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trigpi.c
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trigpi.c
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#include <stdio.h>
#include <stdlib.h>
#include "crlibm.h"
/*
* Correctly rounded trigpi functions
*
* Authors : F. de Dinechin, S. Chevillard, C. Lauter (the latter two
* didn't write a line of this file, but wrote a tool that wrote a
* tool that wrote etc that wrote bits of code related to polynomial
* evaluation.)
*
* This file is part of the crlibm library developed by the Arenaire
* project at Ecole Normale Superieure de Lyon
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU Lesser General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU Lesser General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
*/
#include "crlibm_private.h"
#include "triple-double.h"
#include "trigpi.h"
/* TODO
Use the symmetries of the tables
Actually, use the tables from the standard trigo.
Write decent quick steps. Or hammer Christoph and Sylvain to do so.
*/
/* This ugly bits of code in the beginning are polynomial evaluations
automagically generated and proven by Chevillard and Lauter's
tools
*/
#define sinpiacc_coeff_1h 3.14159265358979311599796346854418516159057617187500000000000000000000000000000000e+00
#define sinpiacc_coeff_1m 1.22464679914735320717376402945839660462569212467758006379625612680683843791484833e-16
#define sinpiacc_coeff_1l -2.87889731599645993207191707893463395148177292198731390393739579574603302514349608e-33
#define sinpiacc_coeff_3h -5.16771278004997025590228076907806098461151123046875000000000000000000000000000000e+00
#define sinpiacc_coeff_3m 2.26656228257550136196266687046492287115561324595258696490418515168130397796630859e-16
#define sinpiacc_coeff_5h 2.55016403987734552316624103696085512638092041015625000000000000000000000000000000e+00
#define sinpiacc_coeff_5m -7.93098961936403945684716222915171282926664203267314023904077657789457589387893677e-17
#define sinpiacc_coeff_7h -5.99264529320792105338000510528218001127243041992187500000000000000000000000000000e-01
#define sinpiacc_coeff_9h 8.21458866130424236740026344705256633460521697998046875000000000000000000000000000e-02
#define sinpiacc_coeff_11h -7.37046804820839888960914976223648409359157085418701171875000000000000000000000000e-03
#define cospiacc_coeff_0h 1.00000000000000000000000000000000000000000000000000000000000000000000000000000000e+00
#define cospiacc_coeff_2h -4.93480220054467899615247006295248866081237792968750000000000000000000000000000000e+00
#define cospiacc_coeff_2m -3.13264775437072047133490817894057799839785556899468543790021612949203699827194214e-16
#define cospiacc_coeff_4h 4.05871212641676848420502210501581430435180664062500000000000000000000000000000000e+00
#define cospiacc_coeff_4m -2.66019969731660223662555032718185048048635055542576743903282476821914315223693848e-16
#define cospiacc_coeff_6h -1.33526276885458949905682857206556946039199829101562500000000000000000000000000000e+00
#define cospiacc_coeff_8h 2.35330630358513925859398341344785876572132110595703125000000000000000000000000000e-01
#define cospiacc_coeff_10h -2.58068327360992909313974763563237502239644527435302734375000000000000000000000000e-02
static void sincospiacc(double *sinpiacc_resh, double *sinpiacc_resm, double *sinpiacc_resl,
double *cospiacc_resh, double *cospiacc_resm, double *cospiacc_resl,
double x) {
double x2h, x2m;
double sinpiacc_t_1_0h;
double sinpiacc_t_2_0h;
double sinpiacc_t_3_0h;
double sinpiacc_t_4_0h;
double sinpiacc_t_5_0h, sinpiacc_t_5_0m;
double sinpiacc_t_6_0h, sinpiacc_t_6_0m;
double sinpiacc_t_7_0h, sinpiacc_t_7_0m;
double sinpiacc_t_8_0h, sinpiacc_t_8_0m;
double sinpiacc_t_9_0h, sinpiacc_t_9_0m, sinpiacc_t_9_0l;
double sinpiacc_t_10_0h, sinpiacc_t_10_0m, sinpiacc_t_10_0l;
double cospiacc_t_1_0h;
double cospiacc_t_2_0h;
double cospiacc_t_3_0h;
double cospiacc_t_4_0h;
double cospiacc_t_5_0h, cospiacc_t_5_0m;
double cospiacc_t_6_0h, cospiacc_t_6_0m;
double cospiacc_t_7_0h, cospiacc_t_7_0m;
double cospiacc_t_8_0h, cospiacc_t_8_0m;
double cospiacc_t_9_0h, cospiacc_t_9_0m, cospiacc_t_9_0l;
Mul12(&x2h,&x2m,x,x);
sinpiacc_t_1_0h = sinpiacc_coeff_11h;
sinpiacc_t_2_0h = sinpiacc_t_1_0h * x2h;
sinpiacc_t_3_0h = sinpiacc_coeff_9h + sinpiacc_t_2_0h;
sinpiacc_t_4_0h = sinpiacc_t_3_0h * x2h;
Add12(sinpiacc_t_5_0h,sinpiacc_t_5_0m,sinpiacc_coeff_7h,sinpiacc_t_4_0h);
MulAdd22(&sinpiacc_t_6_0h,&sinpiacc_t_6_0m,sinpiacc_coeff_5h,sinpiacc_coeff_5m,x2h,x2m,sinpiacc_t_5_0h,sinpiacc_t_5_0m);
MulAdd22(&sinpiacc_t_7_0h,&sinpiacc_t_7_0m,sinpiacc_coeff_3h,sinpiacc_coeff_3m,x2h,x2m,sinpiacc_t_6_0h,sinpiacc_t_6_0m);
Mul22(&sinpiacc_t_8_0h,&sinpiacc_t_8_0m,sinpiacc_t_7_0h,sinpiacc_t_7_0m,x2h,x2m);
Add233Cond(&sinpiacc_t_9_0h,&sinpiacc_t_9_0m,&sinpiacc_t_9_0l,sinpiacc_t_8_0h,sinpiacc_t_8_0m,sinpiacc_coeff_1h,sinpiacc_coeff_1m,sinpiacc_coeff_1l);
Mul133(&sinpiacc_t_10_0h,&sinpiacc_t_10_0m,&sinpiacc_t_10_0l,x,sinpiacc_t_9_0h,sinpiacc_t_9_0m,sinpiacc_t_9_0l);
Renormalize3(sinpiacc_resh,sinpiacc_resm,sinpiacc_resl,sinpiacc_t_10_0h,sinpiacc_t_10_0m,sinpiacc_t_10_0l);
cospiacc_t_1_0h = cospiacc_coeff_10h;
cospiacc_t_2_0h = cospiacc_t_1_0h * x2h;
cospiacc_t_3_0h = cospiacc_coeff_8h + cospiacc_t_2_0h;
cospiacc_t_4_0h = cospiacc_t_3_0h * x2h;
Add12(cospiacc_t_5_0h,cospiacc_t_5_0m,cospiacc_coeff_6h,cospiacc_t_4_0h);
MulAdd22(&cospiacc_t_6_0h,&cospiacc_t_6_0m,cospiacc_coeff_4h,cospiacc_coeff_4m,x2h,x2m,cospiacc_t_5_0h,cospiacc_t_5_0m);
MulAdd22(&cospiacc_t_7_0h,&cospiacc_t_7_0m,cospiacc_coeff_2h,cospiacc_coeff_2m,x2h,x2m,cospiacc_t_6_0h,cospiacc_t_6_0m);
Mul22(&cospiacc_t_8_0h,&cospiacc_t_8_0m,cospiacc_t_7_0h,cospiacc_t_7_0m,x2h,x2m);
Add123(&cospiacc_t_9_0h,&cospiacc_t_9_0m,&cospiacc_t_9_0l,cospiacc_coeff_0h,cospiacc_t_8_0h,cospiacc_t_8_0m);
*cospiacc_resh = cospiacc_t_9_0h; *cospiacc_resm = cospiacc_t_9_0m; *cospiacc_resl = cospiacc_t_9_0l;
}
/* Comment on comparing sa, ca, sy and cy
either index=0, then sa=0 and ca=1, therefore t2=0, and the Add33 will be exact
or index !=0, and
-eps1 < sy < eps1 (but sy may be negative)
sa > eps1 (sa>0)
1-eps2 < cy < 1
ca < 1-eps2
therefore
sacy = t2 >0
casy = t1 may be negative
abs(t1) <= abs(t2)
Unfortunately we need a stronger condition to use the Add33
*/
static void sinpi_accurate(double *rh, double *rm, double *rl,
double y, int index, int quadrant)
{
double syh, sym, syl, cyh, cym, cyl, sah, sam, sal, cah, cam, cal;
double t1h, t1m, t1l, t2h, t2m, t2l;
sincospiacc(&syh, &sym, &syl, &cyh, &cym, &cyl, y);
sah=sincosTable[index].sh;
cah=sincosTable[index].ch;
sam=sincosTable[index].sm;
cam=sincosTable[index].cm;
sal=sincosTable[index].sl;
cal=sincosTable[index].cl;
if(quadrant==0 || quadrant==2) {
/* compute sy*ca+sa*cy : t1 = sy*ca, t2 = sa*cy*/
Mul33(&t1h,&t1m,&t1l, syh,sym,syl, cah,cam,cal);
Mul33(&t2h,&t2m,&t2l, sah,sam,sal, cyh,cym,cyl);
Add33Cond(rh, rm, rl, t2h,t2m,t2l, t1h,t1m,t1l);
}
else {
/* compute cy*ca - sa*sy : t1 = cy*ca, t2 = sa*sy */
Mul33(&t1h,&t1m,&t1l, cyh,cym,cyl, cah,cam,cal);
Mul33(&t2h,&t2m,&t2l, sah,sam,sal, syh,sym,syl);
Add33Cond(rh, rm, rl, t1h,t1m,t1l, -t2h,-t2m,-t2l);
}
if (quadrant>=2) {
*rh = -*rh;
*rm = -*rm;
*rl = -*rl;
}
};
#define sinpiquick_coeff_1h 3.14159265358979311599796346854418516159057617187500000000000000000000000000000000e+00
#define sinpiquick_coeff_1m 1.22464971683184787123862072310058851157814368464452070561776508839102461934089661e-16
#define sinpiquick_coeff_3h -5.16771278004997025590228076907806098461151123046875000000000000000000000000000000e+00
#define sinpiquick_coeff_5h 2.55016403989992213041659852024167776107788085937500000000000000000000000000000000e+00
#define sinpiquick_coeff_7h -5.99263913290728922333983064163476228713989257812500000000000000000000000000000000e-01
#define cospiquick_coeff_0h 1.00000000000000000000000000000000000000000000000000000000000000000000000000000000e+00
#define cospiquick_coeff_2h -4.93480220054467899615247006295248866081237792968750000000000000000000000000000000e+00
#define cospiquick_coeff_4h 4.05871212632582167856298838160000741481781005859375000000000000000000000000000000e+00
#define cospiquick_coeff_6h -1.33525456323720947970912220625905320048332214355468750000000000000000000000000000e+00
static void cospi_accurate(double *rh, double *rm, double *rl,
double y, int index, int quadrant)
{
double syh, sym, syl, cyh, cym, cyl, sah, sam, sal, cah, cam, cal;
double t1h, t1m, t1l, t2h, t2m, t2l;
sincospiacc(&syh, &sym, &syl, &cyh, &cym, &cyl, y);
sah=sincosTable[index].sh;
cah=sincosTable[index].ch;
sam=sincosTable[index].sm;
cam=sincosTable[index].cm;
sal=sincosTable[index].sl;
cal=sincosTable[index].cl;
if(quadrant==0 || quadrant==2) {
/* compute cy*ca - sa*sy : t1 = cy*ca, t2 = sa*sy */
Mul33(&t1h,&t1m,&t1l, cyh,cym,cyl, cah,cam,cal);
Mul33(&t2h,&t2m,&t2l, sah,sam,sal, syh,sym,syl);
Add33Cond(rh, rm, rl, t1h,t1m,t1l, -t2h,-t2m,-t2l);
}
else {
/* compute sy*ca+sa*cy : t1 = sy*ca, t2 = sa*cy*/
Mul33(&t1h,&t1m,&t1l, syh,sym,syl, cah,cam,cal);
Mul33(&t2h,&t2m,&t2l, sah,sam,sal, cyh,cym,cyl);
Add33Cond(rh, rm, rl, t2h,t2m,t2l, t1h,t1m,t1l);
}
if (quadrant==1 || quadrant==2) {
*rh = -*rh;
*rm = -*rm;
*rl = -*rl;
}
};
/* This one can clearly be improved. It was set up in less than one hour */
void sinpiquick(double *rh, double *rm, double x, int index, int quadrant) {
double x2h, x2m;
double sinpiquick_t_1_0h;
double sinpiquick_t_2_0h;
double sinpiquick_t_3_0h;
double sinpiquick_t_4_0h;
double sinpiquick_t_5_0h, sinpiquick_t_5_0m;
double sinpiquick_t_6_0h, sinpiquick_t_6_0m;
double sinpiquick_t_7_0h, sinpiquick_t_7_0m;
double syh, sym;
double cospiquick_t_1_0h;
double cospiquick_t_2_0h;
double cospiquick_t_3_0h;
double cospiquick_t_4_0h;
double cospiquick_t_5_0h, cospiquick_t_5_0m;
double cospiquick_t_6_0h, cospiquick_t_6_0m;
double cospiquick_t_7_0h, cospiquick_t_7_0m;
double cyh, cym;
double t1h, t1m, t2h, t2m, sah, sam, cah,cam;
Mul12(&x2h,&x2m,x,x);
sah=sincosTable[index].sh;
cah=sincosTable[index].ch;
sam=sincosTable[index].sm;
cam=sincosTable[index].cm;
sinpiquick_t_1_0h = sinpiquick_coeff_7h;
sinpiquick_t_2_0h = sinpiquick_t_1_0h * x2h;
sinpiquick_t_3_0h = sinpiquick_coeff_5h + sinpiquick_t_2_0h;
sinpiquick_t_4_0h = sinpiquick_t_3_0h * x2h;
Add12(sinpiquick_t_5_0h,sinpiquick_t_5_0m,sinpiquick_coeff_3h,sinpiquick_t_4_0h);
MulAdd22(&sinpiquick_t_6_0h,&sinpiquick_t_6_0m,sinpiquick_coeff_1h,sinpiquick_coeff_1m,x2h,x2m,sinpiquick_t_5_0h,sinpiquick_t_5_0m);
Mul122(&sinpiquick_t_7_0h,&sinpiquick_t_7_0m,x,sinpiquick_t_6_0h,sinpiquick_t_6_0m);
syh = sinpiquick_t_7_0h; sym = sinpiquick_t_7_0m;
cospiquick_t_1_0h = cospiquick_coeff_6h;
cospiquick_t_2_0h = cospiquick_t_1_0h * x2h;
cospiquick_t_3_0h = cospiquick_coeff_4h + cospiquick_t_2_0h;
cospiquick_t_4_0h = cospiquick_t_3_0h * x2h;
Add12(cospiquick_t_5_0h,cospiquick_t_5_0m,cospiquick_coeff_2h,cospiquick_t_4_0h);
Mul22(&cospiquick_t_6_0h,&cospiquick_t_6_0m,cospiquick_t_5_0h,cospiquick_t_5_0m,x2h,x2m);
Add122(&cospiquick_t_7_0h,&cospiquick_t_7_0m,cospiquick_coeff_0h,cospiquick_t_6_0h,cospiquick_t_6_0m);
cyh = cospiquick_t_7_0h; cym = cospiquick_t_7_0m;
/* Here comes the hand-written, unproven yet code */
if(quadrant==0 || quadrant==2) {
/* compute sy*ca+sa*cy : t1 = sy*ca, t2 = sa*cy*/
Mul22(&t1h,&t1m, syh,sym, cah,cam);
Mul22(&t2h,&t2m, sah,sam, cyh,cym);
Add22Cond(rh, rm, t2h,t2m, t1h,t1m);
}
else {
/* compute cy*ca - sa*sy : t1 = cy*ca, t2 = sa*sy */
Mul22(&t1h,&t1m, cyh,cym, cah,cam);
Mul22(&t2h,&t2m, sah,sam, syh,sym);
Add22Cond(rh, rm, t1h,t1m, -t2h,-t2m);
}
if (quadrant>=2) {
*rh = -*rh;
*rm = -*rm;
}
}
double sinpi_rn(double x){
double xs, y,u, rh, rm, rl, sign,absx;
db_number xdb, t;
int32_t xih, absxih, index, quadrant;
if (x<0) absx = -x; else absx = x;
xdb.d = x;
xs = x*128.0;
/* argument reduction */
if(absx> TWOTO42 ) { /* x is very large, let us first subtract a large integer from it */
t.d = xs;
t.i[LO] =0; /* remove the low part. The point is somewhere there since x > 2^42.
So what remains in t is an FP integer almost as large as x */
xs = xs-t.d; /* we are going to throw away the int part anyway */
}
t.d = TWOTO5251 + xs;
u = t.d - TWOTO5251;
y = xs - u;
index = t.i[LO] & 0x3f;
quadrant = (t.i[LO] & 0xff) >>6;
/* Special case tests come late because the conversion FP to int is slow */
xih = xdb.i[HI];
absxih = xih & 0x7fffffff;
if (xih>>31) sign=-1.; else sign=1.; /* consider the sign bit */
if(index==0 && y==0.0 && ((quadrant&1)==0)) return sign*0.0; /*signed, inspired by LIA-2 */
y = y * INV128;
/* SPECIAL CASES: x=(Nan, Inf) sin(pi*x)=Nan */
if (absxih>=0x7ff00000) {
xdb.l=0xfff8000000000000LL;
return xdb.d - xdb.d;
}
if(absxih>=0x43300000) /* 2^52, which entails that x is an integer */
return sign*0.0; /*signed */
if(absxih<=0x3E000000) /*2^{-31}*/ {
if (absxih<0x01700000) { /* 2^{-1000} : Get rid of possible subnormals */
/* in this case, SCS computation, accurate to 2^-210 which is provably enough */
scs_t result;
scs_set_d(result, x );
scs_mul(result, PiSCS_ptr, result);
scs_get_d(&rh, result);
return rh;
}
/* First step for Pi*x. TODO: FMA-based optimisation */
const double DekkerConst = 134217729.; /* 2^27 +1 */
double tt, xh, xl;
/* Splitting of x. Both xh and xl have at least 26 consecutive LSB zeroes */
tt = x*DekkerConst;
xh = (x-tt)+tt;
xl = x-xh;
Add12(rh,rl, xh*PIHH, (xl*PIHH + xh*PIHM) + (xh*PIM + xl*PIHM) );
if(rh == (rh + (rl * PIX_RNCST_SIN)))
return rh;
}
/* Fall here either if we have a large input, or if we have a small
input and the rounding test fails. */
sinpiquick(&rh, &rm, y, index, quadrant);
if (rh==rh+1.00001*rm) /* See trigpiquick.gappa. This first step is ridiculously too accurate */
return rh;
sinpi_accurate(&rh, &rm, &rl, y, index, quadrant);
ReturnRoundToNearest3(rh,rm,rl);
}
double sinpi_rd(double x){
double xs, y,u, rh, rm, rl, sign,absx;
db_number xdb, t;
int32_t xih, absxih, index, quadrant;
if (x<0) absx = -x; else absx = x;
xdb.d = x;
xs = x*128.0;
/* argument reduction */
if(absx> TWOTO42 ) { /* x is very large, let us first subtract a large integer from it */
t.d = xs;
t.i[LO] =0; /* remove the low part. The point is somewhere there since x > 2^42.
So what remains in t is an FP integer almost as large as x */
xs = xs-t.d; /* we are going to throw away the int part anyway */
}
t.d = TWOTO5251 + xs;
u = t.d - TWOTO5251;
y = xs - u;
index = t.i[LO] & 0x3f;
quadrant = (t.i[LO] & 0xff) >>6;
/* Special case tests come late because the conversion FP to int is slow */
xih = xdb.i[HI];
absxih = xih & 0x7fffffff;
if (xih>>31) sign=-1.; else sign=1.; /* consider the sign bit */
if(index==0 && y==0.0 && ((quadrant&1)==0)) return -0.0; /*signed, inspired by LIA-2 */
y = y * INV128;
/* SPECIAL CASES: x=(Nan, Inf) sin(pi*x)=Nan */
if (absxih>=0x7ff00000) {
xdb.l=0xfff8000000000000LL;
return xdb.d - xdb.d;
}
if(absxih>=0x43300000) /* 2^52, which entails that x is an integer */
return sign*0.0; /*signed */
if(absxih<=0x3E000000) /*2^{-31}*/ {
if (absxih<0x01700000) { /* 2^{-1000} : Get rid of possible subnormals */
/* in this case, SCS computation, accurate to 2^-210 which is provably enough */
scs_t result;
scs_set_d(result, x );
scs_mul(result, PiSCS_ptr, result);
scs_get_d_minf(&rh, result);
return rh;
}
/* First step for Pi*x. TODO: FMA-based optimisation */
const double DekkerConst = 134217729.; /* 2^27 +1 */
double tt, xh, xl;
/* Splitting of x. Both xh and xl have at least 26 consecutive LSB zeroes */
tt = x*DekkerConst;
xh = (x-tt)+tt;
xl = x-xh;
Add12(rh,rl, xh*PIHH, (xl*PIHH + xh*PIHM) + (xh*PIM + xl*PIHM) );
TEST_AND_RETURN_RD(rh,rl,PIX_EPS_SIN);
}
/* Fall here either if we have a large input, or if we have a small
input and the rounding test fails. */
sinpi_accurate(&rh, &rm, &rl, y, index, quadrant);
ReturnRoundDownwards3(rh,rm,rl);
};
double sinpi_ru(double x){
double xs, y,u, rh, rm, rl, sign,absx;
db_number xdb, t;
int32_t xih, absxih, index, quadrant;
if (x<0) absx = -x; else absx = x;
xdb.d = x;
xs = x*128.0;
/* argument reduction */
if(absx> TWOTO42 ) { /* x is very large, let us first subtract a large integer from it */
t.d = xs;
t.i[LO] =0; /* remove the low part. The point is somewhere there since x > 2^42.
So what remains in t is an FP integer almost as large as x */
xs = xs-t.d; /* we are going to throw away the int part anyway */
}
t.d = TWOTO5251 + xs;
u = t.d - TWOTO5251;
y = xs - u;
index = t.i[LO] & 0x3f;
quadrant = (t.i[LO] & 0xff) >>6;
/* Special case tests come late because the conversion FP to int is slow */
xih = xdb.i[HI];
absxih = xih & 0x7fffffff;
if (xih>>31) sign=-1.; else sign=1.; /* consider the sign bit */
if(index==0 && y==0.0 && ((quadrant&1)==0)) return +0.0; /*signed, inspired by LIA-2 */
y = y * INV128;
/* SPECIAL CASES: x=(Nan, Inf) sin(pi*x)=Nan */
if (absxih>=0x7ff00000) {
xdb.l=0xfff8000000000000LL;
return xdb.d - xdb.d;
}
if(absxih>=0x43300000) /* 2^52, which entails that x is an integer */
return sign*0.0; /*signed */
if(absxih<=0x3E000000) /*2^{-31}*/ {
if (absxih<0x01700000) { /* 2^{-1000} : Get rid of possible subnormals */
/* in this case, SCS computation, accurate to 2^-210 which is provably enough */
scs_t result;
scs_set_d(result, x );
scs_mul(result, PiSCS_ptr, result);
scs_get_d_pinf(&rh, result);
return rh;
}
/* First step for Pi*x. TODO: FMA-based optimisation */
const double DekkerConst = 134217729.; /* 2^27 +1 */
double tt, xh, xl;
/* Splitting of x. Both xh and xl have at least 26 consecutive LSB zeroes */
tt = x*DekkerConst;
xh = (x-tt)+tt;
xl = x-xh;
Add12(rh,rl, xh*PIHH, (xl*PIHH + xh*PIHM) + (xh*PIM + xl*PIHM) );
TEST_AND_RETURN_RU(rh,rl,PIX_EPS_SIN);
}
/* Fall here either if we have a large input, or if we have a small
input and the rounding test fails. */
sinpi_accurate(&rh, &rm, &rl, y, index, quadrant);
ReturnRoundUpwards3(rh,rm,rl);
};
double sinpi_rz(double x){
double xs, y,u, rh, rm, rl, sign,absx;
db_number xdb, t;
int32_t xih, absxih, index, quadrant;
if (x<0) absx = -x; else absx = x;
xdb.d = x;
xs = x*128.0;
/* argument reduction */
if(absx> TWOTO42 ) { /* x is very large, let us first subtract a large integer from it */
t.d = xs;
t.i[LO] =0; /* remove the low part. The point is somewhere there since x > 2^42.
So what remains in t is an FP integer almost as large as x */
xs = xs-t.d; /* we are going to throw away the int part anyway */
}
t.d = TWOTO5251 + xs;
u = t.d - TWOTO5251;
y = xs - u;
index = t.i[LO] & 0x3f;
quadrant = (t.i[LO] & 0xff) >>6;
/* Special case tests come late because the conversion FP to int is slow */
xih = xdb.i[HI];
absxih = xih & 0x7fffffff;
if (xih>>31) sign=-1.; else sign=1.; /* consider the sign bit */
if(index==0 && y==0.0 && ((quadrant&1)==0)) return sign*0.0; /*signed, inspired by LIA-2 */
y = y * INV128;
/* SPECIAL CASES: x=(Nan, Inf) sin(pi*x)=Nan */
if (absxih>=0x7ff00000) {
xdb.l=0xfff8000000000000LL;
return xdb.d - xdb.d;
}
if(absxih>=0x43300000) /* 2^52, which entails that x is an integer */
return sign*0.0; /*signed */
if(absxih<=0x3E000000) /*2^{-31}*/ {
if (absxih<0x01700000) { /* 2^{-1000} : Get rid of possible subnormals */
/* in this case, SCS computation, accurate to 2^-210 which is provably enough */
scs_t result;
scs_set_d(result, x );
scs_mul(result, PiSCS_ptr, result);
scs_get_d_zero(&rh, result);
return rh;
}
/* First step for Pi*x. TODO: FMA-based optimisation */
const double DekkerConst = 134217729.; /* 2^27 +1 */
double tt, xh, xl;
/* Splitting of x. Both xh and xl have at least 26 consecutive LSB zeroes */
tt = x*DekkerConst;
xh = (x-tt)+tt;
xl = x-xh;
Add12(rh,rl, xh*PIHH, (xl*PIHH + xh*PIHM) + (xh*PIM + xl*PIHM) );
TEST_AND_RETURN_RZ(rh,rl,PIX_EPS_SIN);
}
/* Fall here either if we have a large input, or if we have a small
input and the rounding test fails. */
sinpi_accurate(&rh, &rm, &rl, y, index, quadrant);
ReturnRoundTowardsZero3(rh,rm,rl);
};
/* to nearest */
double cospi_rn(double x){
double xs, y,u, rh, rm, rl, absx;
db_number xdb, t;
int32_t xih, absxih, index, quadrant;
if (x<0) absx =-x; else absx=x;
xdb.d=x;
xs = x*128.0;
/* argument reduction.
We do it before the special case tests for performance,
it might compute garbage for inf, very large inputs, etc */
if(absx> TWOTO42 ) { /* 2^42, x is very large, let us first subtract a large integer from it */
t.d = xs;
t.i[LO] =0; /* remove the low part, in which was the coma since x > 2^42.
So what remains in t is an FP integer almost as large as x */
xs = xs-t.d; /* we are going to throw away the int part anyway */
}
t.d = TWOTO5251 + xs;
u = t.d - TWOTO5251;
y = xs - u;
y = y * INV128;
index = t.i[LO] & 0x3f;
quadrant = (t.i[LO] & 0xff) >>6;
/* Special case tests come late because the conversion FP to int is slow */
xih = xdb.i[HI];
absxih = xih & 0x7fffffff;
/* SPECIAL CASES: x=(Nan, Inf) cos(pi*x)=Nan */
if (absxih>=0x7ff00000) {
xdb.l=0xfff8000000000000LL;
return xdb.d - xdb.d;
}
if(absxih>=0x43400000) /* 2^53, which entails that x is an even integer */
return 1.0;
if(index==0 && y==0. && ((quadrant&1)==1)) return +0.;
/* Always +0, inpired by LIA2; We do not have cos(x+pi) == - cos(x)
in this case */
if(index==0 && y==0. && quadrant==0) return 1.;
if(index==0 && y==0. && quadrant==2) return -1.;
if (absxih<0x3E26A09E) /* sqrt(2^-53)/4 */
return 1.0;
/* printf("\n\nint part = %f frac part = %f index=%d quadrant=%d \n", u, y, index, quadrant);
*/
cospi_accurate(&rh, &rm, &rl, y, index, quadrant);
ReturnRoundToNearest3(rh,rm,rl);
};
double cospi_rd(double x){
double xs, y,u, rh, rm, rl, absx;
db_number xdb, t;
int32_t xih, absxih, index, quadrant;
if (x<0) absx =-x; else absx=x;
xdb.d=x;
xs = x*128.0;
/* argument reduction.
We do it before the special case tests for performance,
it might compute garbage for inf, very large inputs, etc */
if(absx> TWOTO42 ) { /* 2^42, x is very large, let us first subtract a large integer from it */
t.d = xs;
t.i[LO] =0; /* remove the low part, in which was the coma since x > 2^42.
So what remains in t is an FP integer almost as large as x */
xs = xs-t.d; /* we are going to throw away the int part anyway */
}
t.d = TWOTO5251 + xs;
u = t.d - TWOTO5251;
y = xs - u;
y = y * INV128;
index = t.i[LO] & 0x3f;
quadrant = (t.i[LO] & 0xff) >>6;
/* Special case tests come late because the conversion FP to int is slow */
xih = xdb.i[HI];
absxih = xih & 0x7fffffff;
/* SPECIAL CASES: x=(Nan, Inf) cos(pi*x)=Nan */
if (absxih>=0x7ff00000) {
xdb.l=0xfff8000000000000LL;
return xdb.d - xdb.d;
}
if(absxih>=0x43400000) /* 2^53, which entails that x is an even integer */
return 1.0; /*signed */
if(index==0 && y==0. && ((quadrant&1)==1)) return -0.;
if(index==0 && y==0. && quadrant==0) return 1.;
if(index==0 && y==0. && quadrant==2) return -1.;
if (absxih<0x3E200000) /* 2^-29 */
return 0.9999999999999998889776975374843459576368331909179687500; /* 1-2^-53 */
/* Always +0, inpired by LIA2; We do not have cos(x+pi) == - cos(x)
in this case */
cospi_accurate(&rh, &rm, &rl, y, index, quadrant);
ReturnRoundDownwards3(rh,rm,rl);
};
double cospi_ru(double x){
double xs, y,u, rh, rm, rl, absx;
db_number xdb, t;
int32_t xih, absxih, index, quadrant;
if (x<0) absx =-x; else absx=x;
xdb.d=x;
xs = x*128.0;
/* argument reduction.
We do it before the special case tests for performance,
it might compute garbage for inf, very large inputs, etc */
if(absx> TWOTO42 ) { /* 2^42, x is very large, let us first subtract a large integer from it */
t.d = xs;
t.i[LO] =0; /* remove the low part, in which was the coma since x > 2^42.
So what remains in t is an FP integer almost as large as x */
xs = xs-t.d; /* we are going to throw away the int part anyway */
}
t.d = TWOTO5251 + xs;
u = t.d - TWOTO5251;
y = xs - u;
y = y * INV128;
index = t.i[LO] & 0x3f;
quadrant = (t.i[LO] & 0xff) >>6;
/* Special case tests come late because the conversion FP to int is slow */
xih = xdb.i[HI];
absxih = xih & 0x7fffffff;
/* SPECIAL CASES: x=(Nan, Inf) cos(pi*x)=Nan */
if (absxih>=0x7ff00000) {
xdb.l=0xfff8000000000000LL;
return xdb.d - xdb.d;
}
if(absxih>=0x43400000) /* 2^53, which entails that x is an even integer */
return 1.0; /*signed */
if(index==0 && y==0. && quadrant==0) return 1.;
if(index==0 && y==0. && quadrant==2) return -1.;
if(index==0 && y==0. && ((quadrant&1)==1)) return +0.;
/* Always +0, inpired by LIA2; We do not have cos(x+pi) == - cos(x)
in this case */
if (absxih<0x3E200000) /* 2^-29 */
return 1;
cospi_accurate(&rh, &rm, &rl, y, index, quadrant);
ReturnRoundUpwards3(rh,rm,rl);
};
double cospi_rz(double x){
double xs, y,u, rh, rm, rl, absx;
db_number xdb, t;
int32_t xih, absxih, index, quadrant;
if (x<0) absx =-x; else absx=x;
xdb.d=x;
xs = x*128.0;
/* argument reduction.
We do it before the special case tests for performance,
it might compute garbage for inf, very large inputs, etc */
if(absx> TWOTO42 ) { /* 2^42, x is very large, let us first subtract a large integer from it */
t.d = xs;
t.i[LO] =0; /* remove the low part, in which was the coma since x > 2^42.
So what remains in t is an FP integer almost as large as x */
xs = xs-t.d; /* we are going to throw away the int part anyway */
}
t.d = TWOTO5251 + xs;
u = t.d - TWOTO5251;
y = xs - u;
y = y * INV128;
index = t.i[LO] & 0x3f;
quadrant = (t.i[LO] & 0xff) >>6;
/* Special case tests come late because the conversion FP to int is slow */
xih = xdb.i[HI];
absxih = xih & 0x7fffffff;
/* SPECIAL CASES: x=(Nan, Inf) cos(pi*x)=Nan */
if (absxih>=0x7ff00000) {
xdb.l=0xfff8000000000000LL;
return xdb.d - xdb.d;
}
if(absxih>=0x43400000) /* 2^53, which entails that x is an even integer */
return 1.0; /*signed */
if(index==0 && y==0. && ((quadrant&1)==1)) return +0.;
/* Always +0, inpired by LIA2; We do not have cos(x+pi) == - cos(x)
in this case */
if(index==0 && y==0. && quadrant==0) return 1.;
if(index==0 && y==0. && quadrant==2) return -1.;
if (absxih<0x3E200000) /* 2^-29 */
return 0.9999999999999998889776975374843459576368331909179687500; /* 1-2^-53 */
cospi_accurate(&rh, &rm, &rl, y, index, quadrant);
ReturnRoundTowardsZero3(rh,rm,rl);
};
/* tangent of pi times x */
double tanpi_rn(double x){
double xs, y,u, rh, rm, rl, ch,cm,cl, ich,icm,icl, sh,sm,sl, sign,absx;
db_number xdb, t;
int32_t xih, absxih, index, quadrant;
if (x<0) absx = -x; else absx = x;
xdb.d = x;
xs = x*128.0;
/* argument reduction */
if(absx> TWOTO42 ) { /* x is very large, let us first subtract a large integer from it */
t.d = xs;
t.i[LO] =0; /* remove the low part. The point is somewhere there since x > 2^42.
So what remains in t is an FP integer almost as large as x */
xs = xs-t.d; /* we are going to throw away the int part anyway */
}
t.d = TWOTO5251 + xs;
u = t.d - TWOTO5251;
y = xs - u;
index = t.i[LO] & 0x3f;
quadrant = (t.i[LO] & 0xff) >>6;
/* Special case tests come late because the conversion FP to int is slow */
xih = xdb.i[HI];
absxih = xih & 0x7fffffff;
if (xih>>31) sign=-1.; else sign=1.; /* consider the sign bit */
if(index==0 && y==0.0 && ((quadrant&1)==0)) return sign*0.0; /*signed, inspired by LIA-2 */
/* TODO ? No test for Pi/4. Such a value will lauch the accurate phase. */
y = y * INV128;
/* SPECIAL CASES: x=(Nan, Inf) sin(pi*x)=Nan */
if (absxih>=0x7ff00000) {
xdb.l=0xfff8000000000000LL;
return xdb.d - xdb.d;
}
if(absxih>=0x43300000) /* 2^52, which entails that x is an integer */
return sign*0.0; /*signed */
if(absxih<=0x3E000000) /*2^{-31}*/ {
if (absxih<0x01700000) { /* 2^{-1000} : Get rid of possible subnormals */
/* in this case, SCS computation, accurate to 2^-210 which is provably enough */
scs_t result;
scs_set_d(result, x );
scs_mul(result, PiSCS_ptr, result);
scs_get_d(&rh, result);
return rh;
}
/* First step for Pi*x. TODO: FMA-based optimisation */
const double DekkerConst = 134217729.; /* 2^27 +1 */
double tt, xh, xl;
/* Splitting of x. Both xh and xl have at least 26 consecutive LSB zeroes */
tt = x*DekkerConst;
xh = (x-tt)+tt;
xl = x-xh;
Add12(rh,rl, xh*PIHH, (xl*PIHH + xh*PIHM) + (xh*PIM + xl*PIHM) );
if(rh == (rh + (rl * PIX_RNCST_TAN)))
return rh;
}
/* Fall here either if we have a large input, or if we have a small
input and the rounding test fails. */
cospi_accurate(&ch, &cm, &cl, y, index, quadrant);
Recpr33(&ich, &icm, &icl, ch, cm, cl);
sinpi_accurate(&sh, &sm, &sl, y, index, quadrant);
Mul33(&rh,&rm,&rl, sh,sm,sl, ich,icm,icl);
ReturnRoundToNearest3(rh,rm,rl);
};
double tanpi_rd(double x){
double xs, y,u, rh, rm, rl, ch,cm,cl, ich,icm,icl, sh,sm,sl, sign,absx;
db_number xdb, t;
int32_t xih, absxih, index, quadrant;
if (x<0) absx = -x; else absx = x;
xdb.d = x;
xs = x*128.0;
/* argument reduction */
if(absx> TWOTO42 ) { /* x is very large, let us first subtract a large integer from it */
t.d = xs;
t.i[LO] =0; /* remove the low part. The point is somewhere there since x > 2^42.
So what remains in t is an FP integer almost as large as x */
xs = xs-t.d; /* we are going to throw away the int part anyway */
}
t.d = TWOTO5251 + xs;
u = t.d - TWOTO5251;
y = xs - u;
index = t.i[LO] & 0x3f;
quadrant = (t.i[LO] & 0xff) >>6;
/* Special case tests come late because the conversion FP to int is slow */
xih = xdb.i[HI];
absxih = xih & 0x7fffffff;
if (xih>>31) sign=-1.; else sign=1.; /* consider the sign bit */
if(index==0 && y==0.0 && ((quadrant&1)==0)) return sign*0.0; /*signed, inspired by LIA-2 */
y = y * INV128;
/* SPECIAL CASES: x=(Nan, Inf) sin(pi*x)=Nan */
if (absxih>=0x7ff00000) {
xdb.l=0xfff8000000000000LL;
return xdb.d - xdb.d;
}
if(absxih>=0x43300000) /* 2^52, which entails that x is an integer */
return sign*0.0; /*signed */
if(absxih<=0x3E000000) /*2^{-31}*/ {
if (absxih<0x01700000) { /* 2^{-1000} : Get rid of possible subnormals */
/* in this case, SCS computation, accurate to 2^-210 which is provably enough */
scs_t result;
scs_set_d(result, x );
scs_mul(result, PiSCS_ptr, result);
scs_get_d_minf(&rh, result);
return rh;
}
/* First step for Pi*x. TODO: FMA-based optimisation */
const double DekkerConst = 134217729.; /* 2^27 +1 */
double tt, xh, xl;
/* Splitting of x. Both xh and xl have at least 26 consecutive LSB zeroes */