.file "wm_sqrt.S" /* * 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: src/sys/gnu/i386/fpemul/wm_sqrt.s,v 1.9.2.1 2000/07/07 00:38:42 obrien Exp $ * $DragonFly: src/sys/i386/gnu/fpemul/Attic/wm_sqrt.s,v 1.3 2003/08/07 21:17:21 dillon 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 "fpu_asm.h" .data /* Local storage: */ ALIGN_DATA 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 ENTRY(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