/* SPDX-License-Identifier: GPL-2.0 */
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.file "wm_sqrt.S"
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/*---------------------------------------------------------------------------+
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| wm_sqrt.S |
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| |
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| Fixed point arithmetic square root evaluation. |
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| Copyright (C) 1992,1993,1995,1997 |
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| W. Metzenthen, 22 Parker St, Ormond, Vic 3163, |
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| Australia. E-mail billm@suburbia.net |
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| |
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| Call from C as: |
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| int wm_sqrt(FPU_REG *n, unsigned int control_word) |
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| |
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+---------------------------------------------------------------------------*/
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/*---------------------------------------------------------------------------+
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| wm_sqrt(FPU_REG *n, unsigned int control_word) |
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| returns the square root of n in n. |
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| |
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| Use Newton's method to compute the square root of a number, which must |
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| be in the range [1.0 .. 4.0), to 64 bits accuracy. |
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| Does not check the sign or tag of the argument. |
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| Sets the exponent, but not the sign or tag of the result. |
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| |
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| The guess is kept in %esi:%edi |
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+---------------------------------------------------------------------------*/
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#include "exception.h"
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#include "fpu_emu.h"
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#ifndef NON_REENTRANT_FPU
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/* Local storage on the stack: */
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#define FPU_accum_3 -4(%ebp) /* ms word */
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#define FPU_accum_2 -8(%ebp)
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#define FPU_accum_1 -12(%ebp)
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#define FPU_accum_0 -16(%ebp)
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/*
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* The de-normalised argument:
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* sq_2 sq_1 sq_0
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* b b b b b b b ... b b b b b b .... b b b b 0 0 0 ... 0
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* ^ binary point here
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*/
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#define FPU_fsqrt_arg_2 -20(%ebp) /* ms word */
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#define FPU_fsqrt_arg_1 -24(%ebp)
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#define FPU_fsqrt_arg_0 -28(%ebp) /* ls word, at most the ms bit is set */
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#else
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/* Local storage in a static area: */
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.data
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.align 4,0
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FPU_accum_3:
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.long 0 /* ms word */
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FPU_accum_2:
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.long 0
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FPU_accum_1:
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.long 0
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FPU_accum_0:
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.long 0
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/* The de-normalised argument:
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sq_2 sq_1 sq_0
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b b b b b b b ... b b b b b b .... b b b b 0 0 0 ... 0
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^ binary point here
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*/
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FPU_fsqrt_arg_2:
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.long 0 /* ms word */
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FPU_fsqrt_arg_1:
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.long 0
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FPU_fsqrt_arg_0:
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.long 0 /* ls word, at most the ms bit is set */
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#endif /* NON_REENTRANT_FPU */
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.text
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SYM_FUNC_START(wm_sqrt)
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pushl %ebp
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movl %esp,%ebp
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#ifndef NON_REENTRANT_FPU
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subl $28,%esp
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#endif /* NON_REENTRANT_FPU */
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pushl %esi
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pushl %edi
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pushl %ebx
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movl PARAM1,%esi
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movl SIGH(%esi),%eax
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movl SIGL(%esi),%ecx
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xorl %edx,%edx
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/* We use a rough linear estimate for the first guess.. */
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cmpw EXP_BIAS,EXP(%esi)
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jnz sqrt_arg_ge_2
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shrl $1,%eax /* arg is in the range [1.0 .. 2.0) */
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rcrl $1,%ecx
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rcrl $1,%edx
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sqrt_arg_ge_2:
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/* From here on, n is never accessed directly again until it is
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replaced by the answer. */
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movl %eax,FPU_fsqrt_arg_2 /* ms word of n */
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movl %ecx,FPU_fsqrt_arg_1
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movl %edx,FPU_fsqrt_arg_0
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/* Make a linear first estimate */
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shrl $1,%eax
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addl $0x40000000,%eax
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movl $0xaaaaaaaa,%ecx
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mull %ecx
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shll %edx /* max result was 7fff... */
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testl $0x80000000,%edx /* but min was 3fff... */
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jnz sqrt_prelim_no_adjust
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movl $0x80000000,%edx /* round up */
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sqrt_prelim_no_adjust:
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movl %edx,%esi /* Our first guess */
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/* We have now computed (approx) (2 + x) / 3, which forms the basis
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for a few iterations of Newton's method */
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movl FPU_fsqrt_arg_2,%ecx /* ms word */
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/*
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* From our initial estimate, three iterations are enough to get us
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* to 30 bits or so. This will then allow two iterations at better
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* precision to complete the process.
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*/
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/* Compute (g + n/g)/2 at each iteration (g is the guess). */
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shrl %ecx /* Doing this first will prevent a divide */
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/* overflow later. */
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movl %ecx,%edx /* msw of the arg / 2 */
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divl %esi /* current estimate */
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shrl %esi /* divide by 2 */
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addl %eax,%esi /* the new estimate */
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movl %ecx,%edx
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divl %esi
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shrl %esi
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addl %eax,%esi
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movl %ecx,%edx
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divl %esi
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shrl %esi
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addl %eax,%esi
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/*
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* Now that an estimate accurate to about 30 bits has been obtained (in %esi),
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* we improve it to 60 bits or so.
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*
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* The strategy from now on is to compute new estimates from
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* guess := guess + (n - guess^2) / (2 * guess)
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*/
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/* First, find the square of the guess */
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movl %esi,%eax
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mull %esi
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/* guess^2 now in %edx:%eax */
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movl FPU_fsqrt_arg_1,%ecx
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subl %ecx,%eax
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movl FPU_fsqrt_arg_2,%ecx /* ms word of normalized n */
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sbbl %ecx,%edx
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jnc sqrt_stage_2_positive
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/* Subtraction gives a negative result,
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negate the result before division. */
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notl %edx
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notl %eax
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addl $1,%eax
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adcl $0,%edx
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divl %esi
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movl %eax,%ecx
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movl %edx,%eax
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divl %esi
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jmp sqrt_stage_2_finish
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sqrt_stage_2_positive:
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divl %esi
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movl %eax,%ecx
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movl %edx,%eax
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divl %esi
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notl %ecx
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notl %eax
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addl $1,%eax
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adcl $0,%ecx
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sqrt_stage_2_finish:
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sarl $1,%ecx /* divide by 2 */
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rcrl $1,%eax
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/* Form the new estimate in %esi:%edi */
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movl %eax,%edi
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addl %ecx,%esi
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jnz sqrt_stage_2_done /* result should be [1..2) */
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#ifdef PARANOID
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/* It should be possible to get here only if the arg is ffff....ffff */
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cmpl $0xffffffff,FPU_fsqrt_arg_1
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jnz sqrt_stage_2_error
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#endif /* PARANOID */
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/* The best rounded result. */
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xorl %eax,%eax
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decl %eax
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movl %eax,%edi
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movl %eax,%esi
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movl $0x7fffffff,%eax
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jmp sqrt_round_result
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#ifdef PARANOID
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sqrt_stage_2_error:
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pushl EX_INTERNAL|0x213
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call EXCEPTION
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#endif /* PARANOID */
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sqrt_stage_2_done:
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/* Now the square root has been computed to better than 60 bits. */
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/* Find the square of the guess. */
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movl %edi,%eax /* ls word of guess */
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mull %edi
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movl %edx,FPU_accum_1
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movl %esi,%eax
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mull %esi
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movl %edx,FPU_accum_3
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movl %eax,FPU_accum_2
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movl %edi,%eax
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mull %esi
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addl %eax,FPU_accum_1
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adcl %edx,FPU_accum_2
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adcl $0,FPU_accum_3
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/* movl %esi,%eax */
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/* mull %edi */
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addl %eax,FPU_accum_1
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adcl %edx,FPU_accum_2
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adcl $0,FPU_accum_3
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/* guess^2 now in FPU_accum_3:FPU_accum_2:FPU_accum_1 */
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movl FPU_fsqrt_arg_0,%eax /* get normalized n */
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subl %eax,FPU_accum_1
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movl FPU_fsqrt_arg_1,%eax
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sbbl %eax,FPU_accum_2
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movl FPU_fsqrt_arg_2,%eax /* ms word of normalized n */
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sbbl %eax,FPU_accum_3
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jnc sqrt_stage_3_positive
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/* Subtraction gives a negative result,
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negate the result before division */
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notl FPU_accum_1
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notl FPU_accum_2
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notl FPU_accum_3
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addl $1,FPU_accum_1
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adcl $0,FPU_accum_2
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#ifdef PARANOID
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adcl $0,FPU_accum_3 /* This must be zero */
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jz sqrt_stage_3_no_error
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sqrt_stage_3_error:
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pushl EX_INTERNAL|0x207
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call EXCEPTION
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sqrt_stage_3_no_error:
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#endif /* PARANOID */
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movl FPU_accum_2,%edx
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movl FPU_accum_1,%eax
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divl %esi
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movl %eax,%ecx
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movl %edx,%eax
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divl %esi
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sarl $1,%ecx /* divide by 2 */
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rcrl $1,%eax
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/* prepare to round the result */
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addl %ecx,%edi
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adcl $0,%esi
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jmp sqrt_stage_3_finished
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sqrt_stage_3_positive:
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movl FPU_accum_2,%edx
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movl FPU_accum_1,%eax
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divl %esi
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movl %eax,%ecx
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movl %edx,%eax
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divl %esi
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sarl $1,%ecx /* divide by 2 */
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rcrl $1,%eax
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/* prepare to round the result */
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notl %eax /* Negate the correction term */
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notl %ecx
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addl $1,%eax
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adcl $0,%ecx /* carry here ==> correction == 0 */
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adcl $0xffffffff,%esi
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addl %ecx,%edi
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adcl $0,%esi
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sqrt_stage_3_finished:
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/*
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* The result in %esi:%edi:%esi should be good to about 90 bits here,
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* and the rounding information here does not have sufficient accuracy
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* in a few rare cases.
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*/
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cmpl $0xffffffe0,%eax
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ja sqrt_near_exact_x
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cmpl $0x00000020,%eax
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jb sqrt_near_exact
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cmpl $0x7fffffe0,%eax
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jb sqrt_round_result
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cmpl $0x80000020,%eax
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jb sqrt_get_more_precision
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sqrt_round_result:
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/* Set up for rounding operations */
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movl %eax,%edx
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movl %esi,%eax
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movl %edi,%ebx
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movl PARAM1,%edi
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movw EXP_BIAS,EXP(%edi) /* Result is in [1.0 .. 2.0) */
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jmp fpu_reg_round
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sqrt_near_exact_x:
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/* First, the estimate must be rounded up. */
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addl $1,%edi
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adcl $0,%esi
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sqrt_near_exact:
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/*
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* This is an easy case because x^1/2 is monotonic.
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* We need just find the square of our estimate, compare it
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* with the argument, and deduce whether our estimate is
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* above, below, or exact. We use the fact that the estimate
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* is known to be accurate to about 90 bits.
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*/
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movl %edi,%eax /* ls word of guess */
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mull %edi
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movl %edx,%ebx /* 2nd ls word of square */
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movl %eax,%ecx /* ls word of square */
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movl %edi,%eax
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mull %esi
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addl %eax,%ebx
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addl %eax,%ebx
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#ifdef PARANOID
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cmp $0xffffffb0,%ebx
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jb sqrt_near_exact_ok
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cmp $0x00000050,%ebx
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ja sqrt_near_exact_ok
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pushl EX_INTERNAL|0x214
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call EXCEPTION
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sqrt_near_exact_ok:
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#endif /* PARANOID */
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or %ebx,%ebx
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js sqrt_near_exact_small
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jnz sqrt_near_exact_large
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or %ebx,%edx
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jnz sqrt_near_exact_large
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/* Our estimate is exactly the right answer */
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xorl %eax,%eax
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jmp sqrt_round_result
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sqrt_near_exact_small:
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/* Our estimate is too small */
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movl $0x000000ff,%eax
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jmp sqrt_round_result
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sqrt_near_exact_large:
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/* Our estimate is too large, we need to decrement it */
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subl $1,%edi
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sbbl $0,%esi
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movl $0xffffff00,%eax
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jmp sqrt_round_result
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sqrt_get_more_precision:
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/* This case is almost the same as the above, except we start
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with an extra bit of precision in the estimate. */
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stc /* The extra bit. */
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rcll $1,%edi /* Shift the estimate left one bit */
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rcll $1,%esi
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movl %edi,%eax /* ls word of guess */
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mull %edi
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movl %edx,%ebx /* 2nd ls word of square */
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movl %eax,%ecx /* ls word of square */
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movl %edi,%eax
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mull %esi
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addl %eax,%ebx
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addl %eax,%ebx
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/* Put our estimate back to its original value */
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stc /* The ms bit. */
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rcrl $1,%esi /* Shift the estimate left one bit */
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rcrl $1,%edi
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#ifdef PARANOID
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cmp $0xffffff60,%ebx
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jb sqrt_more_prec_ok
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cmp $0x000000a0,%ebx
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ja sqrt_more_prec_ok
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pushl EX_INTERNAL|0x215
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call EXCEPTION
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sqrt_more_prec_ok:
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#endif /* PARANOID */
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or %ebx,%ebx
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js sqrt_more_prec_small
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jnz sqrt_more_prec_large
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or %ebx,%ecx
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jnz sqrt_more_prec_large
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/* Our estimate is exactly the right answer */
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movl $0x80000000,%eax
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jmp sqrt_round_result
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sqrt_more_prec_small:
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/* Our estimate is too small */
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movl $0x800000ff,%eax
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jmp sqrt_round_result
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sqrt_more_prec_large:
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/* Our estimate is too large */
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movl $0x7fffff00,%eax
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jmp sqrt_round_result
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SYM_FUNC_END(wm_sqrt)
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