/* Limited measurements show no results worse than 64 bit precision
except for the results for arguments close to 2^63, where the
precision of the result sometimes degrades to about 63.9 bits */
static int trig_arg(FPU_REG *st0_ptr, int even)
{
FPU_REG tmp;
u_char tmptag;
unsigned long long q;
int old_cw = control_word, saved_status = partial_status;
int tag, st0_tag = TAG_Valid;
if ( exponent(st0_ptr) >= 63 )
{
partial_status |= SW_C2; /* Reduction incomplete. */
return -1;
}
control_word &= ~CW_RC;
control_word |= RC_CHOP;
setpositive(st0_ptr);
tag = FPU_u_div(st0_ptr, &CONST_PI2, &tmp, PR_64_BITS | RC_CHOP | 0x3f,
SIGN_POS);
FPU_round_to_int(&tmp, tag); /* Fortunately, this can't overflow
to 2^64 */
q = significand(&tmp);
if ( q )
{
rem_kernel(significand(st0_ptr),
&significand(&tmp),
significand(&CONST_PI2),
q, exponent(st0_ptr) - exponent(&CONST_PI2));
setexponent16(&tmp, exponent(&CONST_PI2));
st0_tag = FPU_normalize(&tmp);
FPU_copy_to_reg0(&tmp, st0_tag);
}
if ( (even && !(q & 1)) || (!even && (q & 1)) )
{
st0_tag = FPU_sub(REV|LOADED|TAG_Valid, (int)&CONST_PI2, FULL_PRECISION);
#ifdef BETTER_THAN_486
/* So far, the results are exact but based upon a 64 bit
precision approximation to pi/2. The technique used
now is equivalent to using an approximation to pi/2 which
is accurate to about 128 bits. */
if ( (exponent(st0_ptr) <= exponent(&CONST_PI2extra) + 64) || (q > 1) )
{
/* This code gives the effect of having pi/2 to better than
128 bits precision. */
significand(&tmp) = q + 1;
setexponent16(&tmp, 63);
FPU_normalize(&tmp);
tmptag =
FPU_u_mul(&CONST_PI2extra, &tmp, &tmp, FULL_PRECISION, SIGN_POS,
exponent(&CONST_PI2extra) + exponent(&tmp));
setsign(&tmp, getsign(&CONST_PI2extra));
st0_tag = FPU_add(&tmp, tmptag, 0, FULL_PRECISION);
if ( signnegative(st0_ptr) )
{
/* CONST_PI2extra is negative, so the result of the addition
can be negative. This means that the argument is actually
in a different quadrant. The correction is always < pi/2,
so it can't overflow into yet another quadrant. */
setpositive(st0_ptr);
q++;
}
}
#endif /* BETTER_THAN_486 */
}
#ifdef BETTER_THAN_486
else
{
/* So far, the results are exact but based upon a 64 bit
precision approximation to pi/2. The technique used
now is equivalent to using an approximation to pi/2 which
is accurate to about 128 bits. */
if ( ((q > 0) && (exponent(st0_ptr) <= exponent(&CONST_PI2extra) + 64))
|| (q > 1) )
{
/* This code gives the effect of having p/2 to better than
128 bits precision. */
significand(&tmp) = q;
setexponent16(&tmp, 63);
FPU_normalize(&tmp); /* This must return TAG_Valid */
tmptag = FPU_u_mul(&CONST_PI2extra, &tmp, &tmp, FULL_PRECISION,
SIGN_POS,
exponent(&CONST_PI2extra) + exponent(&tmp));
setsign(&tmp, getsign(&CONST_PI2extra));
st0_tag = FPU_sub(LOADED|(tmptag & 0x0f), (int)&tmp,
FULL_PRECISION);
if ( (exponent(st0_ptr) == exponent(&CONST_PI2)) &&
((st0_ptr->sigh > CONST_PI2.sigh)
|| ((st0_ptr->sigh == CONST_PI2.sigh)
&& (st0_ptr->sigl > CONST_PI2.sigl))) )
{
/* CONST_PI2extra is negative, so the result of the
subtraction can be larger than pi/2. This means
that the argument is actually in a different quadrant.
The correction is always < pi/2, so it can't overflow
into yet another quadrant. */
st0_tag = FPU_sub(REV|LOADED|TAG_Valid, (int)&CONST_PI2,
FULL_PRECISION);
q++;
}
}
}
#endif /* BETTER_THAN_486 */
FPU_settag0(st0_tag);
control_word = old_cw;
partial_status = saved_status & ~SW_C2; /* Reduction complete. */
return (q & 3) | even;
}
/*--- poly_tan() ------------------------------------------------------------+
| |
+---------------------------------------------------------------------------*/
void poly_tan(FPU_REG const *arg, FPU_REG *result)
{
long int exponent;
int invert;
Xsig argSq, argSqSq, accumulatoro, accumulatore, accum,
argSignif, fix_up;
unsigned long adj;
exponent = arg->exp - EXP_BIAS;
#ifdef PARANOID
if ( arg->sign != 0 ) /* Can't hack a number < 0.0 */
{ arith_invalid(result); return; } /* Need a positive number */
#endif PARANOID
/* Split the problem into two domains, smaller and larger than pi/4 */
if ( (exponent == 0) || ((exponent == -1) && (arg->sigh > 0xc90fdaa2)) )
{
/* The argument is greater than (approx) pi/4 */
invert = 1;
accum.lsw = 0;
XSIG_LL(accum) = significand(arg);
if ( exponent == 0 )
{
/* The argument is >= 1.0 */
/* Put the binary point at the left. */
XSIG_LL(accum) <<= 1;
}
/* pi/2 in hex is: 1.921fb54442d18469 898CC51701B839A2 52049C1 */
XSIG_LL(accum) = 0x921fb54442d18469LL - XSIG_LL(accum);
argSignif.lsw = accum.lsw;
XSIG_LL(argSignif) = XSIG_LL(accum);
exponent = -1 + norm_Xsig(&argSignif);
}
else
{
invert = 0;
argSignif.lsw = 0;
XSIG_LL(accum) = XSIG_LL(argSignif) = significand(arg);
if ( exponent < -1 )
{
/* shift the argument right by the required places */
if ( shrx(&XSIG_LL(accum), -1-exponent) >= 0x80000000U )
XSIG_LL(accum) ++; /* round up */
}
}
XSIG_LL(argSq) = XSIG_LL(accum); argSq.lsw = accum.lsw;
mul_Xsig_Xsig(&argSq, &argSq);
XSIG_LL(argSqSq) = XSIG_LL(argSq); argSqSq.lsw = argSq.lsw;
mul_Xsig_Xsig(&argSqSq, &argSqSq);
/* Compute the negative terms for the numerator polynomial */
accumulatoro.msw = accumulatoro.midw = accumulatoro.lsw = 0;
polynomial_Xsig(&accumulatoro, &XSIG_LL(argSqSq), oddnegterm, HiPOWERon-1);
mul_Xsig_Xsig(&accumulatoro, &argSq);
negate_Xsig(&accumulatoro);
/* Add the positive terms */
polynomial_Xsig(&accumulatoro, &XSIG_LL(argSqSq), oddplterm, HiPOWERop-1);
/* Compute the positive terms for the denominator polynomial */
accumulatore.msw = accumulatore.midw = accumulatore.lsw = 0;
polynomial_Xsig(&accumulatore, &XSIG_LL(argSqSq), evenplterm, HiPOWERep-1);
mul_Xsig_Xsig(&accumulatore, &argSq);
negate_Xsig(&accumulatore);
/* Add the negative terms */
polynomial_Xsig(&accumulatore, &XSIG_LL(argSqSq), evennegterm, HiPOWERen-1);
/* Multiply by arg^2 */
mul64_Xsig(&accumulatore, &XSIG_LL(argSignif));
mul64_Xsig(&accumulatore, &XSIG_LL(argSignif));
/* de-normalize and divide by 2 */
shr_Xsig(&accumulatore, -2*(1+exponent) + 1);
negate_Xsig(&accumulatore); /* This does 1 - accumulator */
/* Now find the ratio. */
if ( accumulatore.msw == 0 )
{
/* accumulatoro must contain 1.0 here, (actually, 0) but it
really doesn't matter what value we use because it will
have negligible effect in later calculations
*/
XSIG_LL(accum) = 0x8000000000000000LL;
accum.lsw = 0;
}
else
{
div_Xsig(&accumulatoro, &accumulatore, &accum);
}
/* Multiply by 1/3 * arg^3 */
mul64_Xsig(&accum, &XSIG_LL(argSignif));
mul64_Xsig(&accum, &XSIG_LL(argSignif));
mul64_Xsig(&accum, &XSIG_LL(argSignif));
mul64_Xsig(&accum, &twothirds);
shr_Xsig(&accum, -2*(exponent+1));
/* tan(arg) = arg + accum */
add_two_Xsig(&accum, &argSignif, &exponent);
if ( invert )
{
/* We now have the value of tan(pi_2 - arg) where pi_2 is an
approximation for pi/2
*/
/* The next step is to fix the answer to compensate for the
error due to the approximation used for pi/2
*/
/* This is (approx) delta, the error in our approx for pi/2
(see above). It has an exponent of -65
*/
XSIG_LL(fix_up) = 0x898cc51701b839a2LL;
fix_up.lsw = 0;
if ( exponent == 0 )
adj = 0xffffffff; /* We want approx 1.0 here, but
this is close enough. */
else if ( exponent > -30 )
{
adj = accum.msw >> -(exponent+1); /* tan */
mul_32_32(adj, adj, &adj); /* tan^2 */
}
else
void poly_atan(FPU_REG *st0_ptr, u_char st0_tag,
FPU_REG *st1_ptr, u_char st1_tag)
{
u_char transformed, inverted, sign1, sign2;
int exponent;
long int dummy_exp;
Xsig accumulator, Numer, Denom, accumulatore, argSignif, argSq, argSqSq;
u_char tag;
sign1 = getsign(st0_ptr);
sign2 = getsign(st1_ptr);
if (st0_tag == TAG_Valid) {
exponent = exponent(st0_ptr);
} else {
FPU_to_exp16(st0_ptr, st0_ptr);
exponent = exponent16(st0_ptr);
}
if (st1_tag == TAG_Valid) {
exponent -= exponent(st1_ptr);
} else {
FPU_to_exp16(st1_ptr, st1_ptr);
exponent -= exponent16(st1_ptr);
}
if ((exponent < 0) || ((exponent == 0) &&
((st0_ptr->sigh < st1_ptr->sigh) ||
((st0_ptr->sigh == st1_ptr->sigh) &&
(st0_ptr->sigl < st1_ptr->sigl))))) {
inverted = 1;
Numer.lsw = Denom.lsw = 0;
XSIG_LL(Numer) = significand(st0_ptr);
XSIG_LL(Denom) = significand(st1_ptr);
} else {
inverted = 0;
exponent = -exponent;
Numer.lsw = Denom.lsw = 0;
XSIG_LL(Numer) = significand(st1_ptr);
XSIG_LL(Denom) = significand(st0_ptr);
}
div_Xsig(&Numer, &Denom, &argSignif);
exponent += norm_Xsig(&argSignif);
if ((exponent >= -1)
|| ((exponent == -2) && (argSignif.msw > 0xd413ccd0))) {
transformed = 1;
if (exponent >= 0) {
#ifdef PARANOID
if (!((exponent == 0) &&
(argSignif.lsw == 0) && (argSignif.midw == 0) &&
(argSignif.msw == 0x80000000))) {
EXCEPTION(EX_INTERNAL | 0x104);
return;
}
#endif
argSignif.msw = 0;
} else {
Numer.lsw = Denom.lsw = argSignif.lsw;
XSIG_LL(Numer) = XSIG_LL(Denom) = XSIG_LL(argSignif);
if (exponent < -1)
shr_Xsig(&Numer, -1 - exponent);
negate_Xsig(&Numer);
shr_Xsig(&Denom, -exponent);
Denom.msw |= 0x80000000;
div_Xsig(&Numer, &Denom, &argSignif);
exponent = -1 + norm_Xsig(&argSignif);
}
} else {
transformed = 0;
}
argSq.lsw = argSignif.lsw;
argSq.midw = argSignif.midw;
argSq.msw = argSignif.msw;
mul_Xsig_Xsig(&argSq, &argSq);
argSqSq.lsw = argSq.lsw;
argSqSq.midw = argSq.midw;
argSqSq.msw = argSq.msw;
mul_Xsig_Xsig(&argSqSq, &argSqSq);
accumulatore.lsw = argSq.lsw;
XSIG_LL(accumulatore) = XSIG_LL(argSq);
shr_Xsig(&argSq, 2 * (-1 - exponent - 1));
shr_Xsig(&argSqSq, 4 * (-1 - exponent - 1));
accumulator.msw = accumulator.midw = accumulator.lsw = 0;
polynomial_Xsig(&accumulator, &XSIG_LL(argSqSq),
oddplterms, HIPOWERop - 1);
mul64_Xsig(&accumulator, &XSIG_LL(argSq));
negate_Xsig(&accumulator);
polynomial_Xsig(&accumulator, &XSIG_LL(argSqSq), oddnegterms,
HIPOWERon - 1);
negate_Xsig(&accumulator);
add_two_Xsig(&accumulator, &fixedpterm, &dummy_exp);
mul64_Xsig(&accumulatore, &denomterm);
shr_Xsig(&accumulatore, 1 + 2 * (-1 - exponent));
accumulatore.msw |= 0x80000000;
div_Xsig(&accumulator, &accumulatore, &accumulator);
mul_Xsig_Xsig(&accumulator, &argSignif);
mul_Xsig_Xsig(&accumulator, &argSq);
shr_Xsig(&accumulator, 3);
negate_Xsig(&accumulator);
add_Xsig_Xsig(&accumulator, &argSignif);
if (transformed) {
shr_Xsig(&accumulator, -1 - exponent);
negate_Xsig(&accumulator);
add_Xsig_Xsig(&accumulator, &pi_signif);
exponent = -1;
}
if (inverted) {
shr_Xsig(&accumulator, -exponent);
negate_Xsig(&accumulator);
add_Xsig_Xsig(&accumulator, &pi_signif);
exponent = 0;
}
if (sign1) {
shr_Xsig(&accumulator, 1 - exponent);
negate_Xsig(&accumulator);
add_Xsig_Xsig(&accumulator, &pi_signif);
exponent = 1;
}
exponent += round_Xsig(&accumulator);
significand(st1_ptr) = XSIG_LL(accumulator);
setexponent16(st1_ptr, exponent);
tag = FPU_round(st1_ptr, 1, 0, FULL_PRECISION, sign2);
FPU_settagi(1, tag);
set_precision_flag_up();
}
/*--- poly_l2() -------------------------------------------------------------+
| Base 2 logarithm by a polynomial approximation. |
+---------------------------------------------------------------------------*/
void poly_l2(FPU_REG *st0_ptr, FPU_REG *st1_ptr, u_char st1_sign)
{
s32 exponent, expon, expon_expon;
Xsig accumulator, expon_accum, yaccum;
u_char sign, argsign;
FPU_REG x;
int tag;
exponent = exponent16(st0_ptr);
/* From st0_ptr, make a number > sqrt(2)/2 and < sqrt(2) */
if ( st0_ptr->sigh > (unsigned)0xb504f334 )
{
/* Treat as sqrt(2)/2 < st0_ptr < 1 */
significand(&x) = - significand(st0_ptr);
setexponent16(&x, -1);
exponent++;
argsign = SIGN_NEG;
}
else
{
/* Treat as 1 <= st0_ptr < sqrt(2) */
x.sigh = st0_ptr->sigh - 0x80000000;
x.sigl = st0_ptr->sigl;
setexponent16(&x, 0);
argsign = SIGN_POS;
}
tag = FPU_normalize_nuo(&x, 0);
if ( tag == TAG_Zero )
{
expon = 0;
accumulator.msw = accumulator.midw = accumulator.lsw = 0;
}
else
{
log2_kernel(&x, argsign, &accumulator, &expon);
}
if ( exponent < 0 )
{
sign = SIGN_NEG;
exponent = -exponent;
}
else
sign = SIGN_POS;
expon_accum.msw = exponent; expon_accum.midw = expon_accum.lsw = 0;
if ( exponent )
{
expon_expon = 31 + norm_Xsig(&expon_accum);
shr_Xsig(&accumulator, expon_expon - expon);
if ( sign ^ argsign )
negate_Xsig(&accumulator);
add_Xsig_Xsig(&accumulator, &expon_accum);
}
else
{
expon_expon = expon;
sign = argsign;
}
yaccum.lsw = 0; XSIG_LL(yaccum) = significand(st1_ptr);
mul_Xsig_Xsig(&accumulator, &yaccum);
expon_expon += round_Xsig(&accumulator);
if ( accumulator.msw == 0 )
{
FPU_copy_to_reg1(&CONST_Z, TAG_Zero);
return;
}
significand(st1_ptr) = XSIG_LL(accumulator);
setexponent16(st1_ptr, expon_expon + exponent16(st1_ptr) + 1);
tag = FPU_round(st1_ptr, 1, 0, FULL_PRECISION, sign ^ st1_sign);
FPU_settagi(1, tag);
set_precision_flag_up(); /* 80486 appears to always do this */
return;
}
/*--- log2_kernel() ---------------------------------------------------------+
| Base 2 logarithm by a polynomial approximation. |
| log2(x+1) |
+---------------------------------------------------------------------------*/
static void log2_kernel(FPU_REG const *arg, u_char argsign, Xsig *accum_result,
s32 *expon)
{
s32 exponent, adj;
u64 Xsq;
Xsig accumulator, Numer, Denom, argSignif, arg_signif;
exponent = exponent16(arg);
Numer.lsw = Denom.lsw = 0;
XSIG_LL(Numer) = XSIG_LL(Denom) = significand(arg);
if ( argsign == SIGN_POS )
{
shr_Xsig(&Denom, 2 - (1 + exponent));
Denom.msw |= 0x80000000;
div_Xsig(&Numer, &Denom, &argSignif);
}
else
{
shr_Xsig(&Denom, 1 - (1 + exponent));
negate_Xsig(&Denom);
if ( Denom.msw & 0x80000000 )
{
div_Xsig(&Numer, &Denom, &argSignif);
exponent ++;
}
else
{
/* Denom must be 1.0 */
argSignif.lsw = Numer.lsw; argSignif.midw = Numer.midw;
argSignif.msw = Numer.msw;
}
}
#ifndef PECULIAR_486
/* Should check here that |local_arg| is within the valid range */
if ( exponent >= -2 )
{
if ( (exponent > -2) ||
(argSignif.msw > (unsigned)0xafb0ccc0) )
{
/* The argument is too large */
}
}
#endif /* PECULIAR_486 */
arg_signif.lsw = argSignif.lsw; XSIG_LL(arg_signif) = XSIG_LL(argSignif);
adj = norm_Xsig(&argSignif);
accumulator.lsw = argSignif.lsw; XSIG_LL(accumulator) = XSIG_LL(argSignif);
mul_Xsig_Xsig(&accumulator, &accumulator);
shr_Xsig(&accumulator, 2*(-1 - (1 + exponent + adj)));
Xsq = XSIG_LL(accumulator);
if ( accumulator.lsw & 0x80000000 )
Xsq++;
accumulator.msw = accumulator.midw = accumulator.lsw = 0;
/* Do the basic fixed point polynomial evaluation */
polynomial_Xsig(&accumulator, &Xsq, logterms, HIPOWER-1);
mul_Xsig_Xsig(&accumulator, &argSignif);
shr_Xsig(&accumulator, 6 - adj);
mul32_Xsig(&arg_signif, leadterm);
add_two_Xsig(&accumulator, &arg_signif, &exponent);
*expon = exponent + 1;
accum_result->lsw = accumulator.lsw;
accum_result->midw = accumulator.midw;
accum_result->msw = accumulator.msw;
}
/*--- poly_sine() -----------------------------------------------------------+
| |
+---------------------------------------------------------------------------*/
void poly_sine(FPU_REG const *arg, FPU_REG *result)
{
int exponent, echange;
Xsig accumulator, argSqrd, argTo4;
unsigned long fix_up, adj;
unsigned long long fixed_arg;
#ifdef PARANOID
if ( arg->tag == TW_Zero )
{
/* Return 0.0 */
reg_move(&CONST_Z, result);
return;
}
#endif PARANOID
exponent = arg->exp - EXP_BIAS;
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) && (arg->sigh <= 0xe21240aa)) )
{
/* The argument is <= 0.88309101259 */
argSqrd.msw = arg->sigh; argSqrd.midw = arg->sigl; argSqrd.lsw = 0;
mul64_Xsig(&argSqrd, &significand(arg));
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(arg));
mul64_Xsig(&accumulator, &significand(arg));
mul64_Xsig(&accumulator, &significand(arg));
/* Divide by four, FPU_REG compatible, etc */
exponent = 3*exponent + EXP_BIAS;
/* The minimum exponent difference is 3 */
shr_Xsig(&accumulator, arg->exp - exponent);
negate_Xsig(&accumulator);
XSIG_LL(accumulator) += significand(arg);
echange = round_Xsig(&accumulator);
result->exp = arg->exp + echange;
}
else
{
/* The argument is > 0.88309101259 */
/* We use sin(arg) = cos(pi/2-arg) */
fixed_arg = significand(arg);
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;
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: */
mul_32_32(0x898cc517, argSqrd.msw, &adj);
fix_up -= adj/6;
}
mul_32_32(fix_up, LL_MSW(fixed_arg), &fix_up);
adj = accumulator.lsw; /* temp save */
accumulator.lsw -= fix_up;
if ( accumulator.lsw > adj )
XSIG_LL(accumulator) --;
echange = round_Xsig(&accumulator);
result->exp = EXP_BIAS - 1 + echange;
}
significand(result) = XSIG_LL(accumulator);
result->tag = TW_Valid;
result->sign = arg->sign;
#ifdef PARANOID
if ( (result->exp >= EXP_BIAS)
&& (significand(result) > 0x8000000000000000LL) )
{
EXCEPTION(EX_INTERNAL|0x150);
}
#endif PARANOID
}
/*--- poly_cos() ------------------------------------------------------------+
| |
+---------------------------------------------------------------------------*/
void poly_cos(FPU_REG const *arg, FPU_REG *result)
{
long int exponent, exp2, echange;
Xsig accumulator, argSqrd, fix_up, argTo4;
unsigned long adj;
unsigned long long fixed_arg;
#ifdef PARANOID
if ( arg->tag == TW_Zero )
{
/* Return 1.0 */
reg_move(&CONST_1, result);
return;
}
if ( (arg->exp > EXP_BIAS)
|| ((arg->exp == EXP_BIAS)
&& (significand(arg) > 0xc90fdaa22168c234LL)) )
{
EXCEPTION(EX_Invalid);
reg_move(&CONST_QNaN, result);
return;
}
#endif PARANOID
exponent = arg->exp - EXP_BIAS;
accumulator.lsw = accumulator.midw = accumulator.msw = 0;
if ( (exponent < -1) || ((exponent == -1) && (arg->sigh <= 0xb00d6f54)) )
{
/* arg is < 0.687705 */
argSqrd.msw = arg->sigh; argSqrd.midw = arg->sigl; argSqrd.lsw = 0;
mul64_Xsig(&argSqrd, &significand(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_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(arg));
mul64_Xsig(&accumulator, &significand(arg));
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 */
reg_move(&CONST_1, result);
}
else
{
significand(result) = XSIG_LL(accumulator);
/* will be a valid positive nr with expon = -1 */
*(short *)&(result->sign) = 0;
result->exp = EXP_BIAS - 1;
}
}
else
{
fixed_arg = significand(arg);
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;
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: */
mul_32_32(0x898cc517, argSqrd.msw, &adj);
fix_up.msw -= adj/2;
mul_32_32(0x898cc517, argTo4.msw, &adj);
fix_up.msw += adj/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);
result->exp = exp2 + EXP_BIAS + echange;
*(short *)&(result->sign) = 0; /* Is a valid positive nr */
significand(result) = XSIG_LL(accumulator);
}
#ifdef PARANOID
if ( (result->exp >= EXP_BIAS)
&& (significand(result) > 0x8000000000000000LL) )
{
EXCEPTION(EX_INTERNAL|0x151);
}
#endif PARANOID
}
/*--- poly_atan() -----------------------------------------------------------+
| |
+---------------------------------------------------------------------------*/
void poly_atan(FPU_REG *st0_ptr, u_char st0_tag,
FPU_REG *st1_ptr, u_char st1_tag)
{
u_char transformed, inverted, sign1, sign2;
int exponent;
long int dummy_exp;
Xsig accumulator, Numer, Denom, accumulatore, argSignif, argSq, argSqSq;
u_char tag;
sign1 = getsign(st0_ptr);
sign2 = getsign(st1_ptr);
if (st0_tag == TAG_Valid) {
exponent = exponent(st0_ptr);
} else {
/* This gives non-compatible stack contents... */
FPU_to_exp16(st0_ptr, st0_ptr);
exponent = exponent16(st0_ptr);
}
if (st1_tag == TAG_Valid) {
exponent -= exponent(st1_ptr);
} else {
/* This gives non-compatible stack contents... */
FPU_to_exp16(st1_ptr, st1_ptr);
exponent -= exponent16(st1_ptr);
}
if ((exponent < 0) || ((exponent == 0) &&
((st0_ptr->sigh < st1_ptr->sigh) ||
((st0_ptr->sigh == st1_ptr->sigh) &&
(st0_ptr->sigl < st1_ptr->sigl))))) {
inverted = 1;
Numer.lsw = Denom.lsw = 0;
XSIG_LL(Numer) = significand(st0_ptr);
XSIG_LL(Denom) = significand(st1_ptr);
} else {
inverted = 0;
exponent = -exponent;
Numer.lsw = Denom.lsw = 0;
XSIG_LL(Numer) = significand(st1_ptr);
XSIG_LL(Denom) = significand(st0_ptr);
}
div_Xsig(&Numer, &Denom, &argSignif);
exponent += norm_Xsig(&argSignif);
if ((exponent >= -1)
|| ((exponent == -2) && (argSignif.msw > 0xd413ccd0))) {
/* The argument is greater than sqrt(2)-1 (=0.414213562...) */
/* Convert the argument by an identity for atan */
transformed = 1;
if (exponent >= 0) {
#ifdef PARANOID
if (!((exponent == 0) &&
(argSignif.lsw == 0) && (argSignif.midw == 0) &&
(argSignif.msw == 0x80000000))) {
EXCEPTION(EX_INTERNAL | 0x104); /* There must be a logic error */
return;
}
#endif /* PARANOID */
argSignif.msw = 0; /* Make the transformed arg -> 0.0 */
} else {
Numer.lsw = Denom.lsw = argSignif.lsw;
XSIG_LL(Numer) = XSIG_LL(Denom) = XSIG_LL(argSignif);
if (exponent < -1)
shr_Xsig(&Numer, -1 - exponent);
negate_Xsig(&Numer);
shr_Xsig(&Denom, -exponent);
Denom.msw |= 0x80000000;
div_Xsig(&Numer, &Denom, &argSignif);
exponent = -1 + norm_Xsig(&argSignif);
}
} else {
transformed = 0;
}
argSq.lsw = argSignif.lsw;
argSq.midw = argSignif.midw;
argSq.msw = argSignif.msw;
mul_Xsig_Xsig(&argSq, &argSq);
argSqSq.lsw = argSq.lsw;
argSqSq.midw = argSq.midw;
argSqSq.msw = argSq.msw;
mul_Xsig_Xsig(&argSqSq, &argSqSq);
accumulatore.lsw = argSq.lsw;
XSIG_LL(accumulatore) = XSIG_LL(argSq);
shr_Xsig(&argSq, 2 * (-1 - exponent - 1));
shr_Xsig(&argSqSq, 4 * (-1 - exponent - 1));
/* Now have argSq etc with binary point at the left
.1xxxxxxxx */
/* Do the basic fixed point polynomial evaluation */
accumulator.msw = accumulator.midw = accumulator.lsw = 0;
polynomial_Xsig(&accumulator, &XSIG_LL(argSqSq),
oddplterms, HIPOWERop - 1);
mul64_Xsig(&accumulator, &XSIG_LL(argSq));
negate_Xsig(&accumulator);
polynomial_Xsig(&accumulator, &XSIG_LL(argSqSq), oddnegterms,
HIPOWERon - 1);
negate_Xsig(&accumulator);
add_two_Xsig(&accumulator, &fixedpterm, &dummy_exp);
mul64_Xsig(&accumulatore, &denomterm);
shr_Xsig(&accumulatore, 1 + 2 * (-1 - exponent));
accumulatore.msw |= 0x80000000;
div_Xsig(&accumulator, &accumulatore, &accumulator);
mul_Xsig_Xsig(&accumulator, &argSignif);
mul_Xsig_Xsig(&accumulator, &argSq);
shr_Xsig(&accumulator, 3);
negate_Xsig(&accumulator);
add_Xsig_Xsig(&accumulator, &argSignif);
if (transformed) {
/* compute pi/4 - accumulator */
shr_Xsig(&accumulator, -1 - exponent);
negate_Xsig(&accumulator);
add_Xsig_Xsig(&accumulator, &pi_signif);
exponent = -1;
}
if (inverted) {
/* compute pi/2 - accumulator */
shr_Xsig(&accumulator, -exponent);
negate_Xsig(&accumulator);
add_Xsig_Xsig(&accumulator, &pi_signif);
exponent = 0;
}
if (sign1) {
/* compute pi - accumulator */
shr_Xsig(&accumulator, 1 - exponent);
negate_Xsig(&accumulator);
add_Xsig_Xsig(&accumulator, &pi_signif);
exponent = 1;
}
exponent += round_Xsig(&accumulator);
significand(st1_ptr) = XSIG_LL(accumulator);
setexponent16(st1_ptr, exponent);
tag = FPU_round(st1_ptr, 1, 0, FULL_PRECISION, sign2);
FPU_settagi(1, tag);
set_precision_flag_up(); /* We do not really know if up or down,
use this as the default. */
}
