COMMENT ⊗   VALID 00008 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	.TITLE ARITH
C00004 00003	ASIN - FLOATING POINT, SINGLE PRECISION ARC-SINE FUNCTION
C00005 00004	ACOS - FLOATING POINT, SINGLE PRECISION ARC-COSINE FUNCTION
C00006 00005	ATAN2 - FLOATING POINT, SINGLE PRECISION ARC-TANGENT WITH TWO ARGUMENTS
C00009 00006	"SNCOS" - SINE/COSINE FUNCTION USING TABLE LOOKUP
C00012 00007	SINE/COSINE LOOK UP TABLES
C00016 00008	ARCTANGENT LOOK UP TABLES
C00020 ENDMK
C⊗;
.TITLE ARITH

;"SQRTF" - FLOATING POINT SQUARE ROOT

;COMPUTES THE SQUARE ROOT OF THE NUMBER LOADED INTO AC0 BY USING
;A LINEAR APPROXIMATION AND PERFORMING ONE CONVERGENCE INTERATION.
;THE ANSWER IS RETURNED IN AC0 AND IS ACCURATE TO APPROMIATELY
;+ OR - .0076%.   A SAMPLE CALLING SEQUENCE FOLLOWS:
;
;			LDF	NUMBER,AC0
;			JSR	PC,SQRTF
;
;EXECUTION TIME FOR THIS ROUTINE IS APPROXIMATELY 110 MICRO-SECONDS.

;REGISTERS USED:
;	AC0 PASSES ARGUMENTS AND AC1 IS GARBAGED

SQRTF:	ABSF	AC0		;ARG MUST BE > 0
	MOV	R0,-(SP)
	STEXP	AC0,R0		;EXPONENT ← EXPONENT/2
	STF	AC0,AC1
	INC	R0
	ASR	R0
	BCS	1$		
	LDEXP	#-1,AC1		;1/4 ≤ ARG < 1/2
	MULF	ASM,AC1		;LINEAR APPROXIMATION
	ADDF	BSM,AC1
	BR	2$
1$:	LDEXP	#0,AC1		;1/2 ≤ ARG ≤ 1
	MULF	ALG,AC1
	ADDF	BLG,AC1
2$:	LDEXP	R0,AC1		;SQRT ← ((ARG/GUESS)+GUESS)*.5
	DIVF	AC1,AC0
	MOV	(SP)+,R0
	ADDF	AC1,AC0
	MULF	#40000,AC0
	RTS	PC
	
ASM:	.FLT2	0.8125000
ALG:	.FLT2	0.5781250
BSM:	.FLT2	0.3027340
BLG:	.FLT2	0.4218750

;END OF "SQRTF"
;ASIN - FLOATING POINT, SINGLE PRECISION ARC-SINE FUNCTION

;COMPUTES THE ARCSINE OF AN VALUE USING THE FOLLOWING ALGORITHM
;
;	ASIN(X) = ATAN2(X,SQRT(1-X**2))
;
;WHERE X IS IN THE RANGE -1.0 TO +1.0.  THE ARGUMENT X MUST BE 
;LOADED INTO AC0 BEFORE THE CALL TO ASIN.  AFTER EXECUTION ASIN
;RETURNS THE ARCSINE VALUE IN AC0 ( IN DEGREES ).

;REGISTERS USED:
;
;	AC0 PASSES ARGUMENTS
;	AC1,AC2  GARBAGED

ASIN: 	STF	AC0,AC2		;COMPUTE SQRT(1-SIN↑2)
	MULF	AC0,AC0
	NEGF	AC0
	ADDF	#40200,AC0
	JSR	PC,SQRTF
	STF	AC0,AC1
	STF	AC2,AC0
	JSR	PC,ATAN2 	;ARC TANGENT( SIN, SQRT(1-SIN↑2) )
	RTS	PC

;END OF "ASIN"
;ACOS - FLOATING POINT, SINGLE PRECISION ARC-COSINE FUNCTION

;COMPUTES THE ARC-COSINE OF AN VALUE USING THE FOLLOWING ALGORITHM
;
;	ACOS(X) = ATAN2( SQRT(1-COSX↑2),COS X )
;
;WHERE X IS IN THE RANGE -1.0 TO +1.0.  THE ARGUMENT X MUST BE 
;LOADED INTO AC0 BEFORE THE CALL TO ACOS.  AFTER EXECUTION ACOS
;RETURNS THE ARC-COSINE VALUE IN AC0 ( IN DEGREES ).

;REGISTERS USED:
;
;	AC0 PASSES ARGUMENTS
;	AC1,AC2  GARBAGED


ACOS: 	STF	AC0,AC2		;COMPUTE SQRT(1-COS↑2)
	MULF	AC0,AC0
	NEGF	AC0
	ADDF	#40200,AC0
	JSR	PC,SQRTF
	STF	AC2,AC1
	JSR	PC,ATAN2 	;ARC TANGENT( SQRT(1-COS↑2), COS )
	RTS	PC

;END OF "ACOS"
;ATAN2 - FLOATING POINT, SINGLE PRECISION ARC-TANGENT WITH TWO ARGUMENTS

;COMPUTES THE ARC-TANGENT OF A/B USING A TABLE LOOK UP SCHEME.  SINCE
;TWO ARGUMENTS ARE USED SINGULARITIES ARE AVOIDED AT MULTIPLES OF PI/2 
;AND THERE IS NO AMBIGUITY CONCERNING QUADRANTS.  THE ARGUMENT A MUST
;BE LOADED INTO AC0 AND B INTO AC1 BEFORE CALLING ATAN2.  AFTER EXECUTION,
;ATAN RETURNS THE ARC-TANGENT IN DEGREES IN AC0.  RUN TIME IS APPROXIMATELY
;140 MICRO SECONDS.

;REGISTERS USED:
;
;	AC0,AC1 PASSES ARGUMENTS
;	AC2 GARBAGED

SSIN==1		;SINE IS NEGATIVE
SCOS==2		;COSINE IS NEGATIVE
CMPANG==4	;COMPLEMENTARY ANGLE

ATAN2: 	MOV	R0,-(SP)	;SAVE REGISTERS
	MOV	R1,-(SP)
	CLR	R0
	TSTF	AC0		;SIN > 0 ?
	CFCC
	BLT	SINNEG
	BGT	SINPOS
	TSTF	AC1		;SIN = 0, COS > 0?
	CFCC
	BPL	.+6
    	LDF	#42064,AC0	;THETA = 180 DEGREES
JRET:	MOV	(SP)+,R1
	MOV	(SP)+,R0
	RTS	PC

SINNEG:	NEGF	AC0		;ABS(SIN)
	BIS	#SSIN,R0  	;INDICATE SIN < 0
SINPOS:	TSTF	AC1		;COS > 0 ?
	CFCC
	BLT	COSNEG
	BGT	COSPOS
	LDF	#41664,AC0	;THETA = 90 DEGREES
	BIT	#SSIN,R0
	BEQ	JRET
	NEGF	AC0
	BR	JRET

COSNEG:	NEGF	AC1		;ABS(COS)
	BIS	#SCOS,R0 	;INDICATE COS < 0
COSPOS:	CMPF	AC0,AC1
	CFCC
	BGT	1$		;MUST EXCHANGE
	BNE	2$
	LDF	#41464,AC0	;THETA = 45 DEGREES
	BR	4$

1$:	DIVF	AC0,AC1
	BIS	#CMPANG,R0
	STF	AC1,AC0
	BR	3$

2$:	DIVF	AC1,AC0
3$:	MODF	#42000,AC0	;GET INDEX, INTERPOLATION FACTOR
	STCFI	AC1,R1		;INDEX
	ASH	#2,R1
	LDF	ARCTAN+4(R1),AC2	;INTERPOLATE
	LDF	ARCTAN(R1),AC1
	SUBF	AC1,AC2
	MULF	AC0,AC2
	STF	AC1,AC0
	ADDF	AC2,AC0

4$:	BIT	#CMPANG,R0	;COMPLEMENT?
	BEQ	5$
	SUBF	#41664,AC0
	NEGF	AC0
5$:	BIT	#SCOS,R0	;COS THETA < 0?
	BEQ	6$
	SUBF	#42064,AC0
	NEGF	AC0
6$:	BIT	#SSIN,R0	;SIN THETA < 0?
	BEQ	JRET
	NEGF	R0
	BR	JRET

;END OF "ATAN2"
;"SNCOS" - SINE/COSINE FUNCTION USING TABLE LOOKUP

;THIS PROGRAM CALCULATES BOTH THE SINE AND THE COSINE OF A ANGLE USING
;A TABLE LOOP UP PROCEDURE.  THE IMPLEMENTED APPROXIMATION EQUATIONS
;ARE AS FOLLOWS:
;		SIN(X) = SIN(A) + (B/I)*[SIN(A+I)-SIN(A)]
;		COS(X) = COS(A) + (B/I)*[COS(A+I)-COS(A)]
;	WHERE
;		I = 90/128 DEGREES
;		A = INTEGER(X*128/90)
;		B = REMAINDER(X*128/90)
;
;THE ANGLE SHOULD MUST BE IN DEGREES AND MUST BE LOADED INTO AC0 BEFORE
;CALLING THIS ROUTINE.  ON EXITING, THE SIN IS LEFT IN AC0 AND THE COSINE
;IS LEFT IN AC1.  EXECUTION TIME IS APPROXIMATELY 120 MICRO-SECONDS.

;REGISTERS USED:
;	AC0,AC1 PASS ARGUMENTS AND ARE ALTERED
;	AC2,AC3 ARE GARBAGED

SNCOS:	MODF	3$,AC0		;SEPARATE INDEX AND FRACTION
	MOV	R1,-(SP)	;SAVE REGISTERS
      	MOV	R0,-(SP)	
	STCFI	AC1,R0		;INDEX INTO LOOKUP TABLE
	MOV	R0,R1		;SAVE SIGN OF ANGLE
	BIC	#177600,R0
	ASH	#2,R0
	LDF	SINTAB+4(R0),AC1	;SIN(A+I)
	LDF	SINTAB(R0),AC2	;SIN(A)
	SUBF	AC2,AC1
	ASR	R1
	MULF	AC0,AC1		;(B/I)*[SIN(A+1)-SIN(A)]
	SWAB	R1
	ADDF	AC2,AC1		;NOW HAVE ABS(SIN(X))
	NEG	R0		;ENTER TABLE FROM OPPOSITE END
	LDF	COSTAB-4(R0),AC2	;COS(A+I)
	LDF	COSTAB(R0),AC3	;COS(A)
	SUBF	AC3,AC2
	MULF	AC0,AC2		;(B/I)*[COS(A+I)-COS(A)]
	MOV	(SP)+,R0	;DON'T NEED THIS ANY MORE
	BIT	#40000,R1	;WHICH IS THE SIN?
	ADDF	AC3,AC2		;NOW HAVE ABS(COS(X))
	BEQ	1$
	STF	AC2,AC0		;SWITCH SIN ↔ COSINE
	BR	2$
1$:	STF	AC1,AC0		;SAVE SIN, COSINE
	STF	AC2,AC1
2$:	TST	R1    		;SIN ← -SIN IF X IN QUAD 3 OR 4
	BPL	.+4
	NEGF	AC0
	ADD	#40000,R1   	;COS ← -COS IF QUADRANT 2 OR 3
	BPL	.+4
	NEGF	AC1
	MOV	(SP)+,R1
	RTS	PC

3$:	.FLT2	1.422222	;128/90 = 1.422222    

;END OF "SNCOS"
;SINE/COSINE LOOK UP TABLES

SINTAB:	.WORD	     0,     0, 36511,  7220, 36711,  5260, 37026,141454
	.WORD	 37110,175460, 37173, 25564, 37226,124405, 37257,133200
	.WORD	 37310,136466, 37341,136056, 37372,131163, 37411,147606
	.WORD	 37426, 40203, 37442,125666, 37457, 10242, 37473, 67317
	.WORD	 37507,142702, 37524, 12401, 37540, 56023, 37554,115177
	.WORD	 37570,147714, 37602, 76700, 37610,107223, 37616,115042
	.WORD	 37624,120062, 37632,120206, 37640,115345, 37646,107422
	.WORD	 37654, 76324, 37662, 61757, 37670, 42052, 37676, 16512
	.WORD	 37703,167425, 37711,134523, 37717, 75712, 37725, 33101
	.WORD	 37732,164201, 37740,111117, 37746, 31565, 37753,145673
	.WORD	 37761, 55352, 37766,160313, 37774, 56447, 40000,163744
	.WORD	 40003,116075, 40006, 45603, 40010,172633, 40013,115153
	.WORD	 40016, 34732, 40020,151715, 40023, 64052, 40025,173331
	.WORD	 40030, 77700, 40033,  1306, 40035, 77721, 40037,173313
	.WORD	 40042, 63631, 40044,151045, 40047, 33126, 40051,112025
	.WORD	 40053,165512, 40056, 35736, 40060,102673, 40062,144311
	.WORD	 40065,  2363, 40067, 35043, 40071, 64102, 40073,107473
	.WORD	 40075,127371, 40077,143547, 40101,154160, 40103,161001
	.WORD	 40105,162003, 40107,157145, 40111,150422, 40113,135770
	.WORD	 40115,117402, 40117, 75037, 40121, 46475, 40123, 14111
	.WORD	 40124,155461, 40126,112745, 40130, 44123, 40131,171152
	.WORD	 40133,112032, 40135, 26523, 40136,137005, 40140, 43041
	.WORD	 40141,142630, 40143, 36132, 40144,125131, 40146,  7610
	.WORD	 40147, 65730, 40150,137474, 40152,  4647, 40153, 45414
	.WORD	 40154,101537, 40155,131223, 40156,154236, 40157,172563
	.WORD	 40161,  4410, 40162, 11522, 40163, 12110, 40164,  5735
	.WORD	 40164,175013, 40165,157306, 40166,135007, 40167,105705
	.WORD	 40170, 51770, 40171, 11230, 40171,143635, 40172, 71402
	.WORD	 40173, 12276, 40173,126315, 40174, 35450, 40174,137711
	.WORD	 40175, 35254, 40175,125713, 40176, 11443, 40176, 70260
	.WORD	 40176,142155, 40177,  7130, 40177, 47155, 40177,102253
	.WORD	 40177,130417, 40177,151627, 40177,166103, 40177,175421
COSTAB:	.WORD	 40200,     0
;ARCTANGENT LOOK UP TABLES

ARCTAN:	.WORD	     0,     0, 37745, 26657, 40145, 25033, 40253,155433
	.WORD	 40345, 15712, 40417, 25252, 40453,141371, 40510, 52757
	.WORD	 40544,161252, 40600,132045, 40616,171370, 40635, 26536
	.WORD	 40653, 61353, 40671,111461, 40707,136703, 40725,161067
	.WORD	 40744,    42, 40762, 13434, 41000, 11537, 41007, 13517
	.WORD	 41016, 13456, 41025, 11311, 41034,  4757, 41042,176160
	.WORD	 41051,165034, 41060,151305, 41067,133074, 41076,112127
	.WORD	 41105, 66347, 41114, 37702, 41123,  6276, 41131,151661
	.WORD	 41140,112164, 41147, 47337, 41156,  1313, 41164,130024
	.WORD	 41173, 53224, 41200,175425, 41204, 43531, 41207,110006
	.WORD	 41212,152413, 41216, 13131, 41221, 51741, 41224,106624
	.WORD	 41227,141545, 41232,172506, 41236, 21451, 41241, 46403
	.WORD	 41244, 71310, 41247,112153, 41252,130743, 41255,145445
	.WORD	 41260,160046, 41263,170336, 41266,176504, 41272,  2517
	.WORD	 41275,  4367, 41300,  4065, 41303,  1402, 41305,174527
	.WORD	 41310,165457, 41313,154203, 41316,140516, 41321,122614
	.WORD	 41324,102472, 41327, 60121, 41332, 33320, 41335,  4263
	.WORD	 41337,152767, 41342,117232, 41345, 61232, 41350, 20767
	.WORD	 41352,156256, 41355,111276, 41360, 42050, 41362,170354
	.WORD	 41365,114410, 41370, 36176, 41372,155520, 41375, 72574
	.WORD	 41400,  2603, 41401, 46770, 41402,112035, 41403,153763
	.WORD	 41405, 14574, 41406, 54272, 41407,112655, 41410,150127
	.WORD	 41412,  4272, 41413, 37330, 41414, 71263, 41415,122115
	.WORD	 41416,151647, 41420,   304, 41421, 25647, 41422, 52122
	.WORD	 41423, 75306, 41424,117410, 41425,140431, 41426,160374
	.WORD	 41427,177264, 41431, 15103, 41432, 31655, 41433, 45365
	.WORD	 41434, 60034, 41435, 71450, 41436,102032, 41437,111366
	.WORD	 41440,117676, 41441,125167, 41442,131443, 41443,134706
	.WORD	 41444,137143, 41445,140375, 41446,140631, 41447,140070
	.WORD	 41450,136337, 41451,133621, 41452,130121, 41453,123444
	.WORD	 41454,116014, 41455,107415, 41456,100053, 41457, 67551
	.WORD	 41460, 56313, 41461, 44125, 41462, 31011, 41463, 14753
	.WORD	 41464,     0


;END OF "ARITH"