void exp_td_accurate(double *polyTblh, double *polyTblm, double *polyTbll,
double rh, double rm, double rl,
double tbl1h, double tbl1m, double tbl1l,
double tbl2h, double tbl2m, double tbl2l) {
double highPoly, highPolyMulth, highPolyMultm, highPolyMultl;
double rhSquareh, rhSquarel, rhSquareHalfh, rhSquareHalfl;
double rhCubeh, rhCubem, rhCubel;
double t1h, t1l, t2h, t2l, t3h, t3l, t4h, t4l, t5, t6;
double lowPolyh, lowPolym, lowPolyl;
double ph, pm, pl, phnorm, pmnorm, rmlMultPh, rmlMultPl;
double qh, ql, fullPolyh, fullPolym, fullPolyl;
double polyWithTbl1h, polyWithTbl1m, polyWithTbl1l;
double polyAddOneh,polyAddOnem,polyAddOnel;
double polyWithTablesh, polyWithTablesm, polyWithTablesl;
#if EVAL_PERF
crlibm_second_step_taken++;
#endif
#if defined(PROCESSOR_HAS_FMA) && !defined(AVOID_FMA)
highPoly = FMA(FMA(accPolyC7,rh,accPolyC6),rh,accPolyC5);
#else
highPoly = accPolyC5 + rh * (accPolyC6 + rh * accPolyC7);
#endif
Mul12(&t1h,&t1l,rh,highPoly);
Add22(&t2h,&t2l,accPolyC4h,accPolyC4l,t1h,t1l);
Mul22(&t3h,&t3l,rh,0,t2h,t2l);
Add22(&t4h,&t4l,accPolyC3h,accPolyC3l,t3h,t3l);
Mul12(&rhSquareh,&rhSquarel,rh,rh);
Mul23(&rhCubeh,&rhCubem,&rhCubel,rh,0,rhSquareh,rhSquarel);
rhSquareHalfh = 0.5 * rhSquareh;
rhSquareHalfl = 0.5 * rhSquarel;
Renormalize3(&lowPolyh,&lowPolym,&lowPolyl,rh,rhSquareHalfh,rhSquareHalfl);
Mul233(&highPolyMulth,&highPolyMultm,&highPolyMultl,t4h,t4l,rhCubeh,rhCubem,rhCubel);
Add33(&ph,&pm,&pl,lowPolyh,lowPolym,lowPolyl,highPolyMulth,highPolyMultm,highPolyMultl);
Add12(phnorm,pmnorm,ph,pm);
Mul22(&rmlMultPh,&rmlMultPl,rm,rl,phnorm,pmnorm);
Add22(&qh,&ql,rm,rl,rmlMultPh,rmlMultPl);
Add233Cond(&fullPolyh,&fullPolym,&fullPolyl,qh,ql,ph,pm,pl);
Add12(polyAddOneh,t5,1,fullPolyh);
Add12Cond(polyAddOnem,t6,t5,fullPolym);
polyAddOnel = t6 + fullPolyl;
Mul33(&polyWithTbl1h,&polyWithTbl1m,&polyWithTbl1l,tbl1h,tbl1m,tbl1l,polyAddOneh,polyAddOnem,polyAddOnel);
Mul33(&polyWithTablesh,&polyWithTablesm,&polyWithTablesl,
tbl2h,tbl2m,tbl2l,
polyWithTbl1h,polyWithTbl1m,polyWithTbl1l);
Renormalize3(polyTblh,polyTblm,polyTbll,polyWithTablesh,polyWithTablesm,polyWithTablesl);
}
/*************************************************************
*************************************************************
* ROUNDED TO NEAREST *
*************************************************************
*************************************************************/
double log_rn(double x){
db_number xdb, yhdb;
double yh, yl, ed, ri, logih, logim, logil, yrih, yril, th, zh, zl;
double ph, pl, pm, log2edh, log2edl, log2edm, logTabPolyh, logTabPolyl, logh, logm, logl;
int E, index;
double zhSquare, zhCube, zhSquareHalf;
double p35, p46, p36;
double pUpper;
double zhSquareHalfPlusZl;
double zhFour;
double logyh, logym, logyl;
double loghover, logmover, loglover;
E=0;
xdb.d=x;
/* Filter cases */
if (xdb.i[HI] < 0x00100000){ /* x < 2^(-1022) */
if (((xdb.i[HI] & 0x7fffffff)|xdb.i[LO])==0){
return -1.0/0.0;
} /* log(+/-0) = -Inf */
if (xdb.i[HI] < 0){
return (x-x)/0.0; /* log(-x) = Nan */
}
/* Subnormal number */
E = -52;
xdb.d *= two52; /* make x a normal number */
}
if (xdb.i[HI] >= 0x7ff00000){
return x+x; /* Inf or Nan */
}
/* Extract exponent and mantissa
Do range reduction,
yielding to E holding the exponent and
y the mantissa between sqrt(2)/2 and sqrt(2)
*/
E += (xdb.i[HI]>>20)-1023; /* extract the exponent */
index = (xdb.i[HI] & 0x000fffff);
xdb.i[HI] = index | 0x3ff00000; /* do exponent = 0 */
index = (index + (1<<(20-L-1))) >> (20-L);
/* reduce such that sqrt(2)/2 < xdb.d < sqrt(2) */
if (index >= MAXINDEX){ /* corresponds to xdb>sqrt(2)*/
xdb.i[HI] -= 0x00100000;
E++;
}
yhdb.i[HI] = xdb.i[HI];
yhdb.i[LO] = 0;
yh = yhdb.d;
yl = xdb.d - yh;
index = index & INDEXMASK;
/* Cast integer E into double ed for multiplication later */
ed = (double) E;
/*
Read tables:
Read one float for ri
Read the first two doubles for -log(r_i) (out of three)
Organization of the table:
one struct entry per index, the struct entry containing
r, logih, logim and logil in this order
*/
ri = argredtable[index].ri;
/*
Actually we don't need the logarithm entries now
Move the following two lines to the eventual reconstruction
As long as we don't have any if in the following code, we can overlap
memory access with calculations
*/
logih = argredtable[index].logih;
logim = argredtable[index].logim;
/* Do range reduction:
zh + zl = y * ri - 1.0 exactly
Exactness is assured by use of two part yh + yl and 21 bit ri and Add12
Discard zl for higher monome degrees
*/
yrih = yh * ri;
yril = yl * ri;
th = yrih - 1.0;
Add12Cond(zh, zl, th, yril);
/* Polynomial approximation */
zhSquare = zh * zh; /* 1 */
p35 = p_coeff_3h + zhSquare * p_coeff_5h; /* 3 */
p46 = p_coeff_4h + zhSquare * p_coeff_6h; /* 3 */
zhCube = zhSquare * zh; /* 2 */
zhSquareHalf = p_coeff_2h * zhSquare; /* 2 */
zhFour = zhSquare * zhSquare; /* 2 */
p36 = zhCube * p35 + zhFour * p46; /* 4 */
zhSquareHalfPlusZl = zhSquareHalf + zl; /* 3 */
pUpper = zhSquareHalfPlusZl + p36; /* 5 */
Add12(ph,pl,zh,pUpper); /* 8 */
/* Reconstruction
Read logih and logim in the tables (already done)
Compute log(x) = E * log(2) + log(1+z) - log(ri)
i.e. log(x) = ed * (log2h + log2m) + (ph + pl) + (logih + logim) + delta
Carry out everything in double double precision
*/
/*
We store log2 as log2h + log2m + log2l where log2h and log2m have 12 trailing zeros
Multiplication of ed (double E) and log2h is thus exact
The overall accuracy of log2h + log2m + log2l is 53 * 3 - 24 = 135 which
is enough for the accurate phase
The accuracy suffices also for the quick phase: 53 * 2 - 24 = 82
Nevertheless the storage with trailing zeros implies an overlap of the tabulated
triple double values. We have to take it into account for the accurate phase
basic procedures for addition and multiplication
The OCcondition on the next Add12 is verified as log2m is smaller than log2h
and both are scaled by ed
*/
Add12(log2edh, log2edl, log2h * ed, log2m * ed);
/* Add logih and logim to ph and pl */
Add22(&logTabPolyh, &logTabPolyl, logih, logim, ph, pl);
/* Add log2edh + log2edl to logTabPolyh + logTabPolyl */
Add22(&logh, &logm, log2edh, log2edl, logTabPolyh, logTabPolyl);
/* Rounding test and possible return or call to the accurate function */
if(logh == (logh + (logm * RNROUNDCST)))
return logh;
else
{
logil = argredtable[index].logil;
p_accu(&ph, &pm, &pl, zh, zl);
Add33(&logyh, &logym, &logyl, logih, logim, logil, ph, pm, pl);
log2edh = log2h * ed;
log2edm = log2m * ed;
log2edl = log2l * ed;
Add33(&loghover, &logmover, &loglover, log2edh, log2edm, log2edl, logyh, logym, logyl);
Renormalize3(&logh,&logm,&logl,loghover,logmover,loglover);
ReturnRoundToNearest3(logh, logm, logl);
} /* Accurate phase launched */
}
interval j_log(interval x)
{
interval res;
int roundable;
int cs_inf=0; int cs_sup=0;
double x_inf,x_sup;
x_inf=LOW(x);
x_sup=UP(x);
double res_inf, res_sup, res_simple_inf, res_simple_sup;
db_number xdb_sup;
double y_sup, ed_sup, ri_sup, logih_sup, logim_sup, yrih_sup, yril_sup, th_sup, zh_sup, zl_sup;
double polyHorner_sup, zhSquareh_sup, zhSquarel_sup, polyUpper_sup, zhSquareHalfh_sup, zhSquareHalfl_sup;
double t1h_sup, t1l_sup, t2h_sup, t2l_sup, ph_sup, pl_sup, log2edh_sup, log2edl_sup, logTabPolyh_sup, logTabPolyl_sup, logh_sup, logm_sup, logl_sup, roundcst;
int E_sup, index_sup;
db_number xdb_inf;
double y_inf, ed_inf, ri_inf, logih_inf, logim_inf, yrih_inf, yril_inf, th_inf, zh_inf, zl_inf;
double polyHorner_inf, zhSquareh_inf, zhSquarel_inf, polyUpper_inf, zhSquareHalfh_inf, zhSquareHalfl_inf;
double t1h_inf, t1l_inf, t2h_inf, t2l_inf, ph_inf, pl_inf, log2edh_inf, log2edl_inf, logTabPolyh_inf, logTabPolyl_inf, logh_inf, logm_inf, logl_inf;
int E_inf, index_inf;
E_inf=0;
xdb_inf.d=x_inf;
E_sup=0;
xdb_sup.d=x_sup;
if (__builtin_expect(
(x_inf == 1.0) || (!(x_inf<=x_sup)) || (xdb_sup.i[HI] < 0)
|| (xdb_inf.i[HI] < 0x00100000) || (((xdb_inf.i[HI] & 0x7fffffff)|xdb_inf.i[LO])==0)
|| (xdb_inf.i[HI] < 0)
|| (xdb_inf.i[HI] >= 0x7ff00000)
|| (x_sup == 1.0)
|| (xdb_sup.i[HI] < 0x00100000)
|| (((xdb_sup.i[HI] & 0x7fffffff)|xdb_sup.i[LO])==0)
|| (xdb_sup.i[HI] < 0)
|| (xdb_sup.i[HI] >= 0x7ff00000)
|| ((xdb_inf.d<00) && (xdb_sup.d>0) )
,FALSE))
{
if (!(x_inf<=x_sup)) RETURN_EMPTY_INTERVAL;
if (xdb_sup.i[HI] < 0) RETURN_EMPTY_INTERVAL;
if ((xdb_inf.d<00) && (xdb_sup.d>0) )
{
ASSIGN_LOW(res,-1.0/0.0);
ASSIGN_UP(res,log_ru(UP(x)));
return res;
}
ASSIGN_LOW(res,log_rd(LOW(x)));
ASSIGN_UP(res,log_ru(UP(x)));
return res;
}
/* Extract exponent and mantissa
Do range reduction,
yielding to E holding the exponent and
y the mantissa between sqrt(2)/2 and sqrt(2)
*/
E_inf += (xdb_inf.i[HI]>>20)-1023; /* extract the exponent */
E_sup += (xdb_sup.i[HI]>>20)-1023; /* extract the exponent */
index_inf = (xdb_inf.i[HI] & 0x000fffff);
index_sup = (xdb_sup.i[HI] & 0x000fffff);
xdb_inf.i[HI] = index_inf | 0x3ff00000; /* do exponent = 0 */
xdb_sup.i[HI] = index_sup | 0x3ff00000; /* do exponent = 0 */
index_inf = (index_inf + (1<<(20-L-1))) >> (20-L);
index_sup = (index_sup + (1<<(20-L-1))) >> (20-L);
/* reduce such that sqrt(2)/2 < xdb.d < sqrt(2) */
if (index_inf >= MAXINDEX){ /* corresponds to xdb>sqrt(2)*/
xdb_inf.i[HI] -= 0x00100000;
E_inf++;
}
/* reduce such that sqrt(2)/2 < xdb.d < sqrt(2) */
if (index_sup >= MAXINDEX){ /* corresponds to xdb>sqrt(2)*/
xdb_sup.i[HI] -= 0x00100000;
E_sup++;
}
y_inf = xdb_inf.d;
y_sup = xdb_sup.d;
index_inf = index_inf & INDEXMASK;
index_sup = index_sup & INDEXMASK;
/* Cast integer E into double ed for multiplication later */
ed_inf = (double) E_inf;
ed_sup = (double) E_sup;
/*
Read tables:
Read one float for ri
Read the first two doubles for -log(r_i) (out of three)
Organization of the table:
one struct entry per index, the struct entry containing
r, logih, logim and logil in this order
*/
ri_inf = argredtable[index_inf].ri;
ri_sup = argredtable[index_sup].ri;
/*
Actually we don't need the logarithm entries now
Move the following two lines to the eventual reconstruction
As long as we don't have any if in the following code, we can overlap
memory access with calculations
*/
logih_inf = argredtable[index_inf].logih;
logih_sup = argredtable[index_sup].logih;
logim_inf = argredtable[index_inf].logim;
logim_sup = argredtable[index_sup].logim;
/* Do range reduction:
zh + zl = y * ri - 1.0 correctly
Correctness is assured by use of Mul12 and Add12
even if we don't force ri to have its' LSBs set to zero
Discard zl for higher monome degrees
*/
Mul12(&yrih_inf, &yril_inf, y_inf, ri_inf);
Mul12(&yrih_sup, &yril_sup, y_sup, ri_sup);
th_inf = yrih_inf - 1.0;
th_sup = yrih_sup - 1.0;
/* Do range reduction:
zh + zl = y * ri - 1.0 correctly
Correctness is assured by use of Mul12 and Add12
even if we don't force ri to have its' LSBs set to zero
Discard zl for higher monome degrees
*/
Add12Cond(zh_inf, zl_inf, th_inf, yril_inf);
Add12Cond(zh_sup, zl_sup, th_sup, yril_sup);
/*
Polynomial evaluation
Use a 7 degree polynomial
Evaluate the higher 5 terms in double precision (-7 * 3 = -21) using Horner's scheme
Evaluate the lower 3 terms (the last is 0) in double double precision accounting also for zl
using an ad hoc method
*/
#if defined(PROCESSOR_HAS_FMA) && !defined(AVOID_FMA)
polyHorner_inf = FMA(FMA(FMA(FMA(c7,zh_inf,c6),zh_inf,c5),zh_inf,c4),zh_inf,c3);
polyHorner_sup = FMA(FMA(FMA(FMA(c7,zh_sup,c6),zh_sup,c5),zh_sup,c4),zh_sup,c3);
#else
polyHorner_inf = c3 + zh_inf * (c4 + zh_inf * (c5 + zh_inf * (c6 + zh_inf * c7)));
polyHorner_sup = c3 + zh_sup * (c4 + zh_sup * (c5 + zh_sup * (c6 + zh_sup * c7)));
#endif
Mul12(&zhSquareh_inf, &zhSquarel_inf, zh_inf, zh_inf);
Mul12(&zhSquareh_sup, &zhSquarel_sup, zh_sup, zh_sup);
polyUpper_inf = polyHorner_inf * (zh_inf * zhSquareh_inf);
polyUpper_sup = polyHorner_sup * (zh_sup * zhSquareh_sup);
zhSquareHalfh_inf = zhSquareh_inf * -0.5;
zhSquareHalfh_sup = zhSquareh_sup * -0.5;
zhSquareHalfl_inf = zhSquarel_inf * -0.5;
zhSquareHalfl_sup = zhSquarel_sup * -0.5;
Add12(t1h_inf, t1l_inf, polyUpper_inf, -1 * (zh_inf * zl_inf));
Add12(t1h_sup, t1l_sup, polyUpper_sup, -1 * (zh_sup * zl_sup));
Add22(&t2h_inf, &t2l_inf, zh_inf, zl_inf, zhSquareHalfh_inf, zhSquareHalfl_inf);
Add22(&t2h_sup, &t2l_sup, zh_sup, zl_sup, zhSquareHalfh_sup, zhSquareHalfl_sup);
Add22(&ph_inf, &pl_inf, t2h_inf, t2l_inf, t1h_inf, t1l_inf);
Add22(&ph_sup, &pl_sup, t2h_sup, t2l_sup, t1h_sup, t1l_sup);
/* Reconstruction
Read logih and logim in the tables (already done)
Compute log(x) = E * log(2) + log(1+z) - log(ri)
i.e. log(x) = ed * (log2h + log2m) + (ph + pl) + (logih + logim) + delta
Carry out everything in double double precision
*/
/*
We store log2 as log2h + log2m + log2l where log2h and log2m have 12 trailing zeros
Multiplication of ed (double E) and log2h is thus correct
The overall accuracy of log2h + log2m + log2l is 53 * 3 - 24 = 135 which
is enough for the accurate phase
The accuracy suffices also for the quick phase: 53 * 2 - 24 = 82
Nevertheless the storage with trailing zeros implies an overlap of the tabulated
triple double values. We have to take it into account for the accurate phase
basic procedures for addition and multiplication
The condition on the next Add12 is verified as log2m is smaller than log2h
and both are scaled by ed
*/
Add12(log2edh_inf, log2edl_inf, log2h * ed_inf, log2m * ed_inf);
/*
We store log2 as log2h + log2m + log2l where log2h and log2m have 12 trailing zeros
Multiplication of ed (double E) and log2h is thus correct
The overall accuracy of log2h + log2m + log2l is 53 * 3 - 24 = 135 which
is enough for the accurate phase
The accuracy suffices also for the quick phase: 53 * 2 - 24 = 82
Nevertheless the storage with trailing zeros implies an overlap of the tabulated
triple double values. We have to take it into account for the accurate phase
basic procedures for addition and multiplication
The condition on the next Add12 is verified as log2m is smaller than log2h
and both are scaled by ed
*/
Add12(log2edh_sup, log2edl_sup, log2h * ed_sup, log2m * ed_sup);
/* Add logih and logim to ph and pl
We must use conditioned Add22 as logih can move over ph
*/
Add22Cond(&logTabPolyh_inf, &logTabPolyl_inf, logih_inf, logim_inf, ph_inf, pl_inf);
/* Add log2edh + log2edl to logTabPolyh + logTabPolyl */
Add22Cond(&logh_inf, &logm_inf, log2edh_inf, log2edl_inf, logTabPolyh_inf, logTabPolyl_inf);
/* Add logih and logim to ph and pl
We must use conditioned Add22 as logih can move over ph
*/
Add22Cond(&logTabPolyh_sup, &logTabPolyl_sup, logih_sup, logim_sup, ph_sup, pl_sup);
/* Add log2edh + log2edl to logTabPolyh + logTabPolyl */
Add22Cond(&logh_sup, &logm_sup, log2edh_sup, log2edl_sup, logTabPolyh_sup, logTabPolyl_sup);
/* Rounding test and eventual return or call to the accurate function */
roundcst = RDROUNDCST1;
if(cs_inf) { res_inf=res_simple_inf; }
if(cs_sup) { res_sup=res_simple_sup; }
//TEST_AND_COPY_RDRU_LOG(roundable,res_inf,logh_inf,logm_inf,res_sup,logh_sup,logm_sup,roundcst);
//#define TEST_AND_COPY_RDRU_LOG(__cond__, __res_inf__, __yh_inf__, __yl_inf__, __res_sup__, __yh_sup__, __yl_sup__, __eps__)
db_number yh_inf, yl_inf, u53_inf, yh_sup, yl_sup, u53_sup;
int yh_inf_neg, yl_inf_neg, yh_sup_neg, yl_sup_neg;
int rd_ok, ru_ok;
double save_res_inf=res_inf;
yh_inf.d = logh_inf; yl_inf.d = logm_inf;
yh_inf_neg = (yh_inf.i[HI] & 0x80000000);
yl_inf_neg = (yl_inf.i[HI] & 0x80000000);
yh_inf.l = yh_inf.l & 0x7fffffffffffffffLL; /* compute the absolute value*/
yl_inf.l = yl_inf.l & 0x7fffffffffffffffLL; /* compute the absolute value*/
u53_inf.l = (yh_inf.l & ULL(7ff0000000000000)) + ULL(0010000000000000);
yh_sup.d = logh_sup; yl_sup.d = logm_sup;
yh_sup_neg = (yh_sup.i[HI] & 0x80000000);
yl_sup_neg = (yl_sup.i[HI] & 0x80000000);
yh_sup.l = yh_sup.l & 0x7fffffffffffffffLL; /* compute the absolute value*/
yl_sup.l = yl_sup.l & 0x7fffffffffffffffLL; /* compute the absolute value*/
u53_sup.l = (yh_sup.l & ULL(7ff0000000000000)) + ULL(0010000000000000);
roundable = 0;
rd_ok=(yl_inf.d > roundcst * u53_inf.d);
ru_ok=(yl_sup.d > roundcst * u53_sup.d);
if(yl_inf_neg) { /* The case yl==0 is filtered by the above test*/
/* return next down */
yh_inf.d = logh_inf;
if(yh_inf_neg) yh_inf.l++; else yh_inf.l--; /* Beware: fails for zero */
res_inf = yh_inf.d ;
}
else {
res_inf = logh_inf;
}
if(!yl_sup_neg) { /* The case yl==0 is filtered by the above test*/
/* return next up */
yh_sup.d = logh_sup;
if(yh_sup_neg) yh_sup.l--; else yh_sup.l++; /* Beware: fails for zero */
res_sup = yh_sup.d ;
}
else {
res_sup = logh_sup;
}
if(save_res_inf==-1.0/0.0) res_inf=-1.0/0.0;
if(rd_ok && ru_ok){
ASSIGN_LOW(res,res_inf);
ASSIGN_UP(res,res_sup);
return(res);
}
else if (rd_ok){
roundable=1;
}
else if (ru_ok){
roundable=2;
}
#if DEBUG
printf("Going for Accurate Phase for x=%1.50e\n",x);
#endif
if (roundable==1)
{
log_td_accurate(&logh_sup, &logm_sup, &logl_sup, E_sup, ed_sup, index_sup, zh_sup, zl_sup, logih_sup, logim_sup);
RoundUpwards3(&res_sup, logh_sup, logm_sup, logl_sup);
}
if (roundable==2)
{
log_td_accurate(&logh_inf, &logm_inf, &logl_inf, E_inf, ed_inf, index_inf, zh_inf, zl_inf, logih_inf, logim_inf);
RoundDownwards3(&res_inf, logh_inf, logm_inf, logl_inf);
}
if (roundable==0)
{
log_td_accurate(&logh_inf, &logm_inf, &logl_inf, E_inf, ed_inf, index_inf, zh_inf, zl_inf, logih_inf, logim_inf);
RoundDownwards3(&res_inf, logh_inf, logm_inf, logl_inf);
log_td_accurate(&logh_sup, &logm_sup, &logl_sup, E_sup, ed_sup, index_sup, zh_sup, zl_sup, logih_sup, logim_sup);
RoundUpwards3(&res_sup, logh_sup, logm_sup, logl_sup);
}
ASSIGN_LOW(res,res_inf);
ASSIGN_UP(res,res_sup);
return res;
}
/*************************************************************
*************************************************************
* ROUNDED TOWARDS ZERO *
*************************************************************
*************************************************************/
double log_rz(double x) {
db_number xdb;
double y, ed, ri, logih, logim, yrih, yril, th, zh, zl;
double polyHorner, zhSquareh, zhSquarel, polyUpper, zhSquareHalfh, zhSquareHalfl;
double t1h, t1l, t2h, t2l, ph, pl, log2edh, log2edl, logTabPolyh, logTabPolyl, logh, logm, logl, roundcst;
int E, index;
if (x == 1.0) return 0.0; /* This the only case in which the image under log of a double is a double. */
E=0;
xdb.d=x;
/* Filter cases */
if (xdb.i[HI] < 0x00100000){ /* x < 2^(-1022) */
if (((xdb.i[HI] & 0x7fffffff)|xdb.i[LO])==0){
return -1.0/0.0;
} /* log(+/-0) = -Inf */
if (xdb.i[HI] < 0){
return (x-x)/0; /* log(-x) = Nan */
}
/* Subnormal number */
E = -52;
xdb.d *= two52; /* make x a normal number */
}
if (xdb.i[HI] >= 0x7ff00000){
return x+x; /* Inf or Nan */
}
/* Extract exponent and mantissa
Do range reduction,
yielding to E holding the exponent and
y the mantissa between sqrt(2)/2 and sqrt(2)
*/
E += (xdb.i[HI]>>20)-1023; /* extract the exponent */
index = (xdb.i[HI] & 0x000fffff);
xdb.i[HI] = index | 0x3ff00000; /* do exponent = 0 */
index = (index + (1<<(20-L-1))) >> (20-L);
/* reduce such that sqrt(2)/2 < xdb.d < sqrt(2) */
if (index >= MAXINDEX){ /* corresponds to xdb>sqrt(2)*/
xdb.i[HI] -= 0x00100000;
E++;
}
y = xdb.d;
index = index & INDEXMASK;
/* Cast integer E into double ed for multiplication later */
ed = (double) E;
/*
Read tables:
Read one float for ri
Read the first two doubles for -log(r_i) (out of three)
Organization of the table:
one struct entry per index, the struct entry containing
r, logih, logim and logil in this order
*/
ri = argredtable[index].ri;
/*
Actually we don't need the logarithm entries now
Move the following two lines to the eventual reconstruction
As long as we don't have any if in the following code, we can overlap
memory access with calculations
*/
logih = argredtable[index].logih;
logim = argredtable[index].logim;
/* Do range reduction:
zh + zl = y * ri - 1.0 correctly
Correctness is assured by use of Mul12 and Add12
even if we don't force ri to have its' LSBs set to zero
Discard zl for higher monome degrees
*/
Mul12(&yrih, &yril, y, ri);
th = yrih - 1.0;
Add12Cond(zh, zl, th, yril);
/*
Polynomial evaluation
Use a 7 degree polynomial
Evaluate the higher 5 terms in double precision (-7 * 3 = -21) using Horner's scheme
Evaluate the lower 3 terms (the last is 0) in double double precision accounting also for zl
using an ad hoc method
*/
#if defined(PROCESSOR_HAS_FMA) && !defined(AVOID_FMA)
polyHorner = FMA(FMA(FMA(FMA(c7,zh,c6),zh,c5),zh,c4),zh,c3);
#else
polyHorner = c3 + zh * (c4 + zh * (c5 + zh * (c6 + zh * c7)));
#endif
Mul12(&zhSquareh, &zhSquarel, zh, zh);
polyUpper = polyHorner * (zh * zhSquareh);
zhSquareHalfh = zhSquareh * -0.5;
zhSquareHalfl = zhSquarel * -0.5;
Add12(t1h, t1l, polyUpper, -1 * (zh * zl));
Add22(&t2h, &t2l, zh, zl, zhSquareHalfh, zhSquareHalfl);
Add22(&ph, &pl, t2h, t2l, t1h, t1l);
/* Reconstruction
Read logih and logim in the tables (already done)
Compute log(x) = E * log(2) + log(1+z) - log(ri)
i.e. log(x) = ed * (log2h + log2m) + (ph + pl) + (logih + logim) + delta
Carry out everything in double double precision
*/
/*
We store log2 as log2h + log2m + log2l where log2h and log2m have 12 trailing zeros
Multiplication of ed (double E) and log2h is thus correct
The overall accuracy of log2h + log2m + log2l is 53 * 3 - 24 = 135 which
is enough for the accurate phase
The accuracy suffices also for the quick phase: 53 * 2 - 24 = 82
Nevertheless the storage with trailing zeros implies an overlap of the tabulated
triple double values. We have to take it into account for the accurate phase
basic procedures for addition and multiplication
The condition on the next Add12 is verified as log2m is smaller than log2h
and both are scaled by ed
*/
Add12(log2edh, log2edl, log2h * ed, log2m * ed);
/* Add logih and logim to ph and pl
We must use conditioned Add22 as logih can move over ph
*/
Add22Cond(&logTabPolyh, &logTabPolyl, logih, logim, ph, pl);
/* Add log2edh + log2edl to logTabPolyh + logTabPolyl */
Add22Cond(&logh, &logm, log2edh, log2edl, logTabPolyh, logTabPolyl);
/* Rounding test and eventual return or call to the accurate function */
if(E==0)
roundcst = RDROUNDCST1;
else
roundcst = RDROUNDCST2;
TEST_AND_RETURN_RZ(logh, logm, roundcst);
#if DEBUG
printf("Going for Accurate Phase for x=%1.50e\n",x);
#endif
log_td_accurate(&logh, &logm, &logl, E, ed, index, zh, zl, logih, logim);
ReturnRoundTowardsZero3(logh, logm, logl);
}
interval j_exp(interval x)
{
interval res;
double x_inf, x_sup;
double rh_sup, rm_sup, rl_sup, tbl1h_sup, tbl1m_sup, tbl1l_sup;
double tbl2h_sup, tbl2m_sup, tbl2l_sup;
double xMultLog2InvMult2L_sup, shiftedXMult_sup, kd_sup;
double msLog2Div2LMultKh_sup, msLog2Div2LMultKm_sup, msLog2Div2LMultKl_sup;
double t1_sup, t2_sup, polyTblh_sup, polyTblm_sup, polyTbll_sup;
db_number shiftedXMultdb_sup, xdb_sup, resdb_sup;
int k_sup, M_sup, index1_sup, index2_sup, xIntHi_sup, mightBeDenorm_sup, roundable;
double t8_sup, t9_sup, t10_sup, t11_sup, t12_sup, t13_sup;
double rhSquare_sup, rhSquareHalf_sup, rhC3_sup, rhFour_sup, monomialCube_sup;
double highPoly_sup, highPolyWithSquare_sup, monomialFour_sup;
double tablesh_sup, tablesl_sup;
double s1_sup, s2_sup, s3_sup, s4_sup, s5_sup;
double res_sup;
double rh_inf, rm_inf, rl_inf, tbl1h_inf, tbl1m_inf, tbl1l_inf;
double tbl2h_inf, tbl2m_inf, tbl2l_inf;
double xMultLog2InvMult2L_inf, shiftedXMult_inf, kd_inf;
double msLog2Div2LMultKh_inf, msLog2Div2LMultKm_inf, msLog2Div2LMultKl_inf;
double t1_inf, t2_inf, polyTblh_inf, polyTblm_inf, polyTbll_inf;
db_number shiftedXMultdb_inf, xdb_inf, resdb_inf;
int k_inf, M_inf, index1_inf, index2_inf, xIntHi_inf, mightBeDenorm_inf;
double t8_inf, t9_inf, t10_inf, t11_inf, t12_inf, t13_inf;
double rhSquare_inf, rhSquareHalf_inf, rhC3_inf, rhFour_inf, monomialCube_inf;
double highPoly_inf, highPolyWithSquare_inf, monomialFour_inf;
double tablesh_inf, tablesl_inf;
double s1_inf, s2_inf, s3_inf, s4_inf, s5_inf;
double res_inf;
double res_simple_inf, res_simple_sup;
int infDone=0; int supDone=0;
x_inf=LOW(x);
x_sup=UP(x);
/* Argument reduction and filtering for special cases */
/* Compute k as a double and as an int */
xdb_sup.d = x_sup;
xdb_inf.d = x_inf;
xMultLog2InvMult2L_sup = x_sup * log2InvMult2L;
xMultLog2InvMult2L_inf = x_inf * log2InvMult2L;
shiftedXMult_sup = xMultLog2InvMult2L_sup + shiftConst;
shiftedXMult_inf = xMultLog2InvMult2L_inf + shiftConst;
kd_sup = shiftedXMult_sup - shiftConst;
kd_inf = shiftedXMult_inf - shiftConst;
shiftedXMultdb_sup.d = shiftedXMult_sup;
shiftedXMultdb_inf.d = shiftedXMult_inf;
/* Special cases tests */
xIntHi_sup = xdb_sup.i[HI];
mightBeDenorm_sup = 0;
/* Special cases tests */
xIntHi_inf = xdb_inf.i[HI];
mightBeDenorm_inf = 0;
if ( __builtin_expect(
((xIntHi_sup & 0x7ff00000) == 0)
|| (((xIntHi_sup & 0x7ff00000) == 0) && (x_sup == 0.0))
|| (((xIntHi_sup & 0x7ff00000) == 0) && (x_sup < 0.0))
|| (((xIntHi_sup & 0x7fffffff) >= OVRUDRFLWSMPLBOUND) && ((xIntHi_sup & 0x7fffffff) >= 0x7ff00000))
|| (((xIntHi_sup & 0x7fffffff) >= OVRUDRFLWSMPLBOUND) && ((xIntHi_sup & 0x7fffffff) >= 0x7ff00000) && (((xIntHi_sup & 0x000fffff) | xdb_sup.i[LO]) != 0))
|| (((xIntHi_sup & 0x7fffffff) >= OVRUDRFLWSMPLBOUND) && ((xIntHi_sup & 0x7fffffff) >= 0x7ff00000) && ((xIntHi_sup & 0x80000000)==0))
|| (((xIntHi_sup & 0x7fffffff) >= OVRUDRFLWSMPLBOUND) && (x_sup > OVRFLWBOUND))
|| (((xIntHi_sup & 0x7fffffff) >= OVRUDRFLWSMPLBOUND) && (x_sup <= UNDERFLWBOUND))
|| (((xIntHi_sup & 0x7fffffff) >= OVRUDRFLWSMPLBOUND) && (x_sup <= DENORMBOUND))
|| ((xIntHi_inf & 0x7ff00000) == 0)
|| (((xIntHi_inf & 0x7ff00000) == 0) && (x_inf == 0.0))
|| (((xIntHi_inf & 0x7ff00000) == 0) && (x_inf > 0.0))
|| (((xIntHi_inf & 0x7fffffff) >= OVRUDRFLWSMPLBOUND) && ((xIntHi_inf & 0x7fffffff) >= 0x7ff00000))
|| (((xIntHi_inf & 0x7fffffff) >= OVRUDRFLWSMPLBOUND) && ((xIntHi_inf & 0x7fffffff) >= 0x7ff00000) && (((xIntHi_inf & 0x000fffff) | xdb_inf.i[LO]) != 0))
|| (((xIntHi_inf & 0x7fffffff) >= OVRUDRFLWSMPLBOUND) && ((xIntHi_inf & 0x7fffffff) >= 0x7ff00000) && ((xIntHi_inf & 0x80000000)==0))
|| (((xIntHi_inf & 0x7fffffff) >= OVRUDRFLWSMPLBOUND) && (x_inf > OVRFLWBOUND))
|| (((xIntHi_inf & 0x7fffffff) >= OVRUDRFLWSMPLBOUND) && (x_inf <= UNDERFLWBOUND))
|| (((xIntHi_inf & 0x7fffffff) >= OVRUDRFLWSMPLBOUND) && (x_inf <= DENORMBOUND))
,FALSE))
{
ASSIGN_LOW(res,exp_rd(LOW(x)));
ASSIGN_UP(res,exp_ru(UP(x)));
return res;
}
/* Test if argument is a denormal or zero */
/* If we are here, we are sure to be neither +/- Inf nor NaN nor overflowed nor denormalized in the argument
but we might be denormalized in the result
We continue the argument reduction for the quick phase and table reads for both phases
*/
Mul12(&s1_sup,&s2_sup,msLog2Div2Lh,kd_sup);
Mul12(&s1_inf,&s2_inf,msLog2Div2Lh,kd_inf);
s3_sup = kd_sup * msLog2Div2Lm;
s3_inf = kd_inf * msLog2Div2Lm;
s4_sup = s2_sup + s3_sup;
s4_inf = s2_inf + s3_inf;
s5_sup = x_sup + s1_sup;
s5_inf = x_inf + s1_inf;
Add12Cond(rh_sup,rm_sup,s5_sup,s4_sup);
Add12Cond(rh_inf,rm_inf,s5_inf,s4_inf);
k_sup = shiftedXMultdb_sup.i[LO];
k_inf = shiftedXMultdb_inf.i[LO];
M_sup = k_sup >> L;
M_inf = k_inf >> L;
index1_sup = k_sup & INDEXMASK1;
index1_inf = k_inf & INDEXMASK1;
index2_sup = (k_sup & INDEXMASK2) >> LHALF;
index2_inf = (k_inf & INDEXMASK2) >> LHALF;
/* Table reads */
tbl1h_sup = twoPowerIndex1[index1_sup].hi;
tbl1h_inf = twoPowerIndex1[index1_inf].hi;
tbl1m_sup = twoPowerIndex1[index1_sup].mi;
tbl1m_inf = twoPowerIndex1[index1_inf].mi;
tbl2h_sup = twoPowerIndex2[index2_sup].hi;
tbl2h_inf = twoPowerIndex2[index2_inf].hi;
tbl2m_sup = twoPowerIndex2[index2_sup].mi;
tbl2m_inf = twoPowerIndex2[index2_inf].mi;
/* No more underflow nor denormal is possible. There may be the case where
M is 1024 and the value 2^M is to be multiplied may be less than 1
So the final result will be normalized and representable by the multiplication must be
made in 2 steps
*/
/* Quick phase starts here */
rhSquare_sup = rh_sup * rh_sup;
rhSquare_inf = rh_inf * rh_inf;
rhC3_sup = c3 * rh_sup;
rhC3_inf = c3 * rh_inf;
rhSquareHalf_sup = 0.5 * rhSquare_sup;
rhSquareHalf_inf = 0.5 * rhSquare_inf;
monomialCube_sup = rhC3_sup * rhSquare_sup;
monomialCube_inf = rhC3_inf * rhSquare_inf;
rhFour_sup = rhSquare_sup * rhSquare_sup;
rhFour_inf = rhSquare_inf * rhSquare_inf;
monomialFour_sup = c4 * rhFour_sup;
monomialFour_inf = c4 * rhFour_inf;
highPoly_sup = monomialCube_sup + monomialFour_sup;
highPoly_inf = monomialCube_inf + monomialFour_inf;
highPolyWithSquare_sup = rhSquareHalf_sup + highPoly_sup;
highPolyWithSquare_inf = rhSquareHalf_inf + highPoly_inf;
Mul22(&tablesh_sup,&tablesl_sup,tbl1h_sup,tbl1m_sup,tbl2h_sup,tbl2m_sup);
Mul22(&tablesh_inf,&tablesl_inf,tbl1h_inf,tbl1m_inf,tbl2h_inf,tbl2m_inf);
t8_sup = rm_sup + highPolyWithSquare_sup;
t8_inf = rm_inf + highPolyWithSquare_inf;
t9_sup = rh_sup + t8_sup;
t9_inf = rh_inf + t8_inf;
t10_sup = tablesh_sup * t9_sup;
t10_inf = tablesh_inf * t9_inf;
Add12(t11_sup,t12_sup,tablesh_sup,t10_sup);
Add12(t11_inf,t12_inf,tablesh_inf,t10_inf);
t13_sup = t12_sup + tablesl_sup;
t13_inf = t12_inf + tablesl_inf;
Add12(polyTblh_sup,polyTblm_sup,t11_sup,t13_sup);
Add12(polyTblh_inf,polyTblm_inf,t11_inf,t13_inf);
/* Rounding test
Since we know that the result of the final multiplication with 2^M
will always be representable, we can do the rounding test on the
factors and multiply only the final result.
We implement the multiplication in integer computations to overcome
the problem of the non-representability of 2^1024 if M = 1024
*/
if (infDone==1) res_inf=res_simple_inf;
if (supDone==1) res_sup=res_simple_sup;
// TEST_AND_COPY_RDRU_EXP(roundable,infDone,supDone,res_inf,polyTblh_inf,polyTblm_inf,res_sup,polyTblh_sup,polyTblm_sup,RDROUNDCST);
db_number yh_inf, yl_inf, u53_inf, yh_sup, yl_sup, u53_sup;
int yh_inf_neg, yl_inf_neg, yh_sup_neg, yl_sup_neg;
int rd_ok, ru_ok;
double save_res_inf=res_inf;
double save_res_sup=res_sup;
yh_inf.d = polyTblh_inf; yl_inf.d = polyTblm_inf;
yh_inf_neg = (yh_inf.i[HI] & 0x80000000);
yl_inf_neg = (yl_inf.i[HI] & 0x80000000);
yh_inf.l = yh_inf.l & 0x7fffffffffffffffLL; /* compute the absolute value*/
yl_inf.l = yl_inf.l & 0x7fffffffffffffffLL; /* compute the absolute value*/
u53_inf.l = (yh_inf.l & ULL(7ff0000000000000)) + ULL(0010000000000000);
yh_sup.d = polyTblh_sup; yl_sup.d = polyTblm_sup;
yh_sup_neg = (yh_sup.i[HI] & 0x80000000);
yl_sup_neg = (yl_sup.i[HI] & 0x80000000);
yh_sup.l = yh_sup.l & 0x7fffffffffffffffLL; /* compute the absolute value*/
yl_sup.l = yl_sup.l & 0x7fffffffffffffffLL; /* compute the absolute value*/
u53_sup.l = (yh_sup.l & ULL(7ff0000000000000)) + ULL(0010000000000000);
roundable = 0;
rd_ok=(yl_inf.d > RDROUNDCST * u53_inf.d);
ru_ok=(yl_sup.d > RDROUNDCST * u53_sup.d);
if(yl_inf_neg) { /* The case yl==0 is filtered by the above test*/
/* return next down */
yh_inf.d = polyTblh_inf;
if(yh_inf_neg) yh_inf.l++; else yh_inf.l--; /* Beware: fails for zero */
res_inf = yh_inf.d;
}
else {
res_inf = polyTblh_inf;
}
if(!yl_sup_neg) { /* The case yl==0 is filtered by the above test*/
/* return next up */
yh_sup.d = polyTblh_sup;
if(yh_sup_neg) yh_sup.l--; else yh_sup.l++; /* Beware: fails for zero */
res_sup = yh_sup.d;
}
else {
res_sup = polyTblh_sup;
}
if(infDone) res_inf=save_res_inf;
if(supDone) res_sup=save_res_sup;
if(rd_ok && ru_ok){
roundable=3;
}
else if (rd_ok){
roundable=1;
}
else if (ru_ok){
roundable=2;
}
resdb_inf.d = res_inf;
resdb_sup.d = res_sup;
if (roundable==3)
{
if (infDone==0){
resdb_inf.i[HI] += M_inf << 20;
}
ASSIGN_LOW(res,resdb_inf.d);
if (supDone==0){
resdb_sup.i[HI] += M_sup << 20;
}
ASSIGN_UP(res,resdb_sup.d);
return res;
}
if(roundable==1)
{
if(infDone==0){
resdb_inf.i[HI] += M_inf << 20;
}
ASSIGN_LOW(res,resdb_inf.d);
if(supDone==0){
/* Rest of argument reduction for accurate phase */
Mul133(&msLog2Div2LMultKh_sup,&msLog2Div2LMultKm_sup,&msLog2Div2LMultKl_sup,kd_sup,msLog2Div2Lh,msLog2Div2Lm,msLog2Div2Ll);
t1_sup = x_sup + msLog2Div2LMultKh_sup;
Add12Cond(rh_sup,t2_sup,t1_sup,msLog2Div2LMultKm_sup);
Add12Cond(rm_sup,rl_sup,t2_sup,msLog2Div2LMultKl_sup);
/* Table reads for accurate phase */
tbl1l_sup = twoPowerIndex1[index1_sup].lo;
tbl2l_sup = twoPowerIndex2[index2_sup].lo;
/* Call accurate phase */
exp_td_accurate(&polyTblh_sup, &polyTblm_sup, &polyTbll_sup, rh_sup, rm_sup, rl_sup, tbl1h_sup, tbl1m_sup, tbl1l_sup, tbl2h_sup, tbl2m_sup, tbl2l_sup);
/* Since the final multiplication is exact, we can do the final rounding before multiplying
We overcome this way also the cases where the final result is not underflowed whereas the
lower parts of the intermediate final result are.
*/
RoundUpwards3(&res_sup,polyTblh_sup,polyTblm_sup,polyTbll_sup);
/* Final multiplication with 2^M
We implement the multiplication in integer computations to overcome
the problem of the non-representability of 2^1024 if M = 1024
*/
resdb_sup.d = res_sup;
resdb_sup.i[HI] += M_sup << 20;
}
ASSIGN_UP(res,resdb_sup.d);
return res;
} /* Accurate phase launched after rounding test*/
if (roundable==2) {
if (infDone==0){
/* Rest of argument reduction for accurate phase */
Mul133(&msLog2Div2LMultKh_inf,&msLog2Div2LMultKm_inf,&msLog2Div2LMultKl_inf,kd_inf,msLog2Div2Lh,msLog2Div2Lm,msLog2Div2Ll);
t1_inf = x_inf + msLog2Div2LMultKh_inf;
Add12Cond(rh_inf,t2_inf,t1_inf,msLog2Div2LMultKm_inf);
Add12Cond(rm_inf,rl_inf,t2_inf,msLog2Div2LMultKl_inf);
/* Table reads for accurate phase */
tbl1l_inf = twoPowerIndex1[index1_inf].lo;
tbl2l_inf = twoPowerIndex2[index2_inf].lo;
/* Call accurate phase */
exp_td_accurate(&polyTblh_inf, &polyTblm_inf, &polyTbll_inf, rh_inf, rm_inf, rl_inf, tbl1h_inf, tbl1m_inf, tbl1l_inf, tbl2h_inf, tbl2m_inf, tbl2l_inf);
/* Since the final multiplication is exact, we can do the final rounding before multiplying
We overcome this way also the cases where the final result is not underflowed whereas the
lower parts of the intermediate final result are.
*/
RoundDownwards3(&res_inf,polyTblh_inf,polyTblm_inf,polyTbll_inf);
/* Final multiplication with 2^M
We implement the multiplication in integer computations to overcome
the problem of the non-representability of 2^1024 if M = 1024
*/
resdb_inf.d = res_inf;
resdb_inf.i[HI] += M_inf << 20;
}
ASSIGN_LOW(res,resdb_inf.d);
if(supDone==0){
resdb_sup.i[HI] += M_sup << 20;
}
ASSIGN_UP(res,resdb_sup.d);
return res;
} /* Accurate phase launched after rounding test*/
if(roundable==0)
{
if(supDone==0){
/* Rest of argument reduction for accurate phase */
Mul133(&msLog2Div2LMultKh_sup,&msLog2Div2LMultKm_sup,&msLog2Div2LMultKl_sup,kd_sup,msLog2Div2Lh,msLog2Div2Lm,msLog2Div2Ll);
t1_sup = x_sup + msLog2Div2LMultKh_sup;
Add12Cond(rh_sup,t2_sup,t1_sup,msLog2Div2LMultKm_sup);
Add12Cond(rm_sup,rl_sup,t2_sup,msLog2Div2LMultKl_sup);
/* Table reads for accurate phase */
tbl1l_sup = twoPowerIndex1[index1_sup].lo;
tbl2l_sup = twoPowerIndex2[index2_sup].lo;
/* Call accurate phase */
exp_td_accurate(&polyTblh_sup, &polyTblm_sup, &polyTbll_sup, rh_sup, rm_sup, rl_sup, tbl1h_sup, tbl1m_sup, tbl1l_sup, tbl2h_sup, tbl2m_sup, tbl2l_sup);
/* Since the final multiplication is exact, we can do the final rounding before multiplying
We overcome this way also the cases where the final result is not underflowed whereas the
lower parts of the intermediate final result are.
*/
RoundUpwards3(&res_sup,polyTblh_sup,polyTblm_sup,polyTbll_sup);
/* Final multiplication with 2^M
We implement the multiplication in integer computations to overcome
the problem of the non-representability of 2^1024 if M = 1024
*/
resdb_sup.d = res_sup;
resdb_sup.i[HI] += M_sup << 20;
}
ASSIGN_UP(res,resdb_sup.d);
if (infDone==0){
/* Rest of argument reduction for accurate phase */
Mul133(&msLog2Div2LMultKh_inf,&msLog2Div2LMultKm_inf,&msLog2Div2LMultKl_inf,kd_inf,msLog2Div2Lh,msLog2Div2Lm,msLog2Div2Ll);
t1_inf = x_inf + msLog2Div2LMultKh_inf;
Add12Cond(rh_inf,t2_inf,t1_inf,msLog2Div2LMultKm_inf);
Add12Cond(rm_inf,rl_inf,t2_inf,msLog2Div2LMultKl_inf);
/* Table reads for accurate phase */
tbl1l_inf = twoPowerIndex1[index1_inf].lo;
tbl2l_inf = twoPowerIndex2[index2_inf].lo;
/* Call accurate phase */
exp_td_accurate(&polyTblh_inf, &polyTblm_inf, &polyTbll_inf, rh_inf, rm_inf, rl_inf, tbl1h_inf, tbl1m_inf, tbl1l_inf, tbl2h_inf, tbl2m_inf, tbl2l_inf);
/* Since the final multiplication is exact, we can do the final rounding before multiplying
We overcome this way also the cases where the final result is not underflowed whereas the
lower parts of the intermediate final result are.
*/
RoundDownwards3(&res_inf,polyTblh_inf,polyTblm_inf,polyTbll_inf);
/* Final multiplication with 2^M
We implement the multiplication in integer computations to overcome
the problem of the non-representability of 2^1024 if M = 1024
*/
resdb_inf.d = res_inf;
resdb_inf.i[HI] += M_inf << 20;
}
ASSIGN_LOW(res,resdb_inf.d);
return res;
} /* Accurate phase launched after rounding test*/
return res;
}
/*************************************************************
*************************************************************
* ROUNDED DOWNWARDS *
*************************************************************
*************************************************************/
double exp_rd(double x) {
double rh, rm, rl, tbl1h, tbl1m, tbl1l;
double tbl2h, tbl2m, tbl2l;
double xMultLog2InvMult2L, shiftedXMult, kd;
double msLog2Div2LMultKh, msLog2Div2LMultKm, msLog2Div2LMultKl;
double t1, t2, t3, t4, polyTblh, polyTblm, polyTbll;
db_number shiftedXMultdb, twoPowerMdb, xdb, t4db, t4db2, resdb;
int k, M, index1, index2, xIntHi, mightBeDenorm, roundable;
double t5, t6, t7, t8, t9, t10, t11, t12, t13;
double rhSquare, rhSquareHalf, rhC3, rhFour, monomialCube;
double highPoly, highPolyWithSquare, monomialFour;
double tablesh, tablesl;
double s1, s2, s3, s4, s5;
double res;
/* Argument reduction and filtering for special cases */
/* Compute k as a double and as an int */
xdb.d = x;
xMultLog2InvMult2L = x * log2InvMult2L;
shiftedXMult = xMultLog2InvMult2L + shiftConst;
kd = shiftedXMult - shiftConst;
shiftedXMultdb.d = shiftedXMult;
/* Special cases tests */
xIntHi = xdb.i[HI];
mightBeDenorm = 0;
/* Test if argument is a denormal or zero */
if ((xIntHi & 0x7ff00000) == 0) {
/* If the argument is exactly zero, we just return 1.0
which is the mathematical image of the function
*/
if (x == 0.0) return 1.0;
/* If the argument is a positive denormal, we
must return 1.0 and raise the inexact flag.
*/
if (x > 0.0) return 1.0 + SMALLEST;
/* Otherwise, we return 1.0 - 1ulp since
exp(-greatest denorm) > 1.0 - 1ulp
We must do the addition dynamically for
raising the inexact flag.
*/
return 1.0 + mTwoM53;
}
/* Test if argument is greater than approx. 709 in magnitude */
if ((xIntHi & 0x7fffffff) >= OVRUDRFLWSMPLBOUND) {
/* If we are here, the result might be overflowed, underflowed, inf, or NaN */
/* Test if +/- Inf or NaN */
if ((xIntHi & 0x7fffffff) >= 0x7ff00000) {
/* Either NaN or Inf in this case since exponent is maximal */
/* Test if NaN: mantissa is not 0 */
if (((xIntHi & 0x000fffff) | xdb.i[LO]) != 0) {
/* x = NaN, return NaN */
return x + x;
} else {
/* +/- Inf */
/* Test sign */
if ((xIntHi & 0x80000000)==0)
/* x = +Inf, return +Inf */
return x;
else
/* x = -Inf, return 0 */
return 0;
} /* End which in NaN, Inf */
} /* End NaN or Inf ? */
/* If we are here, we might be overflowed, denormalized or underflowed in the result
but there is no special case (NaN, Inf) left */
/* Test if actually overflowed */
if (x > OVRFLWBOUND) {
/* We would be overflowed but as we are rounding downwards
the nearest number lesser than the exact result is the greatest
normal. In any case, we must raise the inexact flag.
*/
return LARGEST * (1.0 + SMALLEST);
}
/* Test if surely underflowed */
if (x <= UNDERFLWBOUND) {
/* We are actually sure to be underflowed and not denormalized any more
(at least where computing makes sense); since we are in the round
upwards case, we return the smallest denormal possible.
*/
return SMALLEST * SMALLEST;
}
/* Test if possibly denormalized */
if (x <= DENORMBOUND) {
/* We know now that we are not sure to be normalized in the result
We just set an internal flag for a further test
*/
mightBeDenorm = 1;
}
} /* End might be a special case */
/* If we are here, we are sure to be neither +/- Inf nor NaN nor overflowed nor denormalized in the argument
but we might be denormalized in the result
We continue the argument reduction for the quick phase and table reads for both phases
*/
Mul12(&s1,&s2,msLog2Div2Lh,kd);
s3 = kd * msLog2Div2Lm;
s4 = s2 + s3;
s5 = x + s1;
Add12Cond(rh,rm,s5,s4);
k = shiftedXMultdb.i[LO];
M = k >> L;
index1 = k & INDEXMASK1;
index2 = (k & INDEXMASK2) >> LHALF;
/* Table reads */
tbl1h = twoPowerIndex1[index1].hi;
tbl1m = twoPowerIndex1[index1].mi;
tbl2h = twoPowerIndex2[index2].hi;
tbl2m = twoPowerIndex2[index2].mi;
/* Test now if it is sure to launch the quick phase because no denormalized result is possible */
if (mightBeDenorm == 1) {
/* The result might be denormalized, we launch the accurate phase in all cases */
/* Rest of argument reduction for accurate phase */
Mul133(&msLog2Div2LMultKh,&msLog2Div2LMultKm,&msLog2Div2LMultKl,kd,msLog2Div2Lh,msLog2Div2Lm,msLog2Div2Ll);
t1 = x + msLog2Div2LMultKh;
Add12Cond(rh,t2,t1,msLog2Div2LMultKm);
Add12Cond(rm,rl,t2,msLog2Div2LMultKl);
/* Table reads for accurate phase */
tbl1l = twoPowerIndex1[index1].lo;
tbl2l = twoPowerIndex2[index2].lo;
/* Call accurate phase */
exp_td_accurate(&polyTblh, &polyTblm, &polyTbll, rh, rm, rl, tbl1h, tbl1m, tbl1l, tbl2h, tbl2m, tbl2l);
/* Final rounding and multiplication with 2^M
We first multiply the highest significant byte by 2^M in two steps
and adjust it then depending on the lower significant parts.
We cannot multiply directly by 2^M since M is less than -1022.
We first multiply by 2^(-1000) and then by 2^(M+1000).
*/
t3 = polyTblh * twoPowerM1000;
/* Form now twoPowerM with adjusted M */
twoPowerMdb.i[LO] = 0;
twoPowerMdb.i[HI] = (M + 2023) << 20;
/* Multiply with the rest of M, the result will be denormalized */
t4 = t3 * twoPowerMdb.d;
/* For x86, force the compiler to pass through memory for having the right rounding */
t4db.d = t4; /* Do not #if-ify this line, we need the copy */
#if defined(CRLIBM_TYPECPU_AMD64) || defined(CRLIBM_TYPECPU_X86)
t4db2.i[HI] = t4db.i[HI];
t4db2.i[LO] = t4db.i[LO];
t4 = t4db2.d;
#endif
/* Remultiply by 2^(-M) for manipulating the rounding error and the lower significant parts */
M *= -1;
twoPowerMdb.i[LO] = 0;
twoPowerMdb.i[HI] = (M + 23) << 20;
t5 = t4 * twoPowerMdb.d;
t6 = t5 * twoPower1000;
t7 = polyTblh - t6;
/* The rounding can be decided using the sign of the arithmetical sum of the
round-to-nearest-error (i.e. t7) and the lower part(s) of the final result.
We add first the lower parts and add the result to the error in t7. We have to
keep in mind that everything is scaled by 2^(-M).
t8 can never be exactly 0 since we filter out the cases where the image of the
function is algebraic and the implementation is exacter than the TMD worst case.
*/
polyTblm = polyTblm + polyTbll;
t8 = t7 + polyTblm;
/* Since we are rounding downwards, the round-to-nearest-rounding result in t4 is
equal to the final result if the rounding error (i.e. the error plus the lower parts)
is positive, i.e. if the rounding-to-nearest was downwards.
*/
if (t8 > 0.0) return t4;
/* If we are here, we must adjust the final result by +1ulp
Relying on the fact that the exponential is always positive, we can simplify this
adjustment
*/
t4db.l--;
return t4db.d;
} /* End accurate phase launched as there might be a denormalized result */
/* No more underflow nor denormal is possible. There may be the case where
M is 1024 and the value 2^M is to be multiplied may be less than 1
So the final result will be normalized and representable by the multiplication must be
made in 2 steps
*/
/* Quick phase starts here */
rhSquare = rh * rh;
rhC3 = c3 * rh;
rhSquareHalf = 0.5 * rhSquare;
monomialCube = rhC3 * rhSquare;
rhFour = rhSquare * rhSquare;
monomialFour = c4 * rhFour;
highPoly = monomialCube + monomialFour;
highPolyWithSquare = rhSquareHalf + highPoly;
Mul22(&tablesh,&tablesl,tbl1h,tbl1m,tbl2h,tbl2m);
t8 = rm + highPolyWithSquare;
t9 = rh + t8;
t10 = tablesh * t9;
Add12(t11,t12,tablesh,t10);
t13 = t12 + tablesl;
Add12(polyTblh,polyTblm,t11,t13);
/* Rounding test
Since we know that the result of the final multiplication with 2^M
will always be representable, we can do the rounding test on the
factors and multiply only the final result.
We implement the multiplication in integer computations to overcome
the problem of the non-representability of 2^1024 if M = 1024
*/
TEST_AND_COPY_RD(roundable,res,polyTblh,polyTblm,RDROUNDCST);
if (roundable) {
resdb.d = res;
resdb.i[HI] += M << 20;
return resdb.d;
} else {
/* Rest of argument reduction for accurate phase */
Mul133(&msLog2Div2LMultKh,&msLog2Div2LMultKm,&msLog2Div2LMultKl,kd,msLog2Div2Lh,msLog2Div2Lm,msLog2Div2Ll);
t1 = x + msLog2Div2LMultKh;
Add12Cond(rh,t2,t1,msLog2Div2LMultKm);
Add12Cond(rm,rl,t2,msLog2Div2LMultKl);
/* Table reads for accurate phase */
tbl1l = twoPowerIndex1[index1].lo;
tbl2l = twoPowerIndex2[index2].lo;
/* Call accurate phase */
exp_td_accurate(&polyTblh, &polyTblm, &polyTbll, rh, rm, rl, tbl1h, tbl1m, tbl1l, tbl2h, tbl2m, tbl2l);
/* Since the final multiplication is exact, we can do the final rounding before multiplying
We overcome this way also the cases where the final result is not underflowed whereas the
lower parts of the intermediate final result are.
*/
RoundDownwards3(&res,polyTblh,polyTblm,polyTbll);
/* Final multiplication with 2^M
We implement the multiplication in integer computations to overcome
the problem of the non-representability of 2^1024 if M = 1024
*/
resdb.d = res;
resdb.i[HI] += M << 20;
return resdb.d;
} /* Accurate phase launched after rounding test*/
}
/*************************************************************
*************************************************************
* ROUNDED TO NEAREST *
*************************************************************
*************************************************************/
double exp_rn(double x){
double rh, rm, rl, tbl1h, tbl1m, tbl1l;
double tbl2h, tbl2m, tbl2l;
double xMultLog2InvMult2L, shiftedXMult, kd;
double msLog2Div2LMultKh, msLog2Div2LMultKm, msLog2Div2LMultKl;
double t1, t2, t3, t4, polyTblh, polyTblm, polyTbll;
db_number shiftedXMultdb, twoPowerMdb, xdb, t4db, t4db2, polyTblhdb, resdb;
int k, M, index1, index2, xIntHi, mightBeDenorm;
double t5, t6, t7, t8, t9, t10, t11, t12, t13;
double rhSquare, rhSquareHalf, rhC3, rhFour, monomialCube;
double highPoly, highPolyWithSquare, monomialFour;
double tablesh, tablesl;
double s1, s2, s3, s4, s5;
double res;
/* Argument reduction and filtering for special cases */
/* Compute k as a double and as an int */
xdb.d = x;
xMultLog2InvMult2L = x * log2InvMult2L;
shiftedXMult = xMultLog2InvMult2L + shiftConst;
kd = shiftedXMult - shiftConst;
shiftedXMultdb.d = shiftedXMult;
/* Special cases tests */
xIntHi = xdb.i[HI];
mightBeDenorm = 0;
/* Test if argument is a denormal or zero */
if ((xIntHi & 0x7ff00000) == 0) {
/* We are in the RN case, return 1.0 in all cases */
return 1.0;
}
/* Test if argument is greater than approx. 709 in magnitude */
if ((xIntHi & 0x7fffffff) >= OVRUDRFLWSMPLBOUND) {
/* If we are here, the result might be overflowed, underflowed, inf, or NaN */
/* Test if +/- Inf or NaN */
if ((xIntHi & 0x7fffffff) >= 0x7ff00000) {
/* Either NaN or Inf in this case since exponent is maximal */
/* Test if NaN: mantissa is not 0 */
if (((xIntHi & 0x000fffff) | xdb.i[LO]) != 0) {
/* x = NaN, return NaN */
return x + x;
} else {
/* +/- Inf */
/* Test sign */
if ((xIntHi & 0x80000000)==0)
/* x = +Inf, return +Inf */
return x;
else
/* x = -Inf, return 0 */
return 0;
} /* End which in NaN, Inf */
} /* End NaN or Inf ? */
/* If we are here, we might be overflowed, denormalized or underflowed in the result
but there is no special case (NaN, Inf) left */
/* Test if actually overflowed */
if (x > OVRFLWBOUND) {
/* We are actually overflowed in the result */
return LARGEST * LARGEST;
}
/* Test if surely underflowed */
if (x <= UNDERFLWBOUND) {
/* We are actually sure to be underflowed and not denormalized any more
So we return 0 and raise the inexact flag */
return SMALLEST * SMALLEST;
}
/* Test if possibly denormalized */
if (x <= DENORMBOUND) {
/* We know now that we are not sure to be normalized in the result
We just set an internal flag for a further test
*/
mightBeDenorm = 1;
}
} /* End might be a special case */
/* If we are here, we are sure to be neither +/- Inf nor NaN nor overflowed nor denormalized in the argument
but we might be denormalized in the result
We continue the argument reduction for the quick phase and table reads for both phases
*/
Mul12(&s1,&s2,msLog2Div2Lh,kd);
s3 = kd * msLog2Div2Lm;
s4 = s2 + s3;
s5 = x + s1;
Add12Cond(rh,rm,s5,s4);
k = shiftedXMultdb.i[LO];
M = k >> L;
index1 = k & INDEXMASK1;
index2 = (k & INDEXMASK2) >> LHALF;
/* Table reads */
tbl1h = twoPowerIndex1[index1].hi;
tbl1m = twoPowerIndex1[index1].mi;
tbl2h = twoPowerIndex2[index2].hi;
tbl2m = twoPowerIndex2[index2].mi;
/* Test now if it is sure to launch the quick phase because no denormalized result is possible */
if (mightBeDenorm == 1) {
/* The result might be denormalized, we launch the accurate phase in all cases */
/* Rest of argument reduction for accurate phase */
Mul133(&msLog2Div2LMultKh,&msLog2Div2LMultKm,&msLog2Div2LMultKl,kd,msLog2Div2Lh,msLog2Div2Lm,msLog2Div2Ll);
t1 = x + msLog2Div2LMultKh;
Add12Cond(rh,t2,t1,msLog2Div2LMultKm);
Add12Cond(rm,rl,t2,msLog2Div2LMultKl);
/* Table reads for accurate phase */
tbl1l = twoPowerIndex1[index1].lo;
tbl2l = twoPowerIndex2[index2].lo;
/* Call accurate phase */
exp_td_accurate(&polyTblh, &polyTblm, &polyTbll, rh, rm, rl, tbl1h, tbl1m, tbl1l, tbl2h, tbl2m, tbl2l);
/* Final rounding and multiplication with 2^M
We first multiply the highest significant byte by 2^M in two steps
and adjust it then depending on the lower significant parts.
We cannot multiply directly by 2^M since M is less than -1022.
We first multiply by 2^(-1000) and then by 2^(M+1000).
*/
t3 = polyTblh * twoPowerM1000;
/* Form now twoPowerM with adjusted M */
twoPowerMdb.i[LO] = 0;
twoPowerMdb.i[HI] = (M + 2023) << 20;
/* Multiply with the rest of M, the result will be denormalized */
t4 = t3 * twoPowerMdb.d;
/* For x86, force the compiler to pass through memory for having the right rounding */
t4db.d = t4; /* Do not #if-ify this line, we need the copy */
#if defined(CRLIBM_TYPECPU_AMD64) || defined(CRLIBM_TYPECPU_X86)
t4db2.i[HI] = t4db.i[HI];
t4db2.i[LO] = t4db.i[LO];
t4 = t4db2.d;
#endif
/* Remultiply by 2^(-M) for manipulating the rounding error and the lower significant parts */
M *= -1;
twoPowerMdb.i[LO] = 0;
twoPowerMdb.i[HI] = (M + 23) << 20;
t5 = t4 * twoPowerMdb.d;
t6 = t5 * twoPower1000;
t7 = polyTblh - t6;
/* The rounding decision is made at 1/2 ulp of a denormal, i.e. at 2^(-1075)
We construct this number and by comparing with it we get to know
whether we are in a difficult rounding case or not. If not we just return
the known result. Otherwise we continue with further tests.
*/
twoPowerMdb.i[LO] = 0;
twoPowerMdb.i[HI] = (M - 52) << 20;
if (ABS(t7) != twoPowerMdb.d) return t4;
/* If we are here, we are in a difficult rounding case */
/* We have to adjust the result iff the sign of the error on
rounding 2^M * polyTblh (which must be an ulp of a denormal)
and polyTblm +arith polyTbll is the same which means that
the error made was greater than an ulp of an denormal.
*/
polyTblm = polyTblm + polyTbll;
if (t7 > 0.0) {
if (polyTblm > 0.0) {
t4db.l++;
return t4db.d;
} else return t4;
} else {
if (polyTblm < 0.0) {
t4db.l--;
return t4db.d;
} else return t4;
}
} /* End accurate phase launched as there might be a denormalized result */
/* No more underflow nor denormal is possible. There may be the case where
M is 1024 and the value 2^M is to be multiplied may be less than 1
So the final result will be normalized and representable by the multiplication must be
made in 2 steps
*/
/* Quick phase starts here */
rhSquare = rh * rh;
rhC3 = c3 * rh;
rhSquareHalf = 0.5 * rhSquare;
monomialCube = rhC3 * rhSquare;
rhFour = rhSquare * rhSquare;
monomialFour = c4 * rhFour;
highPoly = monomialCube + monomialFour;
highPolyWithSquare = rhSquareHalf + highPoly;
Mul22(&tablesh,&tablesl,tbl1h,tbl1m,tbl2h,tbl2m);
t8 = rm + highPolyWithSquare;
t9 = rh + t8;
t10 = tablesh * t9;
Add12(t11,t12,tablesh,t10);
t13 = t12 + tablesl;
Add12(polyTblh,polyTblm,t11,t13);
/* Rounding test
Since we know that the result of the final multiplication with 2^M
will always be representable, we can do the rounding test on the
factors and multiply only the final result.
We implement the multiplication in integer computations to overcome
the problem of the non-representability of 2^1024 if M = 1024
*/
if(polyTblh == (polyTblh + (polyTblm * ROUNDCST))) {
polyTblhdb.d = polyTblh;
polyTblhdb.i[HI] += M << 20;
return polyTblhdb.d;
} else
{
/* Rest of argument reduction for accurate phase */
Mul133(&msLog2Div2LMultKh,&msLog2Div2LMultKm,&msLog2Div2LMultKl,kd,msLog2Div2Lh,msLog2Div2Lm,msLog2Div2Ll);
t1 = x + msLog2Div2LMultKh;
Add12Cond(rh,t2,t1,msLog2Div2LMultKm);
Add12Cond(rm,rl,t2,msLog2Div2LMultKl);
/* Table reads for accurate phase */
tbl1l = twoPowerIndex1[index1].lo;
tbl2l = twoPowerIndex2[index2].lo;
/* Call accurate phase */
exp_td_accurate(&polyTblh, &polyTblm, &polyTbll, rh, rm, rl, tbl1h, tbl1m, tbl1l, tbl2h, tbl2m, tbl2l);
/* Since the final multiplication is exact, we can do the final rounding before multiplying
We overcome this way also the cases where the final result is not underflowed whereas the
lower parts of the intermediate final result are.
*/
RoundToNearest3(&res,polyTblh,polyTblm,polyTbll);
/* Final multiplication with 2^M
We implement the multiplication in integer computations to overcome
the problem of the non-representability of 2^1024 if M = 1024
*/
resdb.d = res;
resdb.i[HI] += M << 20;
return resdb.d;
} /* Accurate phase launched after rounding test*/
}
static void do_cosh(double x, double* preshi, double* preslo) {
int k;
db_number y;
double ch_hi, ch_lo, sh_hi, sh_lo;/* cosh(x) = (ch_hi + ch_lo)*(cosh(k*ln(2)) + (sh_hi + sh_lo)*(sinh(k*ln(2))) */
db_number table_index_float;
int table_index;
double temp_hi, temp_lo, temp;/* some temporary variables */
double b_hi, b_lo,b_ca_hi, b_ca_lo, b_sa_hi, b_sa_lo;
double ca_hi, ca_lo, sa_hi, sa_lo; /*will be the tabulated values */
double tcb_hi, tsb_hi; /*results of polynomial approximations*/
double square_b_hi;
double ch_2_pk_hi, ch_2_pk_lo, ch_2_mk_hi, ch_2_mk_lo;
double sh_2_pk_hi, sh_2_pk_lo, sh_2_mk_hi, sh_2_mk_lo;
db_number two_p_plus_k, two_p_minus_k; /* 2^(k-1) + 2^(-k-1) */
/* First range reduction*/
DOUBLE2INT(k, x * inv_ln_2.d)
if (k != 0) { /* b_hi+b_lo = x - (ln2_hi + ln2_lo) * k */
temp_hi = x - ln2_hi.d * k;
temp_lo = -ln2_lo.d * k;
Add12Cond(b_hi, b_lo, temp_hi, temp_lo);
}
else {
b_hi = x;
b_lo = 0.;
}
/*we'll construct 2 constants for the last reconstruction */
two_p_plus_k.i[LO] = 0;
two_p_plus_k.i[HI] = (k-1+1023) << 20;
two_p_minus_k.i[LO] = 0;
two_p_minus_k.i[HI] = (-k-1+1023) << 20;
/* at this stage, we've done the first range reduction : we have b_hi + b_lo between -ln(2)/2 and ln(2)/2 */
/* now we can do the second range reduction */
/* we'll get the 8 leading bits of b_hi */
table_index_float.d = b_hi + two_43_44.d;
/*this add do the float equivalent of a rotation to the right, since -0.5 <= b_hi <= 0.5*/
table_index = table_index_float.i[LO];/* -89 <= table_index <= 89 */
table_index_float.d -= two_43_44.d;
table_index += bias; /* to have only positive values */
b_hi -= table_index_float.d;/* to remove the 8 leading bits*/
/* since b_hi was between -2^-1 and 2^1, we now have b_hi between -2^-9 and 2^-9 */
y.d = b_hi;
/* first, y² */
square_b_hi = b_hi * b_hi;
/* effective computation of the polynomial approximation */
if (((y.i[HI])&(0x7FFFFFFF)) < (two_minus_30.i[HI])) {
tcb_hi = 0;
tsb_hi = 0;
}
else {
/* second, cosh(y) = y² * (1/2 + y² * (1/24 + y² * 1/720)) */
tcb_hi = (square_b_hi)* (c2.d + square_b_hi * (c4.d + square_b_hi * c6.d));
tsb_hi = square_b_hi * (s3.d + square_b_hi * (s5.d + square_b_hi * s7.d));
}
if( table_index != bias) {
/* we get the tabulated the tabulated values */
ca_hi = cosh_sinh_table[table_index][0].d;
ca_lo = cosh_sinh_table[table_index][1].d;
sa_hi = cosh_sinh_table[table_index][2].d;
sa_lo = cosh_sinh_table[table_index][3].d;
/* first reconstruction of the cosh (corresponding to the second range reduction) */
Mul12(&b_sa_hi,&b_sa_lo, sa_hi, b_hi);
temp = ((((((ca_lo + (b_hi * sa_lo)) + b_lo * sa_hi) + b_sa_lo) + (b_sa_hi * tsb_hi)) + ca_hi * tcb_hi) + b_sa_hi);
Add12Cond(ch_hi, ch_lo, ca_hi, temp);
/* first reconstruction for the sinh (corresponding to the second range reduction) */
}
else {
Add12Cond(ch_hi, ch_lo, (double) 1, tcb_hi);
}
if(k != 0) {
if( table_index != bias) {
/* first reconstruction for the sinh (corresponding to the second range reduction) */
Mul12(&b_ca_hi , &b_ca_lo, ca_hi, b_hi);
temp = (((((sa_lo + (b_lo * ca_hi)) + (b_hi * ca_lo)) + b_ca_lo) + (sa_hi*tcb_hi)) + (b_ca_hi * tsb_hi));
Add12(temp_hi, temp_lo, b_ca_hi, temp);
Add22Cond(&sh_hi, &sh_lo, sa_hi, (double) 0, temp_hi, temp_lo);
}
else {
Add12Cond(sh_hi, sh_lo, b_hi, tsb_hi * b_hi + b_lo);
}
if((k < 35) && (k > -35) )
{
ch_2_pk_hi = ch_hi * two_p_plus_k.d;
ch_2_pk_lo = ch_lo * two_p_plus_k.d;
ch_2_mk_hi = ch_hi * two_p_minus_k.d;
ch_2_mk_lo = ch_lo * two_p_minus_k.d;
sh_2_pk_hi = sh_hi * two_p_plus_k.d;
sh_2_pk_lo = sh_lo * two_p_plus_k.d;
sh_2_mk_hi = - sh_hi * two_p_minus_k.d;
sh_2_mk_lo = - sh_lo * two_p_minus_k.d;
Add22Cond(preshi, preslo, ch_2_mk_hi, ch_2_mk_lo, sh_2_mk_hi, sh_2_mk_lo);
Add22Cond(&ch_2_mk_hi, &ch_2_mk_lo , sh_2_pk_hi, sh_2_pk_lo, *preshi, *preslo);
Add22Cond(preshi, preslo, ch_2_pk_hi, ch_2_pk_lo, ch_2_mk_hi, ch_2_mk_lo);
}
else if (k >= 35)
{
ch_2_pk_hi = ch_hi * two_p_plus_k.d;
ch_2_pk_lo = ch_lo * two_p_plus_k.d;
sh_2_pk_hi = sh_hi * two_p_plus_k.d;
sh_2_pk_lo = sh_lo * two_p_plus_k.d;
Add22Cond(preshi, preslo, ch_2_pk_hi, ch_2_pk_lo, sh_2_pk_hi, sh_2_pk_lo);
}
else /* if (k <= -35) */
{
ch_2_mk_hi = ch_hi * two_p_minus_k.d;
ch_2_mk_lo = ch_lo * two_p_minus_k.d;
sh_2_mk_hi = - sh_hi * two_p_minus_k.d;
sh_2_mk_lo = - sh_lo * two_p_minus_k.d;
Add22Cond(preshi, preslo, ch_2_mk_hi, ch_2_mk_lo, sh_2_mk_hi, sh_2_mk_lo);
}
}
else {
*preshi = ch_hi;
*preslo = ch_lo;
}
return;
}
