1993-03-21 12:45:37 +03:00
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; Copyright (c) 1985 Regents of the University of California.
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; All rights reserved.
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;
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; Redistribution and use in source and binary forms, with or without
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; modification, are permitted provided that the following conditions
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; are met:
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; 1. Redistributions of source code must retain the above copyright
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; notice, this list of conditions and the following disclaimer.
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; 2. Redistributions in binary form must reproduce the above copyright
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; notice, this list of conditions and the following disclaimer in the
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; documentation and/or other materials provided with the distribution.
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; 3. All advertising materials mentioning features or use of this software
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; must display the following acknowledgement:
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; This product includes software developed by the University of
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; California, Berkeley and its contributors.
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; 4. Neither the name of the University nor the names of its contributors
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; may be used to endorse or promote products derived from this software
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; without specific prior written permission.
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;
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; THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
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; ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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; IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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; ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
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; FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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; DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
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; OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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; HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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; LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
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; OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
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; SUCH DAMAGE.
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;
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1993-08-14 17:42:09 +04:00
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;_sccsid:
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;.asciz "from: @(#)sqrt.s 5.4 (Berkeley) 10/9/90"
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_rcsid:
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.asciz "$Id: sqrt.S,v 1.3 1993/08/14 13:43:52 mycroft Exp $"
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1993-03-21 12:45:37 +03:00
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; double sqrt(x)
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; double x;
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; IEEE double precision sqrt
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; code in NSC assembly by K.C. Ng
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; 12/13/85
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;
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; Method:
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; Use Kahan's trick to get 8 bits initial approximation
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; by integer shift and add/subtract. Then three Newton
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; iterations to get down error to within one ulp. Finally
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; twiddle the last bit to make to correctly rounded
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; according to the rounding mode.
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;
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.vers 2
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.text
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.align 2
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.globl _sqrt
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_sqrt:
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enter [r3,r4,r5,r6,r7],44
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movl f4,tos
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movl f6,tos
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movd 2146435072,r2 ; r2 = 0x7ff00000
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movl 8(fp),f0 ; f2 = x
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movd 12(fp),r3 ; r3 = high part of x
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movd r3,r4 ; make a copy of high part of x in r4
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andd r2,r3 ; r3 become the bias exponent of x
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cmpd r2,r3 ; if r3 = 0x7ff00000 then x is INF or NAN
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bne L22
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; to see if x is INF
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movd 8(fp),r0 ; r0 = low part of x
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movd r4,r1 ; r1 is high part of x again
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andd 0xfff00000,r1 ; mask off the sign and exponent of x
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ord r0,r1 ; or with low part, if 0 then x is INF
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cmpqd 0,r1 ;
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bne L1 ; not 0; therefore x is NaN; return x.
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cmpqd 0,r4 ; now x is Inf, is it +inf?
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blt L1 ; +INF, return x
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; -INF, return NaN by doing 0/0
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nan: movl 0f0.0,f0 ;
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divl f0,f0
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br L1
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L22: ; now x is finite
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cmpl 0f0.0,f0 ; x = 0 ?
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beq L1 ; return x if x is +0 or -0
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cmpqd 0,r4 ; Is x < 0 ?
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bgt nan ; if x < 0 return NaN
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movqd 0,r5 ; r5 == scalx initialize to zero
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cmpqd 0,r3 ; is x is subnormal ? (r3 is the exponent)
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bne L21 ; if x is normal, goto L21
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movl L30,f2 ; f2 = 2**54
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mull f2,f0 ; scale up x by 2**54
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subd 0x1b00000,r5 ; off set the scale factor by -27 in exponent
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L21:
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; now x is normal
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; notations:
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; r1 == copy of fsr
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; r2 == mask of e inexact enable flag
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; r3 == mask of i inexact flag
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; r4 == mask of r rounding mode
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; r5 == x's scale factor (already defined)
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movd 0x20,r2
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movd 0x40,r3
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movd 0x180,r4
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sfsr r0 ; store fsr to r0
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movd r0,r1 ; make a copy of fsr to r1
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bicd [5,6,7,8],r0 ; clear e,i, and set r to round to nearest
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lfsr r0
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; begin to compute sqrt(x)
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movl f0,-8(fp)
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movd -4(fp),r0 ; r0 the high part of modified x
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lshd -1,r0 ; r0 >> 1
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addd 0x1ff80000,r0 ; add correction to r0 ...got 5 bits approx.
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movd r0,r6
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lshd -13,r6 ; r6 = r0>>-15
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andd 0x7c,r6 ; obtain 4*leading 5 bits of r0
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addrd L29,r7 ; r7 = address of L29 = table[0]
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addd r6,r7 ; r6 = address of L29[r6] = table[r6]
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subd 0(r7),r0 ; r0 = r0 - table[r6]
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movd r0,-4(fp)
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movl -8(fp),f2 ; now f2 = y approximate sqrt(f0) to 8 bits
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movl 0f0.5,f6 ; f6 = 0.5
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movl f0,f4
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divl f2,f4 ; t = x/y
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addl f4,f2 ; y = y + x/y
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mull f6,f2 ; y = 0.5(y+x/y) got 17 bits approx.
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movl f0,f4
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divl f2,f4 ; t = x/y
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addl f4,f2 ; y = y + x/y
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mull f6,f2 ; y = 0.5(y+x/y) got 35 bits approx.
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movl f0,f4
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divl f2,f4 ; t = x/y
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subl f2,f4 ; t = x/y - y
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mull f6,f4 ; t = 0.5(x/y-y)
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addl f4,f2 ; y = y + 0.5(x/y -y)
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; now y approx. sqrt(x) to within 1 ulp
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; twiddle last bit to force y correctly rounded
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movd r1,r0 ; restore the old fsr
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bicd [6,7,8],r0 ; clear inexact bit but retain inexact enable
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ord 0x80,r0 ; set rounding mode to round to zero
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lfsr r0
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movl f0,f4
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divl f2,f4 ; f4 = x/y
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sfsr r0
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andd r3,r0 ; get the inexact flag
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cmpqd 0,r0
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bne L18
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; if x/y exact, then ...
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cmpl f2,f4 ; if y == x/y
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beq L2
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movl f4,-8(fp)
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subd 1,-8(fp)
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subcd 0,-4(fp)
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movl -8(fp),f4 ; f4 = f4 - ulp
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L18:
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bicd [6],r1
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ord r3,r1 ; set inexact flag in r1
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andd r1,r4 ; r4 = the old rounding mode
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cmpqd 0,r4 ; round to nearest?
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bne L17
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movl f4,-8(fp)
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addd 1,-8(fp)
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addcd 0,-4(fp)
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movl -8(fp),f4 ; f4 = f4 + ulp
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br L16
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L17:
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cmpd 0x100,r4 ; round to positive inf ?
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bne L16
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movl f4,-8(fp)
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addd 1,-8(fp)
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addcd 0,-4(fp)
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movl -8(fp),f4 ; f4 = f4 + ulp
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movl f2,-8(fp)
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addd 1,-8(fp)
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addcd 0,-4(fp)
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movl -8(fp),f2 ; f2 = f2 + ulp
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L16:
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addl f4,f2 ; y = y + t
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subd 0x100000,r5 ; scalx = scalx - 1
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L2:
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movl f2,-8(fp)
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addd r5,-4(fp)
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movl -8(fp),f0
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lfsr r1
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L1:
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movl tos,f6
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movl tos,f4
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exit [r3,r4,r5,r6,r7]
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ret 0
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.data
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L28: .byte 64,40,35,41,115,113,114,116,46,99
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.byte 9,49,46,49,32,40,117,99,98,46
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.byte 101,108,101,102,117,110,116,41,32,57
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.byte 47,49,57,47,56,53,0
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L29: .blkb 4
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.double 1204
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.double 3062
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.double 5746
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.double 9193
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.double 13348
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.double 18162
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.double 23592
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.double 29598
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.double 36145
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.double 43202
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.double 50740
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.double 58733
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.double 67158
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.double 75992
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.double 85215
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.double 83599
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.double 71378
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.double 60428
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.double 50647
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.double 41945
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.double 34246
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.double 27478
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.double 21581
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.double 16499
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.double 12183
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.double 8588
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.double 5674
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.double 3403
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.double 1742
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.double 661
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.double 130
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L30: .blkb 4
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.double 1129316352 ;L30: .double 0,0x43500000
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L31: .blkb 4
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.double 0x1ff00000
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L32: .blkb 4
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.double 0x5ff00000
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