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Diffstat (limited to 'sys/powerpc/fpu/fpu_sqrt.c')
| -rw-r--r-- | sys/powerpc/fpu/fpu_sqrt.c | 411 |
1 files changed, 411 insertions, 0 deletions
diff --git a/sys/powerpc/fpu/fpu_sqrt.c b/sys/powerpc/fpu/fpu_sqrt.c new file mode 100644 index 000000000000..7d6cdcb1b72e --- /dev/null +++ b/sys/powerpc/fpu/fpu_sqrt.c @@ -0,0 +1,411 @@ +/* $NetBSD: fpu_sqrt.c,v 1.4 2005/12/11 12:18:42 christos Exp $ */ + +/*- + * SPDX-License-Identifier: BSD-3-Clause + * + * Copyright (c) 1992, 1993 + * The Regents of the University of California. All rights reserved. + * + * This software was developed by the Computer Systems Engineering group + * at Lawrence Berkeley Laboratory under DARPA contract BG 91-66 and + * contributed to Berkeley. + * + * 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, Lawrence Berkeley Laboratory. + * + * 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. 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. + */ + +/* + * Perform an FPU square root (return sqrt(x)). + */ + +#include <sys/types.h> +#include <sys/systm.h> + +#include <machine/fpu.h> + +#include <powerpc/fpu/fpu_arith.h> +#include <powerpc/fpu/fpu_emu.h> + +/* + * Our task is to calculate the square root of a floating point number x0. + * This number x normally has the form: + * + * exp + * x = mant * 2 (where 1 <= mant < 2 and exp is an integer) + * + * This can be left as it stands, or the mantissa can be doubled and the + * exponent decremented: + * + * exp-1 + * x = (2 * mant) * 2 (where 2 <= 2 * mant < 4) + * + * If the exponent `exp' is even, the square root of the number is best + * handled using the first form, and is by definition equal to: + * + * exp/2 + * sqrt(x) = sqrt(mant) * 2 + * + * If exp is odd, on the other hand, it is convenient to use the second + * form, giving: + * + * (exp-1)/2 + * sqrt(x) = sqrt(2 * mant) * 2 + * + * In the first case, we have + * + * 1 <= mant < 2 + * + * and therefore + * + * sqrt(1) <= sqrt(mant) < sqrt(2) + * + * while in the second case we have + * + * 2 <= 2*mant < 4 + * + * and therefore + * + * sqrt(2) <= sqrt(2*mant) < sqrt(4) + * + * so that in any case, we are sure that + * + * sqrt(1) <= sqrt(n * mant) < sqrt(4), n = 1 or 2 + * + * or + * + * 1 <= sqrt(n * mant) < 2, n = 1 or 2. + * + * This root is therefore a properly formed mantissa for a floating + * point number. The exponent of sqrt(x) is either exp/2 or (exp-1)/2 + * as above. This leaves us with the problem of finding the square root + * of a fixed-point number in the range [1..4). + * + * Though it may not be instantly obvious, the following square root + * algorithm works for any integer x of an even number of bits, provided + * that no overflows occur: + * + * let q = 0 + * for k = NBITS-1 to 0 step -1 do -- for each digit in the answer... + * x *= 2 -- multiply by radix, for next digit + * if x >= 2q + 2^k then -- if adding 2^k does not + * x -= 2q + 2^k -- exceed the correct root, + * q += 2^k -- add 2^k and adjust x + * fi + * done + * sqrt = q / 2^(NBITS/2) -- (and any remainder is in x) + * + * If NBITS is odd (so that k is initially even), we can just add another + * zero bit at the top of x. Doing so means that q is not going to acquire + * a 1 bit in the first trip around the loop (since x0 < 2^NBITS). If the + * final value in x is not needed, or can be off by a factor of 2, this is + * equivalant to moving the `x *= 2' step to the bottom of the loop: + * + * for k = NBITS-1 to 0 step -1 do if ... fi; x *= 2; done + * + * and the result q will then be sqrt(x0) * 2^floor(NBITS / 2). + * (Since the algorithm is destructive on x, we will call x's initial + * value, for which q is some power of two times its square root, x0.) + * + * If we insert a loop invariant y = 2q, we can then rewrite this using + * C notation as: + * + * q = y = 0; x = x0; + * for (k = NBITS; --k >= 0;) { + * #if (NBITS is even) + * x *= 2; + * #endif + * t = y + (1 << k); + * if (x >= t) { + * x -= t; + * q += 1 << k; + * y += 1 << (k + 1); + * } + * #if (NBITS is odd) + * x *= 2; + * #endif + * } + * + * If x0 is fixed point, rather than an integer, we can simply alter the + * scale factor between q and sqrt(x0). As it happens, we can easily arrange + * for the scale factor to be 2**0 or 1, so that sqrt(x0) == q. + * + * In our case, however, x0 (and therefore x, y, q, and t) are multiword + * integers, which adds some complication. But note that q is built one + * bit at a time, from the top down, and is not used itself in the loop + * (we use 2q as held in y instead). This means we can build our answer + * in an integer, one word at a time, which saves a bit of work. Also, + * since 1 << k is always a `new' bit in q, 1 << k and 1 << (k+1) are + * `new' bits in y and we can set them with an `or' operation rather than + * a full-blown multiword add. + * + * We are almost done, except for one snag. We must prove that none of our + * intermediate calculations can overflow. We know that x0 is in [1..4) + * and therefore the square root in q will be in [1..2), but what about x, + * y, and t? + * + * We know that y = 2q at the beginning of each loop. (The relation only + * fails temporarily while y and q are being updated.) Since q < 2, y < 4. + * The sum in t can, in our case, be as much as y+(1<<1) = y+2 < 6, and. + * Furthermore, we can prove with a bit of work that x never exceeds y by + * more than 2, so that even after doubling, 0 <= x < 8. (This is left as + * an exercise to the reader, mostly because I have become tired of working + * on this comment.) + * + * If our floating point mantissas (which are of the form 1.frac) occupy + * B+1 bits, our largest intermediary needs at most B+3 bits, or two extra. + * In fact, we want even one more bit (for a carry, to avoid compares), or + * three extra. There is a comment in fpu_emu.h reminding maintainers of + * this, so we have some justification in assuming it. + */ +struct fpn * +fpu_sqrt(struct fpemu *fe) +{ + struct fpn *x = &fe->fe_f1; + u_int bit, q, tt; + u_int x0, x1, x2, x3; + u_int y0, y1, y2, y3; + u_int d0, d1, d2, d3; + int e; + FPU_DECL_CARRY; + + /* + * Take care of special cases first. In order: + * + * sqrt(NaN) = NaN + * sqrt(+0) = +0 + * sqrt(-0) = -0 + * sqrt(x < 0) = NaN (including sqrt(-Inf)) + * sqrt(+Inf) = +Inf + * + * Then all that remains are numbers with mantissas in [1..2). + */ + DPRINTF(FPE_REG, ("fpu_sqer:\n")); + DUMPFPN(FPE_REG, x); + DPRINTF(FPE_REG, ("=>\n")); + if (ISNAN(x)) { + fe->fe_cx |= FPSCR_VXSNAN; + DUMPFPN(FPE_REG, x); + return (x); + } + if (ISZERO(x)) { + fe->fe_cx |= FPSCR_ZX; + x->fp_class = FPC_INF; + DUMPFPN(FPE_REG, x); + return (x); + } + if (x->fp_sign) { + fe->fe_cx |= FPSCR_VXSQRT; + return (fpu_newnan(fe)); + } + if (ISINF(x)) { + DUMPFPN(FPE_REG, x); + return (x); + } + + /* + * Calculate result exponent. As noted above, this may involve + * doubling the mantissa. We will also need to double x each + * time around the loop, so we define a macro for this here, and + * we break out the multiword mantissa. + */ +#ifdef FPU_SHL1_BY_ADD +#define DOUBLE_X { \ + FPU_ADDS(x3, x3, x3); FPU_ADDCS(x2, x2, x2); \ + FPU_ADDCS(x1, x1, x1); FPU_ADDC(x0, x0, x0); \ +} +#else +#define DOUBLE_X { \ + x0 = (x0 << 1) | (x1 >> 31); x1 = (x1 << 1) | (x2 >> 31); \ + x2 = (x2 << 1) | (x3 >> 31); x3 <<= 1; \ +} +#endif +#if (FP_NMANT & 1) != 0 +# define ODD_DOUBLE DOUBLE_X +# define EVEN_DOUBLE /* nothing */ +#else +# define ODD_DOUBLE /* nothing */ +# define EVEN_DOUBLE DOUBLE_X +#endif + x0 = x->fp_mant[0]; + x1 = x->fp_mant[1]; + x2 = x->fp_mant[2]; + x3 = x->fp_mant[3]; + e = x->fp_exp; + if (e & 1) /* exponent is odd; use sqrt(2mant) */ + DOUBLE_X; + /* THE FOLLOWING ASSUMES THAT RIGHT SHIFT DOES SIGN EXTENSION */ + x->fp_exp = e >> 1; /* calculates (e&1 ? (e-1)/2 : e/2 */ + + /* + * Now calculate the mantissa root. Since x is now in [1..4), + * we know that the first trip around the loop will definitely + * set the top bit in q, so we can do that manually and start + * the loop at the next bit down instead. We must be sure to + * double x correctly while doing the `known q=1.0'. + * + * We do this one mantissa-word at a time, as noted above, to + * save work. To avoid `(1U << 31) << 1', we also do the top bit + * outside of each per-word loop. + * + * The calculation `t = y + bit' breaks down into `t0 = y0, ..., + * t3 = y3, t? |= bit' for the appropriate word. Since the bit + * is always a `new' one, this means that three of the `t?'s are + * just the corresponding `y?'; we use `#define's here for this. + * The variable `tt' holds the actual `t?' variable. + */ + + /* calculate q0 */ +#define t0 tt + bit = FP_1; + EVEN_DOUBLE; + /* if (x >= (t0 = y0 | bit)) { */ /* always true */ + q = bit; + x0 -= bit; + y0 = bit << 1; + /* } */ + ODD_DOUBLE; + while ((bit >>= 1) != 0) { /* for remaining bits in q0 */ + EVEN_DOUBLE; + t0 = y0 | bit; /* t = y + bit */ + if (x0 >= t0) { /* if x >= t then */ + x0 -= t0; /* x -= t */ + q |= bit; /* q += bit */ + y0 |= bit << 1; /* y += bit << 1 */ + } + ODD_DOUBLE; + } + x->fp_mant[0] = q; +#undef t0 + + /* calculate q1. note (y0&1)==0. */ +#define t0 y0 +#define t1 tt + q = 0; + y1 = 0; + bit = 1 << 31; + EVEN_DOUBLE; + t1 = bit; + FPU_SUBS(d1, x1, t1); + FPU_SUBC(d0, x0, t0); /* d = x - t */ + if ((int)d0 >= 0) { /* if d >= 0 (i.e., x >= t) then */ + x0 = d0, x1 = d1; /* x -= t */ + q = bit; /* q += bit */ + y0 |= 1; /* y += bit << 1 */ + } + ODD_DOUBLE; + while ((bit >>= 1) != 0) { /* for remaining bits in q1 */ + EVEN_DOUBLE; /* as before */ + t1 = y1 | bit; + FPU_SUBS(d1, x1, t1); + FPU_SUBC(d0, x0, t0); + if ((int)d0 >= 0) { + x0 = d0, x1 = d1; + q |= bit; + y1 |= bit << 1; + } + ODD_DOUBLE; + } + x->fp_mant[1] = q; +#undef t1 + + /* calculate q2. note (y1&1)==0; y0 (aka t0) is fixed. */ +#define t1 y1 +#define t2 tt + q = 0; + y2 = 0; + bit = 1 << 31; + EVEN_DOUBLE; + t2 = bit; + FPU_SUBS(d2, x2, t2); + FPU_SUBCS(d1, x1, t1); + FPU_SUBC(d0, x0, t0); + if ((int)d0 >= 0) { + x0 = d0, x1 = d1, x2 = d2; + q = bit; + y1 |= 1; /* now t1, y1 are set in concrete */ + } + ODD_DOUBLE; + while ((bit >>= 1) != 0) { + EVEN_DOUBLE; + t2 = y2 | bit; + FPU_SUBS(d2, x2, t2); + FPU_SUBCS(d1, x1, t1); + FPU_SUBC(d0, x0, t0); + if ((int)d0 >= 0) { + x0 = d0, x1 = d1, x2 = d2; + q |= bit; + y2 |= bit << 1; + } + ODD_DOUBLE; + } + x->fp_mant[2] = q; +#undef t2 + + /* calculate q3. y0, t0, y1, t1 all fixed; y2, t2, almost done. */ +#define t2 y2 +#define t3 tt + q = 0; + y3 = 0; + bit = 1 << 31; + EVEN_DOUBLE; + t3 = bit; + FPU_SUBS(d3, x3, t3); + FPU_SUBCS(d2, x2, t2); + FPU_SUBCS(d1, x1, t1); + FPU_SUBC(d0, x0, t0); + if ((int)d0 >= 0) { + x0 = d0, x1 = d1, x2 = d2; x3 = d3; + q = bit; + y2 |= 1; + } + ODD_DOUBLE; + while ((bit >>= 1) != 0) { + EVEN_DOUBLE; + t3 = y3 | bit; + FPU_SUBS(d3, x3, t3); + FPU_SUBCS(d2, x2, t2); + FPU_SUBCS(d1, x1, t1); + FPU_SUBC(d0, x0, t0); + if ((int)d0 >= 0) { + x0 = d0, x1 = d1, x2 = d2; x3 = d3; + q |= bit; + y3 |= bit << 1; + } + ODD_DOUBLE; + } + x->fp_mant[3] = q; + + /* + * The result, which includes guard and round bits, is exact iff + * x is now zero; any nonzero bits in x represent sticky bits. + */ + x->fp_sticky = x0 | x1 | x2 | x3; + DUMPFPN(FPE_REG, x); + return (x); +} |