits of the argument (in DAC.HI) must be in the range from 10 to 99. I convert them to an index between $02 and $13 by shifting the first digit over three, and adding one if the second digit is 5 or more. In other words, 10-14 become $02, $15-19 become $03, on up to $95-99 becoming $13. Then I use that value as an index into a table which gives a good approximation to the first two digits of the square root. For example, any number between .10 and .19999...9 will get a first approximation of .35. I store those two digits into DAC.HI, letting the remaining digits stay as they were. This method gives a first approximation which in the worst case still has at least the first digit correct.
It turns out the worst case is for numbers with odd exponents and the mantissa=1, such as 1 (which is .1*10^1), 100 (which is .1*10^3), and so on. Even in this worst case, four iterations give 20 digits of precision.
The end result of these changes is a faster and shorter program which is more accurate. Here is the new listing:
1000 *SAVE S.NEW SQR ROUTINE 1010 *-------------------------------- 1020 * SQR (DAC) 1030 *-------------------------------- 1040 ERR.SQ JMP AS.ILLERR ILLEGAL QUANTITY 1050 DP.SQR.0 RTS 1060 DP.SQR LDA DAC.EXPONENT 1070 BEQ DP.SQR.0 SQR(0)=0 1080 LDA DAC.SIGN 1090 BMI ERR.SQ MUST BE POSITIVE 1100 JSR MOVE.DAC.TEMP3 SAVE X 1110 *---APPROX. ROOT OF .1 - 1------- 1120 LDA DAC.HI CONVERT TWO DIGITS TO BINARY 1130 AND #$0F SAVE LO DIGIT 1140 CMP #5 01234 OR 56789 1150 PHP SAVE ANSWER 1160 LDA DAC.HI GET HI DIGIT 1170 LSR 1180 LSR 1190 LSR 1200 LSR $01...$09 1210 PLP 01234 OR 56789 1220 ROL $02...$13 1230 TAX 1240 LDA SQR.TBL,X 1250 STA DAC.HI 1260 *---TAKE HALF OF EXPONENT-------- 1270 LDA DAC.EXPONENT 1280 SEC 1290 SBC #$40 REMOVE OFFSET 1300 ROR DIVIDE BY TWO (KEEP SIGN) 1310 PHP SAVE ODD/EVEN BIT 1320 CLC 1330 ADC #$C0 RE-BIAS EXPONENT 1340 STA DAC.EXPONENT 1350 PLP 1360 BCC .1 EVEN, DON'T MULT BY SQR(10) 1370 *---ADJUST APPROX FOR ODD EXP---- 1380 LDA #CON.SQR10 1390 LDY /CON.SQR10 1400 JSR MOVE.YA.ARG.1 1410 JSR DMULT 1420 *---THREE NEWTON ITERATIONS------ 1430 .1 LDA #3 1440 STA TEMP3 1450 .2 JSR MOVE.DAC.TEMP2 TEMP2 = Y 1460 JSR MOVE.TEMP3.ARG GET X 1470 JSR DDIV X/Y 1480 JSR MOVE.TEMP2.ARG 1490 JSR DADD X/Y+Y 1500 LDA #CON.HALF 1510 LDY /CON.HALF 1520 JSR MOVE.YA.ARG.1 1530 JSR DMULT (X/Y+Y)/2 1540 DEC TEMP3 ANY MORE? 1550 BNE .2 ...YES 1560 *---ONE MORE NEWTON ITERATION---- 1570 JSR MOVE.DAC.TEMP2 TEMP2 = Y 1580 JSR MOVE.TEMP3.ARG GET X 1590 JSR DDIV X/Y 1600 JSR MOVE.TEMP2.ARG 1610 LDA #$FF 1620 STA ARG.SIGN 1630 JSR DADD X/Y-Y 1640 LDA #CON.HALF 1650 LDY /CON.HALF 1660 JSR MOVE.YA.ARG.1 1670 JSR DMULT (X/Y-Y)/2 1680 JSR MOVE.TEMP2.ARG 1690 JMP DADD Y + (X/Y-Y)/2 1700 *-------------------------------- 1710 SQR.TBL .EQ *-2 (NO ENTRIES AT 0...1) 1720 .HS 35.42.47.52.57.61.65.69.72 1730 .HS 76.79.82.85.88.91.94.96.99 1740 CON.SQR10 .HS 4131622776601683793320 1750 CON.HALF .HS 4050000000000000000000 1760 *-------------------------------- |
| Improvements to 80-column Monitor Dump | Jan Eugenides |
I found a little bug in the 80-column ASCII monitor dump, as presented in Sept 1983 AAL (page 27,28). It worked great in the 80-column mode, but if I happened to be in 40-column mode when I used the monitor dump command something strange happens.
Some time ago I incorporated the dump and Steve Knouse's monitor patches into an EPROM and installed it in my system. Everything seemed to be working fine, until one day.... I was working on a short Applesoft program, and I went into the monitor in 40-column mode to check a few program bytes. When I returned to Applesoft and listed the program, the first line had been changed. Huh?
I eventually figured out that the problem had to do with the tab to column 60. In 40-column mode this will be 20 characters beyond the bottom of the screen, which is $80C.
The solution was to just print a few spaces rather than attempting to tab. This approach makes for more compatibility among various 80-column devices, too.
While I was at it, I even squeezed a byte out of the code.
[And I squeezed some more, saving a total of 11 bytes. Bob S-C]
Here is the modified routine:
1000 *SAVE S.NEW 80 COL MONITOR DUMP 1010 *-------------------------------- 1020 * TO INSTALL, 1030 * 1. ASSEMBLE THIS PROGRAM 1040 * 2. ENTER THESE MONITOR COMMANDS 1050 * $C083 C083 FCC9<CC9.CEFM 1060 * $FDBE:C9 FC N FDA6:F N FDB0:F 1070 *-------------------------------- 1080 * BY JAN EUGENIDES & BOB S-C 1090 *-------------------------------- 1100 CH .EQ $24 1110 A1 .EQ $3C,3D 1120 A2 .EQ $3E,3F 1130 A4 .EQ $42,43 1140 BUFFER .EQ $2F0 1150 PRBYTE .EQ $FDDA 1160 COUT .EQ $FDED 1170 PRBLNK .EQ $F948 1180 *-------------------------------- 1190 .OR $FCC9 1200 .TA $CC9 1210 *-------------------------------- 1220 PATCH PHA SAVE BYTE 1230 LDA A1 COMPUTE INDEX 1240 AND #$0F 0...F 1250 TAX 1260 PLA GET BYTE AGAIN 1270 STA BUFFER,X SAVE IN BUFFER 1280 JSR PRBYTE PRINT ON SCREEN 1290 INX GET # BYTES THIS LINE 1300 STX A4 SAVE IN A4L 1310 CPX #$10 END OF LINE? 1320 BEQ .1 ...YES, PRINT ASCII CHARS 1330 LDA A1 ...NO, SEE IF END OF RANGE 1340 CMP A2 1350 LDA A1+1 1360 SBC A2+1 1370 BCC .4 ...NO, RETURN 1380 .1 JSR PRBLNK PRINT 3 SPACES 1390 LDX #0 PRINT ASCII CHARS FROM BUFFER 1400 .2 LDA BUFFER,X GET CHAR 1410 ORA #$80 MAKE NORMAL VIDEO 1420 CMP #$A0 TRAP CONTROL CHARS 1430 BCS .3 ...NOT CONTROL CHAR 1440 LDA #$AE ...CTRL, SUBSTITUTE "." 1450 .3 JSR COUT PRINT CHAR 1460 INX NEXT 1470 CPX A4 END OF LIST? 1480 BCC .2 ...NOT YET 1490 .4 RTS RETURN |
Note the directions for installing the routine in a RAM card copy of the monitor, in lines 1020-1060. "$C083 C083 FCC9<CC9.CFFM" write enables the RAM area and copies the dump code over the top of cassette I/O stuff. "$FDBE:C9 FC N FDA6:F N FDB0:F" patches the monitor dump command code to call the new patch, and also patches to print 16 bytes per screen line.
If you want to use this routine in 40-column mode only, change line 1240 from "AND #$0F" to "AND #$07", line 1310 from "CPX #$10" to "CPX #$08", and leave out the patches at FDA6 and FDB0 in the previous paragraph.
| Generating Cross Reference Text File with DISASM | Bob Kovacs |
I received a phone call from Don Lancaster the other day. He had been using DISASM to probe the mysteries of AppleWriter, and was now preparing to document his findings. Although he liked the way DISASM generated a triple cross reference table, he preferred to have it in a form that could be used by his word processor (that is, on a text file). The cross reference table generated by DISASM is normally output to either the screen or a printer, so Don's only alternative was to manually type it into his word processor. There were hundreds of labels....
It turned out that a simple patch to DISASM will do the trick. All that is necessary is to change the JSR PASS2 which normally generates the source code listing to JSR XREF.
The following patch outputs the cross reference table to your file after responding "Y" to the prompt "GENERATE TEXT FILE?":
$09A1:20 F1 0A
Back in the April issue of AAL, I described a method of using EXEC files with DISASM. A patch was required to the "YES/NO" routine to input the response via KEYIN rather than directly from the keyboard. Although the patch I gave in April works, KEYIN uses the Y-register as an index to the screen. My patch did not always wind up in the right place. So I have expanded the patch as follows:
$0C57:EA A4 24 20 18 FD 09 80
I hope that this has not caused any inconvenience.
| Macro Information by Example | Sandy Greenfarb |
The following are three examples of macro use which I have found interesting and informative.
The first example, TEST, shows that you can use parameters in places other than the operand field. In this case, one of the parameters becomes part of an opcode name.
SETD shows how a macro can make more efficient code. If both bytes are the same, there is no need to have two LDA instructions.
MOVD copies two bytes from one variable to another. If you use MOVD to move two bytes one byte higher in RAM, MOVD will reverse the order the bytes are moved so that the data are not clobbered.
1000 *SAVE S.MACRO EXAMPLES 1010 *-------------------------------- 1020 * BY SANDY GREENFARB 1030 *-------------------------------- 1040 * 1050 * PARAMETERS CAN SUBSTITUTE ANYWHERE, 1060 * EVEN IN OPCODES 1070 *-------------------------------- 1080 .MA TEST VALUE,CONDITION,LABEL 1090 CMP ]1 1100 B]2 ]3 1110 .EM 1120 * 1130 >TEST #3,CC,SMALLER 1140 >TEST TYPE,EQ,SAME 1150 * 1160 TYPE .DA #35 1170 SAME NOP 1180 SMALLER NOP 1190 *-------------------------------- 1200 * 1210 * MACROS CAN SIMPLIFY CODE FOR EFFICIENCY 1220 *-------------------------------- 1230 .MA SETD VALUE,VARIABLE 1240 LDA #]1 LO-BYTE 1250 STA ]2 1260 .DO ]1/256*257-]1 ARE LOW AND HI EQUAL? 1270 LDA /]1 1280 .ELSE 1290 * HI = LO-BYTE 1300 .FIN 1310 STA ]2+1 1320 .EM 1330 * 1340 >SETD $1234,VALUE 1350 >SETD $2323,VALUE 1360 * 1370 VALUE .BS 2 1380 *-------------------------------- 1400 * 1410 * MACROS CAN PREVENT PROGRAMMING MISTAKES 1420 * SUCH AS OVER-WRITING WHEN YOU COPY 1430 * ONE VARIABLE INTO ANOTHER. 1440 *-------------------------------- 1450 .MA MOVD VAR1,VAR2 1460 .DO ]2-]1-1 1470 LDA ]1 NO OVERLAP 1480 STA ]2 1490 LDA ]1+1 1500 STA ]2+1 1510 .ELSE 1520 LDA ]1+1 THIS CODE BUILT WHEN THE 1530 STA ]2+1 VARIABLES OVERLAP 1540 LDA ]1 1550 STA ]2 1560 .FIN 1570 .EM 1580 * 1590 >MOVD $11,$22 1600 >MOVD $28,VALUE 1610 >MOVD $11,$12 1620 *-------------------------------- |
| Turning Bit-Masks into Indices | Bob Sander-Cederlof |
A few months ago I presented several ways to turn an index (0-7) into a bit mask (01, 02, 04,...,80). We got a lot of feedback, including some faster and better programs. Bruce Love suggested the possibility of the reverse transformation.
According to Bruce, who is a high school teacher in New Zealand, the method which uses the fewest bytes is the one I show in lines 1390-1450. In order to be fair in comparing different algorithms, I am going to count the RTS opcodes both for bytes and for cycles. With this in mind, Bruce's routine takes 8 bytes and from 16 to 65 cycles. This is certainly the smallest way, and it really is pretty fast.
Bruce mentioned that he had written several other programs to solve the same problem: one used the X-register, took 26 bytes with an average of 33.5 cycles; another without using X or Y took 28 bytes and an average of 39 cycles. Unfortunately, he did not include a copy of either of these.
I worked out four more methods, shown in the listing after Bruce's. I wrote a test driver which is in lines 1000-1310. The test driver calls each routine, printing the results of each, for all possible values of the bit-mask.
The following table summarizes the data for the five algorithms:
# of cycles
bytes min max ave
SMALLEST.WAY 8 16 65 40.5
WAY.WITH.X 26 25 42 33.5
WAY.WITHOUT.X 23 14 30 22
ANOTHER.WAY.W... 32 14 24 18.375
STRAIGHT.TEST... 33 14 27 18.5
If the SMALLEST.WAY is not fast enough, I would probably go with the one named WAY.WITHOUT.X. It is almost as fast as the fastest, and is the shortest of the longer routines. Of course, some of you may come up with better and faster ones....
1000 *SAVE S.MASK --> INDEX 1010 *-------------------------------- 1020 TEST LDY #$01 1030 .1 TYA 1040 JSR $FDDA 1050 TYA 1060 JSR SMALLEST.WAY 1070 JSR HEX 1080 TYA 1090 JSR WAY.WITH.X 1100 JSR HEX 1110 TYA 1120 JSR WAY.WITHOUT.X 1130 JSR HEX 1140 TYA 1150 JSR ANOTHER.WAY.WITHOUT.X 1160 JSR HEX 1170 TYA 1180 JSR STRAIGHT.TESTING.WAY 1190 JSR HEX 1200 JSR $FD8E 1210 TYA 1220 ASL 1230 TAY 1240 BCC .1 1250 RTS 1260 *-------------------------------- 1270 HEX PHA 1280 LDA #"-" 1290 JSR $FDED 1300 PLA 1310 JMP $FDDA 1320 *-------------------------------- 1330 * WAY WITH FEWEST BYTES 1340 * 8 BYTES 1350 * MIN: 16 CYCLES 1360 * MAX: 65 CYCLES 1370 * AVE: 40.5 CYCLES 1380 *-------------------------------- 1390 SMALLEST.WAY 1400 LDX #8 1410 .1 DEX 1420 ASL 1430 BCC .1 1440 TXA 1450 RTS 1460 *-------------------------------- 1470 * FASTER WAY USING X-REGISTER 1480 * 26 BYTES 1490 * MIN: 25 CYCLES 1500 * MAX: 42 CYCLES 1510 * AVE: 33.5 CYCLES 1520 *-------------------------------- 1530 WAY.WITH.X 1540 LDX #0 KEEP INDEX IN X 1550 CMP #$10 80-40-20-10 / 08-04-02-01 1560 BCC .1 ...8,4,2,1 1570 LSR ...80,40,20,10 1580 LSR SHIFT OVER TO 8,4,2,1 1590 LSR 1600 LSR 1610 LDX #4 AND BUMP INDEX BY 4 1620 .1 CMP #$04 08-04 / 02-01 1630 BCC .2 ...2,1 1640 LSR ...8,4 1650 LSR SHIFT OVER TO 2,1 1660 INX AND BUMP INDEX BY 2 1670 INX 1680 .2 LSR 02 / 01 1690 BEQ .3 ...01 1700 INX ...02, BUMP INDEX 1710 .3 TXA GET RESULT 1720 RTS 1730 *-------------------------------- 1740 * WAY WITHOUT USING X-REGISTER 1750 * 23 BYTES 1760 * MIN: 14 CYCLES 1770 * MAX: 30 CYCLES 1780 * AVE: 22 CYCLES 1790 *-------------------------------- 1800 WAY.WITHOUT.X 1810 LSR 40-20-10-08-04-02-01-00 1820 CMP #$04 1830 BCC .2 ...2,1,0 1840 BEQ .3 ...4, SHOULD BE 3 1850 LSR 20-10-08-04 1860 LSR 10-08-04-02 1870 LSR 08-04-02-01 1880 LSR 04-02-01-00 1890 CMP #4 1900 BCC .1 2,1,0 INTO 6,5,4 1910 LDA #2 4 INTO 7 1920 .1 ADC #4 1930 .2 RTS 1940 .3 SBC #1 4 INTO 3 1950 RTS 1960 *-------------------------------- 1970 * ANOTHER WAY WITHOUT X-REGISTER 1980 * 32 BYTES 1990 * MIN: 14 CYCLES 2000 * MAX: 24 CYCLES 2010 * AVE: 18.375 CYCLES 2020 *-------------------------------- 2030 ANOTHER.WAY.WITHOUT.X 2040 CMP #$08 80-40-20-10-08-04-02-01 2050 BCC .5 ...4,2,1 2060 BEQ .4 ...8, SHOULD BE 3 2070 CMP #$40 2080 BCC .2 ...20,10 2090 BEQ .1 ...40 2100 LDA #7 2110 RTS 2120 .1 LDA #6 2130 RTS 2140 .2 CMP #$20 2150 BEQ .3 2160 LDA #4 2170 RTS 2180 .3 LDA #5 2190 RTS 2200 .4 SBC #2 2210 .5 LSR 2220 RTS 2230 *-------------------------------- 2240 * STRAIGHTFORWARD TESTING APPROACH 2250 * 33 BYTES 2260 * MIN: 14 CYCLES 2270 * MAX: 27 CYCLES 2280 * AVE: 18.5 CYCLES 2290 *-------------------------------- 2300 STRAIGHT.TESTING.WAY 2310 CMP #$08 2320 BCC .5 2330 BEQ .4 2340 CMP #$20 2350 BCC .3 2360 BEQ .2 2370 CMP #$80 2380 BCC .1 2390 LDA #7 2400 RTS 2410 .1 LDA #6 2420 RTS 2430 .2 LDA #5 2440 RTS 2450 .3 LDA #4 2460 RTS 2470 .4 LDA #3 2480 RTS 2490 .5 LSR CONVERT 4,2,1 TO 2,1,0 2500 RTS 2510 *-------------------------------- |
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