1|
2|	stan.sa 3.3 7/29/91
3|
4|	The entry point stan computes the tangent of
5|	an input argument;
6|	stand does the same except for denormalized input.
7|
8|	Input: Double-extended number X in location pointed to
9|		by address register a0.
10|
11|	Output: The value tan(X) returned in floating-point register Fp0.
12|
13|	Accuracy and Monotonicity: The returned result is within 3 ulp in
14|		64 significant bit, i.e. within 0.5001 ulp to 53 bits if the
15|		result is subsequently rounded to double precision. The
16|		result is provably monotonic in double precision.
17|
18|	Speed: The program sTAN takes approximately 170 cycles for
19|		input argument X such that |X| < 15Pi, which is the usual
20|		situation.
21|
22|	Algorithm:
23|
24|	1. If |X| >= 15Pi or |X| < 2**(-40), go to 6.
25|
26|	2. Decompose X as X = N(Pi/2) + r where |r| <= Pi/4. Let
27|		k = N mod 2, so in particular, k = 0 or 1.
28|
29|	3. If k is odd, go to 5.
30|
31|	4. (k is even) Tan(X) = tan(r) and tan(r) is approximated by a
32|		rational function U/V where
33|		U = r + r*s*(P1 + s*(P2 + s*P3)), and
34|		V = 1 + s*(Q1 + s*(Q2 + s*(Q3 + s*Q4))),  s = r*r.
35|		Exit.
36|
37|	4. (k is odd) Tan(X) = -cot(r). Since tan(r) is approximated by a
38|		rational function U/V where
39|		U = r + r*s*(P1 + s*(P2 + s*P3)), and
40|		V = 1 + s*(Q1 + s*(Q2 + s*(Q3 + s*Q4))), s = r*r,
41|		-Cot(r) = -V/U. Exit.
42|
43|	6. If |X| > 1, go to 8.
44|
45|	7. (|X|<2**(-40)) Tan(X) = X. Exit.
46|
47|	8. Overwrite X by X := X rem 2Pi. Now that |X| <= Pi, go back to 2.
48|
49
50|		Copyright (C) Motorola, Inc. 1990
51|			All Rights Reserved
52|
53|       For details on the license for this file, please see the
54|       file, README, in this same directory.
55
56|STAN	idnt	2,1 | Motorola 040 Floating Point Software Package
57
58	|section	8
59
60#include "fpsp.h"
61
62BOUNDS1:	.long 0x3FD78000,0x4004BC7E
63TWOBYPI:	.long 0x3FE45F30,0x6DC9C883
64
65TANQ4:	.long 0x3EA0B759,0xF50F8688
66TANP3:	.long 0xBEF2BAA5,0xA8924F04
67
68TANQ3:	.long 0xBF346F59,0xB39BA65F,0x00000000,0x00000000
69
70TANP2:	.long 0x3FF60000,0xE073D3FC,0x199C4A00,0x00000000
71
72TANQ2:	.long 0x3FF90000,0xD23CD684,0x15D95FA1,0x00000000
73
74TANP1:	.long 0xBFFC0000,0x8895A6C5,0xFB423BCA,0x00000000
75
76TANQ1:	.long 0xBFFD0000,0xEEF57E0D,0xA84BC8CE,0x00000000
77
78INVTWOPI: .long 0x3FFC0000,0xA2F9836E,0x4E44152A,0x00000000
79
80TWOPI1:	.long 0x40010000,0xC90FDAA2,0x00000000,0x00000000
81TWOPI2:	.long 0x3FDF0000,0x85A308D4,0x00000000,0x00000000
82
83|--N*PI/2, -32 <= N <= 32, IN A LEADING TERM IN EXT. AND TRAILING
84|--TERM IN SGL. NOTE THAT PI IS 64-BIT LONG, THUS N*PI/2 IS AT
85|--MOST 69 BITS LONG.
86	.global	PITBL
87PITBL:
88  .long  0xC0040000,0xC90FDAA2,0x2168C235,0x21800000
89  .long  0xC0040000,0xC2C75BCD,0x105D7C23,0xA0D00000
90  .long  0xC0040000,0xBC7EDCF7,0xFF523611,0xA1E80000
91  .long  0xC0040000,0xB6365E22,0xEE46F000,0x21480000
92  .long  0xC0040000,0xAFEDDF4D,0xDD3BA9EE,0xA1200000
93  .long  0xC0040000,0xA9A56078,0xCC3063DD,0x21FC0000
94  .long  0xC0040000,0xA35CE1A3,0xBB251DCB,0x21100000
95  .long  0xC0040000,0x9D1462CE,0xAA19D7B9,0xA1580000
96  .long  0xC0040000,0x96CBE3F9,0x990E91A8,0x21E00000
97  .long  0xC0040000,0x90836524,0x88034B96,0x20B00000
98  .long  0xC0040000,0x8A3AE64F,0x76F80584,0xA1880000
99  .long  0xC0040000,0x83F2677A,0x65ECBF73,0x21C40000
100  .long  0xC0030000,0xFB53D14A,0xA9C2F2C2,0x20000000
101  .long  0xC0030000,0xEEC2D3A0,0x87AC669F,0x21380000
102  .long  0xC0030000,0xE231D5F6,0x6595DA7B,0xA1300000
103  .long  0xC0030000,0xD5A0D84C,0x437F4E58,0x9FC00000
104  .long  0xC0030000,0xC90FDAA2,0x2168C235,0x21000000
105  .long  0xC0030000,0xBC7EDCF7,0xFF523611,0xA1680000
106  .long  0xC0030000,0xAFEDDF4D,0xDD3BA9EE,0xA0A00000
107  .long  0xC0030000,0xA35CE1A3,0xBB251DCB,0x20900000
108  .long  0xC0030000,0x96CBE3F9,0x990E91A8,0x21600000
109  .long  0xC0030000,0x8A3AE64F,0x76F80584,0xA1080000
110  .long  0xC0020000,0xFB53D14A,0xA9C2F2C2,0x1F800000
111  .long  0xC0020000,0xE231D5F6,0x6595DA7B,0xA0B00000
112  .long  0xC0020000,0xC90FDAA2,0x2168C235,0x20800000
113  .long  0xC0020000,0xAFEDDF4D,0xDD3BA9EE,0xA0200000
114  .long  0xC0020000,0x96CBE3F9,0x990E91A8,0x20E00000
115  .long  0xC0010000,0xFB53D14A,0xA9C2F2C2,0x1F000000
116  .long  0xC0010000,0xC90FDAA2,0x2168C235,0x20000000
117  .long  0xC0010000,0x96CBE3F9,0x990E91A8,0x20600000
118  .long  0xC0000000,0xC90FDAA2,0x2168C235,0x1F800000
119  .long  0xBFFF0000,0xC90FDAA2,0x2168C235,0x1F000000
120  .long  0x00000000,0x00000000,0x00000000,0x00000000
121  .long  0x3FFF0000,0xC90FDAA2,0x2168C235,0x9F000000
122  .long  0x40000000,0xC90FDAA2,0x2168C235,0x9F800000
123  .long  0x40010000,0x96CBE3F9,0x990E91A8,0xA0600000
124  .long  0x40010000,0xC90FDAA2,0x2168C235,0xA0000000
125  .long  0x40010000,0xFB53D14A,0xA9C2F2C2,0x9F000000
126  .long  0x40020000,0x96CBE3F9,0x990E91A8,0xA0E00000
127  .long  0x40020000,0xAFEDDF4D,0xDD3BA9EE,0x20200000
128  .long  0x40020000,0xC90FDAA2,0x2168C235,0xA0800000
129  .long  0x40020000,0xE231D5F6,0x6595DA7B,0x20B00000
130  .long  0x40020000,0xFB53D14A,0xA9C2F2C2,0x9F800000
131  .long  0x40030000,0x8A3AE64F,0x76F80584,0x21080000
132  .long  0x40030000,0x96CBE3F9,0x990E91A8,0xA1600000
133  .long  0x40030000,0xA35CE1A3,0xBB251DCB,0xA0900000
134  .long  0x40030000,0xAFEDDF4D,0xDD3BA9EE,0x20A00000
135  .long  0x40030000,0xBC7EDCF7,0xFF523611,0x21680000
136  .long  0x40030000,0xC90FDAA2,0x2168C235,0xA1000000
137  .long  0x40030000,0xD5A0D84C,0x437F4E58,0x1FC00000
138  .long  0x40030000,0xE231D5F6,0x6595DA7B,0x21300000
139  .long  0x40030000,0xEEC2D3A0,0x87AC669F,0xA1380000
140  .long  0x40030000,0xFB53D14A,0xA9C2F2C2,0xA0000000
141  .long  0x40040000,0x83F2677A,0x65ECBF73,0xA1C40000
142  .long  0x40040000,0x8A3AE64F,0x76F80584,0x21880000
143  .long  0x40040000,0x90836524,0x88034B96,0xA0B00000
144  .long  0x40040000,0x96CBE3F9,0x990E91A8,0xA1E00000
145  .long  0x40040000,0x9D1462CE,0xAA19D7B9,0x21580000
146  .long  0x40040000,0xA35CE1A3,0xBB251DCB,0xA1100000
147  .long  0x40040000,0xA9A56078,0xCC3063DD,0xA1FC0000
148  .long  0x40040000,0xAFEDDF4D,0xDD3BA9EE,0x21200000
149  .long  0x40040000,0xB6365E22,0xEE46F000,0xA1480000
150  .long  0x40040000,0xBC7EDCF7,0xFF523611,0x21E80000
151  .long  0x40040000,0xC2C75BCD,0x105D7C23,0x20D00000
152  .long  0x40040000,0xC90FDAA2,0x2168C235,0xA1800000
153
154	.set	INARG,FP_SCR4
155
156	.set	TWOTO63,L_SCR1
157	.set	ENDFLAG,L_SCR2
158	.set	N,L_SCR3
159
160	| xref	t_frcinx
161	|xref	t_extdnrm
162
163	.global	stand
164stand:
165|--TAN(X) = X FOR DENORMALIZED X
166
167	bra		t_extdnrm
168
169	.global	stan
170stan:
171	fmovex		(%a0),%fp0	| ...LOAD INPUT
172
173	movel		(%a0),%d0
174	movew		4(%a0),%d0
175	andil		#0x7FFFFFFF,%d0
176
177	cmpil		#0x3FD78000,%d0		| ...|X| >= 2**(-40)?
178	bges		TANOK1
179	bra		TANSM
180TANOK1:
181	cmpil		#0x4004BC7E,%d0		| ...|X| < 15 PI?
182	blts		TANMAIN
183	bra		REDUCEX
184
185
186TANMAIN:
187|--THIS IS THE USUAL CASE, |X| <= 15 PI.
188|--THE ARGUMENT REDUCTION IS DONE BY TABLE LOOK UP.
189	fmovex		%fp0,%fp1
190	fmuld		TWOBYPI,%fp1	| ...X*2/PI
191
192|--HIDE THE NEXT TWO INSTRUCTIONS
193	leal		PITBL+0x200,%a1 | ...TABLE OF N*PI/2, N = -32,...,32
194
195|--FP1 IS NOW READY
196	fmovel		%fp1,%d0		| ...CONVERT TO INTEGER
197
198	asll		#4,%d0
199	addal		%d0,%a1		| ...ADDRESS N*PIBY2 IN Y1, Y2
200
201	fsubx		(%a1)+,%fp0	| ...X-Y1
202|--HIDE THE NEXT ONE
203
204	fsubs		(%a1),%fp0	| ...FP0 IS R = (X-Y1)-Y2
205
206	rorl		#5,%d0
207	andil		#0x80000000,%d0	| ...D0 WAS ODD IFF D0 < 0
208
209TANCONT:
210
211	cmpil		#0,%d0
212	blt		NODD
213
214	fmovex		%fp0,%fp1
215	fmulx		%fp1,%fp1		| ...S = R*R
216
217	fmoved		TANQ4,%fp3
218	fmoved		TANP3,%fp2
219
220	fmulx		%fp1,%fp3		| ...SQ4
221	fmulx		%fp1,%fp2		| ...SP3
222
223	faddd		TANQ3,%fp3	| ...Q3+SQ4
224	faddx		TANP2,%fp2	| ...P2+SP3
225
226	fmulx		%fp1,%fp3		| ...S(Q3+SQ4)
227	fmulx		%fp1,%fp2		| ...S(P2+SP3)
228
229	faddx		TANQ2,%fp3	| ...Q2+S(Q3+SQ4)
230	faddx		TANP1,%fp2	| ...P1+S(P2+SP3)
231
232	fmulx		%fp1,%fp3		| ...S(Q2+S(Q3+SQ4))
233	fmulx		%fp1,%fp2		| ...S(P1+S(P2+SP3))
234
235	faddx		TANQ1,%fp3	| ...Q1+S(Q2+S(Q3+SQ4))
236	fmulx		%fp0,%fp2		| ...RS(P1+S(P2+SP3))
237
238	fmulx		%fp3,%fp1		| ...S(Q1+S(Q2+S(Q3+SQ4)))
239
240
241	faddx		%fp2,%fp0		| ...R+RS(P1+S(P2+SP3))
242
243
244	fadds		#0x3F800000,%fp1	| ...1+S(Q1+...)
245
246	fmovel		%d1,%fpcr		|restore users exceptions
247	fdivx		%fp1,%fp0		|last inst - possible exception set
248
249	bra		t_frcinx
250
251NODD:
252	fmovex		%fp0,%fp1
253	fmulx		%fp0,%fp0		| ...S = R*R
254
255	fmoved		TANQ4,%fp3
256	fmoved		TANP3,%fp2
257
258	fmulx		%fp0,%fp3		| ...SQ4
259	fmulx		%fp0,%fp2		| ...SP3
260
261	faddd		TANQ3,%fp3	| ...Q3+SQ4
262	faddx		TANP2,%fp2	| ...P2+SP3
263
264	fmulx		%fp0,%fp3		| ...S(Q3+SQ4)
265	fmulx		%fp0,%fp2		| ...S(P2+SP3)
266
267	faddx		TANQ2,%fp3	| ...Q2+S(Q3+SQ4)
268	faddx		TANP1,%fp2	| ...P1+S(P2+SP3)
269
270	fmulx		%fp0,%fp3		| ...S(Q2+S(Q3+SQ4))
271	fmulx		%fp0,%fp2		| ...S(P1+S(P2+SP3))
272
273	faddx		TANQ1,%fp3	| ...Q1+S(Q2+S(Q3+SQ4))
274	fmulx		%fp1,%fp2		| ...RS(P1+S(P2+SP3))
275
276	fmulx		%fp3,%fp0		| ...S(Q1+S(Q2+S(Q3+SQ4)))
277
278
279	faddx		%fp2,%fp1		| ...R+RS(P1+S(P2+SP3))
280	fadds		#0x3F800000,%fp0	| ...1+S(Q1+...)
281
282
283	fmovex		%fp1,-(%sp)
284	eoril		#0x80000000,(%sp)
285
286	fmovel		%d1,%fpcr		|restore users exceptions
287	fdivx		(%sp)+,%fp0	|last inst - possible exception set
288
289	bra		t_frcinx
290
291TANBORS:
292|--IF |X| > 15PI, WE USE THE GENERAL ARGUMENT REDUCTION.
293|--IF |X| < 2**(-40), RETURN X OR 1.
294	cmpil		#0x3FFF8000,%d0
295	bgts		REDUCEX
296
297TANSM:
298
299	fmovex		%fp0,-(%sp)
300	fmovel		%d1,%fpcr		 |restore users exceptions
301	fmovex		(%sp)+,%fp0	|last inst - possible exception set
302
303	bra		t_frcinx
304
305
306REDUCEX:
307|--WHEN REDUCEX IS USED, THE CODE WILL INEVITABLY BE SLOW.
308|--THIS REDUCTION METHOD, HOWEVER, IS MUCH FASTER THAN USING
309|--THE REMAINDER INSTRUCTION WHICH IS NOW IN SOFTWARE.
310
311	fmovemx	%fp2-%fp5,-(%a7)	| ...save FP2 through FP5
312	movel		%d2,-(%a7)
313        fmoves         #0x00000000,%fp1
314
315|--If compact form of abs(arg) in d0=$7ffeffff, argument is so large that
316|--there is a danger of unwanted overflow in first LOOP iteration.  In this
317|--case, reduce argument by one remainder step to make subsequent reduction
318|--safe.
319	cmpil	#0x7ffeffff,%d0		|is argument dangerously large?
320	bnes	LOOP
321	movel	#0x7ffe0000,FP_SCR2(%a6)	|yes
322|					;create 2**16383*PI/2
323	movel	#0xc90fdaa2,FP_SCR2+4(%a6)
324	clrl	FP_SCR2+8(%a6)
325	ftstx	%fp0			|test sign of argument
326	movel	#0x7fdc0000,FP_SCR3(%a6)	|create low half of 2**16383*
327|					;PI/2 at FP_SCR3
328	movel	#0x85a308d3,FP_SCR3+4(%a6)
329	clrl   FP_SCR3+8(%a6)
330	fblt	red_neg
331	orw	#0x8000,FP_SCR2(%a6)	|positive arg
332	orw	#0x8000,FP_SCR3(%a6)
333red_neg:
334	faddx  FP_SCR2(%a6),%fp0		|high part of reduction is exact
335	fmovex  %fp0,%fp1		|save high result in fp1
336	faddx  FP_SCR3(%a6),%fp0		|low part of reduction
337	fsubx  %fp0,%fp1			|determine low component of result
338	faddx  FP_SCR3(%a6),%fp1		|fp0/fp1 are reduced argument.
339
340|--ON ENTRY, FP0 IS X, ON RETURN, FP0 IS X REM PI/2, |X| <= PI/4.
341|--integer quotient will be stored in N
342|--Intermediate remainder is 66-bit long; (R,r) in (FP0,FP1)
343
344LOOP:
345	fmovex		%fp0,INARG(%a6)	| ...+-2**K * F, 1 <= F < 2
346	movew		INARG(%a6),%d0
347        movel          %d0,%a1		| ...save a copy of D0
348	andil		#0x00007FFF,%d0
349	subil		#0x00003FFF,%d0	| ...D0 IS K
350	cmpil		#28,%d0
351	bles		LASTLOOP
352CONTLOOP:
353	subil		#27,%d0	 | ...D0 IS L := K-27
354	movel		#0,ENDFLAG(%a6)
355	bras		WORK
356LASTLOOP:
357	clrl		%d0		| ...D0 IS L := 0
358	movel		#1,ENDFLAG(%a6)
359
360WORK:
361|--FIND THE REMAINDER OF (R,r) W.R.T.	2**L * (PI/2). L IS SO CHOSEN
362|--THAT	INT( X * (2/PI) / 2**(L) ) < 2**29.
363
364|--CREATE 2**(-L) * (2/PI), SIGN(INARG)*2**(63),
365|--2**L * (PIby2_1), 2**L * (PIby2_2)
366
367	movel		#0x00003FFE,%d2	| ...BIASED EXPO OF 2/PI
368	subl		%d0,%d2		| ...BIASED EXPO OF 2**(-L)*(2/PI)
369
370	movel		#0xA2F9836E,FP_SCR1+4(%a6)
371	movel		#0x4E44152A,FP_SCR1+8(%a6)
372	movew		%d2,FP_SCR1(%a6)	| ...FP_SCR1 is 2**(-L)*(2/PI)
373
374	fmovex		%fp0,%fp2
375	fmulx		FP_SCR1(%a6),%fp2
376|--WE MUST NOW FIND INT(FP2). SINCE WE NEED THIS VALUE IN
377|--FLOATING POINT FORMAT, THE TWO FMOVE'S	FMOVE.L FP <--> N
378|--WILL BE TOO INEFFICIENT. THE WAY AROUND IT IS THAT
379|--(SIGN(INARG)*2**63	+	FP2) - SIGN(INARG)*2**63 WILL GIVE
380|--US THE DESIRED VALUE IN FLOATING POINT.
381
382|--HIDE SIX CYCLES OF INSTRUCTION
383        movel		%a1,%d2
384        swap		%d2
385	andil		#0x80000000,%d2
386	oril		#0x5F000000,%d2	| ...D2 IS SIGN(INARG)*2**63 IN SGL
387	movel		%d2,TWOTO63(%a6)
388
389	movel		%d0,%d2
390	addil		#0x00003FFF,%d2	| ...BIASED EXPO OF 2**L * (PI/2)
391
392|--FP2 IS READY
393	fadds		TWOTO63(%a6),%fp2	| ...THE FRACTIONAL PART OF FP1 IS ROUNDED
394
395|--HIDE 4 CYCLES OF INSTRUCTION; creating 2**(L)*Piby2_1  and  2**(L)*Piby2_2
396        movew		%d2,FP_SCR2(%a6)
397	clrw           FP_SCR2+2(%a6)
398	movel		#0xC90FDAA2,FP_SCR2+4(%a6)
399	clrl		FP_SCR2+8(%a6)		| ...FP_SCR2 is  2**(L) * Piby2_1
400
401|--FP2 IS READY
402	fsubs		TWOTO63(%a6),%fp2		| ...FP2 is N
403
404	addil		#0x00003FDD,%d0
405        movew		%d0,FP_SCR3(%a6)
406	clrw           FP_SCR3+2(%a6)
407	movel		#0x85A308D3,FP_SCR3+4(%a6)
408	clrl		FP_SCR3+8(%a6)		| ...FP_SCR3 is 2**(L) * Piby2_2
409
410	movel		ENDFLAG(%a6),%d0
411
412|--We are now ready to perform (R+r) - N*P1 - N*P2, P1 = 2**(L) * Piby2_1 and
413|--P2 = 2**(L) * Piby2_2
414	fmovex		%fp2,%fp4
415	fmulx		FP_SCR2(%a6),%fp4		| ...W = N*P1
416	fmovex		%fp2,%fp5
417	fmulx		FP_SCR3(%a6),%fp5		| ...w = N*P2
418	fmovex		%fp4,%fp3
419|--we want P+p = W+w  but  |p| <= half ulp of P
420|--Then, we need to compute  A := R-P   and  a := r-p
421	faddx		%fp5,%fp3			| ...FP3 is P
422	fsubx		%fp3,%fp4			| ...W-P
423
424	fsubx		%fp3,%fp0			| ...FP0 is A := R - P
425        faddx		%fp5,%fp4			| ...FP4 is p = (W-P)+w
426
427	fmovex		%fp0,%fp3			| ...FP3 A
428	fsubx		%fp4,%fp1			| ...FP1 is a := r - p
429
430|--Now we need to normalize (A,a) to  "new (R,r)" where R+r = A+a but
431|--|r| <= half ulp of R.
432	faddx		%fp1,%fp0			| ...FP0 is R := A+a
433|--No need to calculate r if this is the last loop
434	cmpil		#0,%d0
435	bgt		RESTORE
436
437|--Need to calculate r
438	fsubx		%fp0,%fp3			| ...A-R
439	faddx		%fp3,%fp1			| ...FP1 is r := (A-R)+a
440	bra		LOOP
441
442RESTORE:
443        fmovel		%fp2,N(%a6)
444	movel		(%a7)+,%d2
445	fmovemx	(%a7)+,%fp2-%fp5
446
447
448	movel		N(%a6),%d0
449        rorl		#1,%d0
450
451
452	bra		TANCONT
453
454	|end
455