xref: /src/sys/arch/m68k/fpsp/stan.sa

*	MOTOROLA MICROPROCESSOR & MEMORY TECHNOLOGY GROUP
*	M68000 Hi-Performance Microprocessor Division
*	M68040 Software Package 
*
*	M68040 Software Package Copyright (c) 1993, 1994 Motorola Inc.
*	All rights reserved.
*
*	THE SOFTWARE is provided on an "AS IS" basis and without warranty.
*	To the maximum extent permitted by applicable law,
*	MOTOROLA DISCLAIMS ALL WARRANTIES WHETHER EXPRESS OR IMPLIED,
*	INCLUDING IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A
*	PARTICULAR PURPOSE and any warranty against infringement with
*	regard to the SOFTWARE (INCLUDING ANY MODIFIED VERSIONS THEREOF)
*	and any accompanying written materials. 
*
*	To the maximum extent permitted by applicable law,
*	IN NO EVENT SHALL MOTOROLA BE LIABLE FOR ANY DAMAGES WHATSOEVER
*	(INCLUDING WITHOUT LIMITATION, DAMAGES FOR LOSS OF BUSINESS
*	PROFITS, BUSINESS INTERRUPTION, LOSS OF BUSINESS INFORMATION, OR
*	OTHER PECUNIARY LOSS) ARISING OF THE USE OR INABILITY TO USE THE
*	SOFTWARE.  Motorola assumes no responsibility for the maintenance
*	and support of the SOFTWARE.  
*
*	You are hereby granted a copyright license to use, modify, and
*	distribute the SOFTWARE so long as this entire notice is retained
*	without alteration in any modified and/or redistributed versions,
*	and that such modified versions are clearly identified as such.
*	No licenses are granted by implication, estoppel or otherwise
*	under any patents or trademarks of Motorola, Inc.

*
*	stan.sa 3.3 7/29/91
*
*	The entry point stan computes the tangent of
*	an input argument;
*	stand does the same except for denormalized input.
*
*	Input: Double-extended number X in location pointed to
*		by address register a0.
*
*	Output: The value tan(X) returned in floating-point register Fp0.
*
*	Accuracy and Monotonicity: The returned result is within 3 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 program sTAN takes approximately 170 cycles for
*		input argument X such that |X| < 15Pi, which is the the usual
*		situation.
*
*	Algorithm:
*
*	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 2, so in particular, k = 0 or 1.
*
*	3. If k is odd, go to 5.
*
*	4. (k is even) Tan(X) = tan(r) and tan(r) is approximated by a
*		rational function U/V where
*		U = r + r*s*(P1 + s*(P2 + s*P3)), and
*		V = 1 + s*(Q1 + s*(Q2 + s*(Q3 + s*Q4))),  s = r*r.
*		Exit.
*
*	4. (k is odd) Tan(X) = -cot(r). Since tan(r) is approximated by a
*		rational function U/V where
*		U = r + r*s*(P1 + s*(P2 + s*P3)), and
*		V = 1 + s*(Q1 + s*(Q2 + s*(Q3 + s*Q4))), s = r*r,
*		-Cot(r) = -V/U. Exit.
*
*	6. If |X| > 1, go to 8.
*
*	7. (|X|<2**(-40)) Tan(X) = X. Exit.
*
*	8. Overwrite X by X := X rem 2Pi. Now that |X| <= Pi, go back to 2.
*

STAN	IDNT	2,1 Motorola 040 Floating Point Software Package

	section	8

	include fpsp.h

BOUNDS1	DC.L $3FD78000,$4004BC7E
TWOBYPI	DC.L $3FE45F30,$6DC9C883

TANQ4	DC.L $3EA0B759,$F50F8688
TANP3	DC.L $BEF2BAA5,$A8924F04

TANQ3	DC.L $BF346F59,$B39BA65F,$00000000,$00000000

TANP2	DC.L $3FF60000,$E073D3FC,$199C4A00,$00000000

TANQ2	DC.L $3FF90000,$D23CD684,$15D95FA1,$00000000

TANP1	DC.L $BFFC0000,$8895A6C5,$FB423BCA,$00000000

TANQ1	DC.L $BFFD0000,$EEF57E0D,$A84BC8CE,$00000000

INVTWOPI DC.L $3FFC0000,$A2F9836E,$4E44152A,$00000000

TWOPI1	DC.L $40010000,$C90FDAA2,$00000000,$00000000
TWOPI2	DC.L $3FDF0000,$85A308D4,$00000000,$00000000

*--N*PI/2, -32 <= N <= 32, IN A LEADING TERM IN EXT. AND TRAILING
*--TERM IN SGL. NOTE THAT PI IS 64-BIT LONG, THUS N*PI/2 IS AT
*--MOST 69 BITS LONG.
	xdef	PITBL
PITBL:
  DC.L  $C0040000,$C90FDAA2,$2168C235,$21800000
  DC.L  $C0040000,$C2C75BCD,$105D7C23,$A0D00000
  DC.L  $C0040000,$BC7EDCF7,$FF523611,$A1E80000
  DC.L  $C0040000,$B6365E22,$EE46F000,$21480000
  DC.L  $C0040000,$AFEDDF4D,$DD3BA9EE,$A1200000
  DC.L  $C0040000,$A9A56078,$CC3063DD,$21FC0000
  DC.L  $C0040000,$A35CE1A3,$BB251DCB,$21100000
  DC.L  $C0040000,$9D1462CE,$AA19D7B9,$A1580000
  DC.L  $C0040000,$96CBE3F9,$990E91A8,$21E00000
  DC.L  $C0040000,$90836524,$88034B96,$20B00000
  DC.L  $C0040000,$8A3AE64F,$76F80584,$A1880000
  DC.L  $C0040000,$83F2677A,$65ECBF73,$21C40000
  DC.L  $C0030000,$FB53D14A,$A9C2F2C2,$20000000
  DC.L  $C0030000,$EEC2D3A0,$87AC669F,$21380000
  DC.L  $C0030000,$E231D5F6,$6595DA7B,$A1300000
  DC.L  $C0030000,$D5A0D84C,$437F4E58,$9FC00000
  DC.L  $C0030000,$C90FDAA2,$2168C235,$21000000
  DC.L  $C0030000,$BC7EDCF7,$FF523611,$A1680000
  DC.L  $C0030000,$AFEDDF4D,$DD3BA9EE,$A0A00000
  DC.L  $C0030000,$A35CE1A3,$BB251DCB,$20900000
  DC.L  $C0030000,$96CBE3F9,$990E91A8,$21600000
  DC.L  $C0030000,$8A3AE64F,$76F80584,$A1080000
  DC.L  $C0020000,$FB53D14A,$A9C2F2C2,$1F800000
  DC.L  $C0020000,$E231D5F6,$6595DA7B,$A0B00000
  DC.L  $C0020000,$C90FDAA2,$2168C235,$20800000
  DC.L  $C0020000,$AFEDDF4D,$DD3BA9EE,$A0200000
  DC.L  $C0020000,$96CBE3F9,$990E91A8,$20E00000
  DC.L  $C0010000,$FB53D14A,$A9C2F2C2,$1F000000
  DC.L  $C0010000,$C90FDAA2,$2168C235,$20000000
  DC.L  $C0010000,$96CBE3F9,$990E91A8,$20600000
  DC.L  $C0000000,$C90FDAA2,$2168C235,$1F800000
  DC.L  $BFFF0000,$C90FDAA2,$2168C235,$1F000000
  DC.L  $00000000,$00000000,$00000000,$00000000
  DC.L  $3FFF0000,$C90FDAA2,$2168C235,$9F000000
  DC.L  $40000000,$C90FDAA2,$2168C235,$9F800000
  DC.L  $40010000,$96CBE3F9,$990E91A8,$A0600000
  DC.L  $40010000,$C90FDAA2,$2168C235,$A0000000
  DC.L  $40010000,$FB53D14A,$A9C2F2C2,$9F000000
  DC.L  $40020000,$96CBE3F9,$990E91A8,$A0E00000
  DC.L  $40020000,$AFEDDF4D,$DD3BA9EE,$20200000
  DC.L  $40020000,$C90FDAA2,$2168C235,$A0800000
  DC.L  $40020000,$E231D5F6,$6595DA7B,$20B00000
  DC.L  $40020000,$FB53D14A,$A9C2F2C2,$9F800000
  DC.L  $40030000,$8A3AE64F,$76F80584,$21080000
  DC.L  $40030000,$96CBE3F9,$990E91A8,$A1600000
  DC.L  $40030000,$A35CE1A3,$BB251DCB,$A0900000
  DC.L  $40030000,$AFEDDF4D,$DD3BA9EE,$20A00000
  DC.L  $40030000,$BC7EDCF7,$FF523611,$21680000
  DC.L  $40030000,$C90FDAA2,$2168C235,$A1000000
  DC.L  $40030000,$D5A0D84C,$437F4E58,$1FC00000
  DC.L  $40030000,$E231D5F6,$6595DA7B,$21300000
  DC.L  $40030000,$EEC2D3A0,$87AC669F,$A1380000
  DC.L  $40030000,$FB53D14A,$A9C2F2C2,$A0000000
  DC.L  $40040000,$83F2677A,$65ECBF73,$A1C40000
  DC.L  $40040000,$8A3AE64F,$76F80584,$21880000
  DC.L  $40040000,$90836524,$88034B96,$A0B00000
  DC.L  $40040000,$96CBE3F9,$990E91A8,$A1E00000
  DC.L  $40040000,$9D1462CE,$AA19D7B9,$21580000
  DC.L  $40040000,$A35CE1A3,$BB251DCB,$A1100000
  DC.L  $40040000,$A9A56078,$CC3063DD,$A1FC0000
  DC.L  $40040000,$AFEDDF4D,$DD3BA9EE,$21200000
  DC.L  $40040000,$B6365E22,$EE46F000,$A1480000
  DC.L  $40040000,$BC7EDCF7,$FF523611,$21E80000
  DC.L  $40040000,$C2C75BCD,$105D7C23,$20D00000
  DC.L  $40040000,$C90FDAA2,$2168C235,$A1800000

INARG	equ	FP_SCR4

TWOTO63 equ     L_SCR1
ENDFLAG	equ	L_SCR2
N       equ     L_SCR3

	xref	t_frcinx
	xref	t_extdnrm

	xdef	stand
stand:
*--TAN(X) = X FOR DENORMALIZED X

	bra		t_extdnrm

	xdef	stan
stan:
	FMOVE.X		(a0),FP0	...LOAD INPUT

	MOVE.L		(A0),D0
	MOVE.W		4(A0),D0
	ANDI.L		#$7FFFFFFF,D0

	CMPI.L		#$3FD78000,D0		...|X| >= 2**(-40)?
	BGE.B		TANOK1
	BRA.W		TANSM
TANOK1:
	CMPI.L		#$4004BC7E,D0		...|X| < 15 PI?
	BLT.B		TANMAIN
	BRA.W		REDUCEX


TANMAIN:
*--THIS IS THE USUAL CASE, |X| <= 15 PI.
*--THE ARGUMENT REDUCTION IS DONE BY TABLE LOOK UP.
	FMOVE.X		FP0,FP1
	FMUL.D		TWOBYPI,FP1	...X*2/PI

*--HIDE THE NEXT TWO INSTRUCTIONS
	lea.l		PITBL+$200,a1 ...TABLE OF N*PI/2, N = -32,...,32

*--FP1 IS NOW READY
	FMOVE.L		FP1,D0		...CONVERT TO INTEGER

	ASL.L		#4,D0
	ADDA.L		D0,a1		...ADDRESS N*PIBY2 IN Y1, Y2

	FSUB.X		(a1)+,FP0	...X-Y1
*--HIDE THE NEXT ONE

	FSUB.S		(a1),FP0	...FP0 IS R = (X-Y1)-Y2

	ROR.L		#5,D0
	ANDI.L		#$80000000,D0	...D0 WAS ODD IFF D0 < 0

TANCONT:

	CMPI.L		#0,D0
	BLT.W		NODD

	FMOVE.X		FP0,FP1
	FMUL.X		FP1,FP1	 	...S = R*R

	FMOVE.D		TANQ4,FP3
	FMOVE.D		TANP3,FP2

	FMUL.X		FP1,FP3	 	...SQ4
	FMUL.X		FP1,FP2	 	...SP3

	FADD.D		TANQ3,FP3	...Q3+SQ4
	FADD.X		TANP2,FP2	...P2+SP3

	FMUL.X		FP1,FP3	 	...S(Q3+SQ4)
	FMUL.X		FP1,FP2	 	...S(P2+SP3)

	FADD.X		TANQ2,FP3	...Q2+S(Q3+SQ4)
	FADD.X		TANP1,FP2	...P1+S(P2+SP3)

	FMUL.X		FP1,FP3	 	...S(Q2+S(Q3+SQ4))
	FMUL.X		FP1,FP2	 	...S(P1+S(P2+SP3))

	FADD.X		TANQ1,FP3	...Q1+S(Q2+S(Q3+SQ4))
	FMUL.X		FP0,FP2	 	...RS(P1+S(P2+SP3))

	FMUL.X		FP3,FP1	 	...S(Q1+S(Q2+S(Q3+SQ4)))
	

	FADD.X		FP2,FP0	 	...R+RS(P1+S(P2+SP3))
	

	FADD.S		#:3F800000,FP1	...1+S(Q1+...)

	FMOVE.L		d1,fpcr		;restore users exceptions
	FDIV.X		FP1,FP0		;last inst - possible exception set

	bra		t_frcinx

NODD:
	FMOVE.X		FP0,FP1
	FMUL.X		FP0,FP0	 	...S = R*R

	FMOVE.D		TANQ4,FP3
	FMOVE.D		TANP3,FP2

	FMUL.X		FP0,FP3	 	...SQ4
	FMUL.X		FP0,FP2	 	...SP3

	FADD.D		TANQ3,FP3	...Q3+SQ4
	FADD.X		TANP2,FP2	...P2+SP3

	FMUL.X		FP0,FP3	 	...S(Q3+SQ4)
	FMUL.X		FP0,FP2	 	...S(P2+SP3)

	FADD.X		TANQ2,FP3	...Q2+S(Q3+SQ4)
	FADD.X		TANP1,FP2	...P1+S(P2+SP3)

	FMUL.X		FP0,FP3	 	...S(Q2+S(Q3+SQ4))
	FMUL.X		FP0,FP2	 	...S(P1+S(P2+SP3))

	FADD.X		TANQ1,FP3	...Q1+S(Q2+S(Q3+SQ4))
	FMUL.X		FP1,FP2	 	...RS(P1+S(P2+SP3))

	FMUL.X		FP3,FP0	 	...S(Q1+S(Q2+S(Q3+SQ4)))
	

	FADD.X		FP2,FP1	 	...R+RS(P1+S(P2+SP3))
	FADD.S		#:3F800000,FP0	...1+S(Q1+...)
	

	FMOVE.X		FP1,-(sp)
	EORI.L		#$80000000,(sp)

	FMOVE.L		d1,fpcr	 	;restore users exceptions
	FDIV.X		(sp)+,FP0	;last inst - possible exception set

	bra		t_frcinx

TANBORS:
*--IF |X| > 15PI, WE USE THE GENERAL ARGUMENT REDUCTION.
*--IF |X| < 2**(-40), RETURN X OR 1.
	CMPI.L		#$3FFF8000,D0
	BGT.B		REDUCEX

TANSM:

	FMOVE.X		FP0,-(sp)
	FMOVE.L		d1,fpcr		 ;restore users exceptions
	FMOVE.X		(sp)+,FP0	;last inst - posibble exception set

	bra		t_frcinx


REDUCEX:
*--WHEN REDUCEX IS USED, THE CODE WILL INEVITABLY BE SLOW.
*--THIS REDUCTION METHOD, HOWEVER, IS MUCH FASTER THAN USING
*--THE REMAINDER INSTRUCTION WHICH IS NOW IN SOFTWARE.

	FMOVEM.X	FP2-FP5,-(A7)	...save FP2 through FP5
	MOVE.L		D2,-(A7)
        FMOVE.S         #:00000000,FP1

*--If compact form of abs(arg) in d0=$7ffeffff, argument is so large that
*--there is a danger of unwanted overflow in first LOOP iteration.  In this
*--case, reduce argument by one remainder step to make subsequent reduction
*--safe.
	cmpi.l	#$7ffeffff,d0		;is argument dangerously large?
	bne.b	LOOP
	move.l	#$7ffe0000,FP_SCR2(a6)	;yes
*					;create 2**16383*PI/2
	move.l	#$c90fdaa2,FP_SCR2+4(a6)
	clr.l	FP_SCR2+8(a6)
	ftst.x	fp0			;test sign of argument
	move.l	#$7fdc0000,FP_SCR3(a6)	;create low half of 2**16383*
*					;PI/2 at FP_SCR3
	move.l	#$85a308d3,FP_SCR3+4(a6)
	clr.l   FP_SCR3+8(a6)
	fblt.w	red_neg
	or.w	#$8000,FP_SCR2(a6)	;positive arg
	or.w	#$8000,FP_SCR3(a6)
red_neg:
	fadd.x  FP_SCR2(a6),fp0		;high part of reduction is exact
	fmove.x  fp0,fp1		;save high result in fp1
	fadd.x  FP_SCR3(a6),fp0		;low part of reduction
	fsub.x  fp0,fp1			;determine low component of result
	fadd.x  FP_SCR3(a6),fp1		;fp0/fp1 are reduced argument.

*--ON ENTRY, FP0 IS X, ON RETURN, FP0 IS X REM PI/2, |X| <= PI/4.
*--integer quotient will be stored in N
*--Intermeditate remainder is 66-bit long; (R,r) in (FP0,FP1)

LOOP:
	FMOVE.X		FP0,INARG(a6)	...+-2**K * F, 1 <= F < 2
	MOVE.W		INARG(a6),D0
        MOVE.L          D0,A1		...save a copy of D0
	ANDI.L		#$00007FFF,D0
	SUBI.L		#$00003FFF,D0	...D0 IS K
	CMPI.L		#28,D0
	BLE.B		LASTLOOP
CONTLOOP:
	SUBI.L		#27,D0	 ...D0 IS L := K-27
	MOVE.L		#0,ENDFLAG(a6)
	BRA.B		WORK
LASTLOOP:
	CLR.L		D0		...D0 IS L := 0
	MOVE.L		#1,ENDFLAG(a6)

WORK:
*--FIND THE REMAINDER OF (R,r) W.R.T.	2**L * (PI/2). L IS SO CHOSEN
*--THAT	INT( X * (2/PI) / 2**(L) ) < 2**29.

*--CREATE 2**(-L) * (2/PI), SIGN(INARG)*2**(63),
*--2**L * (PIby2_1), 2**L * (PIby2_2)

	MOVE.L		#$00003FFE,D2	...BIASED EXPO OF 2/PI
	SUB.L		D0,D2		...BIASED EXPO OF 2**(-L)*(2/PI)

	MOVE.L		#$A2F9836E,FP_SCR1+4(a6)
	MOVE.L		#$4E44152A,FP_SCR1+8(a6)
	MOVE.W		D2,FP_SCR1(a6)	...FP_SCR1 is 2**(-L)*(2/PI)

	FMOVE.X		FP0,FP2
	FMUL.X		FP_SCR1(a6),FP2
*--WE MUST NOW FIND INT(FP2). SINCE WE NEED THIS VALUE IN
*--FLOATING POINT FORMAT, THE TWO FMOVE'S	FMOVE.L FP <--> N
*--WILL BE TOO INEFFICIENT. THE WAY AROUND IT IS THAT
*--(SIGN(INARG)*2**63	+	FP2) - SIGN(INARG)*2**63 WILL GIVE
*--US THE DESIRED VALUE IN FLOATING POINT.

*--HIDE SIX CYCLES OF INSTRUCTION
        MOVE.L		A1,D2
        SWAP		D2
	ANDI.L		#$80000000,D2
	ORI.L		#$5F000000,D2	...D2 IS SIGN(INARG)*2**63 IN SGL
	MOVE.L		D2,TWOTO63(a6)

	MOVE.L		D0,D2
	ADDI.L		#$00003FFF,D2	...BIASED EXPO OF 2**L * (PI/2)

*--FP2 IS READY
	FADD.S		TWOTO63(a6),FP2	...THE FRACTIONAL PART OF FP1 IS ROUNDED

*--HIDE 4 CYCLES OF INSTRUCTION; creating 2**(L)*Piby2_1  and  2**(L)*Piby2_2
        MOVE.W		D2,FP_SCR2(a6)
	CLR.W           FP_SCR2+2(a6)
	MOVE.L		#$C90FDAA2,FP_SCR2+4(a6)
	CLR.L		FP_SCR2+8(a6)		...FP_SCR2 is  2**(L) * Piby2_1	

*--FP2 IS READY
	FSUB.S		TWOTO63(a6),FP2		...FP2 is N

	ADDI.L		#$00003FDD,D0
        MOVE.W		D0,FP_SCR3(a6)
	CLR.W           FP_SCR3+2(a6)
	MOVE.L		#$85A308D3,FP_SCR3+4(a6)
	CLR.L		FP_SCR3+8(a6)		...FP_SCR3 is 2**(L) * Piby2_2

	MOVE.L		ENDFLAG(a6),D0

*--We are now ready to perform (R+r) - N*P1 - N*P2, P1 = 2**(L) * Piby2_1 and
*--P2 = 2**(L) * Piby2_2
	FMOVE.X		FP2,FP4
	FMul.X		FP_SCR2(a6),FP4		...W = N*P1
	FMove.X		FP2,FP5
	FMul.X		FP_SCR3(a6),FP5		...w = N*P2
	FMove.X		FP4,FP3
*--we want P+p = W+w  but  |p| <= half ulp of P
*--Then, we need to compute  A := R-P   and  a := r-p
	FAdd.X		FP5,FP3			...FP3 is P
	FSub.X		FP3,FP4			...W-P

	FSub.X		FP3,FP0			...FP0 is A := R - P
        FAdd.X		FP5,FP4			...FP4 is p = (W-P)+w

	FMove.X		FP0,FP3			...FP3 A
	FSub.X		FP4,FP1			...FP1 is a := r - p

*--Now we need to normalize (A,a) to  "new (R,r)" where R+r = A+a but
*--|r| <= half ulp of R.
	FAdd.X		FP1,FP0			...FP0 is R := A+a
*--No need to calculate r if this is the last loop
	CMPI.L		#0,D0
	BGT.W		RESTORE

*--Need to calculate r
	FSub.X		FP0,FP3			...A-R
	FAdd.X		FP3,FP1			...FP1 is r := (A-R)+a
	BRA.W		LOOP

RESTORE:
        FMOVE.L		FP2,N(a6)
	MOVE.L		(A7)+,D2
	FMOVEM.X	(A7)+,FP2-FP5

	
	MOVE.L		N(a6),D0
        ROR.L		#1,D0


	BRA.W		TANCONT

	end