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Diffstat (limited to 'arch/i386/math-emu/wm_sqrt.S')
| -rw-r--r-- | arch/i386/math-emu/wm_sqrt.S | 470 |
1 files changed, 470 insertions, 0 deletions
diff --git a/arch/i386/math-emu/wm_sqrt.S b/arch/i386/math-emu/wm_sqrt.S new file mode 100644 index 000000000000..d258f59564e1 --- /dev/null +++ b/arch/i386/math-emu/wm_sqrt.S @@ -0,0 +1,470 @@ + .file "wm_sqrt.S" +/*---------------------------------------------------------------------------+ + | wm_sqrt.S | + | | + | Fixed point arithmetic square root evaluation. | + | | + | Copyright (C) 1992,1993,1995,1997 | + | W. Metzenthen, 22 Parker St, Ormond, Vic 3163, | + | Australia. E-mail billm@suburbia.net | + | | + | Call from C as: | + | int wm_sqrt(FPU_REG *n, unsigned int control_word) | + | | + +---------------------------------------------------------------------------*/ + +/*---------------------------------------------------------------------------+ + | 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 "exception.h" +#include "fpu_emu.h" + + +#ifndef NON_REENTRANT_FPU +/* Local storage on the stack: */ +#define FPU_accum_3 -4(%ebp) /* ms word */ +#define FPU_accum_2 -8(%ebp) +#define FPU_accum_1 -12(%ebp) +#define FPU_accum_0 -16(%ebp) + +/* + * 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 + */ +#define FPU_fsqrt_arg_2 -20(%ebp) /* ms word */ +#define FPU_fsqrt_arg_1 -24(%ebp) +#define FPU_fsqrt_arg_0 -28(%ebp) /* ls word, at most the ms bit is set */ + +#else +/* Local storage in a static area: */ +.data + .align 4,0 +FPU_accum_3: + .long 0 /* ms word */ +FPU_accum_2: + .long 0 +FPU_accum_1: + .long 0 +FPU_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 + */ +FPU_fsqrt_arg_2: + .long 0 /* ms word */ +FPU_fsqrt_arg_1: + .long 0 +FPU_fsqrt_arg_0: + .long 0 /* ls word, at most the ms bit is set */ +#endif /* NON_REENTRANT_FPU */ + + +.text +ENTRY(wm_sqrt) + pushl %ebp + movl %esp,%ebp +#ifndef NON_REENTRANT_FPU + subl $28,%esp +#endif /* NON_REENTRANT_FPU */ + 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.. */ + + cmpw 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,FPU_fsqrt_arg_2 /* ms word of n */ + movl %ecx,FPU_fsqrt_arg_1 + movl %edx,FPU_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 FPU_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 FPU_fsqrt_arg_1,%ecx + subl %ecx,%eax + movl FPU_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,FPU_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,FPU_accum_1 + + movl %esi,%eax + mull %esi + movl %edx,FPU_accum_3 + movl %eax,FPU_accum_2 + + movl %edi,%eax + mull %esi + addl %eax,FPU_accum_1 + adcl %edx,FPU_accum_2 + adcl $0,FPU_accum_3 + +/* movl %esi,%eax */ +/* mull %edi */ + addl %eax,FPU_accum_1 + adcl %edx,FPU_accum_2 + adcl $0,FPU_accum_3 + +/* guess^2 now in FPU_accum_3:FPU_accum_2:FPU_accum_1 */ + + movl FPU_fsqrt_arg_0,%eax /* get normalized n */ + subl %eax,FPU_accum_1 + movl FPU_fsqrt_arg_1,%eax + sbbl %eax,FPU_accum_2 + movl FPU_fsqrt_arg_2,%eax /* ms word of normalized n */ + sbbl %eax,FPU_accum_3 + jnc sqrt_stage_3_positive + +/* Subtraction gives a negative result, + negate the result before division */ + notl FPU_accum_1 + notl FPU_accum_2 + notl FPU_accum_3 + addl $1,FPU_accum_1 + adcl $0,FPU_accum_2 + +#ifdef PARANOID + adcl $0,FPU_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 FPU_accum_2,%edx + movl FPU_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 FPU_accum_2,%edx + movl FPU_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 + movw EXP_BIAS,EXP(%edi) /* Result is in [1.0 .. 2.0) */ + jmp fpu_reg_round + + +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 |