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);
}
static void do_cosh_accurate(int* pexponent,
double* presh, double* presm, double* presl,
double x) {
double exph, expm, expl;
double expph, exppm, exppl;
int exponentm, deltaexponent;
db_number expmh, expmm, expml;
#if EVAL_PERF==1
crlibm_second_step_taken++;
#endif
if(x<0)
x=-x;
if (x > 40.0) { /* then exp(-x) < 2^-118 exp(x) */
exp13(pexponent, presh, presm, presl, x);
}
else {
exp13(pexponent, &expph, &exppm, &exppl, x);
exp13(&exponentm, &(expmh.d), &(expmm.d), &(expml.d), -x);
/* align the mantissas.
The exponent is increased but stays well below overflow threshold */
deltaexponent = exponentm - *pexponent ;
expmh.i[HI] += (deltaexponent) << 20;
expmm.i[HI] += (deltaexponent) << 20;
expml.i[HI] += (deltaexponent) << 20;
Add33(&exph, &expm, &expl, expph, exppm, exppl, expmh.d, expmm.d, expml.d);
Renormalize3(presh,presm,presl, exph, expm, expl);
}
}
static void do_sinh_accurate(int* pexponent,
double* presh, double* presm, double* presl,
double x) {
double exph, expm, expl;
double expph, exppm, exppl, expmh, expmm, expml;
#if EVAL_PERF==1
crlibm_second_step_taken++;
#endif
if(x > 40.0) { /* then exp(-x) < 2^-129 exp(x) */
exp13(pexponent, presh, presm, presl, x);
return;
}
if(x < -40.0) { /* then exp(x) < 2^-129 exp(-x) */
exp13(pexponent, presh, presm, presl, -x);
*presh = -*presh;
*presm = -*presm;
*presl = -*presl;
return;
}
/* Otherwise we are between -40 and 40, and we also know that |x| > 2^-25 */
if(x>0.0) {
expm1_13(&expph, &exppm, &exppl, x);
expm1_13(&expmh, &expmm, &expml, -x);
/* The following is OK because expph and -expmh have the same sign */
Add33(&exph, &expm, &expl, expph, exppm, exppl, -expmh, -expmm, -expml);
Renormalize3(presh,presm,presl, exph, expm, expl);
*pexponent=0;
return;
}
else { /* x<0 */
expm1_13(&expph, &exppm, &exppl, x);
expm1_13(&expmh, &expmm, &expml, -x);
/* The following is OK because expph and -expmh have the same sign */
Add33(&exph, &expm, &expl, -expmh, -expmm, -expml, expph, exppm, exppl);
Renormalize3(presh,presm,presl, exph, expm, expl);
*pexponent=0;
return;
}
}
/*************************************************************
*************************************************************
* 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 */
}
void log_td_accurate(double *logh, double *logm, double *logl, int E, double ed, int index, double zh, double zl, double logih, double logim) {
double highPoly, t1h, t1l, t2h, t2l, t3h, t3l, t4h, t4l, t5h, t5l, t6h, t6l, t7h, t7l, t8h, t8l, t9h, t9l, t10h, t10l, t11h, t11l;
double t12h, t12l, t13h, t13l, t14h, t14l, zSquareh, zSquarem, zSquarel, zCubeh, zCubem, zCubel, higherPolyMultZh, higherPolyMultZm;
double higherPolyMultZl, zSquareHalfh, zSquareHalfm, zSquareHalfl, polyWithSquareh, polyWithSquarem, polyWithSquarel;
double polyh, polym, polyl, logil, logyh, logym, logyl, loghover, logmover, loglover, log2edhover, log2edmover, log2edlover;
double log2edh, log2edm, log2edl;
#if EVAL_PERF
crlibm_second_step_taken++;
#endif
/* Accurate phase:
Argument reduction is already done.
We must return logh, logm and logl representing the intermediate result in 118 bits precision.
We use a 14 degree polynomial, computing the first 3 (the first is 0) coefficients in triple double,
calculating the next 7 coefficients in double double arithmetics and the last in double.
We must account for zl starting with the monome of degree 4 (7^3 + 53 - 7 >> 118); so
double double calculations won't account for it.
*/
/* Start of the horner scheme */
#if defined(PROCESSOR_HAS_FMA) && !defined(AVOID_FMA)
highPoly = FMA(FMA(FMA(FMA(accPolyC14,zh,accPolyC13),zh,accPolyC12),zh,accPolyC11),zh,accPolyC10);
#else
highPoly = accPolyC10 + zh * (accPolyC11 + zh * (accPolyC12 + zh * (accPolyC13 + zh * accPolyC14)));
#endif
/* We want to write
accPolyC3 + zh * (accPoly4 + zh * (accPoly5 + zh * (accPoly6 + zh * (accPoly7 + zh * (accPoly8 + zh * (accPoly9 + zh * highPoly))))));
( t14 t13 t12 t11 t10 t9 t8 t7 t6 t5 t4 t3 t2 t1 )
with all additions and multiplications in double double arithmetics
but we will produce intermediate results labelled t1h/t1l thru t14h/t14l
*/
Mul12(&t1h, &t1l, zh, highPoly);
Add22(&t2h, &t2l, accPolyC9h, accPolyC9l, t1h, t1l);
Mul22(&t3h, &t3l, zh, zl, t2h, t2l);
Add22(&t4h, &t4l, accPolyC8h, accPolyC8l, t3h, t3l);
Mul22(&t5h, &t5l, zh, zl, t4h, t4l);
Add22(&t6h, &t6l, accPolyC7h, accPolyC7l, t5h, t5l);
Mul22(&t7h, &t7l, zh, zl, t6h, t6l);
Add22(&t8h, &t8l, accPolyC6h, accPolyC6l, t7h, t7l);
Mul22(&t9h, &t9l, zh, zl, t8h, t8l);
Add22(&t10h, &t10l, accPolyC5h, accPolyC5l, t9h, t9l);
Mul22(&t11h, &t11l, zh, zl, t10h, t10l);
Add22(&t12h, &t12l, accPolyC4h, accPolyC4l, t11h, t11l);
Mul22(&t13h, &t13l, zh, zl, t12h, t12l);
Add22(&t14h, &t14l, accPolyC3h, accPolyC3l, t13h, t13l);
/* We must now prepare (zh + zl)^2 and (zh + zl)^3 as triple doubles */
Mul23(&zSquareh, &zSquarem, &zSquarel, zh, zl, zh, zl);
Mul233(&zCubeh, &zCubem, &zCubel, zh, zl, zSquareh, zSquarem, zSquarel);
/* We can now multiplicate the middle and higher polynomial by z^3 */
Mul233(&higherPolyMultZh, &higherPolyMultZm, &higherPolyMultZl, t14h, t14l, zCubeh, zCubem, zCubel);
/* Multiply now z^2 by -1/2 (exact op) and add to middle and higher polynomial */
zSquareHalfh = zSquareh * -0.5;
zSquareHalfm = zSquarem * -0.5;
zSquareHalfl = zSquarel * -0.5;
Add33(&polyWithSquareh, &polyWithSquarem, &polyWithSquarel,
zSquareHalfh, zSquareHalfm, zSquareHalfl,
higherPolyMultZh, higherPolyMultZm, higherPolyMultZl);
/* Add now zh and zl to obtain the polynomial evaluation result */
Add233(&polyh, &polym, &polyl, zh, zl, polyWithSquareh, polyWithSquarem, polyWithSquarel);
/* Reconstruct now log(y) = log(1 + z) - log(ri) by adding logih, logim, logil
logil has not been read to the time, do this first
*/
logil = argredtable[index].logil;
Add33(&logyh, &logym, &logyl, logih, logim, logil, polyh, polym, polyl);
/* Multiply log2 with E, i.e. log2h, log2m, log2l by ed
ed is always less than 2^(12) and log2h and log2m are stored with at least 12 trailing zeros
So multiplying naively is correct (up to 134 bits at least)
The final result is thus obtained by adding log2 * E to log(y)
*/
log2edhover = log2h * ed;
log2edmover = log2m * ed;
log2edlover = log2l * ed;
/* It may be necessary to renormalize the tabulated value (multiplied by ed) before adding
the to the log(y)-result
If needed, uncomment the following Renormalize3-Statement and comment out the copies
following it.
*/
/* Renormalize3(&log2edh, &log2edm, &log2edl, log2edhover, log2edmover, log2edlover); */
log2edh = log2edhover;
log2edm = log2edmover;
log2edl = log2edlover;
Add33(&loghover, &logmover, &loglover, log2edh, log2edm, log2edl, logyh, logym, logyl);
/* Since we can not guarantee in each addition and multiplication procedure that
the results are not overlapping, we must renormalize the result before handing
it over to the final rounding
*/
Renormalize3(logh,logm,logl,loghover,logmover,loglover);
}
