xref: /original-bsd/old/libm/libm/VAX/argred.s (revision 2301fdfb) 
1#
2# Copyright (c) 1985 Regents of the University of California.
3#
4# Use and reproduction of this software are granted  in  accordance  with
5# the terms and conditions specified in  the  Berkeley  Software  License
6# Agreement (in particular, this entails acknowledgement of the programs'
7# source, and inclusion of this notice) with the additional understanding
8# that  all  recipients  should regard themselves as participants  in  an
9# ongoing  research  project and hence should  feel  obligated  to report
10# their  experiences (good or bad) with these elementary function  codes,
11# using "sendbug 4bsd-bugs@BERKELEY", to the authors.
12#
13
14# @(#)argred.s	1.1 (Berkeley) 08/21/85
15
16#  libm$argred implements Bob Corbett's argument reduction and
17#  libm$sincos implements Peter Tang's double precision sin/cos.
18#
19#  Note: The two entry points libm$argred and libm$sincos are meant
20#        to be used only by _sin, _cos and _tan.
21#
22# method: true range reduction to [-pi/4,pi/4], P. Tang  &  B. Corbett
23# S. McDonald, April 4,  1985
24#
25	.globl	libm$argred
26	.globl	libm$sincos
27	.text
28	.align	1
29
30libm$argred:
31#
32#  Compare the argument with the largest possible that can
33#  be reduced by table lookup.  r3 := |x|  will be used in  table_lookup .
34#
35	movd	r0,r3
36	bgeq	abs1
37	mnegd	r3,r3
38abs1:
39	cmpd	r3,$0d+4.55530934770520019583e+01
40	blss	small_arg
41	jsb	trigred
42	rsb
43small_arg:
44	jsb	table_lookup
45	rsb
46#
47#  At this point,
48#	   r0  contains the quadrant number, 0, 1, 2, or 3;
49#	r2/r1  contains the reduced argument as a D-format number;
50#  	   r3  contains a F-format extension to the reduced argument;
51#          r4  contains a  0 or 1  corresponding to a  sin or cos  entry.
52#
53libm$sincos:
54#
55#  Compensate for a cosine entry by adding one to the quadrant number.
56#
57	addl2	r4,r0
58#
59#  Polyd clobbers  r5-r0 ;  save  X  in  r7/r6 .
60#  This can be avoided by rewriting  trigred .
61#
62	movd	r1,r6
63#
64#  Likewise, save  alpha  in  r8 .
65#  This can be avoided by rewriting  trigred .
66#
67	movf	r3,r8
68#
69#  Odd or even quadrant?  cosine if odd, sine otherwise.
70#  Save  floor(quadrant/2) in  r9  ; it determines the final sign.
71#
72	rotl	$-1,r0,r9
73	blss	cosine
74sine:
75	muld2	r1,r1		# Xsq = X * X
76	polyd	r1,$7,sin_coef	# Q = P(Xsq) , of deg 7
77	mulf3	$0f3.0,r8,r4	# beta = 3 * alpha
78	mulf2	r0,r4		# beta = Q * beta
79	addf2	r8,r4		# beta = alpha + beta
80	muld2	r6,r0		# S(X) = X * Q
81#	cvtfd	r4,r4		... r5 = 0 after a polyd.
82	addd2	r4,r0		# S(X) = beta + S(X)
83	addd2	r6,r0		# S(X) = X + S(X)
84	brb	done
85cosine:
86	muld2	r6,r6		# Xsq = X * X
87	beql	zero_arg
88	mulf2	r1,r8		# beta = X * alpha
89	polyd	r6,$7,cos_coef	# Q = P'(Xsq) , of deg 7
90	subd3	r0,r8,r0	# beta = beta - Q
91	subw2	$0x80,r6	# Xsq = Xsq / 2
92	addd2	r0,r6		# Xsq = Xsq + beta
93zero_arg:
94	subd3	r6,$0d1.0,r0	# C(X) = 1 - Xsq
95done:
96	blbc	r9,even
97	mnegd	r0,r0
98even:
99	rsb
100
101.data
102.align	2
103
104sin_coef:
105	.double	0d-7.53080332264191085773e-13	# s7 = 2^-29 -1.a7f2504ffc49f8..
106	.double	0d+1.60573519267703489121e-10	# s6 = 2^-21  1.611adaede473c8..
107	.double	0d-2.50520965150706067211e-08	# s5 = 2^-1a -1.ae644921ed8382..
108	.double	0d+2.75573191800593885716e-06	# s4 = 2^-13  1.71de3a4b884278..
109	.double	0d-1.98412698411850507950e-04	# s3 = 2^-0d -1.a01a01a0125e7d..
110	.double	0d+8.33333333333325688985e-03	# s2 = 2^-07  1.11111111110e50
111	.double	0d-1.66666666666666664354e-01	# s1 = 2^-03 -1.55555555555554
112	.double	0d+0.00000000000000000000e+00	# s0 = 0
113
114cos_coef:
115	.double	0d-1.13006966202629430300e-11	# s7 = 2^-25 -1.8D9BA04D1374BE..
116	.double	0d+2.08746646574796004700e-09	# s6 = 2^-1D  1.1EE632650350BA..
117	.double	0d-2.75573073031284417300e-07	# s5 = 2^-16 -1.27E4F31411719E..
118	.double	0d+2.48015872682668025200e-05	# s4 = 2^-10  1.A01A0196B902E8..
119	.double	0d-1.38888888888464709200e-03	# s3 = 2^-0A -1.6C16C16C11FACE..
120	.double	0d+4.16666666666664761400e-02	# s2 = 2^-05  1.5555555555539E
121	.double	0d+0.00000000000000000000e+00	# s1 = 0
122	.double	0d+0.00000000000000000000e+00	# s0 = 0
123
124#
125#  Multiples of  pi/2  expressed as the sum of three doubles,
126#
127#  trailing:	n * pi/2 ,  n = 0, 1, 2, ..., 29
128#			trailing[n] ,
129#
130#  middle:	n * pi/2 ,  n = 0, 1, 2, ..., 29
131#			middle[n]   ,
132#
133#  leading:	n * pi/2 ,  n = 0, 1, 2, ..., 29
134#			leading[n]  ,
135#
136#	where
137#		leading[n]  := (n * pi/2)  rounded,
138#		middle[n]   := (n * pi/2  -  leading[n])  rounded,
139#		trailing[n] := (( n * pi/2 - leading[n]) - middle[n])  rounded .
140
141trailing:
142	.double	0d+0.00000000000000000000e+00	#  0 * pi/2  trailing
143	.double	0d+4.33590506506189049611e-35	#  1 * pi/2  trailing
144	.double	0d+8.67181013012378099223e-35	#  2 * pi/2  trailing
145	.double	0d+1.30077151951856714215e-34	#  3 * pi/2  trailing
146	.double	0d+1.73436202602475619845e-34	#  4 * pi/2  trailing
147	.double	0d-1.68390735624352669192e-34	#  5 * pi/2  trailing
148	.double	0d+2.60154303903713428430e-34	#  6 * pi/2  trailing
149	.double	0d-8.16726343231148352150e-35	#  7 * pi/2  trailing
150	.double	0d+3.46872405204951239689e-34	#  8 * pi/2  trailing
151	.double	0d+3.90231455855570147991e-34	#  9 * pi/2  trailing
152	.double	0d-3.36781471248705338384e-34	# 10 * pi/2  trailing
153	.double	0d-1.06379439835298071785e-33	# 11 * pi/2  trailing
154	.double	0d+5.20308607807426856861e-34	# 12 * pi/2  trailing
155	.double	0d+5.63667658458045770509e-34	# 13 * pi/2  trailing
156	.double	0d-1.63345268646229670430e-34	# 14 * pi/2  trailing
157	.double	0d-1.19986217995610764801e-34	# 15 * pi/2  trailing
158	.double	0d+6.93744810409902479378e-34	# 16 * pi/2  trailing
159	.double	0d-8.03640094449267300110e-34	# 17 * pi/2  trailing
160	.double	0d+7.80462911711140295982e-34	# 18 * pi/2  trailing
161	.double	0d-7.16921993148029483506e-34	# 19 * pi/2  trailing
162	.double	0d-6.73562942497410676769e-34	# 20 * pi/2  trailing
163	.double	0d-6.30203891846791677593e-34	# 21 * pi/2  trailing
164	.double	0d-2.12758879670596143570e-33	# 22 * pi/2  trailing
165	.double	0d+2.53800212047402350390e-33	# 23 * pi/2  trailing
166	.double	0d+1.04061721561485371372e-33	# 24 * pi/2  trailing
167	.double	0d+6.11729905311472319056e-32	# 25 * pi/2  trailing
168	.double	0d+1.12733531691609154102e-33	# 26 * pi/2  trailing
169	.double	0d-3.70049587943078297272e-34	# 27 * pi/2  trailing
170	.double	0d-3.26690537292459340860e-34	# 28 * pi/2  trailing
171	.double	0d-1.14812616507957271361e-34	# 29 * pi/2  trailing
172
173middle:
174	.double	0d+0.00000000000000000000e+00	#  0 * pi/2  middle
175	.double	0d+5.72118872610983179676e-18	#  1 * pi/2  middle
176	.double	0d+1.14423774522196635935e-17	#  2 * pi/2  middle
177	.double	0d-3.83475850529283316309e-17	#  3 * pi/2  middle
178	.double	0d+2.28847549044393271871e-17	#  4 * pi/2  middle
179	.double	0d-2.69052076007086676522e-17	#  5 * pi/2  middle
180	.double	0d-7.66951701058566632618e-17	#  6 * pi/2  middle
181	.double	0d-1.54628301484890040587e-17	#  7 * pi/2  middle
182	.double	0d+4.57695098088786543741e-17	#  8 * pi/2  middle
183	.double	0d+1.07001849766246313192e-16	#  9 * pi/2  middle
184	.double	0d-5.38104152014173353044e-17	# 10 * pi/2  middle
185	.double	0d-2.14622680169080983801e-16	# 11 * pi/2  middle
186	.double	0d-1.53390340211713326524e-16	# 12 * pi/2  middle
187	.double	0d-9.21580002543456677056e-17	# 13 * pi/2  middle
188	.double	0d-3.09256602969780081173e-17	# 14 * pi/2  middle
189	.double	0d+3.03066796603896507006e-17	# 15 * pi/2  middle
190	.double	0d+9.15390196177573087482e-17	# 16 * pi/2  middle
191	.double	0d+1.52771359575124969107e-16	# 17 * pi/2  middle
192	.double	0d+2.14003699532492626384e-16	# 18 * pi/2  middle
193	.double	0d-1.68853170360202329427e-16	# 19 * pi/2  middle
194	.double	0d-1.07620830402834670609e-16	# 20 * pi/2  middle
195	.double	0d+3.97700719404595604379e-16	# 21 * pi/2  middle
196	.double	0d-4.29245360338161967602e-16	# 22 * pi/2  middle
197	.double	0d-3.68013020380794313406e-16	# 23 * pi/2  middle
198	.double	0d-3.06780680423426653047e-16	# 24 * pi/2  middle
199	.double	0d-2.45548340466059054318e-16	# 25 * pi/2  middle
200	.double	0d-1.84316000508691335411e-16	# 26 * pi/2  middle
201	.double	0d-1.23083660551323675053e-16	# 27 * pi/2  middle
202	.double	0d-6.18513205939560162346e-17	# 28 * pi/2  middle
203	.double	0d-6.18980636588357585202e-19	# 29 * pi/2  middle
204
205leading:
206	.double	0d+0.00000000000000000000e+00	#  0 * pi/2  leading
207	.double	0d+1.57079632679489661351e+00	#  1 * pi/2  leading
208	.double	0d+3.14159265358979322702e+00	#  2 * pi/2  leading
209	.double	0d+4.71238898038468989604e+00	#  3 * pi/2  leading
210	.double	0d+6.28318530717958645404e+00	#  4 * pi/2  leading
211	.double	0d+7.85398163397448312306e+00	#  5 * pi/2  leading
212	.double	0d+9.42477796076937979208e+00	#  6 * pi/2  leading
213	.double	0d+1.09955742875642763501e+01	#  7 * pi/2  leading
214	.double	0d+1.25663706143591729081e+01	#  8 * pi/2  leading
215	.double	0d+1.41371669411540694661e+01	#  9 * pi/2  leading
216	.double	0d+1.57079632679489662461e+01	# 10 * pi/2  leading
217	.double	0d+1.72787595947438630262e+01	# 11 * pi/2  leading
218	.double	0d+1.88495559215387595842e+01	# 12 * pi/2  leading
219	.double	0d+2.04203522483336561422e+01	# 13 * pi/2  leading
220	.double	0d+2.19911485751285527002e+01	# 14 * pi/2  leading
221	.double	0d+2.35619449019234492582e+01	# 15 * pi/2  leading
222	.double	0d+2.51327412287183458162e+01	# 16 * pi/2  leading
223	.double	0d+2.67035375555132423742e+01	# 17 * pi/2  leading
224	.double	0d+2.82743338823081389322e+01	# 18 * pi/2  leading
225	.double	0d+2.98451302091030359342e+01	# 19 * pi/2  leading
226	.double	0d+3.14159265358979324922e+01	# 20 * pi/2  leading
227	.double	0d+3.29867228626928286062e+01	# 21 * pi/2  leading
228	.double	0d+3.45575191894877260523e+01	# 22 * pi/2  leading
229	.double	0d+3.61283155162826226103e+01	# 23 * pi/2  leading
230	.double	0d+3.76991118430775191683e+01	# 24 * pi/2  leading
231	.double	0d+3.92699081698724157263e+01	# 25 * pi/2  leading
232	.double	0d+4.08407044966673122843e+01	# 26 * pi/2  leading
233	.double	0d+4.24115008234622088423e+01	# 27 * pi/2  leading
234	.double	0d+4.39822971502571054003e+01	# 28 * pi/2  leading
235	.double	0d+4.55530934770520019583e+01	# 29 * pi/2  leading
236
237twoOverPi:
238	.double	0d+6.36619772367581343076e-01
239	.text
240	.align	1
241
242table_lookup:
243	muld3	r3,twoOverPi,r0
244	cvtrdl	r0,r0			# n = nearest int to ((2/pi)*|x|) rnded
245	mull3	$8,r0,r5
246	subd2	leading(r5),r3		# p = (|x| - leading n*pi/2) exactly
247	subd3	middle(r5),r3,r1	# q = (p - middle  n*pi/2) rounded
248	subd2	r1,r3			# r = (p - q)
249	subd2	middle(r5),r3		# r =  r - middle  n*pi/2
250	subd2	trailing(r5),r3		# r =  r - trailing n*pi/2  rounded
251#
252#  If the original argument was negative,
253#  negate the reduce argument and
254#  adjust the octant/quadrant number.
255#
256	tstw	4(ap)
257	bgeq	abs2
258	mnegf	r1,r1
259	mnegf	r3,r3
260#	subb3	r0,$8,r0	...used for  pi/4  reduction -S.McD
261	subb3	r0,$4,r0
262abs2:
263#
264#  Clear all unneeded octant/quadrant bits.
265#
266#	bicb2	$0xf8,r0	...used for  pi/4  reduction -S.McD
267	bicb2	$0xfc,r0
268	rsb
269#
270#						p.0
271	.text
272	.align	2
273#
274# Only 256 (actually 225) bits of 2/pi are needed for VAX double
275# precision; this was determined by enumerating all the nearest
276# machine integer multiples of pi/2 using continued fractions.
277# (8a8d3673775b7ff7 required the most bits.)		-S.McD
278#
279	.long	0
280	.long	0
281	.long	0xaef1586d
282	.long	0x9458eaf7
283	.long	0x10e4107f
284	.long	0xd8a5664f
285	.long	0x4d377036
286	.long	0x09d5f47d
287	.long	0x91054a7f
288	.long	0xbe60db93
289bits2opi:
290	.long	0x00000028
291	.long	0
292#
293#  Note: wherever you see the word `octant', read `quadrant'.
294#  Currently this code is set up for  pi/2  argument reduction.
295#  By uncommenting/commenting the appropriate lines, it will
296#  also serve as a  pi/4  argument reduction code.
297#
298
299#						p.1
300#  Trigred  preforms argument reduction
301#  for the trigonometric functions.  It
302#  takes one input argument, a D-format
303#  number in  r1/r0 .  The magnitude of
304#  the input argument must be greater
305#  than or equal to  1/2 .  Trigred produces
306#  three results:  the number of the octant
307#  occupied by the argument, the reduced
308#  argument, and an extension of the
309#  reduced argument.  The octant number is
310#  returned in  r0 .  The reduced argument
311#  is returned as a D-format number in
312#  r2/r1 .  An 8 bit extension of the
313#  reduced argument is returned as an
314#  F-format number in r3.
315#						p.2
316trigred:
317#
318#  Save the sign of the input argument.
319#
320	movw	r0,-(sp)
321#
322#  Extract the exponent field.
323#
324	extzv	$7,$7,r0,r2
325#
326#  Convert the fraction part of the input
327#  argument into a quadword integer.
328#
329	bicw2	$0xff80,r0
330	bisb2	$0x80,r0	# -S.McD
331	rotl	$16,r0,r0
332	rotl	$16,r1,r1
333#
334#  If  r1  is negative, add  1  to  r0 .  This
335#  adjustment is made so that the two's
336#  complement multiplications done later
337#  will produce unsigned results.
338#
339	bgeq	posmid
340	incl	r0
341posmid:
342#						p.3
343#
344#  Set  r3  to the address of the first quadword
345#  used to obtain the needed portion of  2/pi .
346#  The address is longword aligned to ensure
347#  efficient access.
348#
349	ashl	$-3,r2,r3
350	bicb2	$3,r3
351	subl3	r3,$bits2opi,r3
352#
353#  Set  r2  to the size of the shift needed to
354#  obtain the correct portion of  2/pi .
355#
356	bicb2	$0xe0,r2
357#						p.4
358#
359#  Move the needed  128  bits of  2/pi  into
360#  r11 - r8 .  Adjust the numbers to allow
361#  for unsigned multiplication.
362#
363	ashq	r2,(r3),r10
364
365	subl2	$4,r3
366	ashq	r2,(r3),r9
367	bgeq	signoff1
368	incl	r11
369signoff1:
370	subl2	$4,r3
371	ashq	r2,(r3),r8
372	bgeq	signoff2
373	incl	r10
374signoff2:
375	subl2	$4,r3
376	ashq	r2,(r3),r7
377	bgeq	signoff3
378	incl	r9
379signoff3:
380#						p.5
381#
382#  Multiply the contents of  r0/r1  by the
383#  slice of  2/pi  in  r11 - r8 .
384#
385	emul	r0,r8,$0,r4
386	emul	r0,r9,r5,r5
387	emul	r0,r10,r6,r6
388
389	emul	r1,r8,$0,r7
390	emul	r1,r9,r8,r8
391	emul	r1,r10,r9,r9
392	emul	r1,r11,r10,r10
393
394	addl2	r4,r8
395	adwc	r5,r9
396	adwc	r6,r10
397#						p.6
398#
399#  If there are more than five leading zeros
400#  after the first two quotient bits or if there
401#  are more than five leading ones after the first
402#  two quotient bits, generate more fraction bits.
403#  Otherwise, branch to code to produce the result.
404#
405	bicl3	$0xc1ffffff,r10,r4
406	beql	more1
407	cmpl	$0x3e000000,r4
408	bneq	result
409more1:
410#						p.7
411#
412#  generate another  32  result bits.
413#
414	subl2	$4,r3
415	ashq	r2,(r3),r5
416	bgeq	signoff4
417
418	emul	r1,r6,$0,r4
419	addl2	r1,r5
420	emul	r0,r6,r5,r5
421	addl2	r0,r6
422	brb	addbits1
423
424signoff4:
425	emul	r1,r6,$0,r4
426	emul	r0,r6,r5,r5
427
428addbits1:
429	addl2	r5,r7
430	adwc	r6,r8
431	adwc	$0,r9
432	adwc	$0,r10
433#						p.8
434#
435#  Check for massive cancellation.
436#
437	bicl3	$0xc0000000,r10,r6
438#	bneq	more2			-S.McD  Test was backwards
439	beql	more2
440	cmpl	$0x3fffffff,r6
441	bneq	result
442more2:
443#						p.9
444#
445#  If massive cancellation has occurred,
446#  generate another  24  result bits.
447#  Testing has shown there will always be
448#  enough bits after this point.
449#
450	subl2	$4,r3
451	ashq	r2,(r3),r5
452	bgeq	signoff5
453
454	emul	r0,r6,r4,r5
455	addl2	r0,r6
456	brb	addbits2
457
458signoff5:
459	emul	r0,r6,r4,r5
460
461addbits2:
462	addl2	r6,r7
463	adwc	$0,r8
464	adwc	$0,r9
465	adwc	$0,r10
466#						p.10
467#
468#  The following code produces the reduced
469#  argument from the product bits contained
470#  in  r10 - r7 .
471#
472result:
473#
474#  Extract the octant number from  r10 .
475#
476#	extzv	$29,$3,r10,r0	...used for  pi/4  reduction -S.McD
477	extzv	$30,$2,r10,r0
478#
479#  Clear the octant bits in  r10 .
480#
481#	bicl2	$0xe0000000,r10	...used for  pi/4  reduction -S.McD
482	bicl2	$0xc0000000,r10
483#
484#  Zero the sign flag.
485#
486	clrl	r5
487#						p.11
488#
489#  Check to see if the fraction is greater than
490#  or equal to one-half.  If it is, add one
491#  to the octant number, set the sign flag
492#  on, and replace the fraction with  1 minus
493#  the fraction.
494#
495#	bitl	$0x10000000,r10		...used for  pi/4  reduction -S.McD
496	bitl	$0x20000000,r10
497	beql	small
498	incl	r0
499	incl	r5
500#	subl3	r10,$0x1fffffff,r10	...used for  pi/4  reduction -S.McD
501	subl3	r10,$0x3fffffff,r10
502	mcoml	r9,r9
503	mcoml	r8,r8
504	mcoml	r7,r7
505small:
506#						p.12
507#
508##  Test whether the first  29  bits of the ...used for  pi/4  reduction -S.McD
509#  Test whether the first  30  bits of the
510#  fraction are zero.
511#
512	tstl	r10
513	beql	tiny
514#
515#  Find the position of the first one bit in  r10 .
516#
517	cvtld	r10,r1
518	extzv	$7,$7,r1,r1
519#
520#  Compute the size of the shift needed.
521#
522	subl3	r1,$32,r6
523#
524#  Shift up the high order  64  bits of the
525#  product.
526#
527	ashq	r6,r9,r10
528	ashq	r6,r8,r9
529	brb	mult
530#						p.13
531#
532#  Test to see if the sign bit of  r9  is on.
533#
534tiny:
535	tstl	r9
536	bgeq	tinier
537#
538#  If it is, shift the product bits up  32  bits.
539#
540	movl	$32,r6
541	movq	r8,r10
542	tstl	r10
543	brb	mult
544#						p.14
545#
546#  Test whether  r9  is zero.  It is probably
547#  impossible for both  r10  and  r9  to be
548#  zero, but until proven to be so, the test
549#  must be made.
550#
551tinier:
552	beql	zero
553#
554#  Find the position of the first one bit in  r9 .
555#
556	cvtld	r9,r1
557	extzv	$7,$7,r1,r1
558#
559#  Compute the size of the shift needed.
560#
561	subl3	r1,$32,r1
562	addl3	$32,r1,r6
563#
564#  Shift up the high order  64  bits of the
565#  product.
566#
567	ashq	r1,r8,r10
568	ashq	r1,r7,r9
569	brb	mult
570#						p.15
571#
572#  The following code sets the reduced
573#  argument to zero.
574#
575zero:
576	clrl	r1
577	clrl	r2
578	clrl	r3
579	brw	return
580#						p.16
581#
582#  At this point,  r0  contains the octant number,
583#  r6  indicates the number of bits the fraction
584#  has been shifted,  r5  indicates the sign of
585#  the fraction,  r11/r10  contain the high order
586#  64  bits of the fraction, and the condition
587#  codes indicate where the sign bit of  r10
588#  is on.  The following code multiplies the
589#  fraction by  pi/2 .
590#
591mult:
592#
593#  Save  r11/r10  in  r4/r1 .		-S.McD
594	movl	r11,r4
595	movl	r10,r1
596#
597#  If the sign bit of  r10  is on, add  1  to  r11 .
598#
599	bgeq	signoff6
600	incl	r11
601signoff6:
602#						p.17
603#
604#  Move  pi/2  into  r3/r2 .
605#
606	movq	$0xc90fdaa22168c235,r2
607#
608#  Multiply the fraction by the portion of  pi/2
609#  in  r2 .
610#
611	emul	r2,r10,$0,r7
612	emul	r2,r11,r8,r7
613#
614#  Multiply the fraction by the portion of  pi/2
615#  in  r3 .
616	emul	r3,r10,$0,r9
617	emul	r3,r11,r10,r10
618#
619#  Add the product bits together.
620#
621	addl2	r7,r9
622	adwc	r8,r10
623	adwc	$0,r11
624#
625#  Compensate for not sign extending  r8  above.-S.McD
626#
627	tstl	r8
628	bgeq	signoff6a
629	decl	r11
630signoff6a:
631#
632#  Compensate for  r11/r10  being unsigned.	-S.McD
633#
634	addl2	r2,r10
635	adwc	r3,r11
636#
637#  Compensate for  r3/r2  being unsigned.	-S.McD
638#
639	addl2	r1,r10
640	adwc	r4,r11
641#						p.18
642#
643#  If the sign bit of  r11  is zero, shift the
644#  product bits up one bit and increment  r6 .
645#
646	blss	signon
647	incl	r6
648	ashq	$1,r10,r10
649	tstl	r9
650	bgeq	signoff7
651	incl	r10
652signoff7:
653signon:
654#						p.19
655#
656#  Shift the  56  most significant product
657#  bits into  r9/r8 .  The sign extension
658#  will be handled later.
659#
660	ashq	$-8,r10,r8
661#
662#  Convert the low order  8  bits of  r10
663#  into an F-format number.
664#
665	cvtbf	r10,r3
666#
667#  If the result of the conversion was
668#  negative, add  1  to  r9/r8 .
669#
670	bgeq	chop
671	incl	r8
672	adwc	$0,r9
673#
674#  If  r9  is now zero, branch to special
675#  code to handle that possibility.
676#
677	beql	carryout
678chop:
679#						p.20
680#
681#  Convert the number in  r9/r8  into
682#  D-format number in  r2/r1 .
683#
684	rotl	$16,r8,r2
685	rotl	$16,r9,r1
686#
687#  Set the exponent field to the appropriate
688#  value.  Note that the extra bits created by
689#  sign extension are now eliminated.
690#
691	subw3	r6,$131,r6
692	insv	r6,$7,$9,r1
693#
694#  Set the exponent field of the F-format
695#  number in  r3  to the appropriate value.
696#
697	tstf	r3
698	beql	return
699#	extzv	$7,$8,r3,r4	-S.McD
700	extzv	$7,$7,r3,r4
701	addw2	r4,r6
702#	subw2	$217,r6		-S.McD
703	subw2	$64,r6
704	insv	r6,$7,$8,r3
705	brb	return
706#						p.21
707#
708#  The following code generates the appropriate
709#  result for the unlikely possibility that
710#  rounding the number in  r9/r8  resulted in
711#  a carry out.
712#
713carryout:
714	clrl	r1
715	clrl	r2
716	subw3	r6,$132,r6
717	insv	r6,$7,$9,r1
718	tstf	r3
719	beql	return
720	extzv	$7,$8,r3,r4
721	addw2	r4,r6
722	subw2	$218,r6
723	insv	r6,$7,$8,r3
724#						p.22
725#
726#  The following code makes an needed
727#  adjustments to the signs of the
728#  results or to the octant number, and
729#  then returns.
730#
731return:
732#
733#  Test if the fraction was greater than or
734#  equal to  1/2 .  If so, negate the reduced
735#  argument.
736#
737	blbc	r5,signoff8
738	mnegf	r1,r1
739	mnegf	r3,r3
740signoff8:
741#						p.23
742#
743#  If the original argument was negative,
744#  negate the reduce argument and
745#  adjust the octant number.
746#
747	tstw	(sp)+
748	bgeq	signoff9
749	mnegf	r1,r1
750	mnegf	r3,r3
751#	subb3	r0,$8,r0	...used for  pi/4  reduction -S.McD
752	subb3	r0,$4,r0
753signoff9:
754#
755#  Clear all unneeded octant bits.
756#
757#	bicb2	$0xf8,r0	...used for  pi/4  reduction -S.McD
758	bicb2	$0xfc,r0
759#
760#  Return.
761#
762	rsb
763