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Annotation of nono/fpe/fpu_div.c, revision 1.1

1.1     ! root        1: /*     $NetBSD: fpu_div.c,v 1.10 2014/01/01 05:23:40 isaki Exp $ */
        !             2: 
        !             3: /*
        !             4:  * Copyright (c) 1992, 1993
        !             5:  *     The Regents of the University of California.  All rights reserved.
        !             6:  *
        !             7:  * This software was developed by the Computer Systems Engineering group
        !             8:  * at Lawrence Berkeley Laboratory under DARPA contract BG 91-66 and
        !             9:  * contributed to Berkeley.
        !            10:  *
        !            11:  * All advertising materials mentioning features or use of this software
        !            12:  * must display the following acknowledgement:
        !            13:  *     This product includes software developed by the University of
        !            14:  *     California, Lawrence Berkeley Laboratory.
        !            15:  *
        !            16:  * Redistribution and use in source and binary forms, with or without
        !            17:  * modification, are permitted provided that the following conditions
        !            18:  * are met:
        !            19:  * 1. Redistributions of source code must retain the above copyright
        !            20:  *    notice, this list of conditions and the following disclaimer.
        !            21:  * 2. Redistributions in binary form must reproduce the above copyright
        !            22:  *    notice, this list of conditions and the following disclaimer in the
        !            23:  *    documentation and/or other materials provided with the distribution.
        !            24:  * 3. Neither the name of the University nor the names of its contributors
        !            25:  *    may be used to endorse or promote products derived from this software
        !            26:  *    without specific prior written permission.
        !            27:  *
        !            28:  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
        !            29:  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
        !            30:  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
        !            31:  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
        !            32:  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
        !            33:  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
        !            34:  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
        !            35:  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
        !            36:  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
        !            37:  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
        !            38:  * SUCH DAMAGE.
        !            39:  *
        !            40:  *     @(#)fpu_div.c   8.1 (Berkeley) 6/11/93
        !            41:  */
        !            42: 
        !            43: /*
        !            44:  * Perform an FPU divide (return x / y).
        !            45:  */
        !            46: 
        !            47: #include "fpu_arith.h"
        !            48: #include "fpu_emulate.h"
        !            49: 
        !            50: /*
        !            51:  * Division of normal numbers is done as follows:
        !            52:  *
        !            53:  * x and y are floating point numbers, i.e., in the form 1.bbbb * 2^e.
        !            54:  * If X and Y are the mantissas (1.bbbb's), the quotient is then:
        !            55:  *
        !            56:  *     q = (X / Y) * 2^((x exponent) - (y exponent))
        !            57:  *
        !            58:  * Since X and Y are both in [1.0,2.0), the quotient's mantissa (X / Y)
        !            59:  * will be in [0.5,2.0).  Moreover, it will be less than 1.0 if and only
        !            60:  * if X < Y.  In that case, it will have to be shifted left one bit to
        !            61:  * become a normal number, and the exponent decremented.  Thus, the
        !            62:  * desired exponent is:
        !            63:  *
        !            64:  *     left_shift = x->fp_mant < y->fp_mant;
        !            65:  *     result_exp = x->fp_exp - y->fp_exp - left_shift;
        !            66:  *
        !            67:  * The quotient mantissa X/Y can then be computed one bit at a time
        !            68:  * using the following algorithm:
        !            69:  *
        !            70:  *     Q = 0;                  -- Initial quotient.
        !            71:  *     R = X;                  -- Initial remainder,
        !            72:  *     if (left_shift)         --   but fixed up in advance.
        !            73:  *             R *= 2;
        !            74:  *     for (bit = FP_NMANT; --bit >= 0; R *= 2) {
        !            75:  *             if (R >= Y) {
        !            76:  *                     Q |= 1 << bit;
        !            77:  *                     R -= Y;
        !            78:  *             }
        !            79:  *     }
        !            80:  *
        !            81:  * The subtraction R -= Y always removes the uppermost bit from R (and
        !            82:  * can sometimes remove additional lower-order 1 bits); this proof is
        !            83:  * left to the reader.
        !            84:  *
        !            85:  * This loop correctly calculates the guard and round bits since they are
        !            86:  * included in the expanded internal representation.  The sticky bit
        !            87:  * is to be set if and only if any other bits beyond guard and round
        !            88:  * would be set.  From the above it is obvious that this is true if and
        !            89:  * only if the remainder R is nonzero when the loop terminates.
        !            90:  *
        !            91:  * Examining the loop above, we can see that the quotient Q is built
        !            92:  * one bit at a time ``from the top down''.  This means that we can
        !            93:  * dispense with the multi-word arithmetic and just build it one word
        !            94:  * at a time, writing each result word when it is done.
        !            95:  *
        !            96:  * Furthermore, since X and Y are both in [1.0,2.0), we know that,
        !            97:  * initially, R >= Y.  (Recall that, if X < Y, R is set to X * 2 and
        !            98:  * is therefore at in [2.0,4.0).)  Thus Q is sure to have bit FP_NMANT-1
        !            99:  * set, and R can be set initially to either X - Y (when X >= Y) or
        !           100:  * 2X - Y (when X < Y).  In addition, comparing R and Y is difficult,
        !           101:  * so we will simply calculate R - Y and see if that underflows.
        !           102:  * This leads to the following revised version of the algorithm:
        !           103:  *
        !           104:  *     R = X;
        !           105:  *     bit = FP_1;
        !           106:  *     D = R - Y;
        !           107:  *     if (D >= 0) {
        !           108:  *             result_exp = x->fp_exp - y->fp_exp;
        !           109:  *             R = D;
        !           110:  *             q = bit;
        !           111:  *             bit >>= 1;
        !           112:  *     } else {
        !           113:  *             result_exp = x->fp_exp - y->fp_exp - 1;
        !           114:  *             q = 0;
        !           115:  *     }
        !           116:  *     R <<= 1;
        !           117:  *     do  {
        !           118:  *             D = R - Y;
        !           119:  *             if (D >= 0) {
        !           120:  *                     q |= bit;
        !           121:  *                     R = D;
        !           122:  *             }
        !           123:  *             R <<= 1;
        !           124:  *     } while ((bit >>= 1) != 0);
        !           125:  *     Q[0] = q;
        !           126:  *     for (i = 1; i < 4; i++) {
        !           127:  *             q = 0, bit = 1 << 31;
        !           128:  *             do {
        !           129:  *                     D = R - Y;
        !           130:  *                     if (D >= 0) {
        !           131:  *                             q |= bit;
        !           132:  *                             R = D;
        !           133:  *                     }
        !           134:  *                     R <<= 1;
        !           135:  *             } while ((bit >>= 1) != 0);
        !           136:  *             Q[i] = q;
        !           137:  *     }
        !           138:  *
        !           139:  * This can be refined just a bit further by moving the `R <<= 1'
        !           140:  * calculations to the front of the do-loops and eliding the first one.
        !           141:  * The process can be terminated immediately whenever R becomes 0, but
        !           142:  * this is relatively rare, and we do not bother.
        !           143:  */
        !           144: 
        !           145: struct fpn *
        !           146: fpu_div(struct fpemu *fe)
        !           147: {
        !           148:        struct fpn *x = &fe->fe_f1, *y = &fe->fe_f2;
        !           149:        uint32_t q, bit;
        !           150:        uint32_t r0, r1, r2, d0, d1, d2, y0, y1, y2;
        !           151:        FPU_DECL_CARRY
        !           152: 
        !           153:        fe->fe_fpsr &= ~FPSR_EXCP; /* clear all exceptions */
        !           154: 
        !           155:        /*
        !           156:         * Since divide is not commutative, we cannot just use ORDER.
        !           157:         * Check either operand for NaN first; if there is at least one,
        !           158:         * order the signalling one (if only one) onto the right, then
        !           159:         * return it.  Otherwise we have the following cases:
        !           160:         *
        !           161:         *      Inf / Inf = NaN, plus NV exception
        !           162:         *      Inf / num = Inf
        !           163:         *      Inf / 0   = Inf
        !           164:         *      0   / Inf = 0
        !           165:         *      0   / num = 0
        !           166:         *      0   / 0   = NaN, plus NV exception
        !           167:         *      num / Inf = 0
        !           168:         *      num / num = num (do the divide)
        !           169:         *      num / 0   = Inf, plus DZ exception
        !           170:         */
        !           171:        if (ISNAN(x) || ISNAN(y)) {
        !           172:                ORDER(x, y);
        !           173:                return (y);
        !           174:        }
        !           175:        if (ISINF(x) || ISZERO(x)) {
        !           176:                if (x->fp_class == y->fp_class)
        !           177:                        return (fpu_newnan(fe));
        !           178:                /* all results at this point use XOR of operand signs */
        !           179:                x->fp_sign ^= y->fp_sign;
        !           180:                return (x);
        !           181:        }
        !           182: 
        !           183:        /* all results at this point use XOR of operand signs */
        !           184:        x->fp_sign ^= y->fp_sign;
        !           185:        if (ISINF(y)) {
        !           186:                x->fp_class = FPC_ZERO;
        !           187:                return (x);
        !           188:        }
        !           189:        if (ISZERO(y)) {
        !           190:                fe->fe_fpsr |= FPSR_DZ;
        !           191:                x->fp_class = FPC_INF;
        !           192:                return (x);
        !           193:        }
        !           194: 
        !           195:        /*
        !           196:         * Macros for the divide.  See comments at top for algorithm.
        !           197:         * Note that we expand R, D, and Y here.
        !           198:         */
        !           199: 
        !           200: #define        SUBTRACT                /* D = R - Y */ \
        !           201:        FPU_SUBS(d2, r2, y2); \
        !           202:        FPU_SUBCS(d1, r1, y1); FPU_SUBC(d0, r0, y0)
        !           203: 
        !           204: #define        NONNEGATIVE             /* D >= 0 */ \
        !           205:        ((int)d0 >= 0)
        !           206: 
        !           207: #ifdef FPU_SHL1_BY_ADD
        !           208: #define        SHL1                    /* R <<= 1 */ \
        !           209:        FPU_ADDS(r2, r2, r2); \
        !           210:        FPU_ADDCS(r1, r1, r1); FPU_ADDC(r0, r0, r0)
        !           211: #else
        !           212: #define        SHL1 \
        !           213:        r0 = (r0 << 1) | (r1 >> 31), r1 = (r1 << 1) | (r2 >> 31), \
        !           214:        r2 <<= 1
        !           215: #endif
        !           216: 
        !           217: #define        LOOP                    /* do ... while (bit >>= 1) */ \
        !           218:        do { \
        !           219:                SHL1; \
        !           220:                SUBTRACT; \
        !           221:                if (NONNEGATIVE) { \
        !           222:                        q |= bit; \
        !           223:                        r0 = d0, r1 = d1, r2 = d2; \
        !           224:                } \
        !           225:        } while ((bit >>= 1) != 0)
        !           226: 
        !           227: #define        WORD(r, i)                      /* calculate r->fp_mant[i] */ \
        !           228:        q = 0; \
        !           229:        bit = 1 << 31; \
        !           230:        LOOP; \
        !           231:        (x)->fp_mant[i] = q
        !           232: 
        !           233:        /* Setup.  Note that we put our result in x. */
        !           234:        r0 = x->fp_mant[0];
        !           235:        r1 = x->fp_mant[1];
        !           236:        r2 = x->fp_mant[2];
        !           237:        y0 = y->fp_mant[0];
        !           238:        y1 = y->fp_mant[1];
        !           239:        y2 = y->fp_mant[2];
        !           240: 
        !           241:        bit = FP_1;
        !           242:        SUBTRACT;
        !           243:        if (NONNEGATIVE) {
        !           244:                x->fp_exp -= y->fp_exp;
        !           245:                r0 = d0, r1 = d1, r2 = d2;
        !           246:                q = bit;
        !           247:                bit >>= 1;
        !           248:        } else {
        !           249:                x->fp_exp -= y->fp_exp + 1;
        !           250:                q = 0;
        !           251:        }
        !           252:        LOOP;
        !           253:        x->fp_mant[0] = q;
        !           254:        WORD(x, 1);
        !           255:        WORD(x, 2);
        !           256:        x->fp_sticky = r0 | r1 | r2;
        !           257: 
        !           258:        return (x);
        !           259: }

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