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log2-td.c
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log2-td.c
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/*
* This function computes log2, correctly rounded,
* using experimental techniques based on triple double arithmetics
THIS IS EXPERIMENTAL SOFTWARE
*
* Author : Christoph Lauter
* christoph.lauter at ens-lyon.fr
*
To have it replace the crlibm log2, do:
gcc -DHAVE_CONFIG_H -I. -fPIC -O2 -c log2-td.c; mv log2-td.o log2_accurate.o; make
*/
#include <stdio.h>
#include <stdlib.h>
#include "crlibm.h"
#include "crlibm_private.h"
#include "triple-double.h"
#include "log2-td.h"
#ifdef BUILD_INTERVAL_FUNCTIONS
#include "interval.h"
#endif
#define AVOID_FMA 0
void log2_td_accurate(double *logb2h, double *logb2m, double *logb2l, 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, logb2hover, logb2mover, logb2lover;
#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);
/* Change logarithm base from natural base to base 2 by multiplying */
Mul233(&logb2hover, &logb2mover, &logb2lover, log2invh, log2invl, loghover, logmover, loglover);
/* 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(logb2h,logb2m,logb2l,logb2hover,logb2mover,logb2lover);
}
/*************************************************************
*************************************************************
* ROUNDED TO NEAREST *
*************************************************************
*************************************************************/
double log2_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, roundcst;
double logb2h, logb2m, logb2l;
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);
/* Change logarithm base from natural base to base 2 by multiplying */
Mul22(&logb2h, &logb2m, log2invh, log2invl, logh, logm);
/* Rounding test and eventual return or call to the accurate function */
if(E==0)
roundcst = ROUNDCST1;
else
roundcst = ROUNDCST2;
if(logb2h == (logb2h + (logb2m * roundcst)))
return logb2h;
else
{
#if DEBUG
printf("Going for Accurate Phase for x=%1.50e\n",x);
#endif
log2_td_accurate(&logb2h, &logb2m, &logb2l, E, ed, index, zh, zl, logih, logim);
ReturnRoundToNearest3(logb2h, logb2m, logb2l);
} /* Accurate phase launched */
}
/*************************************************************
*************************************************************
* ROUNDED UPWARDS *
*************************************************************
*************************************************************/
double log2_ru(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, roundcst;
double logb2h, logb2m, logb2l;
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);
/* Test now if the argument is an exact power of 2
i.e. if the mantissa is exactly 1 (0x0..0 with the implicit bit)
This test is necessary for filtering out the cases where the final
rounding test cannot distinguish between an exact algebraic
number and a hard case to round
*/
if ((index | xdb.i[LO]) == 0) {
/* Handle the "trivial" case for log2:
The argument is an exact power of 2, return thus
just the exponant of the number
*/
return (double) E;
}
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);
/* Change logarithm base from natural base to base 2 by multiplying */
Mul22(&logb2h, &logb2m, log2invh, log2invl, logh, logm);
/* Rounding test and eventual return or call to the accurate function */
if(E==0)
roundcst = RDROUNDCST1;
else
roundcst = RDROUNDCST2;
TEST_AND_RETURN_RU(logb2h, logb2m, roundcst);
#if DEBUG
printf("Going for Accurate Phase for x=%1.50e\n",x);
#endif
log2_td_accurate(&logb2h, &logb2m, &logb2l, E, ed, index, zh, zl, logih, logim);
ReturnRoundUpwards3(logb2h, logb2m, logb2l);
}
/*************************************************************
*************************************************************
* ROUNDED DOWNWARDS *
*************************************************************
*************************************************************/
double log2_rd(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, roundcst;
double logb2h, logb2m, logb2l;
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);
/* Test now if the argument is an exact power of 2
i.e. if the mantissa is exactly 1 (0x0..0 with the implicit bit)
This test is necessary for filtering out the cases where the final
rounding test cannot distinguish between an exact algebraic
number and a hard case to round
*/
if ((index | xdb.i[LO]) == 0) {
/* Handle the "trivial" case for log2:
The argument is an exact power of 2, return thus
just the exponant of the number
*/
return (double) E;
}
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);
/* Change logarithm base from natural base to base 2 by multiplying */
Mul22(&logb2h, &logb2m, log2invh, log2invl, logh, logm);
/* Rounding test and eventual return or call to the accurate function */
if(E==0)
roundcst = RDROUNDCST1;
else
roundcst = RDROUNDCST2;
TEST_AND_RETURN_RD(logb2h, logb2m, roundcst);
#if DEBUG
printf("Going for Accurate Phase for x=%1.50e\n",x);
#endif
log2_td_accurate(&logb2h, &logb2m, &logb2l, E, ed, index, zh, zl, logih, logim);
ReturnRoundDownwards3(logb2h, logb2m, logb2l);
}
/*************************************************************
*************************************************************
* ROUNDED TOWARDS ZERO *
*************************************************************
*************************************************************/
double log2_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, roundcst;
double logb2h, logb2m, logb2l;
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);
/* Test now if the argument is an exact power of 2
i.e. if the mantissa is exactly 1 (0x0..0 with the implicit bit)
This test is necessary for filtering out the cases where the final
rounding test cannot distinguish between an exact algebraic
number and a hard case to round
*/
if ((index | xdb.i[LO]) == 0) {
/* Handle the "trivial" case for log2:
The argument is an exact power of 2, return thus
just the exponant of the number
*/
return (double) E;
}
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);
/* Change logarithm base from natural base to base 2 by multiplying */
Mul22(&logb2h, &logb2m, log2invh, log2invl, logh, logm);
/* Rounding test and eventual return or call to the accurate function */
if(E==0)
roundcst = RDROUNDCST1;
else
roundcst = RDROUNDCST2;
TEST_AND_RETURN_RZ(logb2h, logb2m, roundcst);
#if DEBUG
printf("Going for Accurate Phase for x=%1.50e\n",x);
#endif
log2_td_accurate(&logb2h, &logb2m, &logb2l, E, ed, index, zh, zl, logih, logim);
ReturnRoundTowardsZero3(logb2h, logb2m, logb2l);
}
#ifdef BUILD_INTERVAL_FUNCTIONS
interval j_log2(interval x) {
interval res;
double res_inf, res_sup, restemp_inf, restemp_sup;
int infDone, supDone;
int roundable;
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, roundcst_inf;
double logb2h_inf, logb2m_inf, logb2l_inf;
int E_inf, index_inf;
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, roundcst_sup;
double logb2h_sup, logb2m_sup, logb2l_sup;
int E_sup, index_sup;
double x_inf, x_sup;
x_inf=LOW(x);
x_sup=UP(x);
infDone=0; supDone=0;
E_inf=0;
E_sup=0;
xdb_inf.d=x_inf;
xdb_sup.d=x_sup;
/* Filter cases */
if (xdb_inf.i[HI] < 0x00100000){ /* x < 2^(-1022) */
if (((xdb_inf.i[HI] & 0x7fffffff)|xdb_inf.i[LO])==0){
infDone=1;
restemp_inf = -1.0/0.0;
} /* log(+/-0) = -Inf */
if ((xdb_inf.i[HI] < 0) && (infDone==0)){
infDone=1;
restemp_inf = (x_inf-x_inf)/0; /* log(-x) = Nan */
}
/* Subnormal number */
E_inf = -52;
xdb_inf.d *= two52; /* make x a normal number */
}
if ((xdb_inf.i[HI] >= 0x7ff00000) && (infDone==0)){
infDone=1;
restemp_inf = x_inf+x_inf; /* Inf or Nan */
}
/* Filter cases */
if (xdb_sup.i[HI] < 0x00100000){ /* x < 2^(-1022) */
if (((xdb_sup.i[HI] & 0x7fffffff)|xdb_sup.i[LO])==0){
supDone=1;
restemp_sup = -1.0/0.0;
} /* log(+/-0) = -Inf */
if ((xdb_sup.i[HI] < 0) && (supDone==0)){
supDone=1;
restemp_sup = (x_sup-x_sup)/0; /* log(-x) = Nan */
}
/* Subnormal number */
E_sup = -52;
xdb_sup.d *= two52; /* make x a normal number */