stan.S

|
|	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.
|

|		Copyright (C) Motorola, Inc. 1990
|			All Rights Reserved
|
|       For details on the license for this file, please see the
|       file, README, in this same directory.

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

	|section	8

#include "fpsp.h"

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

TANQ4:	.long 0x3EA0B759,0xF50F8688
TANP3:	.long 0xBEF2BAA5,0xA8924F04

TANQ3:	.long 0xBF346F59,0xB39BA65F,0x00000000,0x00000000

TANP2:	.long 0x3FF60000,0xE073D3FC,0x199C4A00,0x00000000

TANQ2:	.long 0x3FF90000,0xD23CD684,0x15D95FA1,0x00000000

TANP1:	.long 0xBFFC0000,0x8895A6C5,0xFB423BCA,0x00000000

TANQ1:	.long 0xBFFD0000,0xEEF57E0D,0xA84BC8CE,0x00000000

INVTWOPI: .long 0x3FFC0000,0xA2F9836E,0x4E44152A,0x00000000

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

|--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.
	.global	PITBL
PITBL:
  .long  0xC0040000,0xC90FDAA2,0x2168C235,0x21800000
  .long  0xC0040000,0xC2C75BCD,0x105D7C23,0xA0D00000
  .long  0xC0040000,0xBC7EDCF7,0xFF523611,0xA1E80000
  .long  0xC0040000,0xB6365E22,0xEE46F000,0x21480000
  .long  0xC0040000,0xAFEDDF4D,0xDD3BA9EE,0xA1200000
  .long  0xC0040000,0xA9A56078,0xCC3063DD,0x21FC0000
  .long  0xC0040000,0xA35CE1A3,0xBB251DCB,0x21100000
  .long  0xC0040000,0x9D1462CE,0xAA19D7B9,0xA1580000
  .long  0xC0040000,0x96CBE3F9,0x990E91A8,0x21E00000
  .long  0xC0040000,0x90836524,0x88034B96,0x20B00000
  .long  0xC0040000,0x8A3AE64F,0x76F80584,0xA1880000
  .long  0xC0040000,0x83F2677A,0x65ECBF73,0x21C40000
  .long  0xC0030000,0xFB53D14A,0xA9C2F2C2,0x20000000
  .long  0xC0030000,0xEEC2D3A0,0x87AC669F,0x21380000
  .long  0xC0030000,0xE231D5F6,0x6595DA7B,0xA1300000
  .long  0xC0030000,0xD5A0D84C,0x437F4E58,0x9FC00000
  .long  0xC0030000,0xC90FDAA2,0x2168C235,0x21000000
  .long  0xC0030000,0xBC7EDCF7,0xFF523611,0xA1680000
  .long  0xC0030000,0xAFEDDF4D,0xDD3BA9EE,0xA0A00000
  .long  0xC0030000,0xA35CE1A3,0xBB251DCB,0x20900000
  .long  0xC0030000,0x96CBE3F9,0x990E91A8,0x21600000
  .long  0xC0030000,0x8A3AE64F,0x76F80584,0xA1080000
  .long  0xC0020000,0xFB53D14A,0xA9C2F2C2,0x1F800000
  .long  0xC0020000,0xE231D5F6,0x6595DA7B,0xA0B00000
  .long  0xC0020000,0xC90FDAA2,0x2168C235,0x20800000
  .long  0xC0020000,0xAFEDDF4D,0xDD3BA9EE,0xA0200000
  .long  0xC0020000,0x96CBE3F9,0x990E91A8,0x20E00000
  .long  0xC0010000,0xFB53D14A,0xA9C2F2C2,0x1F000000
  .long  0xC0010000,0xC90FDAA2,0x2168C235,0x20000000
  .long  0xC0010000,0x96CBE3F9,0x990E91A8,0x20600000
  .long  0xC0000000,0xC90FDAA2,0x2168C235,0x1F800000
  .long  0xBFFF0000,0xC90FDAA2,0x2168C235,0x1F000000
  .long  0x00000000,0x00000000,0x00000000,0x00000000
  .long  0x3FFF0000,0xC90FDAA2,0x2168C235,0x9F000000
  .long  0x40000000,0xC90FDAA2,0x2168C235,0x9F800000
  .long  0x40010000,0x96CBE3F9,0x990E91A8,0xA0600000
  .long  0x40010000,0xC90FDAA2,0x2168C235,0xA0000000
  .long  0x40010000,0xFB53D14A,0xA9C2F2C2,0x9F000000
  .long  0x40020000,0x96CBE3F9,0x990E91A8,0xA0E00000
  .long  0x40020000,0xAFEDDF4D,0xDD3BA9EE,0x20200000
  .long  0x40020000,0xC90FDAA2,0x2168C235,0xA0800000
  .long  0x40020000,0xE231D5F6,0x6595DA7B,0x20B00000
  .long  0x40020000,0xFB53D14A,0xA9C2F2C2,0x9F800000
  .long  0x40030000,0x8A3AE64F,0x76F80584,0x21080000
  .long  0x40030000,0x96CBE3F9,0x990E91A8,0xA1600000
  .long  0x40030000,0xA35CE1A3,0xBB251DCB,0xA0900000
  .long  0x40030000,0xAFEDDF4D,0xDD3BA9EE,0x20A00000
  .long  0x40030000,0xBC7EDCF7,0xFF523611,0x21680000
  .long  0x40030000,0xC90FDAA2,0x2168C235,0xA1000000
  .long  0x40030000,0xD5A0D84C,0x437F4E58,0x1FC00000
  .long  0x40030000,0xE231D5F6,0x6595DA7B,0x21300000
  .long  0x40030000,0xEEC2D3A0,0x87AC669F,0xA1380000
  .long  0x40030000,0xFB53D14A,0xA9C2F2C2,0xA0000000
  .long  0x40040000,0x83F2677A,0x65ECBF73,0xA1C40000
  .long  0x40040000,0x8A3AE64F,0x76F80584,0x21880000
  .long  0x40040000,0x90836524,0x88034B96,0xA0B00000
  .long  0x40040000,0x96CBE3F9,0x990E91A8,0xA1E00000
  .long  0x40040000,0x9D1462CE,0xAA19D7B9,0x21580000
  .long  0x40040000,0xA35CE1A3,0xBB251DCB,0xA1100000
  .long  0x40040000,0xA9A56078,0xCC3063DD,0xA1FC0000
  .long  0x40040000,0xAFEDDF4D,0xDD3BA9EE,0x21200000
  .long  0x40040000,0xB6365E22,0xEE46F000,0xA1480000
  .long  0x40040000,0xBC7EDCF7,0xFF523611,0x21E80000
  .long  0x40040000,0xC2C75BCD,0x105D7C23,0x20D00000
  .long  0x40040000,0xC90FDAA2,0x2168C235,0xA1800000

	.set	INARG,FP_SCR4

	.set	TWOTO63,L_SCR1
	.set	ENDFLAG,L_SCR2
	.set	N,L_SCR3

	| xref	t_frcinx
	|xref	t_extdnrm

	.global	stand
stand:
|--TAN(X) = X FOR DENORMALIZED X

	bra		t_extdnrm

	.global	stan
stan:
	fmovex		(%a0),%fp0	| ...LOAD INPUT

	movel		(%a0),%d0
	movew		4(%a0),%d0
	andil		#0x7FFFFFFF,%d0

	cmpil		#0x3FD78000,%d0		| ...|X| >= 2**(-40)?
	bges		TANOK1
	bra		TANSM
TANOK1:
	cmpil		#0x4004BC7E,%d0		| ...|X| < 15 PI?
	blts		TANMAIN
	bra		REDUCEX


TANMAIN:
|--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 TWO INSTRUCTIONS
	leal		PITBL+0x200,%a1 | ...TABLE OF N*PI/2, N = -32,...,32

|--FP1 IS NOW READY
	fmovel		%fp1,%d0		| ...CONVERT TO INTEGER

	asll		#4,%d0
	addal		%d0,%a1		| ...ADDRESS N*PIBY2 IN Y1, Y2

	fsubx		(%a1)+,%fp0	| ...X-Y1
|--HIDE THE NEXT ONE

	fsubs		(%a1),%fp0	| ...FP0 IS R = (X-Y1)-Y2

	rorl		#5,%d0
	andil		#0x80000000,%d0	| ...D0 WAS ODD IFF D0 < 0

TANCONT:

	cmpil		#0,%d0
	blt		NODD

	fmovex		%fp0,%fp1
	fmulx		%fp1,%fp1		| ...S = R*R

	fmoved		TANQ4,%fp3
	fmoved		TANP3,%fp2

	fmulx		%fp1,%fp3		| ...SQ4
	fmulx		%fp1,%fp2		| ...SP3

	faddd		TANQ3,%fp3	| ...Q3+SQ4
	faddx		TANP2,%fp2	| ...P2+SP3

	fmulx		%fp1,%fp3		| ...S(Q3+SQ4)
	fmulx		%fp1,%fp2		| ...S(P2+SP3)

	faddx		TANQ2,%fp3	| ...Q2+S(Q3+SQ4)
	faddx		TANP1,%fp2	| ...P1+S(P2+SP3)

	fmulx		%fp1,%fp3		| ...S(Q2+S(Q3+SQ4))
	fmulx		%fp1,%fp2		| ...S(P1+S(P2+SP3))

	faddx		TANQ1,%fp3	| ...Q1+S(Q2+S(Q3+SQ4))
	fmulx		%fp0,%fp2		| ...RS(P1+S(P2+SP3))

	fmulx		%fp3,%fp1		| ...S(Q1+S(Q2+S(Q3+SQ4)))


	faddx		%fp2,%fp0		| ...R+RS(P1+S(P2+SP3))


	fadds		#0x3F800000,%fp1	| ...1+S(Q1+...)

	fmovel		%d1,%fpcr		|restore users exceptions
	fdivx		%fp1,%fp0		|last inst - possible exception set

	bra		t_frcinx

NODD:
	fmovex		%fp0,%fp1
	fmulx		%fp0,%fp0		| ...S = R*R

	fmoved		TANQ4,%fp3
	fmoved		TANP3,%fp2

	fmulx		%fp0,%fp3		| ...SQ4
	fmulx		%fp0,%fp2		| ...SP3

	faddd		TANQ3,%fp3	| ...Q3+SQ4
	faddx		TANP2,%fp2	| ...P2+SP3

	fmulx		%fp0,%fp3		| ...S(Q3+SQ4)
	fmulx		%fp0,%fp2		| ...S(P2+SP3)

	faddx		TANQ2,%fp3	| ...Q2+S(Q3+SQ4)
	faddx		TANP1,%fp2	| ...P1+S(P2+SP3)

	fmulx		%fp0,%fp3		| ...S(Q2+S(Q3+SQ4))
	fmulx		%fp0,%fp2		| ...S(P1+S(P2+SP3))

	faddx		TANQ1,%fp3	| ...Q1+S(Q2+S(Q3+SQ4))
	fmulx		%fp1,%fp2		| ...RS(P1+S(P2+SP3))

	fmulx		%fp3,%fp0		| ...S(Q1+S(Q2+S(Q3+SQ4)))


	faddx		%fp2,%fp1		| ...R+RS(P1+S(P2+SP3))
	fadds		#0x3F800000,%fp0	| ...1+S(Q1+...)


	fmovex		%fp1,-(%sp)
	eoril		#0x80000000,(%sp)

	fmovel		%d1,%fpcr		|restore users exceptions
	fdivx		(%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.
	cmpil		#0x3FFF8000,%d0
	bgts		REDUCEX

TANSM:

	fmovex		%fp0,-(%sp)
	fmovel		%d1,%fpcr		 |restore users exceptions
	fmovex		(%sp)+,%fp0	|last inst - possible 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.

	fmovemx	%fp2-%fp5,-(%a7)	| ...save FP2 through FP5
	movel		%d2,-(%a7)
        fmoves         #0x00000000,%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.
	cmpil	#0x7ffeffff,%d0		|is argument dangerously large?
	bnes	LOOP
	movel	#0x7ffe0000,FP_SCR2(%a6)	|yes
|					;create 2**16383*PI/2
	movel	#0xc90fdaa2,FP_SCR2+4(%a6)
	clrl	FP_SCR2+8(%a6)
	ftstx	%fp0			|test sign of argument
	movel	#0x7fdc0000,FP_SCR3(%a6)	|create low half of 2**16383*
|					;PI/2 at FP_SCR3
	movel	#0x85a308d3,FP_SCR3+4(%a6)
	clrl   FP_SCR3+8(%a6)
	fblt	red_neg
	orw	#0x8000,FP_SCR2(%a6)	|positive arg
	orw	#0x8000,FP_SCR3(%a6)
red_neg:
	faddx  FP_SCR2(%a6),%fp0		|high part of reduction is exact
	fmovex  %fp0,%fp1		|save high result in fp1
	faddx  FP_SCR3(%a6),%fp0		|low part of reduction
	fsubx  %fp0,%fp1			|determine low component of result
	faddx  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
|--Intermediate remainder is 66-bit long; (R,r) in (FP0,FP1)

LOOP:
	fmovex		%fp0,INARG(%a6)	| ...+-2**K * F, 1 <= F < 2
	movew		INARG(%a6),%d0
        movel          %d0,%a1		| ...save a copy of D0
	andil		#0x00007FFF,%d0
	subil		#0x00003FFF,%d0	| ...D0 IS K
	cmpil		#28,%d0
	bles		LASTLOOP
CONTLOOP:
	subil		#27,%d0	 | ...D0 IS L := K-27
	movel		#0,ENDFLAG(%a6)
	bras		WORK
LASTLOOP:
	clrl		%d0		| ...D0 IS L := 0
	movel		#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)

	movel		#0x00003FFE,%d2	| ...BIASED EXPO OF 2/PI
	subl		%d0,%d2		| ...BIASED EXPO OF 2**(-L)*(2/PI)

	movel		#0xA2F9836E,FP_SCR1+4(%a6)
	movel		#0x4E44152A,FP_SCR1+8(%a6)
	movew		%d2,FP_SCR1(%a6)	| ...FP_SCR1 is 2**(-L)*(2/PI)

	fmovex		%fp0,%fp2
	fmulx		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
        movel		%a1,%d2
        swap		%d2
	andil		#0x80000000,%d2
	oril		#0x5F000000,%d2	| ...D2 IS SIGN(INARG)*2**63 IN SGL
	movel		%d2,TWOTO63(%a6)

	movel		%d0,%d2
	addil		#0x00003FFF,%d2	| ...BIASED EXPO OF 2**L * (PI/2)

|--FP2 IS READY
	fadds		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
        movew		%d2,FP_SCR2(%a6)
	clrw           FP_SCR2+2(%a6)
	movel		#0xC90FDAA2,FP_SCR2+4(%a6)
	clrl		FP_SCR2+8(%a6)		| ...FP_SCR2 is  2**(L) * Piby2_1

|--FP2 IS READY
	fsubs		TWOTO63(%a6),%fp2		| ...FP2 is N

	addil		#0x00003FDD,%d0
        movew		%d0,FP_SCR3(%a6)
	clrw           FP_SCR3+2(%a6)
	movel		#0x85A308D3,FP_SCR3+4(%a6)
	clrl		FP_SCR3+8(%a6)		| ...FP_SCR3 is 2**(L) * Piby2_2

	movel		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
	fmovex		%fp2,%fp4
	fmulx		FP_SCR2(%a6),%fp4		| ...W = N*P1
	fmovex		%fp2,%fp5
	fmulx		FP_SCR3(%a6),%fp5		| ...w = N*P2
	fmovex		%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
	faddx		%fp5,%fp3			| ...FP3 is P
	fsubx		%fp3,%fp4			| ...W-P

	fsubx		%fp3,%fp0			| ...FP0 is A := R - P
        faddx		%fp5,%fp4			| ...FP4 is p = (W-P)+w

	fmovex		%fp0,%fp3			| ...FP3 A
	fsubx		%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.
	faddx		%fp1,%fp0			| ...FP0 is R := A+a
|--No need to calculate r if this is the last loop
	cmpil		#0,%d0
	bgt		RESTORE

|--Need to calculate r
	fsubx		%fp0,%fp3			| ...A-R
	faddx		%fp3,%fp1			| ...FP1 is r := (A-R)+a
	bra		LOOP

RESTORE:
        fmovel		%fp2,N(%a6)
	movel		(%a7)+,%d2
	fmovemx	(%a7)+,%fp2-%fp5


	movel		N(%a6),%d0
        rorl		#1,%d0


	bra		TANCONT

	|end