/*************************************************************
*************************************************************
* 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 */
}
/*************************************************************
*************************************************************
* ROUNDED TO NEAREST *
*************************************************************
*************************************************************/
double log_rn(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;
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 = ROUNDCST1;
else
roundcst = ROUNDCST2;
if(logh == (logh + (logm * roundcst)))
return logh;
else
{
#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);
ReturnRoundToNearest3(logh, logm, logl);
} /* Accurate phase launched */
}