ssin.s

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|
|	ssin.sa 3.3 7/29/91
|
|	The entry point sSIN computes the sine of an input argument
|	sCOS computes the cosine, and sSINCOS computes both. The
|	corresponding entry points with a "d" computes the same
|	corresponding function values for denormalized inputs.
|
|	Input: Double-extended number X in location pointed to
|		by address register a0.
|
|	Output: The function value sin(X) or cos(X) returned in Fp0 if SIN or
|		COS is requested. Otherwise, for SINCOS, sin(X) is returned
|		in Fp0, and cos(X) is returned in Fp1.
|
|	Modifies: Fp0 for SIN or COS; both Fp0 and Fp1 for SINCOS.
|
|	Accuracy and Monotonicity: The returned result is within 1 ulp in
|		64 significant bit, i.e. within 0.5001 ulp to 53 bits if the
|		result is subsequently rounded to double precision. The
|		result is provably monotonic in double precision.
|
|	Speed: The programs sSIN and sCOS take approximately 150 cycles for
|		input argument X such that |X| < 15Pi, which is the usual
|		situation. The speed for sSINCOS is approximately 190 cycles.
|
|	Algorithm:
|
|	SIN and COS:
|	1. If SIN is invoked, set AdjN := 0; otherwise, set AdjN := 1.
|
|	2. If |X| >= 15Pi or |X| < 2**(-40), go to 7.
|
|	3. Decompose X as X = N(Pi/2) + r where |r| <= Pi/4. Let
|		k = N mod 4, so in particular, k = 0,1,2,or 3. Overwrite
|		k by k := k + AdjN.
|
|	4. If k is even, go to 6.
|
|	5. (k is odd) Set j := (k-1)/2, sgn := (-1)**j. Return sgn*cos(r)
|		where cos(r) is approximated by an even polynomial in r,
|		1 + r*r*(B1+s*(B2+ ... + s*B8)),	s = r*r.
|		Exit.
|
|	6. (k is even) Set j := k/2, sgn := (-1)**j. Return sgn*sin(r)
|		where sin(r) is approximated by an odd polynomial in r
|		r + r*s*(A1+s*(A2+ ... + s*A7)),	s = r*r.
|		Exit.
|
|	7. If |X| > 1, go to 9.
|
|	8. (|X|<2**(-40)) If SIN is invoked, return X; otherwise return 1.
|
|	9. Overwrite X by X := X rem 2Pi. Now that |X| <= Pi, go back to 3.
|
|	SINCOS:
|	1. If |X| >= 15Pi or |X| < 2**(-40), go to 6.
|
|	2. Decompose X as X = N(Pi/2) + r where |r| <= Pi/4. Let
|		k = N mod 4, so in particular, k = 0,1,2,or 3.
|
|	3. If k is even, go to 5.
|
|	4. (k is odd) Set j1 := (k-1)/2, j2 := j1 (EOR) (k mod 2), i.e.
|		j1 exclusive or with the l.s.b. of k.
|		sgn1 := (-1)**j1, sgn2 := (-1)**j2.
|		SIN(X) = sgn1 * cos(r) and COS(X) = sgn2*sin(r) where
|		sin(r) and cos(r) are computed as odd and even polynomials
|		in r, respectively. Exit
|
|	5. (k is even) Set j1 := k/2, sgn1 := (-1)**j1.
|		SIN(X) = sgn1 * sin(r) and COS(X) = sgn1*cos(r) where
|		sin(r) and cos(r) are computed as odd and even polynomials
|		in r, respectively. Exit
|
|	6. If |X| > 1, go to 8.
|
|	7. (|X|<2**(-40)) SIN(X) = X and COS(X) = 1. Exit.
|
|	8. Overwrite X by X := X rem 2Pi. Now that |X| <= Pi, go back to 2.
|

|		Copyright (C) Motorola, Inc. 1990
|			All Rights Reserved
|
|	THIS IS UNPUBLISHED PROPRIETARY SOURCE CODE OF MOTOROLA 
|	The copyright notice above does not evidence any  
|	actual or intended publication of such source code.

|SSIN	idnt	2,1 | Motorola 040 Floating Point Software Package

	|section	8

	.include "fpsp.h"

BOUNDS1:	.long 0x3FD78000,0x4004BC7E
TWOBYPI:	.long 0x3FE45F30,0x6DC9C883

SINA7:	.long 0xBD6AAA77,0xCCC994F5
SINA6:	.long 0x3DE61209,0x7AAE8DA1

SINA5:	.long 0xBE5AE645,0x2A118AE4
SINA4:	.long 0x3EC71DE3,0xA5341531

SINA3:	.long 0xBF2A01A0,0x1A018B59,0x00000000,0x00000000

SINA2:	.long 0x3FF80000,0x88888888,0x888859AF,0x00000000

SINA1:	.long 0xBFFC0000,0xAAAAAAAA,0xAAAAAA99,0x00000000

COSB8:	.long 0x3D2AC4D0,0xD6011EE3
COSB7:	.long 0xBDA9396F,0x9F45AC19

COSB6:	.long 0x3E21EED9,0x0612C972
COSB5:	.long 0xBE927E4F,0xB79D9FCF

COSB4:	.long 0x3EFA01A0,0x1A01D423,0x00000000,0x00000000

COSB3:	.long 0xBFF50000,0xB60B60B6,0x0B61D438,0x00000000

COSB2:	.long 0x3FFA0000,0xAAAAAAAA,0xAAAAAB5E
COSB1:	.long 0xBF000000

INVTWOPI: .long 0x3FFC0000,0xA2F9836E,0x4E44152A

TWOPI1:	.long 0x40010000,0xC90FDAA2,0x00000000,0x00000000
TWOPI2:	.long 0x3FDF0000,0x85A308D4,0x00000000,0x00000000

	|xref	PITBL

	.set	INARG,FP_SCR4

	.set	X,FP_SCR5
	.set	XDCARE,X+2
	.set	XFRAC,X+4

	.set	RPRIME,FP_SCR1
	.set	SPRIME,FP_SCR2

	.set	POSNEG1,L_SCR1
	.set	TWOTO63,L_SCR1

	.set	ENDFLAG,L_SCR2
	.set	N,L_SCR2

	.set	ADJN,L_SCR3

	| xref	t_frcinx
	|xref	t_extdnrm
	|xref	sto_cos

	.global	ssind
ssind:
|--SIN(X) = X FOR DENORMALIZED X
	bra		t_extdnrm

	.global	scosd
scosd:
|--COS(X) = 1 FOR DENORMALIZED X

	fmoves		#0x3F800000,%fp0
|
|	9D25B Fix: Sometimes the previous fmove.s sets fpsr bits
|
	fmovel		#0,%fpsr
|
	bra		t_frcinx

	.global	ssin
ssin:
|--SET ADJN TO 0
	movel		#0,ADJN(%a6)
	bras		SINBGN

	.global	scos
scos:
|--SET ADJN TO 1
	movel		#1,ADJN(%a6)

SINBGN:
|--SAVE FPCR, FP1. CHECK IF |X| IS TOO SMALL OR LARGE

	fmovex		(%a0),%fp0	| ...LOAD INPUT

	movel		(%a0),%d0
	movew		4(%a0),%d0
	fmovex		%fp0,X(%a6)
	andil		#0x7FFFFFFF,%d0		| ...COMPACTIFY X

	cmpil		#0x3FD78000,%d0		| ...|X| >= 2**(-40)?
	bges		SOK1
	bra		SINSM

SOK1:
	cmpil		#0x4004BC7E,%d0		| ...|X| < 15 PI?
	blts		SINMAIN
	bra		REDUCEX

SINMAIN:
|--THIS IS THE USUAL CASE, |X| <= 15 PI.
|--THE ARGUMENT REDUCTION IS DONE BY TABLE LOOK UP.
	fmovex		%fp0,%fp1
	fmuld		TWOBYPI,%fp1	| ...X*2/PI

|--HIDE THE NEXT THREE INSTRUCTIONS
	lea		PITBL+0x200,%a1 | ...TABLE OF N*PI/2, N = -32,...,32
	

|--FP1 IS NOW READY
	fmovel		%fp1,N(%a6)		| ...CONVERT TO INTEGER

	movel		N(%a6),%d0
	asll		#4,%d0
	addal		%d0,%a1	| ...A1 IS THE ADDRESS OF N*PIBY2
|				...WHICH IS IN TWO PIECES Y1 & Y2

	fsubx		(%a1)+,%fp0	| ...X-Y1
|--HIDE THE NEXT ONE
	fsubs		(%a1),%fp0	| ...FP0 IS R = (X-Y1)-Y2

SINCONT:
|--continuation from REDUCEX

|--GET N+ADJN AND SEE IF SIN(R) OR COS(R) IS NEEDED
	movel		N(%a6),%d0
	addl		ADJN(%a6),%d0	| ...SEE IF D0 IS ODD OR EVEN
	rorl		#1,%d0	| ...D0 WAS ODD IFF D0 IS NEGATIVE
	cmpil		#0,%d0
	blt		COSPOLY

SINPOLY:
|--LET J BE THE LEAST SIG. BIT OF D0, LET SGN := (-1)**J.
|--THEN WE RETURN	SGN*SIN(R). SGN*SIN(R) IS COMPUTED BY
|--R' + R'*S*(A1 + S(A2 + S(A3 + S(A4 + ... + SA7)))), WHERE
|--R' = SGN*R, S=R*R. THIS CAN BE REWRITTEN AS
|--R' + R'*S*( [A1+T(A3+T(A5+TA7))] + [S(A2+T(A4+TA6))])
|--WHERE T=S*S.
|--NOTE THAT A3 THROUGH A7 ARE STORED IN DOUBLE PRECISION
|--WHILE A1 AND A2 ARE IN DOUBLE-EXTENDED FORMAT.
	fmovex		%fp0,X(%a6)	| ...X IS R
	fmulx		%fp0,%fp0	| ...FP0 IS S
|---HIDE THE NEXT TWO WHILE WAITING FOR FP0
	fmoved		SINA7,%fp3
	fmoved		SINA6,%fp2
|--FP0 IS NOW READY
	fmovex		%fp0,%fp1
	fmulx		%fp1,%fp1	| ...FP1 IS T
|--HIDE THE NEXT TWO WHILE WAITING FOR FP1

	rorl		#1,%d0
	andil		#0x80000000,%d0
|				...LEAST SIG. BIT OF D0 IN SIGN POSITION
	eorl		%d0,X(%a6)	| ...X IS NOW R'= SGN*R

	fmulx		%fp1,%fp3	| ...TA7
	fmulx		%fp1,%fp2	| ...TA6

	faddd		SINA5,%fp3 | ...A5+TA7
	faddd		SINA4,%fp2 | ...A4+TA6

	fmulx		%fp1,%fp3	| ...T(A5+TA7)
	fmulx		%fp1,%fp2	| ...T(A4+TA6)

	faddd		SINA3,%fp3 | ...A3+T(A5+TA7)
	faddx		SINA2,%fp2 | ...A2+T(A4+TA6)

	fmulx		%fp3,%fp1	| ...T(A3+T(A5+TA7))

	fmulx		%fp0,%fp2	| ...S(A2+T(A4+TA6))
	faddx		SINA1,%fp1 | ...A1+T(A3+T(A5+TA7))
	fmulx		X(%a6),%fp0	| ...R'*S

	faddx		%fp2,%fp1	| ...[A1+T(A3+T(A5+TA7))]+[S(A2+T(A4+TA6))]
|--FP3 RELEASED, RESTORE NOW AND TAKE SOME ADVANTAGE OF HIDING
|--FP2 RELEASED, RESTORE NOW AND TAKE FULL ADVANTAGE OF HIDING
	

	fmulx		%fp1,%fp0		| ...SIN(R')-R'
|--FP1 RELEASED.

	fmovel		%d1,%FPCR		|restore users exceptions
	faddx		X(%a6),%fp0		|last inst - possible exception set
	bra		t_frcinx


COSPOLY:
|--LET J BE THE LEAST SIG. BIT OF D0, LET SGN := (-1)**J.
|--THEN WE RETURN	SGN*COS(R). SGN*COS(R) IS COMPUTED BY
|--SGN + S'*(B1 + S(B2 + S(B3 + S(B4 + ... + SB8)))), WHERE
|--S=R*R AND S'=SGN*S. THIS CAN BE REWRITTEN AS
|--SGN + S'*([B1+T(B3+T(B5+TB7))] + [S(B2+T(B4+T(B6+TB8)))])
|--WHERE T=S*S.
|--NOTE THAT B4 THROUGH B8 ARE STORED IN DOUBLE PRECISION
|--WHILE B2 AND B3 ARE IN DOUBLE-EXTENDED FORMAT, B1 IS -1/2
|--AND IS THEREFORE STORED AS SINGLE PRECISION.

	fmulx		%fp0,%fp0	| ...FP0 IS S
|---HIDE THE NEXT TWO WHILE WAITING FOR FP0
	fmoved		COSB8,%fp2
	fmoved		COSB7,%fp3
|--FP0 IS NOW READY
	fmovex		%fp0,%fp1
	fmulx		%fp1,%fp1	| ...FP1 IS T
|--HIDE THE NEXT TWO WHILE WAITING FOR FP1
	fmovex		%fp0,X(%a6)	| ...X IS S
	rorl		#1,%d0
	andil		#0x80000000,%d0
|			...LEAST SIG. BIT OF D0 IN SIGN POSITION

	fmulx		%fp1,%fp2	| ...TB8
|--HIDE THE NEXT TWO WHILE WAITING FOR THE XU
	eorl		%d0,X(%a6)	| ...X IS NOW S'= SGN*S
	andil		#0x80000000,%d0

	fmulx		%fp1,%fp3	| ...TB7
|--HIDE THE NEXT TWO WHILE WAITING FOR THE XU
	oril		#0x3F800000,%d0	| ...D0 IS SGN IN SINGLE
	movel		%d0,POSNEG1(%a6)

	faddd		COSB6,%fp2 | ...B6+TB8
	faddd		COSB5,%fp3 | ...B5+TB7

	fmulx		%fp1,%fp2	| ...T(B6+TB8)
	fmulx		%fp1,%fp3	| ...T(B5+TB7)

	faddd		COSB4,%fp2 | ...B4+T(B6+TB8)
	faddx		COSB3,%fp3 | ...B3+T(B5+TB7)

	fmulx		%fp1,%fp2	| ...T(B4+T(B6+TB8))
	fmulx		%fp3,%fp1	| ...T(B3+T(B5+TB7))

	faddx		COSB2,%fp2 | ...B2+T(B4+T(B6+TB8))
	fadds		COSB1,%fp1 | ...B1+T(B3+T(B5+TB7))

	fmulx		%fp2,%fp0	| ...S(B2+T(B4+T(B6+TB8)))
|--FP3 RELEASED, RESTORE NOW AND TAKE SOME ADVANTAGE OF HIDING
|--FP2 RELEASED.
	

	faddx		%fp1,%fp0
|--FP1 RELEASED

	fmulx		X(%a6),%fp0

	fmovel		%d1,%FPCR		|restore users exceptions
	fadds		POSNEG1(%a6),%fp0	|last inst - possible exception set
	bra		t_frcinx


SINBORS:
|--IF |X| > 15PI, WE USE THE GENERAL ARGUMENT REDUCTION.
|--IF |X| < 2**(-40), RETURN X OR 1.
	cmpil		#0x3FFF8000,%d0
	bgts		REDUCEX
        

SINSM:
	movel		ADJN(%a6),%d0
	cmpil		#0,%d0
	bgts		COSTINY

SINTINY:
	movew		#0x0000,XDCARE(%a6)	| ...JUST IN CASE
	fmovel		%d1,%FPCR		|restore users exceptions
	fmovex		X(%a6),%fp0		|last inst - possible exception set
	bra		t_frcinx


COSTINY:
	fmoves		#0x3F800000,%fp0

	fmovel		%d1,%FPCR		|restore users exceptions
	fsubs		#0x00800000,%fp0	|last inst - possible exception set
	bra		t_frcinx