1 /* $NetBSD: fpu_sqrt.c,v 1.4 2005/12/11 12:18:42 christos Exp $ */
2
3 /*-
4 * SPDX-License-Identifier: BSD-3-Clause
5 *
6 * Copyright (c) 1992, 1993
7 * The Regents of the University of California. All rights reserved.
8 *
9 * This software was developed by the Computer Systems Engineering group
10 * at Lawrence Berkeley Laboratory under DARPA contract BG 91-66 and
11 * contributed to Berkeley.
12 *
13 * All advertising materials mentioning features or use of this software
14 * must display the following acknowledgement:
15 * This product includes software developed by the University of
16 * California, Lawrence Berkeley Laboratory.
17 *
18 * Redistribution and use in source and binary forms, with or without
19 * modification, are permitted provided that the following conditions
20 * are met:
21 * 1. Redistributions of source code must retain the above copyright
22 * notice, this list of conditions and the following disclaimer.
23 * 2. Redistributions in binary form must reproduce the above copyright
24 * notice, this list of conditions and the following disclaimer in the
25 * documentation and/or other materials provided with the distribution.
26 * 3. Neither the name of the University nor the names of its contributors
27 * may be used to endorse or promote products derived from this software
28 * without specific prior written permission.
29 *
30 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
31 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
32 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
33 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
34 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
35 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
36 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
37 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
38 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
39 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
40 * SUCH DAMAGE.
41 */
42
43 /*
44 * Perform an FPU square root (return sqrt(x)).
45 */
46
47 #include <sys/types.h>
48 #include <sys/systm.h>
49
50 #include <machine/fpu.h>
51
52 #include <powerpc/fpu/fpu_arith.h>
53 #include <powerpc/fpu/fpu_emu.h>
54
55 /*
56 * Our task is to calculate the square root of a floating point number x0.
57 * This number x normally has the form:
58 *
59 * exp
60 * x = mant * 2 (where 1 <= mant < 2 and exp is an integer)
61 *
62 * This can be left as it stands, or the mantissa can be doubled and the
63 * exponent decremented:
64 *
65 * exp-1
66 * x = (2 * mant) * 2 (where 2 <= 2 * mant < 4)
67 *
68 * If the exponent `exp' is even, the square root of the number is best
69 * handled using the first form, and is by definition equal to:
70 *
71 * exp/2
72 * sqrt(x) = sqrt(mant) * 2
73 *
74 * If exp is odd, on the other hand, it is convenient to use the second
75 * form, giving:
76 *
77 * (exp-1)/2
78 * sqrt(x) = sqrt(2 * mant) * 2
79 *
80 * In the first case, we have
81 *
82 * 1 <= mant < 2
83 *
84 * and therefore
85 *
86 * sqrt(1) <= sqrt(mant) < sqrt(2)
87 *
88 * while in the second case we have
89 *
90 * 2 <= 2*mant < 4
91 *
92 * and therefore
93 *
94 * sqrt(2) <= sqrt(2*mant) < sqrt(4)
95 *
96 * so that in any case, we are sure that
97 *
98 * sqrt(1) <= sqrt(n * mant) < sqrt(4), n = 1 or 2
99 *
100 * or
101 *
102 * 1 <= sqrt(n * mant) < 2, n = 1 or 2.
103 *
104 * This root is therefore a properly formed mantissa for a floating
105 * point number. The exponent of sqrt(x) is either exp/2 or (exp-1)/2
106 * as above. This leaves us with the problem of finding the square root
107 * of a fixed-point number in the range [1..4).
108 *
109 * Though it may not be instantly obvious, the following square root
110 * algorithm works for any integer x of an even number of bits, provided
111 * that no overflows occur:
112 *
113 * let q = 0
114 * for k = NBITS-1 to 0 step -1 do -- for each digit in the answer...
115 * x *= 2 -- multiply by radix, for next digit
116 * if x >= 2q + 2^k then -- if adding 2^k does not
117 * x -= 2q + 2^k -- exceed the correct root,
118 * q += 2^k -- add 2^k and adjust x
119 * fi
120 * done
121 * sqrt = q / 2^(NBITS/2) -- (and any remainder is in x)
122 *
123 * If NBITS is odd (so that k is initially even), we can just add another
124 * zero bit at the top of x. Doing so means that q is not going to acquire
125 * a 1 bit in the first trip around the loop (since x0 < 2^NBITS). If the
126 * final value in x is not needed, or can be off by a factor of 2, this is
127 * equivalant to moving the `x *= 2' step to the bottom of the loop:
128 *
129 * for k = NBITS-1 to 0 step -1 do if ... fi; x *= 2; done
130 *
131 * and the result q will then be sqrt(x0) * 2^floor(NBITS / 2).
132 * (Since the algorithm is destructive on x, we will call x's initial
133 * value, for which q is some power of two times its square root, x0.)
134 *
135 * If we insert a loop invariant y = 2q, we can then rewrite this using
136 * C notation as:
137 *
138 * q = y = 0; x = x0;
139 * for (k = NBITS; --k >= 0;) {
140 * #if (NBITS is even)
141 * x *= 2;
142 * #endif
143 * t = y + (1 << k);
144 * if (x >= t) {
145 * x -= t;
146 * q += 1 << k;
147 * y += 1 << (k + 1);
148 * }
149 * #if (NBITS is odd)
150 * x *= 2;
151 * #endif
152 * }
153 *
154 * If x0 is fixed point, rather than an integer, we can simply alter the
155 * scale factor between q and sqrt(x0). As it happens, we can easily arrange
156 * for the scale factor to be 2**0 or 1, so that sqrt(x0) == q.
157 *
158 * In our case, however, x0 (and therefore x, y, q, and t) are multiword
159 * integers, which adds some complication. But note that q is built one
160 * bit at a time, from the top down, and is not used itself in the loop
161 * (we use 2q as held in y instead). This means we can build our answer
162 * in an integer, one word at a time, which saves a bit of work. Also,
163 * since 1 << k is always a `new' bit in q, 1 << k and 1 << (k+1) are
164 * `new' bits in y and we can set them with an `or' operation rather than
165 * a full-blown multiword add.
166 *
167 * We are almost done, except for one snag. We must prove that none of our
168 * intermediate calculations can overflow. We know that x0 is in [1..4)
169 * and therefore the square root in q will be in [1..2), but what about x,
170 * y, and t?
171 *
172 * We know that y = 2q at the beginning of each loop. (The relation only
173 * fails temporarily while y and q are being updated.) Since q < 2, y < 4.
174 * The sum in t can, in our case, be as much as y+(1<<1) = y+2 < 6, and.
175 * Furthermore, we can prove with a bit of work that x never exceeds y by
176 * more than 2, so that even after doubling, 0 <= x < 8. (This is left as
177 * an exercise to the reader, mostly because I have become tired of working
178 * on this comment.)
179 *
180 * If our floating point mantissas (which are of the form 1.frac) occupy
181 * B+1 bits, our largest intermediary needs at most B+3 bits, or two extra.
182 * In fact, we want even one more bit (for a carry, to avoid compares), or
183 * three extra. There is a comment in fpu_emu.h reminding maintainers of
184 * this, so we have some justification in assuming it.
185 */
186 struct fpn *
fpu_sqrt(struct fpemu * fe)187 fpu_sqrt(struct fpemu *fe)
188 {
189 struct fpn *x = &fe->fe_f1;
190 u_int bit, q, tt;
191 u_int x0, x1, x2, x3;
192 u_int y0, y1, y2, y3;
193 u_int d0, d1, d2, d3;
194 int e;
195 FPU_DECL_CARRY;
196
197 /*
198 * Take care of special cases first. In order:
199 *
200 * sqrt(NaN) = NaN
201 * sqrt(+0) = +0
202 * sqrt(-0) = -0
203 * sqrt(x < 0) = NaN (including sqrt(-Inf))
204 * sqrt(+Inf) = +Inf
205 *
206 * Then all that remains are numbers with mantissas in [1..2).
207 */
208 DPRINTF(FPE_REG, ("fpu_sqer:\n"));
209 DUMPFPN(FPE_REG, x);
210 DPRINTF(FPE_REG, ("=>\n"));
211 if (ISNAN(x)) {
212 fe->fe_cx |= FPSCR_VXSNAN;
213 DUMPFPN(FPE_REG, x);
214 return (x);
215 }
216 if (ISZERO(x)) {
217 fe->fe_cx |= FPSCR_ZX;
218 x->fp_class = FPC_INF;
219 DUMPFPN(FPE_REG, x);
220 return (x);
221 }
222 if (x->fp_sign) {
223 fe->fe_cx |= FPSCR_VXSQRT;
224 return (fpu_newnan(fe));
225 }
226 if (ISINF(x)) {
227 DUMPFPN(FPE_REG, x);
228 return (x);
229 }
230
231 /*
232 * Calculate result exponent. As noted above, this may involve
233 * doubling the mantissa. We will also need to double x each
234 * time around the loop, so we define a macro for this here, and
235 * we break out the multiword mantissa.
236 */
237 #ifdef FPU_SHL1_BY_ADD
238 #define DOUBLE_X { \
239 FPU_ADDS(x3, x3, x3); FPU_ADDCS(x2, x2, x2); \
240 FPU_ADDCS(x1, x1, x1); FPU_ADDC(x0, x0, x0); \
241 }
242 #else
243 #define DOUBLE_X { \
244 x0 = (x0 << 1) | (x1 >> 31); x1 = (x1 << 1) | (x2 >> 31); \
245 x2 = (x2 << 1) | (x3 >> 31); x3 <<= 1; \
246 }
247 #endif
248 #if (FP_NMANT & 1) != 0
249 # define ODD_DOUBLE DOUBLE_X
250 # define EVEN_DOUBLE /* nothing */
251 #else
252 # define ODD_DOUBLE /* nothing */
253 # define EVEN_DOUBLE DOUBLE_X
254 #endif
255 x0 = x->fp_mant[0];
256 x1 = x->fp_mant[1];
257 x2 = x->fp_mant[2];
258 x3 = x->fp_mant[3];
259 e = x->fp_exp;
260 if (e & 1) /* exponent is odd; use sqrt(2mant) */
261 DOUBLE_X;
262 /* THE FOLLOWING ASSUMES THAT RIGHT SHIFT DOES SIGN EXTENSION */
263 x->fp_exp = e >> 1; /* calculates (e&1 ? (e-1)/2 : e/2 */
264
265 /*
266 * Now calculate the mantissa root. Since x is now in [1..4),
267 * we know that the first trip around the loop will definitely
268 * set the top bit in q, so we can do that manually and start
269 * the loop at the next bit down instead. We must be sure to
270 * double x correctly while doing the `known q=1.0'.
271 *
272 * We do this one mantissa-word at a time, as noted above, to
273 * save work. To avoid `(1U << 31) << 1', we also do the top bit
274 * outside of each per-word loop.
275 *
276 * The calculation `t = y + bit' breaks down into `t0 = y0, ...,
277 * t3 = y3, t? |= bit' for the appropriate word. Since the bit
278 * is always a `new' one, this means that three of the `t?'s are
279 * just the corresponding `y?'; we use `#define's here for this.
280 * The variable `tt' holds the actual `t?' variable.
281 */
282
283 /* calculate q0 */
284 #define t0 tt
285 bit = FP_1;
286 EVEN_DOUBLE;
287 /* if (x >= (t0 = y0 | bit)) { */ /* always true */
288 q = bit;
289 x0 -= bit;
290 y0 = bit << 1;
291 /* } */
292 ODD_DOUBLE;
293 while ((bit >>= 1) != 0) { /* for remaining bits in q0 */
294 EVEN_DOUBLE;
295 t0 = y0 | bit; /* t = y + bit */
296 if (x0 >= t0) { /* if x >= t then */
297 x0 -= t0; /* x -= t */
298 q |= bit; /* q += bit */
299 y0 |= bit << 1; /* y += bit << 1 */
300 }
301 ODD_DOUBLE;
302 }
303 x->fp_mant[0] = q;
304 #undef t0
305
306 /* calculate q1. note (y0&1)==0. */
307 #define t0 y0
308 #define t1 tt
309 q = 0;
310 y1 = 0;
311 bit = 1 << 31;
312 EVEN_DOUBLE;
313 t1 = bit;
314 FPU_SUBS(d1, x1, t1);
315 FPU_SUBC(d0, x0, t0); /* d = x - t */
316 if ((int)d0 >= 0) { /* if d >= 0 (i.e., x >= t) then */
317 x0 = d0, x1 = d1; /* x -= t */
318 q = bit; /* q += bit */
319 y0 |= 1; /* y += bit << 1 */
320 }
321 ODD_DOUBLE;
322 while ((bit >>= 1) != 0) { /* for remaining bits in q1 */
323 EVEN_DOUBLE; /* as before */
324 t1 = y1 | bit;
325 FPU_SUBS(d1, x1, t1);
326 FPU_SUBC(d0, x0, t0);
327 if ((int)d0 >= 0) {
328 x0 = d0, x1 = d1;
329 q |= bit;
330 y1 |= bit << 1;
331 }
332 ODD_DOUBLE;
333 }
334 x->fp_mant[1] = q;
335 #undef t1
336
337 /* calculate q2. note (y1&1)==0; y0 (aka t0) is fixed. */
338 #define t1 y1
339 #define t2 tt
340 q = 0;
341 y2 = 0;
342 bit = 1 << 31;
343 EVEN_DOUBLE;
344 t2 = bit;
345 FPU_SUBS(d2, x2, t2);
346 FPU_SUBCS(d1, x1, t1);
347 FPU_SUBC(d0, x0, t0);
348 if ((int)d0 >= 0) {
349 x0 = d0, x1 = d1, x2 = d2;
350 q = bit;
351 y1 |= 1; /* now t1, y1 are set in concrete */
352 }
353 ODD_DOUBLE;
354 while ((bit >>= 1) != 0) {
355 EVEN_DOUBLE;
356 t2 = y2 | bit;
357 FPU_SUBS(d2, x2, t2);
358 FPU_SUBCS(d1, x1, t1);
359 FPU_SUBC(d0, x0, t0);
360 if ((int)d0 >= 0) {
361 x0 = d0, x1 = d1, x2 = d2;
362 q |= bit;
363 y2 |= bit << 1;
364 }
365 ODD_DOUBLE;
366 }
367 x->fp_mant[2] = q;
368 #undef t2
369
370 /* calculate q3. y0, t0, y1, t1 all fixed; y2, t2, almost done. */
371 #define t2 y2
372 #define t3 tt
373 q = 0;
374 y3 = 0;
375 bit = 1 << 31;
376 EVEN_DOUBLE;
377 t3 = bit;
378 FPU_SUBS(d3, x3, t3);
379 FPU_SUBCS(d2, x2, t2);
380 FPU_SUBCS(d1, x1, t1);
381 FPU_SUBC(d0, x0, t0);
382 if ((int)d0 >= 0) {
383 x0 = d0, x1 = d1, x2 = d2; x3 = d3;
384 q = bit;
385 y2 |= 1;
386 }
387 ODD_DOUBLE;
388 while ((bit >>= 1) != 0) {
389 EVEN_DOUBLE;
390 t3 = y3 | bit;
391 FPU_SUBS(d3, x3, t3);
392 FPU_SUBCS(d2, x2, t2);
393 FPU_SUBCS(d1, x1, t1);
394 FPU_SUBC(d0, x0, t0);
395 if ((int)d0 >= 0) {
396 x0 = d0, x1 = d1, x2 = d2; x3 = d3;
397 q |= bit;
398 y3 |= bit << 1;
399 }
400 ODD_DOUBLE;
401 }
402 x->fp_mant[3] = q;
403
404 /*
405 * The result, which includes guard and round bits, is exact iff
406 * x is now zero; any nonzero bits in x represent sticky bits.
407 */
408 x->fp_sticky = x0 | x1 | x2 | x3;
409 DUMPFPN(FPE_REG, x);
410 return (x);
411 }
412