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|
/*---------------------------------------------------------------------------+
| poly_sin.c |
| |
| Computation of an approximation of the sin function and the cosine |
| function by a polynomial. |
| |
| Copyright (C) 1992,1993,1994,1997,1999 |
| W. Metzenthen, 22 Parker St, Ormond, Vic 3163, Australia |
| E-mail billm@melbpc.org.au |
| |
| |
+---------------------------------------------------------------------------*/
#include "exception.h"
#include "reg_constant.h"
#include "fpu_emu.h"
#include "fpu_system.h"
#include "control_w.h"
#include "poly.h"
#define N_COEFF_P 4
#define N_COEFF_N 4
static const unsigned long long pos_terms_l[N_COEFF_P] =
{
0xaaaaaaaaaaaaaaabLL,
0x00d00d00d00cf906LL,
0x000006b99159a8bbLL,
0x000000000d7392e6LL
};
static const unsigned long long neg_terms_l[N_COEFF_N] =
{
0x2222222222222167LL,
0x0002e3bc74aab624LL,
0x0000000b09229062LL,
0x00000000000c7973LL
};
#define N_COEFF_PH 4
#define N_COEFF_NH 4
static const unsigned long long pos_terms_h[N_COEFF_PH] =
{
0x0000000000000000LL,
0x05b05b05b05b0406LL,
0x000049f93edd91a9LL,
0x00000000c9c9ed62LL
};
static const unsigned long long neg_terms_h[N_COEFF_NH] =
{
0xaaaaaaaaaaaaaa98LL,
0x001a01a01a019064LL,
0x0000008f76c68a77LL,
0x0000000000d58f5eLL
};
/*--- poly_sine() -----------------------------------------------------------+
| |
+---------------------------------------------------------------------------*/
void poly_sine(FPU_REG *st0_ptr)
{
int exponent, echange;
Xsig accumulator, argSqrd, argTo4;
unsigned long fix_up, adj;
unsigned long long fixed_arg;
FPU_REG result;
exponent = exponent(st0_ptr);
accumulator.lsw = accumulator.midw = accumulator.msw = 0;
/* Split into two ranges, for arguments below and above 1.0 */
/* The boundary between upper and lower is approx 0.88309101259 */
if ( (exponent < -1) || ((exponent == -1) && (st0_ptr->sigh <= 0xe21240aa)) )
{
/* The argument is <= 0.88309101259 */
argSqrd.msw = st0_ptr->sigh; argSqrd.midw = st0_ptr->sigl; argSqrd.lsw = 0;
mul64_Xsig(&argSqrd, &significand(st0_ptr));
shr_Xsig(&argSqrd, 2*(-1-exponent));
argTo4.msw = argSqrd.msw; argTo4.midw = argSqrd.midw;
argTo4.lsw = argSqrd.lsw;
mul_Xsig_Xsig(&argTo4, &argTo4);
polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), neg_terms_l,
N_COEFF_N-1);
mul_Xsig_Xsig(&accumulator, &argSqrd);
negate_Xsig(&accumulator);
polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), pos_terms_l,
N_COEFF_P-1);
shr_Xsig(&accumulator, 2); /* Divide by four */
accumulator.msw |= 0x80000000; /* Add 1.0 */
mul64_Xsig(&accumulator, &significand(st0_ptr));
mul64_Xsig(&accumulator, &significand(st0_ptr));
mul64_Xsig(&accumulator, &significand(st0_ptr));
/* Divide by four, FPU_REG compatible, etc */
exponent = 3*exponent;
/* The minimum exponent difference is 3 */
shr_Xsig(&accumulator, exponent(st0_ptr) - exponent);
negate_Xsig(&accumulator);
XSIG_LL(accumulator) += significand(st0_ptr);
echange = round_Xsig(&accumulator);
setexponentpos(&result, exponent(st0_ptr) + echange);
}
else
{
/* The argument is > 0.88309101259 */
/* We use sin(st(0)) = cos(pi/2-st(0)) */
fixed_arg = significand(st0_ptr);
if ( exponent == 0 )
{
/* The argument is >= 1.0 */
/* Put the binary point at the left. */
fixed_arg <<= 1;
}
/* pi/2 in hex is: 1.921fb54442d18469 898CC51701B839A2 52049C1 */
fixed_arg = 0x921fb54442d18469LL - fixed_arg;
/* There is a special case which arises due to rounding, to fix here. */
if ( fixed_arg == 0xffffffffffffffffLL )
fixed_arg = 0;
XSIG_LL(argSqrd) = fixed_arg; argSqrd.lsw = 0;
mul64_Xsig(&argSqrd, &fixed_arg);
XSIG_LL(argTo4) = XSIG_LL(argSqrd); argTo4.lsw = argSqrd.lsw;
mul_Xsig_Xsig(&argTo4, &argTo4);
polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), neg_terms_h,
N_COEFF_NH-1);
mul_Xsig_Xsig(&accumulator, &argSqrd);
negate_Xsig(&accumulator);
polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), pos_terms_h,
N_COEFF_PH-1);
negate_Xsig(&accumulator);
mul64_Xsig(&accumulator, &fixed_arg);
mul64_Xsig(&accumulator, &fixed_arg);
shr_Xsig(&accumulator, 3);
negate_Xsig(&accumulator);
add_Xsig_Xsig(&accumulator, &argSqrd);
shr_Xsig(&accumulator, 1);
accumulator.lsw |= 1; /* A zero accumulator here would cause problems */
negate_Xsig(&accumulator);
/* The basic computation is complete. Now fix the answer to
compensate for the error due to the approximation used for
pi/2
*/
/* This has an exponent of -65 */
fix_up = 0x898cc517;
/* The fix-up needs to be improved for larger args */
if ( argSqrd.msw & 0xffc00000 )
{
/* Get about 32 bit precision in these: */
fix_up -= mul_32_32(0x898cc517, argSqrd.msw) / 6;
}
fix_up = mul_32_32(fix_up, LL_MSW(fixed_arg));
adj = accumulator.lsw; /* temp save */
accumulator.lsw -= fix_up;
if ( accumulator.lsw > adj )
XSIG_LL(accumulator) --;
echange = round_Xsig(&accumulator);
setexponentpos(&result, echange - 1);
}
significand(&result) = XSIG_LL(accumulator);
setsign(&result, getsign(st0_ptr));
FPU_copy_to_reg0(&result, TAG_Valid);
#ifdef PARANOID
if ( (exponent(&result) >= 0)
&& (significand(&result) > 0x8000000000000000LL) )
{
EXCEPTION(EX_INTERNAL|0x150);
}
#endif /* PARANOID */
}
/*--- poly_cos() ------------------------------------------------------------+
| |
+---------------------------------------------------------------------------*/
void poly_cos(FPU_REG *st0_ptr)
{
FPU_REG result;
long int exponent, exp2, echange;
Xsig accumulator, argSqrd, fix_up, argTo4;
unsigned long long fixed_arg;
#ifdef PARANOID
if ( (exponent(st0_ptr) > 0)
|| ((exponent(st0_ptr) == 0)
&& (significand(st0_ptr) > 0xc90fdaa22168c234LL)) )
{
EXCEPTION(EX_Invalid);
FPU_copy_to_reg0(&CONST_QNaN, TAG_Special);
return;
}
#endif /* PARANOID */
exponent = exponent(st0_ptr);
accumulator.lsw = accumulator.midw = accumulator.msw = 0;
if ( (exponent < -1) || ((exponent == -1) && (st0_ptr->sigh <= 0xb00d6f54)) )
{
/* arg is < 0.687705 */
argSqrd.msw = st0_ptr->sigh; argSqrd.midw = st0_ptr->sigl;
argSqrd.lsw = 0;
mul64_Xsig(&argSqrd, &significand(st0_ptr));
if ( exponent < -1 )
{
/* shift the argument right by the required places */
shr_Xsig(&argSqrd, 2*(-1-exponent));
}
argTo4.msw = argSqrd.msw; argTo4.midw = argSqrd.midw;
argTo4.lsw = argSqrd.lsw;
mul_Xsig_Xsig(&argTo4, &argTo4);
polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), neg_terms_h,
N_COEFF_NH-1);
mul_Xsig_Xsig(&accumulator, &argSqrd);
negate_Xsig(&accumulator);
polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), pos_terms_h,
N_COEFF_PH-1);
negate_Xsig(&accumulator);
mul64_Xsig(&accumulator, &significand(st0_ptr));
mul64_Xsig(&accumulator, &significand(st0_ptr));
shr_Xsig(&accumulator, -2*(1+exponent));
shr_Xsig(&accumulator, 3);
negate_Xsig(&accumulator);
add_Xsig_Xsig(&accumulator, &argSqrd);
shr_Xsig(&accumulator, 1);
/* It doesn't matter if accumulator is all zero here, the
following code will work ok */
negate_Xsig(&accumulator);
if ( accumulator.lsw & 0x80000000 )
XSIG_LL(accumulator) ++;
if ( accumulator.msw == 0 )
{
/* The result is 1.0 */
FPU_copy_to_reg0(&CONST_1, TAG_Valid);
return;
}
else
{
significand(&result) = XSIG_LL(accumulator);
/* will be a valid positive nr with expon = -1 */
setexponentpos(&result, -1);
}
}
else
{
fixed_arg = significand(st0_ptr);
if ( exponent == 0 )
{
/* The argument is >= 1.0 */
/* Put the binary point at the left. */
fixed_arg <<= 1;
}
/* pi/2 in hex is: 1.921fb54442d18469 898CC51701B839A2 52049C1 */
fixed_arg = 0x921fb54442d18469LL - fixed_arg;
/* There is a special case which arises due to rounding, to fix here. */
if ( fixed_arg == 0xffffffffffffffffLL )
fixed_arg = 0;
exponent = -1;
exp2 = -1;
/* A shift is needed here only for a narrow range of arguments,
i.e. for fixed_arg approx 2^-32, but we pick up more... */
if ( !(LL_MSW(fixed_arg) & 0xffff0000) )
{
fixed_arg <<= 16;
exponent -= 16;
exp2 -= 16;
}
XSIG_LL(argSqrd) = fixed_arg; argSqrd.lsw = 0;
mul64_Xsig(&argSqrd, &fixed_arg);
if ( exponent < -1 )
{
/* shift the argument right by the required places */
shr_Xsig(&argSqrd, 2*(-1-exponent));
}
argTo4.msw = argSqrd.msw; argTo4.midw = argSqrd.midw;
argTo4.lsw = argSqrd.lsw;
mul_Xsig_Xsig(&argTo4, &argTo4);
polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), neg_terms_l,
N_COEFF_N-1);
mul_Xsig_Xsig(&accumulator, &argSqrd);
negate_Xsig(&accumulator);
polynomial_Xsig(&accumulator, &XSIG_LL(argTo4), pos_terms_l,
N_COEFF_P-1);
shr_Xsig(&accumulator, 2); /* Divide by four */
accumulator.msw |= 0x80000000; /* Add 1.0 */
mul64_Xsig(&accumulator, &fixed_arg);
mul64_Xsig(&accumulator, &fixed_arg);
mul64_Xsig(&accumulator, &fixed_arg);
/* Divide by four, FPU_REG compatible, etc */
exponent = 3*exponent;
/* The minimum exponent difference is 3 */
shr_Xsig(&accumulator, exp2 - exponent);
negate_Xsig(&accumulator);
XSIG_LL(accumulator) += fixed_arg;
/* The basic computation is complete. Now fix the answer to
compensate for the error due to the approximation used for
pi/2
*/
/* This has an exponent of -65 */
XSIG_LL(fix_up) = 0x898cc51701b839a2ll;
fix_up.lsw = 0;
/* The fix-up needs to be improved for larger args */
if ( argSqrd.msw & 0xffc00000 )
{
/* Get about 32 bit precision in these: */
fix_up.msw -= mul_32_32(0x898cc517, argSqrd.msw) / 2;
fix_up.msw += mul_32_32(0x898cc517, argTo4.msw) / 24;
}
exp2 += norm_Xsig(&accumulator);
shr_Xsig(&accumulator, 1); /* Prevent overflow */
exp2++;
shr_Xsig(&fix_up, 65 + exp2);
add_Xsig_Xsig(&accumulator, &fix_up);
echange = round_Xsig(&accumulator);
setexponentpos(&result, exp2 + echange);
significand(&result) = XSIG_LL(accumulator);
}
FPU_copy_to_reg0(&result, TAG_Valid);
#ifdef PARANOID
if ( (exponent(&result) >= 0)
&& (significand(&result) > 0x8000000000000000LL) )
{
EXCEPTION(EX_INTERNAL|0x151);
}
#endif /* PARANOID */
}
|
