summaryrefslogtreecommitdiff
path: root/sys/gnu/arch/i386/fpemul/wm_sqrt.s
blob: e8c6d2bef5c18ce7dd981630da000758f968bffb (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
	.file	"wm_sqrt.S"
/*	$OpenBSD: wm_sqrt.s,v 1.4 2004/01/13 18:08:48 espie Exp $	*/
/*
 *  wm_sqrt.S
 *
 * Fixed point arithmetic square root evaluation.
 *
 * Call from C as:
 *   void wm_sqrt(FPU_REG *n, unsigned int control_word)
 *
 *
 * Copyright (C) 1992,1993,1994
 *                       W. Metzenthen, 22 Parker St, Ormond, Vic 3163,
 *                       Australia.  E-mail   billm@vaxc.cc.monash.edu.au
 * All rights reserved.
 *
 * This copyright notice covers the redistribution and use of the
 * FPU emulator developed by W. Metzenthen. It covers only its use
 * in the 386BSD, FreeBSD and NetBSD operating systems. Any other
 * use is not permitted under this copyright.
 *
 * 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 include information specifying
 *    that source code for the emulator is freely available and include
 *    either:
 *      a) an offer to provide the source code for a nominal distribution
 *         fee, or
 *      b) list at least two alternative methods whereby the source
 *         can be obtained, e.g. a publically accessible bulletin board
 *         and an anonymous ftp site from which the software can be
 *         downloaded.
 * 3. All advertising materials specifically mentioning features or use of
 *    this emulator must acknowledge that it was developed by W. Metzenthen.
 * 4. The name of W. Metzenthen may not be used to endorse or promote
 *    products derived from this software without specific prior written
 *    permission.
 *
 * THIS SOFTWARE IS PROVIDED ``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
 * W. METZENTHEN 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.
 *
 *
 * The purpose of this copyright, based upon the Berkeley copyright, is to
 * ensure that the covered software remains freely available to everyone.
 *
 * The software (with necessary differences) is also available, but under
 * the terms of the GNU copyleft, for the Linux operating system and for
 * the djgpp ms-dos extender.
 *
 * W. Metzenthen   June 1994.
 *
 *
 *     $FreeBSD: wm_sqrt.s,v 1.3 1994/06/10 07:45:04 rich Exp $
 *
 */


/*---------------------------------------------------------------------------+
 |  wm_sqrt(FPU_REG *n, unsigned int control_word)                           |
 |    returns the square root of n in n.                                     |
 |                                                                           |
 |  Use Newton's method to compute the square root of a number, which must   |
 |  be in the range  [1.0 .. 4.0),  to 64 bits accuracy.                     |
 |  Does not check the sign or tag of the argument.                          |
 |  Sets the exponent, but not the sign or tag of the result.                |
 |                                                                           |
 |  The guess is kept in %esi:%edi                                           |
 +---------------------------------------------------------------------------*/

#include <machine/asm.h>

#include <gnu/arch/i386/fpemul/exception.h>
#include <gnu/arch/i386/fpemul/fpu_asm.h>


.data
/*
	Local storage:
 */
#ifdef __ELF__
	.align 16,0
#else
	.align 4,0
#endif
accum_3:
	.long	0		/* ms word */
accum_2:
	.long	0
accum_1:
	.long	0
accum_0:
	.long	0

/* The de-normalised argument:
//                  sq_2                  sq_1              sq_0
//        b b b b b b b ... b b b   b b b .... b b b   b 0 0 0 ... 0
//           ^ binary point here */
fsqrt_arg_2:
	.long	0		/* ms word */
fsqrt_arg_1:
	.long	0
fsqrt_arg_0:
	.long	0		/* ls word, at most the ms bit is set */

.text
#ifdef __ELF__
	.align 4,144
#else
	.align 2,144
#endif

.globl _C_LABEL(wm_sqrt)

_C_LABEL(wm_sqrt):
	pushl	%ebp
	movl	%esp,%ebp
	pushl	%esi
	pushl	%edi
	pushl	%ebx

	movl	PARAM1,%esi

	movl	SIGH(%esi),%eax
	movl	SIGL(%esi),%ecx
	xorl	%edx,%edx

/* We use a rough linear estimate for the first guess.. */

	cmpl	EXP_BIAS,EXP(%esi)
	jnz	sqrt_arg_ge_2

	shrl	$1,%eax			/* arg is in the range  [1.0 .. 2.0) */
	rcrl	$1,%ecx
	rcrl	$1,%edx

sqrt_arg_ge_2:
/* From here on, n is never accessed directly again until it is
// replaced by the answer. */

	movl	%eax,fsqrt_arg_2		/* ms word of n */
	movl	%ecx,fsqrt_arg_1
	movl	%edx,fsqrt_arg_0

/* Make a linear first estimate */
	shrl	$1,%eax
	addl	$0x40000000,%eax
	movl	$0xaaaaaaaa,%ecx
	mull	%ecx
	shll	%edx			/* max result was 7fff... */
	testl	$0x80000000,%edx	/* but min was 3fff... */
	jnz	sqrt_prelim_no_adjust

	movl	$0x80000000,%edx	/* round up */

sqrt_prelim_no_adjust:
	movl	%edx,%esi	/* Our first guess */

/* We have now computed (approx)   (2 + x) / 3, which forms the basis
   for a few iterations of Newton's method */

	movl	fsqrt_arg_2,%ecx	/* ms word */

/* From our initial estimate, three iterations are enough to get us
// to 30 bits or so. This will then allow two iterations at better
// precision to complete the process.

// Compute  (g + n/g)/2  at each iteration (g is the guess). */
	shrl	%ecx		/* Doing this first will prevent a divide */
				/* overflow later. */

	movl	%ecx,%edx	/* msw of the arg / 2 */
	divl	%esi		/* current estimate */
	shrl	%esi		/* divide by 2 */
	addl	%eax,%esi	/* the new estimate */

	movl	%ecx,%edx
	divl	%esi
	shrl	%esi
	addl	%eax,%esi

	movl	%ecx,%edx
	divl	%esi
	shrl	%esi
	addl	%eax,%esi

/* Now that an estimate accurate to about 30 bits has been obtained (in %esi),
// we improve it to 60 bits or so.

// The strategy from now on is to compute new estimates from
//      guess := guess + (n - guess^2) / (2 * guess) */

/* First, find the square of the guess */
	movl	%esi,%eax
	mull	%esi
/* guess^2 now in %edx:%eax */

	movl	fsqrt_arg_1,%ecx
	subl	%ecx,%eax
	movl	fsqrt_arg_2,%ecx	/* ms word of normalized n */
	sbbl	%ecx,%edx
	jnc	sqrt_stage_2_positive
/* subtraction gives a negative result
// negate the result before division */
	notl	%edx
	notl	%eax
	addl	$1,%eax
	adcl	$0,%edx

	divl	%esi
	movl	%eax,%ecx

	movl	%edx,%eax
	divl	%esi
	jmp	sqrt_stage_2_finish

sqrt_stage_2_positive:
	divl	%esi
	movl	%eax,%ecx

	movl	%edx,%eax
	divl	%esi

	notl	%ecx
	notl	%eax
	addl	$1,%eax
	adcl	$0,%ecx

sqrt_stage_2_finish:
	sarl	$1,%ecx		/* divide by 2 */
	rcrl	$1,%eax

	/* Form the new estimate in %esi:%edi */
	movl	%eax,%edi
	addl	%ecx,%esi

	jnz	sqrt_stage_2_done	/* result should be [1..2) */

#ifdef PARANOID
/* It should be possible to get here only if the arg is ffff....ffff*/
	cmp	$0xffffffff,fsqrt_arg_1
	jnz	sqrt_stage_2_error
#endif /* PARANOID */

/* The best rounded result.*/
	xorl	%eax,%eax
	decl	%eax
	movl	%eax,%edi
	movl	%eax,%esi
	movl	$0x7fffffff,%eax
	jmp	sqrt_round_result

#ifdef PARANOID
sqrt_stage_2_error:
	pushl	EX_INTERNAL|0x213
	call	EXCEPTION
#endif /* PARANOID */

sqrt_stage_2_done:

/* Now the square root has been computed to better than 60 bits */

/* Find the square of the guess*/
	movl	%edi,%eax		/* ls word of guess*/
	mull	%edi
	movl	%edx,accum_1

	movl	%esi,%eax
	mull	%esi
	movl	%edx,accum_3
	movl	%eax,accum_2

	movl	%edi,%eax
	mull	%esi
	addl	%eax,accum_1
	adcl	%edx,accum_2
	adcl	$0,accum_3

/*	movl	%esi,%eax*/
/*	mull	%edi*/
	addl	%eax,accum_1
	adcl	%edx,accum_2
	adcl	$0,accum_3

/* guess^2 now in accum_3:accum_2:accum_1*/

	movl	fsqrt_arg_0,%eax		/* get normalized n*/
	subl	%eax,accum_1
	movl	fsqrt_arg_1,%eax
	sbbl	%eax,accum_2
	movl	fsqrt_arg_2,%eax		/* ms word of normalized n*/
	sbbl	%eax,accum_3
	jnc	sqrt_stage_3_positive

/* subtraction gives a negative result*/
/* negate the result before division */
	notl	accum_1
	notl	accum_2
	notl	accum_3
	addl	$1,accum_1
	adcl	$0,accum_2

#ifdef PARANOID
	adcl	$0,accum_3	/* This must be zero */
	jz	sqrt_stage_3_no_error

sqrt_stage_3_error:
	pushl	EX_INTERNAL|0x207
	call	EXCEPTION

sqrt_stage_3_no_error:
#endif /* PARANOID */

	movl	accum_2,%edx
	movl	accum_1,%eax
	divl	%esi
	movl	%eax,%ecx

	movl	%edx,%eax
	divl	%esi

	sarl	$1,%ecx		/* divide by 2*/
	rcrl	$1,%eax

	/* prepare to round the result*/

	addl	%ecx,%edi
	adcl	$0,%esi

	jmp	sqrt_stage_3_finished

sqrt_stage_3_positive:
	movl	accum_2,%edx
	movl	accum_1,%eax
	divl	%esi
	movl	%eax,%ecx

	movl	%edx,%eax
	divl	%esi

	sarl	$1,%ecx		/* divide by 2*/
	rcrl	$1,%eax

	/* prepare to round the result*/

	notl	%eax		/* Negate the correction term*/
	notl	%ecx
	addl	$1,%eax
	adcl	$0,%ecx		/* carry here ==> correction == 0*/
	adcl	$0xffffffff,%esi

	addl	%ecx,%edi
	adcl	$0,%esi

sqrt_stage_3_finished:

/* The result in %esi:%edi:%esi should be good to about 90 bits here,
// and the rounding information here does not have sufficient accuracy
// in a few rare cases. */
	cmpl	$0xffffffe0,%eax
	ja	sqrt_near_exact_x

	cmpl	$0x00000020,%eax
	jb	sqrt_near_exact

	cmpl	$0x7fffffe0,%eax
	jb	sqrt_round_result

	cmpl	$0x80000020,%eax
	jb	sqrt_get_more_precision

sqrt_round_result:
/* Set up for rounding operations*/
	movl	%eax,%edx
	movl	%esi,%eax
	movl	%edi,%ebx
	movl	PARAM1,%edi
	movl	EXP_BIAS,EXP(%edi)	/* Result is in  [1.0 .. 2.0)*/
	movl	PARAM2,%ecx
	jmp	FPU_round_sqrt


sqrt_near_exact_x:
/* First, the estimate must be rounded up.*/
	addl	$1,%edi
	adcl	$0,%esi

sqrt_near_exact:
/* This is an easy case because x^1/2 is monotonic.
// We need just find the square of our estimate, compare it
// with the argument, and deduce whether our estimate is
// above, below, or exact. We use the fact that the estimate
// is known to be accurate to about 90 bits. */
	movl	%edi,%eax		/* ls word of guess*/
	mull	%edi
	movl	%edx,%ebx		/* 2nd ls word of square*/
	movl	%eax,%ecx		/* ls word of square*/

	movl	%edi,%eax
	mull	%esi
	addl	%eax,%ebx
	addl	%eax,%ebx

#ifdef PARANOID
	cmp	$0xffffffb0,%ebx
	jb	sqrt_near_exact_ok

	cmp	$0x00000050,%ebx
	ja	sqrt_near_exact_ok

	pushl	EX_INTERNAL|0x214
	call	EXCEPTION

sqrt_near_exact_ok:
#endif /* PARANOID */

	or	%ebx,%ebx
	js	sqrt_near_exact_small

	jnz	sqrt_near_exact_large

	or	%ebx,%edx
	jnz	sqrt_near_exact_large

/* Our estimate is exactly the right answer*/
	xorl	%eax,%eax
	jmp	sqrt_round_result

sqrt_near_exact_small:
/* Our estimate is too small*/
	movl	$0x000000ff,%eax
	jmp	sqrt_round_result
	
sqrt_near_exact_large:
/* Our estimate is too large, we need to decrement it*/
	subl	$1,%edi
	sbbl	$0,%esi
	movl	$0xffffff00,%eax
	jmp	sqrt_round_result


sqrt_get_more_precision:
/* This case is almost the same as the above, except we start*/
/* with an extra bit of precision in the estimate.*/
	stc			/* The extra bit.*/
	rcll	$1,%edi		/* Shift the estimate left one bit*/
	rcll	$1,%esi

	movl	%edi,%eax		/* ls word of guess*/
	mull	%edi
	movl	%edx,%ebx		/* 2nd ls word of square*/
	movl	%eax,%ecx		/* ls word of square*/

	movl	%edi,%eax
	mull	%esi
	addl	%eax,%ebx
	addl	%eax,%ebx

/* Put our estimate back to its original value*/
	stc			/* The ms bit.*/
	rcrl	$1,%esi		/* Shift the estimate left one bit*/
	rcrl	$1,%edi

#ifdef PARANOID
	cmp	$0xffffff60,%ebx
	jb	sqrt_more_prec_ok

	cmp	$0x000000a0,%ebx
	ja	sqrt_more_prec_ok

	pushl	EX_INTERNAL|0x215
	call	EXCEPTION

sqrt_more_prec_ok:
#endif /* PARANOID */

	or	%ebx,%ebx
	js	sqrt_more_prec_small

	jnz	sqrt_more_prec_large

	or	%ebx,%ecx
	jnz	sqrt_more_prec_large

/* Our estimate is exactly the right answer*/
	movl	$0x80000000,%eax
	jmp	sqrt_round_result

sqrt_more_prec_small:
/* Our estimate is too small*/
	movl	$0x800000ff,%eax
	jmp	sqrt_round_result
	
sqrt_more_prec_large:
/* Our estimate is too large*/
	movl	$0x7fffff00,%eax
	jmp	sqrt_round_result