xref: /src/lib/libm/arch/vax/n_argred.S

/*	$NetBSD: n_argred.S,v 1.4.8.1 2000/08/14 21:24:15 ragge Exp $	*/
/*
 * Copyright (c) 1985, 1993
 *	The Regents of the University of California.  All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 * 3. All advertising materials mentioning features or use of this software
 *    must display the following acknowledgement:
 *	This product includes software developed by the University of
 *	California, Berkeley and its contributors.
 * 4. Neither the name of the University nor the names of its contributors
 *    may be used to endorse or promote products derived from this software
 *    without specific prior written permission.
 *
 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
 * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
 * SUCH DAMAGE.
 *
 *	@(#)argred.s	8.1 (Berkeley) 6/4/93
 */

/*
 *  libm_argred implements Bob Corbett's argument reduction and
 *  libm_sincos implements Peter Tang's double precision sin/cos.
 *
 *  Note: The two entry points libm_argred and libm_sincos are meant
 *        to be used only by _sin, _cos and _tan.
 *
 * method: true range reduction to [-pi/4,pi/4], P. Tang  &  B. Corbett
 * S. McDonald, April 4,  1985
 */
#include <machine/asm.h>
	.text

ENTRY(_libm_argred, 0)
/*
 *  Compare the argument with the largest possible that can
 *  be reduced by table lookup.  r3 := |x|  will be used in  table_lookup .
 */
	movd	r0,r3
	bgeq	abs1
	mnegd	r3,r3
abs1:
	cmpd	r3,$0d+4.55530934770520019583e+01
	blss	small_arg
	jsb	trigred
	ret
small_arg:
	jsb	table_lookup
	ret
/*
 *  At this point,
 *	   r0  contains the quadrant number, 0, 1, 2, or 3;
 *	r2/r1  contains the reduced argument as a D-format number;
 *  	   r3  contains a F-format extension to the reduced argument;
 *          r4  contains a  0 or 1  corresponding to a  sin or cos  entry.
 */
ENTRY(_libm_sincos, 0)
/*
 *  Compensate for a cosine entry by adding one to the quadrant number.
 */
	addl2	r4,r0
/*
 *  Polyd clobbers  r5-r0 ;  save  X  in  r7/r6 .
 *  This can be avoided by rewriting  trigred .
 */
	movd	r1,r6
/*
 *  Likewise, save  alpha  in  r8 .
 *  This can be avoided by rewriting  trigred .
 */
	movf	r3,r8
/*
 *  Odd or even quadrant?  cosine if odd, sine otherwise.
 *  Save  floor(quadrant/2) in  r9  ; it determines the final sign.
 */
	rotl	$-1,r0,r9
	blss	cosine
sine:
	muld2	r1,r1		# Xsq = X * X
	cmpw	$0x2480,r1	# [zl] Xsq > 2^-56?
	blss	1f		# [zl] yes, go ahead and do polyd
	clrq	r1		# [zl] work around 11/780 FPA polyd bug
1:
	polyd	r1,$7,sin_coef	# Q = P(Xsq) , of deg 7
	mulf3	$0f3.0,r8,r4	# beta = 3 * alpha
	mulf2	r0,r4		# beta = Q * beta
	addf2	r8,r4		# beta = alpha + beta
	muld2	r6,r0		# S(X) = X * Q
/*	cvtfd	r4,r4		... r5 = 0 after a polyd. */
	addd2	r4,r0		# S(X) = beta + S(X)
	addd2	r6,r0		# S(X) = X + S(X)
	brb	done
cosine:
	muld2	r6,r6		# Xsq = X * X
	beql	zero_arg
	mulf2	r1,r8		# beta = X * alpha
	polyd	r6,$7,cos_coef	/* Q = P'(Xsq) , of deg 7 */
	subd3	r0,r8,r0	# beta = beta - Q
	subw2	$0x80,r6	# Xsq = Xsq / 2
	addd2	r0,r6		# Xsq = Xsq + beta
zero_arg:
	subd3	r6,$0d1.0,r0	# C(X) = 1 - Xsq
done:
	blbc	r9,even
	mnegd	r0,r0
even:
	ret

.data
.align	2

sin_coef:
	.double	0d-7.53080332264191085773e-13	# s7 = 2^-29 -1.a7f2504ffc49f8..
	.double	0d+1.60573519267703489121e-10	# s6 = 2^-21  1.611adaede473c8..
	.double	0d-2.50520965150706067211e-08	# s5 = 2^-1a -1.ae644921ed8382..
	.double	0d+2.75573191800593885716e-06	# s4 = 2^-13  1.71de3a4b884278..
	.double	0d-1.98412698411850507950e-04	# s3 = 2^-0d -1.a01a01a0125e7d..
	.double	0d+8.33333333333325688985e-03	# s2 = 2^-07  1.11111111110e50
	.double	0d-1.66666666666666664354e-01	# s1 = 2^-03 -1.55555555555554
	.double	0d+0.00000000000000000000e+00	# s0 = 0

cos_coef:
	.double	0d-1.13006966202629430300e-11	# s7 = 2^-25 -1.8D9BA04D1374BE..
	.double	0d+2.08746646574796004700e-09	# s6 = 2^-1D  1.1EE632650350BA..
	.double	0d-2.75573073031284417300e-07	# s5 = 2^-16 -1.27E4F31411719E..
	.double	0d+2.48015872682668025200e-05	# s4 = 2^-10  1.A01A0196B902E8..
	.double	0d-1.38888888888464709200e-03	# s3 = 2^-0A -1.6C16C16C11FACE..
	.double	0d+4.16666666666664761400e-02	# s2 = 2^-05  1.5555555555539E
	.double	0d+0.00000000000000000000e+00	# s1 = 0
	.double	0d+0.00000000000000000000e+00	# s0 = 0

/*
 *  Multiples of  pi/2  expressed as the sum of three doubles,
 *
 *  trailing:	n * pi/2 ,  n = 0, 1, 2, ..., 29
 *			trailing[n] ,
 *
 *  middle:	n * pi/2 ,  n = 0, 1, 2, ..., 29
 *			middle[n]   ,
 *
 *  leading:	n * pi/2 ,  n = 0, 1, 2, ..., 29
 *			leading[n]  ,
 *
 *	where
 *		leading[n]  := (n * pi/2)  rounded,
 *		middle[n]   := (n * pi/2  -  leading[n])  rounded,
 *		trailing[n] := (( n * pi/2 - leading[n]) - middle[n])  rounded .
 */
trailing:
	.double	0d+0.00000000000000000000e+00	#  0 * pi/2  trailing
	.double	0d+4.33590506506189049611e-35	#  1 * pi/2  trailing
	.double	0d+8.67181013012378099223e-35	#  2 * pi/2  trailing
	.double	0d+1.30077151951856714215e-34	#  3 * pi/2  trailing
	.double	0d+1.73436202602475619845e-34	#  4 * pi/2  trailing
	.double	0d-1.68390735624352669192e-34	#  5 * pi/2  trailing
	.double	0d+2.60154303903713428430e-34	#  6 * pi/2  trailing
	.double	0d-8.16726343231148352150e-35	#  7 * pi/2  trailing
	.double	0d+3.46872405204951239689e-34	#  8 * pi/2  trailing
	.double	0d+3.90231455855570147991e-34	#  9 * pi/2  trailing
	.double	0d-3.36781471248705338384e-34	# 10 * pi/2  trailing
	.double	0d-1.06379439835298071785e-33	# 11 * pi/2  trailing
	.double	0d+5.20308607807426856861e-34	# 12 * pi/2  trailing
	.double	0d+5.63667658458045770509e-34	# 13 * pi/2  trailing
	.double	0d-1.63345268646229670430e-34	# 14 * pi/2  trailing
	.double	0d-1.19986217995610764801e-34	# 15 * pi/2  trailing
	.double	0d+6.93744810409902479378e-34	# 16 * pi/2  trailing
	.double	0d-8.03640094449267300110e-34	# 17 * pi/2  trailing
	.double	0d+7.80462911711140295982e-34	# 18 * pi/2  trailing
	.double	0d-7.16921993148029483506e-34	# 19 * pi/2  trailing
	.double	0d-6.73562942497410676769e-34	# 20 * pi/2  trailing
	.double	0d-6.30203891846791677593e-34	# 21 * pi/2  trailing
	.double	0d-2.12758879670596143570e-33	# 22 * pi/2  trailing
	.double	0d+2.53800212047402350390e-33	# 23 * pi/2  trailing
	.double	0d+1.04061721561485371372e-33	# 24 * pi/2  trailing
	.double	0d+6.11729905311472319056e-32	# 25 * pi/2  trailing
	.double	0d+1.12733531691609154102e-33	# 26 * pi/2  trailing
	.double	0d-3.70049587943078297272e-34	# 27 * pi/2  trailing
	.double	0d-3.26690537292459340860e-34	# 28 * pi/2  trailing
	.double	0d-1.14812616507957271361e-34	# 29 * pi/2  trailing

middle:
	.double	0d+0.00000000000000000000e+00	#  0 * pi/2  middle
	.double	0d+5.72118872610983179676e-18	#  1 * pi/2  middle
	.double	0d+1.14423774522196635935e-17	#  2 * pi/2  middle
	.double	0d-3.83475850529283316309e-17	#  3 * pi/2  middle
	.double	0d+2.28847549044393271871e-17	#  4 * pi/2  middle
	.double	0d-2.69052076007086676522e-17	#  5 * pi/2  middle
	.double	0d-7.66951701058566632618e-17	#  6 * pi/2  middle
	.double	0d-1.54628301484890040587e-17	#  7 * pi/2  middle
	.double	0d+4.57695098088786543741e-17	#  8 * pi/2  middle
	.double	0d+1.07001849766246313192e-16	#  9 * pi/2  middle
	.double	0d-5.38104152014173353044e-17	# 10 * pi/2  middle
	.double	0d-2.14622680169080983801e-16	# 11 * pi/2  middle
	.double	0d-1.53390340211713326524e-16	# 12 * pi/2  middle
	.double	0d-9.21580002543456677056e-17	# 13 * pi/2  middle
	.double	0d-3.09256602969780081173e-17	# 14 * pi/2  middle
	.double	0d+3.03066796603896507006e-17	# 15 * pi/2  middle
	.double	0d+9.15390196177573087482e-17	# 16 * pi/2  middle
	.double	0d+1.52771359575124969107e-16	# 17 * pi/2  middle
	.double	0d+2.14003699532492626384e-16	# 18 * pi/2  middle
	.double	0d-1.68853170360202329427e-16	# 19 * pi/2  middle
	.double	0d-1.07620830402834670609e-16	# 20 * pi/2  middle
	.double	0d+3.97700719404595604379e-16	# 21 * pi/2  middle
	.double	0d-4.29245360338161967602e-16	# 22 * pi/2  middle
	.double	0d-3.68013020380794313406e-16	# 23 * pi/2  middle
	.double	0d-3.06780680423426653047e-16	# 24 * pi/2  middle
	.double	0d-2.45548340466059054318e-16	# 25 * pi/2  middle
	.double	0d-1.84316000508691335411e-16	# 26 * pi/2  middle
	.double	0d-1.23083660551323675053e-16	# 27 * pi/2  middle
	.double	0d-6.18513205939560162346e-17	# 28 * pi/2  middle
	.double	0d-6.18980636588357585202e-19	# 29 * pi/2  middle

leading:
	.double	0d+0.00000000000000000000e+00	#  0 * pi/2  leading
	.double	0d+1.57079632679489661351e+00	#  1 * pi/2  leading
	.double	0d+3.14159265358979322702e+00	#  2 * pi/2  leading
	.double	0d+4.71238898038468989604e+00	#  3 * pi/2  leading
	.double	0d+6.28318530717958645404e+00	#  4 * pi/2  leading
	.double	0d+7.85398163397448312306e+00	#  5 * pi/2  leading
	.double	0d+9.42477796076937979208e+00	#  6 * pi/2  leading
	.double	0d+1.09955742875642763501e+01	#  7 * pi/2  leading
	.double	0d+1.25663706143591729081e+01	#  8 * pi/2  leading
	.double	0d+1.41371669411540694661e+01	#  9 * pi/2  leading
	.double	0d+1.57079632679489662461e+01	# 10 * pi/2  leading
	.double	0d+1.72787595947438630262e+01	# 11 * pi/2  leading
	.double	0d+1.88495559215387595842e+01	# 12 * pi/2  leading
	.double	0d+2.04203522483336561422e+01	# 13 * pi/2  leading
	.double	0d+2.19911485751285527002e+01	# 14 * pi/2  leading
	.double	0d+2.35619449019234492582e+01	# 15 * pi/2  leading
	.double	0d+2.51327412287183458162e+01	# 16 * pi/2  leading
	.double	0d+2.67035375555132423742e+01	# 17 * pi/2  leading
	.double	0d+2.82743338823081389322e+01	# 18 * pi/2  leading
	.double	0d+2.98451302091030359342e+01	# 19 * pi/2  leading
	.double	0d+3.14159265358979324922e+01	# 20 * pi/2  leading
	.double	0d+3.29867228626928286062e+01	# 21 * pi/2  leading
	.double	0d+3.45575191894877260523e+01	# 22 * pi/2  leading
	.double	0d+3.61283155162826226103e+01	# 23 * pi/2  leading
	.double	0d+3.76991118430775191683e+01	# 24 * pi/2  leading
	.double	0d+3.92699081698724157263e+01	# 25 * pi/2  leading
	.double	0d+4.08407044966673122843e+01	# 26 * pi/2  leading
	.double	0d+4.24115008234622088423e+01	# 27 * pi/2  leading
	.double	0d+4.39822971502571054003e+01	# 28 * pi/2  leading
	.double	0d+4.55530934770520019583e+01	# 29 * pi/2  leading

twoOverPi:
	.double	0d+6.36619772367581343076e-01
	.text
	.align	1

table_lookup:
	muld3	r3,twoOverPi,r0
	cvtrdl	r0,r0			# n = nearest int to ((2/pi)*|x|) rnded
	mull3	$8,r0,r5
	subd2	leading(r5),r3		# p = (|x| - leading n*pi/2) exactly
	subd3	middle(r5),r3,r1	# q = (p - middle  n*pi/2) rounded
	subd2	r1,r3			# r = (p - q)
	subd2	middle(r5),r3		# r =  r - middle  n*pi/2
	subd2	trailing(r5),r3		# r =  r - trailing n*pi/2  rounded
/*
 *  If the original argument was negative,
 *  negate the reduce argument and
 *  adjust the octant/quadrant number.
 */
	tstw	4(ap)
	bgeq	abs2
	mnegf	r1,r1
	mnegf	r3,r3
/*	subb3	r0,$8,r0	...used for  pi/4  reduction -S.McD */
	subb3	r0,$4,r0
abs2:
/*
 *  Clear all unneeded octant/quadrant bits.
 */
/*	bicb2	$0xf8,r0	...used for  pi/4  reduction -S.McD */
	bicb2	$0xfc,r0
	rsb
/*
 *						p.0
 */
	.text
	.align	2
/*
 * Only 256 (actually 225) bits of 2/pi are needed for VAX double
 * precision; this was determined by enumerating all the nearest
 * machine integer multiples of pi/2 using continued fractions.
 * (8a8d3673775b7ff7 required the most bits.)		-S.McD
 */
	.long	0
	.long	0
	.long	0xaef1586d
	.long	0x9458eaf7
	.long	0x10e4107f
	.long	0xd8a5664f
	.long	0x4d377036
	.long	0x09d5f47d
	.long	0x91054a7f
	.long	0xbe60db93
bits2opi:
	.long	0x00000028
	.long	0
/*
 *  Note: wherever you see the word `octant', read `quadrant'.
 *  Currently this code is set up for  pi/2  argument reduction.
 *  By uncommenting/commenting the appropriate lines, it will
 *  also serve as a  pi/4  argument reduction code.
 */

/*						p.1
 *  Trigred  preforms argument reduction
 *  for the trigonometric functions.  It
 *  takes one input argument, a D-format
 *  number in  r1/r0 .  The magnitude of
 *  the input argument must be greater
 *  than or equal to  1/2 .  Trigred produces
 *  three results:  the number of the octant
 *  occupied by the argument, the reduced
 *  argument, and an extension of the
 *  reduced argument.  The octant number is
 *  returned in  r0 .  The reduced argument
 *  is returned as a D-format number in
 *  r2/r1 .  An 8 bit extension of the
 *  reduced argument is returned as an
 *  F-format number in r3.
 *						p.2
 */
trigred:
/*
 *  Save the sign of the input argument.
 */
	movw	r0,-(sp)
/*
 *  Extract the exponent field.
 */
	extzv	$7,$7,r0,r2
/*
 *  Convert the fraction part of the input
 *  argument into a quadword integer.
 */
	bicw2	$0xff80,r0
	bisb2	$0x80,r0	# -S.McD
	rotl	$16,r0,r0
	rotl	$16,r1,r1
/*
 *  If  r1  is negative, add  1  to  r0 .  This
 *  adjustment is made so that the two's
 *  complement multiplications done later
 *  will produce unsigned results.
 */
	bgeq	posmid
	incl	r0
posmid:
/*						p.3
 *
 *  Set  r3  to the address of the first quadword
 *  used to obtain the needed portion of  2/pi .
 *  The address is longword aligned to ensure
 *  efficient access.
 */
	ashl	$-3,r2,r3
	bicb2	$3,r3
	mnegl	r3,r3
	movab	bits2opi[r3],r3
/*
 *  Set  r2  to the size of the shift needed to
 *  obtain the correct portion of  2/pi .
 */
	bicb2	$0xe0,r2
/*						p.4
 *
 *  Move the needed  128  bits of  2/pi  into
 *  r11 - r8 .  Adjust the numbers to allow
 *  for unsigned multiplication.
 */
	ashq	r2,(r3),r10

	subl2	$4,r3
	ashq	r2,(r3),r9
	bgeq	signoff1
	incl	r11
signoff1:
	subl2	$4,r3
	ashq	r2,(r3),r8
	bgeq	signoff2
	incl	r10
signoff2:
	subl2	$4,r3
	ashq	r2,(r3),r7
	bgeq	signoff3
	incl	r9
signoff3:
/*						p.5
 *
 *  Multiply the contents of  r0/r1  by the
 *  slice of  2/pi  in  r11 - r8 .
 */
	emul	r0,r8,$0,r4
	emul	r0,r9,r5,r5
	emul	r0,r10,r6,r6

	emul	r1,r8,$0,r7
	emul	r1,r9,r8,r8
	emul	r1,r10,r9,r9
	emul	r1,r11,r10,r10

	addl2	r4,r8
	adwc	r5,r9
	adwc	r6,r10
/*						p.6
 *
 *  If there are more than five leading zeros
 *  after the first two quotient bits or if there
 *  are more than five leading ones after the first
 *  two quotient bits, generate more fraction bits.
 *  Otherwise, branch to code to produce the result.
 */
	bicl3	$0xc1ffffff,r10,r4
	beql	more1
	cmpl	$0x3e000000,r4
	bneq	result
more1:
/*						p.7
 *
 *  generate another  32  result bits.
 */
	subl2	$4,r3
	ashq	r2,(r3),r5
	bgeq	signoff4

	emul	r1,r6,$0,r4
	addl2	r1,r5
	emul	r0,r6,r5,r5
	addl2	r0,r6
	brb	addbits1

signoff4:
	emul	r1,r6,$0,r4
	emul	r0,r6,r5,r5

addbits1:
	addl2	r5,r7
	adwc	r6,r8
	adwc	$0,r9
	adwc	$0,r10
/*						p.8
 *
 *  Check for massive cancellation.
 */
	bicl3	$0xc0000000,r10,r6
/*	bneq	more2			-S.McD  Test was backwards */
	beql	more2
	cmpl	$0x3fffffff,r6
	bneq	result
more2:
/*						p.9
 *
 *  If massive cancellation has occurred,
 *  generate another  24  result bits.
 *  Testing has shown there will always be
 *  enough bits after this point.
 */
	subl2	$4,r3
	ashq	r2,(r3),r5
	bgeq	signoff5

	emul	r0,r6,r4,r5
	addl2	r0,r6
	brb	addbits2

signoff5:
	emul	r0,r6,r4,r5

addbits2:
	addl2	r6,r7
	adwc	$0,r8
	adwc	$0,r9
	adwc	$0,r10
/*						p.10
 *
 *  The following code produces the reduced
 *  argument from the product bits contained
 *  in  r10 - r7 .
 */
result:
/*
 *  Extract the octant number from  r10 .
 */
/*	extzv	$29,$3,r10,r0	...used for  pi/4  reduction -S.McD */
	extzv	$30,$2,r10,r0
/*
 *  Clear the octant bits in  r10 .
 */
/*	bicl2	$0xe0000000,r10	...used for  pi/4  reduction -S.McD */
	bicl2	$0xc0000000,r10
/*
 *  Zero the sign flag.
 */
	clrl	r5
/*						p.11
 *
 *  Check to see if the fraction is greater than
 *  or equal to one-half.  If it is, add one
 *  to the octant number, set the sign flag
 *  on, and replace the fraction with  1 minus
 *  the fraction.
 */
/*	bitl	$0x10000000,r10		...used for  pi/4  reduction -S.McD */
	bitl	$0x20000000,r10
	beql	small
	incl	r0
	incl	r5
/*	subl3	r10,$0x1fffffff,r10	...used for  pi/4  reduction -S.McD */
	subl3	r10,$0x3fffffff,r10
	mcoml	r9,r9
	mcoml	r8,r8
	mcoml	r7,r7
small:
/*						p.12
 *
 *  Test whether the first  29  bits of the ...used for  pi/4  reduction -S.McD
 *  Test whether the first  30  bits of the
 *  fraction are zero.
 */
	tstl	r10
	beql	tiny
/*
 *  Find the position of the first one bit in  r10 .
 */
	cvtld	r10,r1
	extzv	$7,$7,r1,r1
/*
 *  Compute the size of the shift needed.
 */
	subl3	r1,$32,r6
/*
 *  Shift up the high order  64  bits of the
 *  product.
 */
	ashq	r6,r9,r10
	ashq	r6,r8,r9
	brb	mult
/*						p.13
 *
 *  Test to see if the sign bit of  r9  is on.
 */
tiny:
	tstl	r9
	bgeq	tinier
/*
 *  If it is, shift the product bits up  32  bits.
 */
	movl	$32,r6
	movq	r8,r10
	tstl	r10
	brb	mult
/*						p.14
 *
 *  Test whether  r9  is zero.  It is probably
 *  impossible for both  r10  and  r9  to be
 *  zero, but until proven to be so, the test
 *  must be made.
 */
tinier:
	beql	zero
/*
 *  Find the position of the first one bit in  r9 .
 */
	cvtld	r9,r1
	extzv	$7,$7,r1,r1
/*
 *  Compute the size of the shift needed.
 */
	subl3	r1,$32,r1
	addl3	$32,r1,r6
/*
 *  Shift up the high order  64  bits of the
 *  product.
 */
	ashq	r1,r8,r10
	ashq	r1,r7,r9
	brb	mult
/*						p.15
 *
 *  The following code sets the reduced
 *  argument to zero.
 */
zero:
	clrl	r1
	clrl	r2
	clrl	r3
	brw	return
/*						p.16
 *
 *  At this point,  r0  contains the octant number,
 *  r6  indicates the number of bits the fraction
 *  has been shifted,  r5  indicates the sign of
 *  the fraction,  r11/r10  contain the high order
 *  64  bits of the fraction, and the condition
 *  codes indicate where the sign bit of  r10
 *  is on.  The following code multiplies the
 *  fraction by  pi/2 .
 */
mult:
/*
 *  Save  r11/r10  in  r4/r1 .		-S.McD
 */
	movl	r11,r4
	movl	r10,r1
/*
 *  If the sign bit of  r10  is on, add  1  to  r11 .
 */
	bgeq	signoff6
	incl	r11
signoff6:
/*						p.17
 *
 *  Move  pi/2  into  r3/r2 .
 */
	movq	$0xc90fdaa22168c235,r2
/*
 *  Multiply the fraction by the portion of  pi/2
 *  in  r2 .
 */
	emul	r2,r10,$0,r7
	emul	r2,r11,r8,r7
/*
 *  Multiply the fraction by the portion of  pi/2
 *  in  r3 .
 */
	emul	r3,r10,$0,r9
	emul	r3,r11,r10,r10
/*
 *  Add the product bits together.
 */
	addl2	r7,r9
	adwc	r8,r10
	adwc	$0,r11
/*
 *  Compensate for not sign extending  r8  above.-S.McD
 */
	tstl	r8
	bgeq	signoff6a
	decl	r11
signoff6a:
/*
 *  Compensate for  r11/r10  being unsigned.	-S.McD
 */
	addl2	r2,r10
	adwc	r3,r11
/*
 *  Compensate for  r3/r2  being unsigned.	-S.McD
 */
	addl2	r1,r10
	adwc	r4,r11
/*						p.18
 *
 *  If the sign bit of  r11  is zero, shift the
 *  product bits up one bit and increment  r6 .
 */
	blss	signon
	incl	r6
	ashq	$1,r10,r10
	tstl	r9
	bgeq	signoff7
	incl	r10
signoff7:
signon:
/*						p.19
 *
 *  Shift the  56  most significant product
 *  bits into  r9/r8 .  The sign extension
 *  will be handled later.
 */
	ashq	$-8,r10,r8
/*
 *  Convert the low order  8  bits of  r10
 *  into an F-format number.
 */
	cvtbf	r10,r3
/*
 *  If the result of the conversion was
 *  negative, add  1  to  r9/r8 .
 */
	bgeq	chop
	incl	r8
	adwc	$0,r9
/*
 *  If  r9  is now zero, branch to special
 *  code to handle that possibility.
 */
	beql	carryout
chop:
/*						p.20
 *
 *  Convert the number in  r9/r8  into
 *  D-format number in  r2/r1 .
 */
	rotl	$16,r8,r2
	rotl	$16,r9,r1
/*
 *  Set the exponent field to the appropriate
 *  value.  Note that the extra bits created by
 *  sign extension are now eliminated.
 */
	subw3	r6,$131,r6
	insv	r6,$7,$9,r1
/*
 *  Set the exponent field of the F-format
 *  number in  r3  to the appropriate value.
 */
	tstf	r3
	beql	return
/*	extzv	$7,$8,r3,r4	-S.McD */
	extzv	$7,$7,r3,r4
	addw2	r4,r6
/*	subw2	$217,r6		-S.McD */
	subw2	$64,r6
	insv	r6,$7,$8,r3
	brb	return
/*						p.21
 *
 *  The following code generates the appropriate
 *  result for the unlikely possibility that
 *  rounding the number in  r9/r8  resulted in
 *  a carry out.
 */
carryout:
	clrl	r1
	clrl	r2
	subw3	r6,$132,r6
	insv	r6,$7,$9,r1
	tstf	r3
	beql	return
	extzv	$7,$8,r3,r4
	addw2	r4,r6
	subw2	$218,r6
	insv	r6,$7,$8,r3
/*						p.22
 *
 *  The following code makes an needed
 *  adjustments to the signs of the
 *  results or to the octant number, and
 *  then returns.
 */
return:
/*
 *  Test if the fraction was greater than or
 *  equal to  1/2 .  If so, negate the reduced
 *  argument.
 */
	blbc	r5,signoff8
	mnegf	r1,r1
	mnegf	r3,r3
signoff8:
/*						p.23
 *
 *  If the original argument was negative,
 *  negate the reduce argument and
 *  adjust the octant number.
 */
	tstw	(sp)+
	bgeq	signoff9
	mnegf	r1,r1
	mnegf	r3,r3
/*	subb3	r0,$8,r0	...used for  pi/4  reduction -S.McD */
	subb3	r0,$4,r0
signoff9:
/*
 *  Clear all unneeded octant bits.
 *
 *	bicb2	$0xf8,r0	...used for  pi/4  reduction -S.McD */
	bicb2	$0xfc,r0
/*
 *  Return.
 */
	rsb