* $OpenBSD: stan.sa,v 1.3 2003/11/07 10:36:10 miod Exp $
* $NetBSD: stan.sa,v 1.3 1994/10/26 07:50:10 cgd Exp $
* 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 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:
TST.L 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
CLR.L 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
TST.L 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
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