/* executes single exponent reduction cycle */
static Bit64u remainder_kernel(Bit64u aSig0, Bit64u bSig, int expDiff, Bit64u *zSig0, Bit64u *zSig1)
{
Bit64u term0, term1;
Bit64u aSig1 = 0;
shortShift128Left(aSig1, aSig0, expDiff, &aSig1, &aSig0);
Bit64u q = estimateDiv128To64(aSig1, aSig0, bSig);
mul64To128(bSig, q, &term0, &term1);
sub128(aSig1, aSig0, term0, term1, zSig1, zSig0);
while ((Bit64s)(*zSig1) < 0) {
--q;
add128(*zSig1, *zSig0, 0, bSig, zSig1, zSig0);
}
return q;
}
/* reduce trigonometric function argument using 128-bit precision
M_PI approximation */
static Bit64u argument_reduction_kernel(Bit64u aSig0, int Exp, Bit64u *zSig0, Bit64u *zSig1)
{
Bit64u term0, term1, term2;
Bit64u aSig1 = 0;
shortShift128Left(aSig1, aSig0, Exp, &aSig1, &aSig0);
Bit64u q = estimateDiv128To64(aSig1, aSig0, FLOAT_PI_HI);
mul128By64To192(FLOAT_PI_HI, FLOAT_PI_LO, q, &term0, &term1, &term2);
sub128(aSig1, aSig0, term0, term1, zSig1, zSig0);
while ((Bit64s)(*zSig1) < 0) {
--q;
add192(*zSig1, *zSig0, term2, 0, FLOAT_PI_HI, FLOAT_PI_LO, zSig1, zSig0, &term2);
}
*zSig1 = term2;
return q;
}
/* reduce trigonometric function argument using 128-bit precision
M_PI approximation */
static UINT64 argument_reduction_kernel(UINT64 aSig0, int Exp, UINT64 *zSig0, UINT64 *zSig1)
{
UINT64 term0, term1, term2;
UINT64 aSig1 = 0;
shortShift128Left(aSig1, aSig0, Exp, &aSig1, &aSig0);
UINT64 q = estimateDiv128To64(aSig1, aSig0, FLOAT_PI_HI);
mul128By64To192(FLOAT_PI_HI, FLOAT_PI_LO, q, &term0, &term1, &term2);
sub128(aSig1, aSig0, term0, term1, zSig1, zSig0);
while ((INT64)(*zSig1) < 0) {
--q;
add192(*zSig1, *zSig0, term2, 0, FLOAT_PI_HI, FLOAT_PI_LO, zSig1, zSig0, &term2);
}
*zSig1 = term2;
return q;
}
floatx80 fyl2xp1(floatx80 a, floatx80 b)
{
INT32 aExp, bExp;
UINT64 aSig, bSig, zSig0, zSig1, zSig2;
int aSign, bSign;
aSig = extractFloatx80Frac(a);
aExp = extractFloatx80Exp(a);
aSign = extractFloatx80Sign(a);
bSig = extractFloatx80Frac(b);
bExp = extractFloatx80Exp(b);
bSign = extractFloatx80Sign(b);
int zSign = aSign ^ bSign;
if (aExp == 0x7FFF) {
if ((UINT64) (aSig<<1)
|| ((bExp == 0x7FFF) && (UINT64) (bSig<<1)))
{
return propagateFloatx80NaN(a, b);
}
if (aSign)
{
invalid:
float_raise(float_flag_invalid);
return floatx80_default_nan;
}
else {
if (bExp == 0) {
if (bSig == 0) goto invalid;
float_raise(float_flag_denormal);
}
return packFloatx80(bSign, 0x7FFF, U64(0x8000000000000000));
}
}
if (bExp == 0x7FFF)
{
if ((UINT64) (bSig<<1))
return propagateFloatx80NaN(a, b);
if (aExp == 0) {
if (aSig == 0) goto invalid;
float_raise(float_flag_denormal);
}
return packFloatx80(zSign, 0x7FFF, U64(0x8000000000000000));
}
if (aExp == 0) {
if (aSig == 0) {
if (bSig && (bExp == 0)) float_raise(float_flag_denormal);
return packFloatx80(zSign, 0, 0);
}
float_raise(float_flag_denormal);
normalizeFloatx80Subnormal(aSig, &aExp, &aSig);
}
if (bExp == 0) {
if (bSig == 0) return packFloatx80(zSign, 0, 0);
float_raise(float_flag_denormal);
normalizeFloatx80Subnormal(bSig, &bExp, &bSig);
}
float_raise(float_flag_inexact);
if (aSign && aExp >= 0x3FFF)
return a;
if (aExp >= 0x3FFC) // big argument
{
return fyl2x(floatx80_add(a, floatx80_one), b);
}
// handle tiny argument
if (aExp < EXP_BIAS-70)
{
// first order approximation, return (a*b)/ln(2)
INT32 zExp = aExp + FLOAT_LN2INV_EXP - 0x3FFE;
mul128By64To192(FLOAT_LN2INV_HI, FLOAT_LN2INV_LO, aSig, &zSig0, &zSig1, &zSig2);
if (0 < (INT64) zSig0) {
shortShift128Left(zSig0, zSig1, 1, &zSig0, &zSig1);
--zExp;
}
zExp = zExp + bExp - 0x3FFE;
mul128By64To192(zSig0, zSig1, bSig, &zSig0, &zSig1, &zSig2);
if (0 < (INT64) zSig0) {
shortShift128Left(zSig0, zSig1, 1, &zSig0, &zSig1);
--zExp;
}
return
roundAndPackFloatx80(80, aSign ^ bSign, zExp, zSig0, zSig1);
}
/* ******************************** */
/* using float128 for approximation */
/* ******************************** */
shift128Right(aSig<<1, 0, 16, &zSig0, &zSig1);
float128 x = packFloat128(aSign, aExp, zSig0, zSig1);
x = poly_l2p1(x);
return floatx80_mul(b, float128_to_floatx80(x));
}
floatx80 floatx80_mod(floatx80 a, floatx80 b, float_status *status)
{
flag aSign, zSign;
int32_t aExp, bExp, expDiff;
uint64_t aSig0, aSig1, bSig;
uint64_t qTemp, term0, term1;
aSig0 = extractFloatx80Frac(a);
aExp = extractFloatx80Exp(a);
aSign = extractFloatx80Sign(a);
bSig = extractFloatx80Frac(b);
bExp = extractFloatx80Exp(b);
if (aExp == 0x7FFF) {
if ((uint64_t) (aSig0 << 1)
|| ((bExp == 0x7FFF) && (uint64_t) (bSig << 1))) {
return propagateFloatx80NaN(a, b, status);
}
goto invalid;
}
if (bExp == 0x7FFF) {
if ((uint64_t) (bSig << 1)) {
return propagateFloatx80NaN(a, b, status);
}
return a;
}
if (bExp == 0) {
if (bSig == 0) {
invalid:
float_raise(float_flag_invalid, status);
return floatx80_default_nan(status);
}
normalizeFloatx80Subnormal(bSig, &bExp, &bSig);
}
if (aExp == 0) {
if ((uint64_t) (aSig0 << 1) == 0) {
return a;
}
normalizeFloatx80Subnormal(aSig0, &aExp, &aSig0);
}
bSig |= LIT64(0x8000000000000000);
zSign = aSign;
expDiff = aExp - bExp;
aSig1 = 0;
if (expDiff < 0) {
return a;
}
qTemp = (bSig <= aSig0);
if (qTemp) {
aSig0 -= bSig;
}
expDiff -= 64;
while (0 < expDiff) {
qTemp = estimateDiv128To64(aSig0, aSig1, bSig);
qTemp = (2 < qTemp) ? qTemp - 2 : 0;
mul64To128(bSig, qTemp, &term0, &term1);
sub128(aSig0, aSig1, term0, term1, &aSig0, &aSig1);
shortShift128Left(aSig0, aSig1, 62, &aSig0, &aSig1);
}
expDiff += 64;
if (0 < expDiff) {
qTemp = estimateDiv128To64(aSig0, aSig1, bSig);
qTemp = (2 < qTemp) ? qTemp - 2 : 0;
qTemp >>= 64 - expDiff;
mul64To128(bSig, qTemp << (64 - expDiff), &term0, &term1);
sub128(aSig0, aSig1, term0, term1, &aSig0, &aSig1);
shortShift128Left(0, bSig, 64 - expDiff, &term0, &term1);
while (le128(term0, term1, aSig0, aSig1)) {
++qTemp;
sub128(aSig0, aSig1, term0, term1, &aSig0, &aSig1);
}
}
