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glibc/sysdeps/ieee754/dbl-64/s_sin.c
Paul Eggert 59ba27a63a Replace FSF snail mail address with URLs.
2012-02-09 23:18:22 +00:00

1182 lines
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/*
* IBM Accurate Mathematical Library
* written by International Business Machines Corp.
* Copyright (C) 2001, 2009, 2011 Free Software Foundation
*
* 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 2.1 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 Lesser 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, see <http://www.gnu.org/licenses/>.
*/
/****************************************************************************/
/* */
/* MODULE_NAME:usncs.c */
/* */
/* FUNCTIONS: usin */
/* ucos */
/* slow */
/* slow1 */
/* slow2 */
/* sloww */
/* sloww1 */
/* sloww2 */
/* bsloww */
/* bsloww1 */
/* bsloww2 */
/* cslow2 */
/* csloww */
/* csloww1 */
/* csloww2 */
/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h usncs.h */
/* branred.c sincos32.c dosincos.c mpa.c */
/* sincos.tbl */
/* */
/* An ultimate sin and routine. Given an IEEE double machine number x */
/* it computes the correctly rounded (to nearest) value of sin(x) or cos(x) */
/* Assumption: Machine arithmetic operations are performed in */
/* round to nearest mode of IEEE 754 standard. */
/* */
/****************************************************************************/
#include <errno.h>
#include "endian.h"
#include "mydefs.h"
#include "usncs.h"
#include "MathLib.h"
#include "math_private.h"
#ifndef SECTION
# define SECTION
#endif
extern const union
{
int4 i[880];
double x[440];
} __sincostab attribute_hidden;
static const double
sn3 = -1.66666666666664880952546298448555E-01,
sn5 = 8.33333214285722277379541354343671E-03,
cs2 = 4.99999999999999999999950396842453E-01,
cs4 = -4.16666666666664434524222570944589E-02,
cs6 = 1.38888874007937613028114285595617E-03;
void __dubsin(double x, double dx, double w[]);
void __docos(double x, double dx, double w[]);
double __mpsin(double x, double dx);
double __mpcos(double x, double dx);
double __mpsin1(double x);
double __mpcos1(double x);
static double slow(double x);
static double slow1(double x);
static double slow2(double x);
static double sloww(double x, double dx, double orig);
static double sloww1(double x, double dx, double orig);
static double sloww2(double x, double dx, double orig, int n);
static double bsloww(double x, double dx, double orig, int n);
static double bsloww1(double x, double dx, double orig, int n);
static double bsloww2(double x, double dx, double orig, int n);
int __branred(double x, double *a, double *aa);
static double cslow2(double x);
static double csloww(double x, double dx, double orig);
static double csloww1(double x, double dx, double orig);
static double csloww2(double x, double dx, double orig, int n);
/*******************************************************************/
/* An ultimate sin routine. Given an IEEE double machine number x */
/* it computes the correctly rounded (to nearest) value of sin(x) */
/*******************************************************************/
double
SECTION
__sin(double x){
double xx,res,t,cor,y,s,c,sn,ssn,cs,ccs,xn,a,da,db,eps,xn1,xn2;
#if 0
double w[2];
#endif
mynumber u,v;
int4 k,m,n;
#if 0
int4 nn;
#endif
u.x = x;
m = u.i[HIGH_HALF];
k = 0x7fffffff&m; /* no sign */
if (k < 0x3e500000) /* if x->0 =>sin(x)=x */
return x;
/*---------------------------- 2^-26 < |x|< 0.25 ----------------------*/
else if (k < 0x3fd00000){
xx = x*x;
/*Taylor series */
t = ((((s5.x*xx + s4.x)*xx + s3.x)*xx + s2.x)*xx + s1.x)*(xx*x);
res = x+t;
cor = (x-res)+t;
return (res == res + 1.07*cor)? res : slow(x);
} /* else if (k < 0x3fd00000) */
/*---------------------------- 0.25<|x|< 0.855469---------------------- */
else if (k < 0x3feb6000) {
u.x=(m>0)?big.x+x:big.x-x;
y=(m>0)?x-(u.x-big.x):x+(u.x-big.x);
xx=y*y;
s = y + y*xx*(sn3 +xx*sn5);
c = xx*(cs2 +xx*(cs4 + xx*cs6));
k=u.i[LOW_HALF]<<2;
sn=(m>0)?__sincostab.x[k]:-__sincostab.x[k];
ssn=(m>0)?__sincostab.x[k+1]:-__sincostab.x[k+1];
cs=__sincostab.x[k+2];
ccs=__sincostab.x[k+3];
cor=(ssn+s*ccs-sn*c)+cs*s;
res=sn+cor;
cor=(sn-res)+cor;
return (res==res+1.096*cor)? res : slow1(x);
} /* else if (k < 0x3feb6000) */
/*----------------------- 0.855469 <|x|<2.426265 ----------------------*/
else if (k < 0x400368fd ) {
y = (m>0)? hp0.x-x:hp0.x+x;
if (y>=0) {
u.x = big.x+y;
y = (y-(u.x-big.x))+hp1.x;
}
else {
u.x = big.x-y;
y = (-hp1.x) - (y+(u.x-big.x));
}
xx=y*y;
s = y + y*xx*(sn3 +xx*sn5);
c = xx*(cs2 +xx*(cs4 + xx*cs6));
k=u.i[LOW_HALF]<<2;
sn=__sincostab.x[k];
ssn=__sincostab.x[k+1];
cs=__sincostab.x[k+2];
ccs=__sincostab.x[k+3];
cor=(ccs-s*ssn-cs*c)-sn*s;
res=cs+cor;
cor=(cs-res)+cor;
return (res==res+1.020*cor)? ((m>0)?res:-res) : slow2(x);
} /* else if (k < 0x400368fd) */
/*-------------------------- 2.426265<|x|< 105414350 ----------------------*/
else if (k < 0x419921FB ) {
t = (x*hpinv.x + toint.x);
xn = t - toint.x;
v.x = t;
y = (x - xn*mp1.x) - xn*mp2.x;
n =v.i[LOW_HALF]&3;
da = xn*mp3.x;
a=y-da;
da = (y-a)-da;
eps = ABS(x)*1.2e-30;
switch (n) { /* quarter of unit circle */
case 0:
case 2:
xx = a*a;
if (n) {a=-a;da=-da;}
if (xx < 0.01588) {
/*Taylor series */
t = (((((s5.x*xx + s4.x)*xx + s3.x)*xx + s2.x)*xx + s1.x)*a - 0.5*da)*xx+da;
res = a+t;
cor = (a-res)+t;
cor = (cor>0)? 1.02*cor+eps : 1.02*cor -eps;
return (res == res + cor)? res : sloww(a,da,x);
}
else {
if (a>0)
{m=1;t=a;db=da;}
else
{m=0;t=-a;db=-da;}
u.x=big.x+t;
y=t-(u.x-big.x);
xx=y*y;
s = y + (db+y*xx*(sn3 +xx*sn5));
c = y*db+xx*(cs2 +xx*(cs4 + xx*cs6));
k=u.i[LOW_HALF]<<2;
sn=__sincostab.x[k];
ssn=__sincostab.x[k+1];
cs=__sincostab.x[k+2];
ccs=__sincostab.x[k+3];
cor=(ssn+s*ccs-sn*c)+cs*s;
res=sn+cor;
cor=(sn-res)+cor;
cor = (cor>0)? 1.035*cor+eps : 1.035*cor-eps;
return (res==res+cor)? ((m)?res:-res) : sloww1(a,da,x);
}
break;
case 1:
case 3:
if (a<0)
{a=-a;da=-da;}
u.x=big.x+a;
y=a-(u.x-big.x)+da;
xx=y*y;
k=u.i[LOW_HALF]<<2;
sn=__sincostab.x[k];
ssn=__sincostab.x[k+1];
cs=__sincostab.x[k+2];
ccs=__sincostab.x[k+3];
s = y + y*xx*(sn3 +xx*sn5);
c = xx*(cs2 +xx*(cs4 + xx*cs6));
cor=(ccs-s*ssn-cs*c)-sn*s;
res=cs+cor;
cor=(cs-res)+cor;
cor = (cor>0)? 1.025*cor+eps : 1.025*cor-eps;
return (res==res+cor)? ((n&2)?-res:res) : sloww2(a,da,x,n);
break;
}
} /* else if (k < 0x419921FB ) */
/*---------------------105414350 <|x|< 281474976710656 --------------------*/
else if (k < 0x42F00000 ) {
t = (x*hpinv.x + toint.x);
xn = t - toint.x;
v.x = t;
xn1 = (xn+8.0e22)-8.0e22;
xn2 = xn - xn1;
y = ((((x - xn1*mp1.x) - xn1*mp2.x)-xn2*mp1.x)-xn2*mp2.x);
n =v.i[LOW_HALF]&3;
da = xn1*pp3.x;
t=y-da;
da = (y-t)-da;
da = (da - xn2*pp3.x) -xn*pp4.x;
a = t+da;
da = (t-a)+da;
eps = 1.0e-24;
switch (n) {
case 0:
case 2:
xx = a*a;
if (n) {a=-a;da=-da;}
if (xx < 0.01588) {
/* Taylor series */
t = (((((s5.x*xx + s4.x)*xx + s3.x)*xx + s2.x)*xx + s1.x)*a - 0.5*da)*xx+da;
res = a+t;
cor = (a-res)+t;
cor = (cor>0)? 1.02*cor+eps : 1.02*cor -eps;
return (res == res + cor)? res : bsloww(a,da,x,n);
}
else {
if (a>0) {m=1;t=a;db=da;}
else {m=0;t=-a;db=-da;}
u.x=big.x+t;
y=t-(u.x-big.x);
xx=y*y;
s = y + (db+y*xx*(sn3 +xx*sn5));
c = y*db+xx*(cs2 +xx*(cs4 + xx*cs6));
k=u.i[LOW_HALF]<<2;
sn=__sincostab.x[k];
ssn=__sincostab.x[k+1];
cs=__sincostab.x[k+2];
ccs=__sincostab.x[k+3];
cor=(ssn+s*ccs-sn*c)+cs*s;
res=sn+cor;
cor=(sn-res)+cor;
cor = (cor>0)? 1.035*cor+eps : 1.035*cor-eps;
return (res==res+cor)? ((m)?res:-res) : bsloww1(a,da,x,n);
}
break;
case 1:
case 3:
if (a<0)
{a=-a;da=-da;}
u.x=big.x+a;
y=a-(u.x-big.x)+da;
xx=y*y;
k=u.i[LOW_HALF]<<2;
sn=__sincostab.x[k];
ssn=__sincostab.x[k+1];
cs=__sincostab.x[k+2];
ccs=__sincostab.x[k+3];
s = y + y*xx*(sn3 +xx*sn5);
c = xx*(cs2 +xx*(cs4 + xx*cs6));
cor=(ccs-s*ssn-cs*c)-sn*s;
res=cs+cor;
cor=(cs-res)+cor;
cor = (cor>0)? 1.025*cor+eps : 1.025*cor-eps;
return (res==res+cor)? ((n&2)?-res:res) : bsloww2(a,da,x,n);
break;
}
} /* else if (k < 0x42F00000 ) */
/* -----------------281474976710656 <|x| <2^1024----------------------------*/
else if (k < 0x7ff00000) {
n = __branred(x,&a,&da);
switch (n) {
case 0:
if (a*a < 0.01588) return bsloww(a,da,x,n);
else return bsloww1(a,da,x,n);
break;
case 2:
if (a*a < 0.01588) return bsloww(-a,-da,x,n);
else return bsloww1(-a,-da,x,n);
break;
case 1:
case 3:
return bsloww2(a,da,x,n);
break;
}
} /* else if (k < 0x7ff00000 ) */
/*--------------------- |x| > 2^1024 ----------------------------------*/
else {
if (k == 0x7ff00000 && u.i[LOW_HALF] == 0)
__set_errno (EDOM);
return x / x;
}
return 0; /* unreachable */
}
/*******************************************************************/
/* An ultimate cos routine. Given an IEEE double machine number x */
/* it computes the correctly rounded (to nearest) value of cos(x) */
/*******************************************************************/
double
SECTION
__cos(double x)
{
double y,xx,res,t,cor,s,c,sn,ssn,cs,ccs,xn,a,da,db,eps,xn1,xn2;
mynumber u,v;
int4 k,m,n;
u.x = x;
m = u.i[HIGH_HALF];
k = 0x7fffffff&m;
if (k < 0x3e400000 ) return 1.0; /* |x|<2^-27 => cos(x)=1 */
else if (k < 0x3feb6000 ) {/* 2^-27 < |x| < 0.855469 */
y=ABS(x);
u.x = big.x+y;
y = y-(u.x-big.x);
xx=y*y;
s = y + y*xx*(sn3 +xx*sn5);
c = xx*(cs2 +xx*(cs4 + xx*cs6));
k=u.i[LOW_HALF]<<2;
sn=__sincostab.x[k];
ssn=__sincostab.x[k+1];
cs=__sincostab.x[k+2];
ccs=__sincostab.x[k+3];
cor=(ccs-s*ssn-cs*c)-sn*s;
res=cs+cor;
cor=(cs-res)+cor;
return (res==res+1.020*cor)? res : cslow2(x);
} /* else if (k < 0x3feb6000) */
else if (k < 0x400368fd ) {/* 0.855469 <|x|<2.426265 */;
y=hp0.x-ABS(x);
a=y+hp1.x;
da=(y-a)+hp1.x;
xx=a*a;
if (xx < 0.01588) {
t = (((((s5.x*xx + s4.x)*xx + s3.x)*xx + s2.x)*xx + s1.x)*a - 0.5*da)*xx+da;
res = a+t;
cor = (a-res)+t;
cor = (cor>0)? 1.02*cor+1.0e-31 : 1.02*cor -1.0e-31;
return (res == res + cor)? res : csloww(a,da,x);
}
else {
if (a>0) {m=1;t=a;db=da;}
else {m=0;t=-a;db=-da;}
u.x=big.x+t;
y=t-(u.x-big.x);
xx=y*y;
s = y + (db+y*xx*(sn3 +xx*sn5));
c = y*db+xx*(cs2 +xx*(cs4 + xx*cs6));
k=u.i[LOW_HALF]<<2;
sn=__sincostab.x[k];
ssn=__sincostab.x[k+1];
cs=__sincostab.x[k+2];
ccs=__sincostab.x[k+3];
cor=(ssn+s*ccs-sn*c)+cs*s;
res=sn+cor;
cor=(sn-res)+cor;
cor = (cor>0)? 1.035*cor+1.0e-31 : 1.035*cor-1.0e-31;
return (res==res+cor)? ((m)?res:-res) : csloww1(a,da,x);
}
} /* else if (k < 0x400368fd) */
else if (k < 0x419921FB ) {/* 2.426265<|x|< 105414350 */
t = (x*hpinv.x + toint.x);
xn = t - toint.x;
v.x = t;
y = (x - xn*mp1.x) - xn*mp2.x;
n =v.i[LOW_HALF]&3;
da = xn*mp3.x;
a=y-da;
da = (y-a)-da;
eps = ABS(x)*1.2e-30;
switch (n) {
case 1:
case 3:
xx = a*a;
if (n == 1) {a=-a;da=-da;}
if (xx < 0.01588) {
t = (((((s5.x*xx + s4.x)*xx + s3.x)*xx + s2.x)*xx + s1.x)*a - 0.5*da)*xx+da;
res = a+t;
cor = (a-res)+t;
cor = (cor>0)? 1.02*cor+eps : 1.02*cor -eps;
return (res == res + cor)? res : csloww(a,da,x);
}
else {
if (a>0) {m=1;t=a;db=da;}
else {m=0;t=-a;db=-da;}
u.x=big.x+t;
y=t-(u.x-big.x);
xx=y*y;
s = y + (db+y*xx*(sn3 +xx*sn5));
c = y*db+xx*(cs2 +xx*(cs4 + xx*cs6));
k=u.i[LOW_HALF]<<2;
sn=__sincostab.x[k];
ssn=__sincostab.x[k+1];
cs=__sincostab.x[k+2];
ccs=__sincostab.x[k+3];
cor=(ssn+s*ccs-sn*c)+cs*s;
res=sn+cor;
cor=(sn-res)+cor;
cor = (cor>0)? 1.035*cor+eps : 1.035*cor-eps;
return (res==res+cor)? ((m)?res:-res) : csloww1(a,da,x);
}
break;
case 0:
case 2:
if (a<0) {a=-a;da=-da;}
u.x=big.x+a;
y=a-(u.x-big.x)+da;
xx=y*y;
k=u.i[LOW_HALF]<<2;
sn=__sincostab.x[k];
ssn=__sincostab.x[k+1];
cs=__sincostab.x[k+2];
ccs=__sincostab.x[k+3];
s = y + y*xx*(sn3 +xx*sn5);
c = xx*(cs2 +xx*(cs4 + xx*cs6));
cor=(ccs-s*ssn-cs*c)-sn*s;
res=cs+cor;
cor=(cs-res)+cor;
cor = (cor>0)? 1.025*cor+eps : 1.025*cor-eps;
return (res==res+cor)? ((n)?-res:res) : csloww2(a,da,x,n);
break;
}
} /* else if (k < 0x419921FB ) */
else if (k < 0x42F00000 ) {
t = (x*hpinv.x + toint.x);
xn = t - toint.x;
v.x = t;
xn1 = (xn+8.0e22)-8.0e22;
xn2 = xn - xn1;
y = ((((x - xn1*mp1.x) - xn1*mp2.x)-xn2*mp1.x)-xn2*mp2.x);
n =v.i[LOW_HALF]&3;
da = xn1*pp3.x;
t=y-da;
da = (y-t)-da;
da = (da - xn2*pp3.x) -xn*pp4.x;
a = t+da;
da = (t-a)+da;
eps = 1.0e-24;
switch (n) {
case 1:
case 3:
xx = a*a;
if (n==1) {a=-a;da=-da;}
if (xx < 0.01588) {
t = (((((s5.x*xx + s4.x)*xx + s3.x)*xx + s2.x)*xx + s1.x)*a - 0.5*da)*xx+da;
res = a+t;
cor = (a-res)+t;
cor = (cor>0)? 1.02*cor+eps : 1.02*cor -eps;
return (res == res + cor)? res : bsloww(a,da,x,n);
}
else {
if (a>0) {m=1;t=a;db=da;}
else {m=0;t=-a;db=-da;}
u.x=big.x+t;
y=t-(u.x-big.x);
xx=y*y;
s = y + (db+y*xx*(sn3 +xx*sn5));
c = y*db+xx*(cs2 +xx*(cs4 + xx*cs6));
k=u.i[LOW_HALF]<<2;
sn=__sincostab.x[k];
ssn=__sincostab.x[k+1];
cs=__sincostab.x[k+2];
ccs=__sincostab.x[k+3];
cor=(ssn+s*ccs-sn*c)+cs*s;
res=sn+cor;
cor=(sn-res)+cor;
cor = (cor>0)? 1.035*cor+eps : 1.035*cor-eps;
return (res==res+cor)? ((m)?res:-res) : bsloww1(a,da,x,n);
}
break;
case 0:
case 2:
if (a<0) {a=-a;da=-da;}
u.x=big.x+a;
y=a-(u.x-big.x)+da;
xx=y*y;
k=u.i[LOW_HALF]<<2;
sn=__sincostab.x[k];
ssn=__sincostab.x[k+1];
cs=__sincostab.x[k+2];
ccs=__sincostab.x[k+3];
s = y + y*xx*(sn3 +xx*sn5);
c = xx*(cs2 +xx*(cs4 + xx*cs6));
cor=(ccs-s*ssn-cs*c)-sn*s;
res=cs+cor;
cor=(cs-res)+cor;
cor = (cor>0)? 1.025*cor+eps : 1.025*cor-eps;
return (res==res+cor)? ((n)?-res:res) : bsloww2(a,da,x,n);
break;
}
} /* else if (k < 0x42F00000 ) */
else if (k < 0x7ff00000) {/* 281474976710656 <|x| <2^1024 */
n = __branred(x,&a,&da);
switch (n) {
case 1:
if (a*a < 0.01588) return bsloww(-a,-da,x,n);
else return bsloww1(-a,-da,x,n);
break;
case 3:
if (a*a < 0.01588) return bsloww(a,da,x,n);
else return bsloww1(a,da,x,n);
break;
case 0:
case 2:
return bsloww2(a,da,x,n);
break;
}
} /* else if (k < 0x7ff00000 ) */
else {
if (k == 0x7ff00000 && u.i[LOW_HALF] == 0)
__set_errno (EDOM);
return x / x; /* |x| > 2^1024 */
}
return 0;
}
/************************************************************************/
/* Routine compute sin(x) for 2^-26 < |x|< 0.25 by Taylor with more */
/* precision and if still doesn't accurate enough by mpsin or dubsin */
/************************************************************************/
static double
SECTION
slow(double x) {
static const double th2_36 = 206158430208.0; /* 1.5*2**37 */
double y,x1,x2,xx,r,t,res,cor,w[2];
x1=(x+th2_36)-th2_36;
y = aa.x*x1*x1*x1;
r=x+y;
x2=x-x1;
xx=x*x;
t = (((((s5.x*xx + s4.x)*xx + s3.x)*xx + s2.x)*xx + bb.x)*xx + 3.0*aa.x*x1*x2)*x +aa.x*x2*x2*x2;
t=((x-r)+y)+t;
res=r+t;
cor = (r-res)+t;
if (res == res + 1.0007*cor) return res;
else {
__dubsin(ABS(x),0,w);
if (w[0] == w[0]+1.000000001*w[1]) return (x>0)?w[0]:-w[0];
else return (x>0)?__mpsin(x,0):-__mpsin(-x,0);
}
}
/*******************************************************************************/
/* Routine compute sin(x) for 0.25<|x|< 0.855469 by __sincostab.tbl and Taylor */
/* and if result still doesn't accurate enough by mpsin or dubsin */
/*******************************************************************************/
static double
SECTION
slow1(double x) {
mynumber u;
double sn,ssn,cs,ccs,s,c,w[2],y,y1,y2,c1,c2,xx,cor,res;
static const double t22 = 6291456.0;
int4 k;
y=ABS(x);
u.x=big.x+y;
y=y-(u.x-big.x);
xx=y*y;
s = y*xx*(sn3 +xx*sn5);
c = xx*(cs2 +xx*(cs4 + xx*cs6));
k=u.i[LOW_HALF]<<2;
sn=__sincostab.x[k]; /* Data */
ssn=__sincostab.x[k+1]; /* from */
cs=__sincostab.x[k+2]; /* tables */
ccs=__sincostab.x[k+3]; /* __sincostab.tbl */
y1 = (y+t22)-t22;
y2 = y - y1;
c1 = (cs+t22)-t22;
c2=(cs-c1)+ccs;
cor=(ssn+s*ccs+cs*s+c2*y+c1*y2)-sn*c;
y=sn+c1*y1;
cor = cor+((sn-y)+c1*y1);
res=y+cor;
cor=(y-res)+cor;
if (res == res+1.0005*cor) return (x>0)?res:-res;
else {
__dubsin(ABS(x),0,w);
if (w[0] == w[0]+1.000000005*w[1]) return (x>0)?w[0]:-w[0];
else return (x>0)?__mpsin(x,0):-__mpsin(-x,0);
}
}
/**************************************************************************/
/* Routine compute sin(x) for 0.855469 <|x|<2.426265 by __sincostab.tbl */
/* and if result still doesn't accurate enough by mpsin or dubsin */
/**************************************************************************/
static double
SECTION
slow2(double x) {
mynumber u;
double sn,ssn,cs,ccs,s,c,w[2],y,y1,y2,e1,e2,xx,cor,res,del;
static const double t22 = 6291456.0;
int4 k;
y=ABS(x);
y = hp0.x-y;
if (y>=0) {
u.x = big.x+y;
y = y-(u.x-big.x);
del = hp1.x;
}
else {
u.x = big.x-y;
y = -(y+(u.x-big.x));
del = -hp1.x;
}
xx=y*y;
s = y*xx*(sn3 +xx*sn5);
c = y*del+xx*(cs2 +xx*(cs4 + xx*cs6));
k=u.i[LOW_HALF]<<2;
sn=__sincostab.x[k];
ssn=__sincostab.x[k+1];
cs=__sincostab.x[k+2];
ccs=__sincostab.x[k+3];
y1 = (y+t22)-t22;
y2 = (y - y1)+del;
e1 = (sn+t22)-t22;
e2=(sn-e1)+ssn;
cor=(ccs-cs*c-e1*y2-e2*y)-sn*s;
y=cs-e1*y1;
cor = cor+((cs-y)-e1*y1);
res=y+cor;
cor=(y-res)+cor;
if (res == res+1.0005*cor) return (x>0)?res:-res;
else {
y=ABS(x)-hp0.x;
y1=y-hp1.x;
y2=(y-y1)-hp1.x;
__docos(y1,y2,w);
if (w[0] == w[0]+1.000000005*w[1]) return (x>0)?w[0]:-w[0];
else return (x>0)?__mpsin(x,0):-__mpsin(-x,0);
}
}
/***************************************************************************/
/* Routine compute sin(x+dx) (Double-Length number) where x is small enough*/
/* to use Taylor series around zero and (x+dx) */
/* in first or third quarter of unit circle.Routine receive also */
/* (right argument) the original value of x for computing error of */
/* result.And if result not accurate enough routine calls mpsin1 or dubsin */
/***************************************************************************/
static double
SECTION
sloww(double x,double dx, double orig) {
static const double th2_36 = 206158430208.0; /* 1.5*2**37 */
double y,x1,x2,xx,r,t,res,cor,w[2],a,da,xn;
union {int4 i[2]; double x;} v;
int4 n;
x1=(x+th2_36)-th2_36;
y = aa.x*x1*x1*x1;
r=x+y;
x2=(x-x1)+dx;
xx=x*x;
t = (((((s5.x*xx + s4.x)*xx + s3.x)*xx + s2.x)*xx + bb.x)*xx + 3.0*aa.x*x1*x2)*x +aa.x*x2*x2*x2+dx;
t=((x-r)+y)+t;
res=r+t;
cor = (r-res)+t;
cor = (cor>0)? 1.0005*cor+ABS(orig)*3.1e-30 : 1.0005*cor-ABS(orig)*3.1e-30;
if (res == res + cor) return res;
else {
(x>0)? __dubsin(x,dx,w) : __dubsin(-x,-dx,w);
cor = (w[1]>0)? 1.000000001*w[1] + ABS(orig)*1.1e-30 : 1.000000001*w[1] - ABS(orig)*1.1e-30;
if (w[0] == w[0]+cor) return (x>0)?w[0]:-w[0];
else {
t = (orig*hpinv.x + toint.x);
xn = t - toint.x;
v.x = t;
y = (orig - xn*mp1.x) - xn*mp2.x;
n =v.i[LOW_HALF]&3;
da = xn*pp3.x;
t=y-da;
da = (y-t)-da;
y = xn*pp4.x;
a = t - y;
da = ((t-a)-y)+da;
if (n&2) {a=-a; da=-da;}
(a>0)? __dubsin(a,da,w) : __dubsin(-a,-da,w);
cor = (w[1]>0)? 1.000000001*w[1] + ABS(orig)*1.1e-40 : 1.000000001*w[1] - ABS(orig)*1.1e-40;
if (w[0] == w[0]+cor) return (a>0)?w[0]:-w[0];
else return __mpsin1(orig);
}
}
}
/***************************************************************************/
/* Routine compute sin(x+dx) (Double-Length number) where x in first or */
/* third quarter of unit circle.Routine receive also (right argument) the */
/* original value of x for computing error of result.And if result not */
/* accurate enough routine calls mpsin1 or dubsin */
/***************************************************************************/
static double
SECTION
sloww1(double x, double dx, double orig) {
mynumber u;
double sn,ssn,cs,ccs,s,c,w[2],y,y1,y2,c1,c2,xx,cor,res;
static const double t22 = 6291456.0;
int4 k;
y=ABS(x);
u.x=big.x+y;
y=y-(u.x-big.x);
dx=(x>0)?dx:-dx;
xx=y*y;
s = y*xx*(sn3 +xx*sn5);
c = xx*(cs2 +xx*(cs4 + xx*cs6));
k=u.i[LOW_HALF]<<2;
sn=__sincostab.x[k];
ssn=__sincostab.x[k+1];
cs=__sincostab.x[k+2];
ccs=__sincostab.x[k+3];
y1 = (y+t22)-t22;
y2 = (y - y1)+dx;
c1 = (cs+t22)-t22;
c2=(cs-c1)+ccs;
cor=(ssn+s*ccs+cs*s+c2*y+c1*y2-sn*y*dx)-sn*c;
y=sn+c1*y1;
cor = cor+((sn-y)+c1*y1);
res=y+cor;
cor=(y-res)+cor;
cor = (cor>0)? 1.0005*cor+3.1e-30*ABS(orig) : 1.0005*cor-3.1e-30*ABS(orig);
if (res == res + cor) return (x>0)?res:-res;
else {
__dubsin(ABS(x),dx,w);
cor = (w[1]>0)? 1.000000005*w[1]+1.1e-30*ABS(orig) : 1.000000005*w[1]-1.1e-30*ABS(orig);
if (w[0] == w[0]+cor) return (x>0)?w[0]:-w[0];
else return __mpsin1(orig);
}
}
/***************************************************************************/
/* Routine compute sin(x+dx) (Double-Length number) where x in second or */
/* fourth quarter of unit circle.Routine receive also the original value */
/* and quarter(n= 1or 3)of x for computing error of result.And if result not*/
/* accurate enough routine calls mpsin1 or dubsin */
/***************************************************************************/
static double
SECTION
sloww2(double x, double dx, double orig, int n) {
mynumber u;
double sn,ssn,cs,ccs,s,c,w[2],y,y1,y2,e1,e2,xx,cor,res;
static const double t22 = 6291456.0;
int4 k;
y=ABS(x);
u.x=big.x+y;
y=y-(u.x-big.x);
dx=(x>0)?dx:-dx;
xx=y*y;
s = y*xx*(sn3 +xx*sn5);
c = y*dx+xx*(cs2 +xx*(cs4 + xx*cs6));
k=u.i[LOW_HALF]<<2;
sn=__sincostab.x[k];
ssn=__sincostab.x[k+1];
cs=__sincostab.x[k+2];
ccs=__sincostab.x[k+3];
y1 = (y+t22)-t22;
y2 = (y - y1)+dx;
e1 = (sn+t22)-t22;
e2=(sn-e1)+ssn;
cor=(ccs-cs*c-e1*y2-e2*y)-sn*s;
y=cs-e1*y1;
cor = cor+((cs-y)-e1*y1);
res=y+cor;
cor=(y-res)+cor;
cor = (cor>0)? 1.0005*cor+3.1e-30*ABS(orig) : 1.0005*cor-3.1e-30*ABS(orig);
if (res == res + cor) return (n&2)?-res:res;
else {
__docos(ABS(x),dx,w);
cor = (w[1]>0)? 1.000000005*w[1]+1.1e-30*ABS(orig) : 1.000000005*w[1]-1.1e-30*ABS(orig);
if (w[0] == w[0]+cor) return (n&2)?-w[0]:w[0];
else return __mpsin1(orig);
}
}
/***************************************************************************/
/* Routine compute sin(x+dx) or cos(x+dx) (Double-Length number) where x */
/* is small enough to use Taylor series around zero and (x+dx) */
/* in first or third quarter of unit circle.Routine receive also */
/* (right argument) the original value of x for computing error of */
/* result.And if result not accurate enough routine calls other routines */
/***************************************************************************/
static double
SECTION
bsloww(double x,double dx, double orig,int n) {
static const double th2_36 = 206158430208.0; /* 1.5*2**37 */
double y,x1,x2,xx,r,t,res,cor,w[2];
#if 0
double a,da,xn;
union {int4 i[2]; double x;} v;
#endif
x1=(x+th2_36)-th2_36;
y = aa.x*x1*x1*x1;
r=x+y;
x2=(x-x1)+dx;
xx=x*x;
t = (((((s5.x*xx + s4.x)*xx + s3.x)*xx + s2.x)*xx + bb.x)*xx + 3.0*aa.x*x1*x2)*x +aa.x*x2*x2*x2+dx;
t=((x-r)+y)+t;
res=r+t;
cor = (r-res)+t;
cor = (cor>0)? 1.0005*cor+1.1e-24 : 1.0005*cor-1.1e-24;
if (res == res + cor) return res;
else {
(x>0)? __dubsin(x,dx,w) : __dubsin(-x,-dx,w);
cor = (w[1]>0)? 1.000000001*w[1] + 1.1e-24 : 1.000000001*w[1] - 1.1e-24;
if (w[0] == w[0]+cor) return (x>0)?w[0]:-w[0];
else return (n&1)?__mpcos1(orig):__mpsin1(orig);
}
}
/***************************************************************************/
/* Routine compute sin(x+dx) or cos(x+dx) (Double-Length number) where x */
/* in first or third quarter of unit circle.Routine receive also */
/* (right argument) the original value of x for computing error of result.*/
/* And if result not accurate enough routine calls other routines */
/***************************************************************************/
static double
SECTION
bsloww1(double x, double dx, double orig,int n) {
mynumber u;
double sn,ssn,cs,ccs,s,c,w[2],y,y1,y2,c1,c2,xx,cor,res;
static const double t22 = 6291456.0;
int4 k;
y=ABS(x);
u.x=big.x+y;
y=y-(u.x-big.x);
dx=(x>0)?dx:-dx;
xx=y*y;
s = y*xx*(sn3 +xx*sn5);
c = xx*(cs2 +xx*(cs4 + xx*cs6));
k=u.i[LOW_HALF]<<2;
sn=__sincostab.x[k];
ssn=__sincostab.x[k+1];
cs=__sincostab.x[k+2];
ccs=__sincostab.x[k+3];
y1 = (y+t22)-t22;
y2 = (y - y1)+dx;
c1 = (cs+t22)-t22;
c2=(cs-c1)+ccs;
cor=(ssn+s*ccs+cs*s+c2*y+c1*y2-sn*y*dx)-sn*c;
y=sn+c1*y1;
cor = cor+((sn-y)+c1*y1);
res=y+cor;
cor=(y-res)+cor;
cor = (cor>0)? 1.0005*cor+1.1e-24 : 1.0005*cor-1.1e-24;
if (res == res + cor) return (x>0)?res:-res;
else {
__dubsin(ABS(x),dx,w);
cor = (w[1]>0)? 1.000000005*w[1]+1.1e-24: 1.000000005*w[1]-1.1e-24;
if (w[0] == w[0]+cor) return (x>0)?w[0]:-w[0];
else return (n&1)?__mpcos1(orig):__mpsin1(orig);
}
}
/***************************************************************************/
/* Routine compute sin(x+dx) or cos(x+dx) (Double-Length number) where x */
/* in second or fourth quarter of unit circle.Routine receive also the */
/* original value and quarter(n= 1or 3)of x for computing error of result. */
/* And if result not accurate enough routine calls other routines */
/***************************************************************************/
static double
SECTION
bsloww2(double x, double dx, double orig, int n) {
mynumber u;
double sn,ssn,cs,ccs,s,c,w[2],y,y1,y2,e1,e2,xx,cor,res;
static const double t22 = 6291456.0;
int4 k;
y=ABS(x);
u.x=big.x+y;
y=y-(u.x-big.x);
dx=(x>0)?dx:-dx;
xx=y*y;
s = y*xx*(sn3 +xx*sn5);
c = y*dx+xx*(cs2 +xx*(cs4 + xx*cs6));
k=u.i[LOW_HALF]<<2;
sn=__sincostab.x[k];
ssn=__sincostab.x[k+1];
cs=__sincostab.x[k+2];
ccs=__sincostab.x[k+3];
y1 = (y+t22)-t22;
y2 = (y - y1)+dx;
e1 = (sn+t22)-t22;
e2=(sn-e1)+ssn;
cor=(ccs-cs*c-e1*y2-e2*y)-sn*s;
y=cs-e1*y1;
cor = cor+((cs-y)-e1*y1);
res=y+cor;
cor=(y-res)+cor;
cor = (cor>0)? 1.0005*cor+1.1e-24 : 1.0005*cor-1.1e-24;
if (res == res + cor) return (n&2)?-res:res;
else {
__docos(ABS(x),dx,w);
cor = (w[1]>0)? 1.000000005*w[1]+1.1e-24 : 1.000000005*w[1]-1.1e-24;
if (w[0] == w[0]+cor) return (n&2)?-w[0]:w[0];
else return (n&1)?__mpsin1(orig):__mpcos1(orig);
}
}
/************************************************************************/
/* Routine compute cos(x) for 2^-27 < |x|< 0.25 by Taylor with more */
/* precision and if still doesn't accurate enough by mpcos or docos */
/************************************************************************/
static double
SECTION
cslow2(double x) {
mynumber u;
double sn,ssn,cs,ccs,s,c,w[2],y,y1,y2,e1,e2,xx,cor,res;
static const double t22 = 6291456.0;
int4 k;
y=ABS(x);
u.x = big.x+y;
y = y-(u.x-big.x);
xx=y*y;
s = y*xx*(sn3 +xx*sn5);
c = xx*(cs2 +xx*(cs4 + xx*cs6));
k=u.i[LOW_HALF]<<2;
sn=__sincostab.x[k];
ssn=__sincostab.x[k+1];
cs=__sincostab.x[k+2];
ccs=__sincostab.x[k+3];
y1 = (y+t22)-t22;
y2 = y - y1;
e1 = (sn+t22)-t22;
e2=(sn-e1)+ssn;
cor=(ccs-cs*c-e1*y2-e2*y)-sn*s;
y=cs-e1*y1;
cor = cor+((cs-y)-e1*y1);
res=y+cor;
cor=(y-res)+cor;
if (res == res+1.0005*cor)
return res;
else {
y=ABS(x);
__docos(y,0,w);
if (w[0] == w[0]+1.000000005*w[1]) return w[0];
else return __mpcos(x,0);
}
}
/***************************************************************************/
/* Routine compute cos(x+dx) (Double-Length number) where x is small enough*/
/* to use Taylor series around zero and (x+dx) .Routine receive also */
/* (right argument) the original value of x for computing error of */
/* result.And if result not accurate enough routine calls other routines */
/***************************************************************************/
static double
SECTION
csloww(double x,double dx, double orig) {
static const double th2_36 = 206158430208.0; /* 1.5*2**37 */
double y,x1,x2,xx,r,t,res,cor,w[2],a,da,xn;
union {int4 i[2]; double x;} v;
int4 n;
x1=(x+th2_36)-th2_36;
y = aa.x*x1*x1*x1;
r=x+y;
x2=(x-x1)+dx;
xx=x*x;
/* Taylor series */
t = (((((s5.x*xx + s4.x)*xx + s3.x)*xx + s2.x)*xx + bb.x)*xx + 3.0*aa.x*x1*x2)*x +aa.x*x2*x2*x2+dx;
t=((x-r)+y)+t;
res=r+t;
cor = (r-res)+t;
cor = (cor>0)? 1.0005*cor+ABS(orig)*3.1e-30 : 1.0005*cor-ABS(orig)*3.1e-30;
if (res == res + cor) return res;
else {
(x>0)? __dubsin(x,dx,w) : __dubsin(-x,-dx,w);
cor = (w[1]>0)? 1.000000001*w[1] + ABS(orig)*1.1e-30 : 1.000000001*w[1] - ABS(orig)*1.1e-30;
if (w[0] == w[0]+cor) return (x>0)?w[0]:-w[0];
else {
t = (orig*hpinv.x + toint.x);
xn = t - toint.x;
v.x = t;
y = (orig - xn*mp1.x) - xn*mp2.x;
n =v.i[LOW_HALF]&3;
da = xn*pp3.x;
t=y-da;
da = (y-t)-da;
y = xn*pp4.x;
a = t - y;
da = ((t-a)-y)+da;
if (n==1) {a=-a; da=-da;}
(a>0)? __dubsin(a,da,w) : __dubsin(-a,-da,w);
cor = (w[1]>0)? 1.000000001*w[1] + ABS(orig)*1.1e-40 : 1.000000001*w[1] - ABS(orig)*1.1e-40;
if (w[0] == w[0]+cor) return (a>0)?w[0]:-w[0];
else return __mpcos1(orig);
}
}
}
/***************************************************************************/
/* Routine compute sin(x+dx) (Double-Length number) where x in first or */
/* third quarter of unit circle.Routine receive also (right argument) the */
/* original value of x for computing error of result.And if result not */
/* accurate enough routine calls other routines */
/***************************************************************************/
static double
SECTION
csloww1(double x, double dx, double orig) {
mynumber u;
double sn,ssn,cs,ccs,s,c,w[2],y,y1,y2,c1,c2,xx,cor,res;
static const double t22 = 6291456.0;
int4 k;
y=ABS(x);
u.x=big.x+y;
y=y-(u.x-big.x);
dx=(x>0)?dx:-dx;
xx=y*y;
s = y*xx*(sn3 +xx*sn5);
c = xx*(cs2 +xx*(cs4 + xx*cs6));
k=u.i[LOW_HALF]<<2;
sn=__sincostab.x[k];
ssn=__sincostab.x[k+1];
cs=__sincostab.x[k+2];
ccs=__sincostab.x[k+3];
y1 = (y+t22)-t22;
y2 = (y - y1)+dx;
c1 = (cs+t22)-t22;
c2=(cs-c1)+ccs;
cor=(ssn+s*ccs+cs*s+c2*y+c1*y2-sn*y*dx)-sn*c;
y=sn+c1*y1;
cor = cor+((sn-y)+c1*y1);
res=y+cor;
cor=(y-res)+cor;
cor = (cor>0)? 1.0005*cor+3.1e-30*ABS(orig) : 1.0005*cor-3.1e-30*ABS(orig);
if (res == res + cor) return (x>0)?res:-res;
else {
__dubsin(ABS(x),dx,w);
cor = (w[1]>0)? 1.000000005*w[1]+1.1e-30*ABS(orig) : 1.000000005*w[1]-1.1e-30*ABS(orig);
if (w[0] == w[0]+cor) return (x>0)?w[0]:-w[0];
else return __mpcos1(orig);
}
}
/***************************************************************************/
/* Routine compute sin(x+dx) (Double-Length number) where x in second or */
/* fourth quarter of unit circle.Routine receive also the original value */
/* and quarter(n= 1or 3)of x for computing error of result.And if result not*/
/* accurate enough routine calls other routines */
/***************************************************************************/
static double
SECTION
csloww2(double x, double dx, double orig, int n) {
mynumber u;
double sn,ssn,cs,ccs,s,c,w[2],y,y1,y2,e1,e2,xx,cor,res;
static const double t22 = 6291456.0;
int4 k;
y=ABS(x);
u.x=big.x+y;
y=y-(u.x-big.x);
dx=(x>0)?dx:-dx;
xx=y*y;
s = y*xx*(sn3 +xx*sn5);
c = y*dx+xx*(cs2 +xx*(cs4 + xx*cs6));
k=u.i[LOW_HALF]<<2;
sn=__sincostab.x[k];
ssn=__sincostab.x[k+1];
cs=__sincostab.x[k+2];
ccs=__sincostab.x[k+3];
y1 = (y+t22)-t22;
y2 = (y - y1)+dx;
e1 = (sn+t22)-t22;
e2=(sn-e1)+ssn;
cor=(ccs-cs*c-e1*y2-e2*y)-sn*s;
y=cs-e1*y1;
cor = cor+((cs-y)-e1*y1);
res=y+cor;
cor=(y-res)+cor;
cor = (cor>0)? 1.0005*cor+3.1e-30*ABS(orig) : 1.0005*cor-3.1e-30*ABS(orig);
if (res == res + cor) return (n)?-res:res;
else {
__docos(ABS(x),dx,w);
cor = (w[1]>0)? 1.000000005*w[1]+1.1e-30*ABS(orig) : 1.000000005*w[1]-1.1e-30*ABS(orig);
if (w[0] == w[0]+cor) return (n)?-w[0]:w[0];
else return __mpcos1(orig);
}
}
#ifndef __cos
weak_alias (__cos, cos)
# ifdef NO_LONG_DOUBLE
strong_alias (__cos, __cosl)
weak_alias (__cos, cosl)
# endif
#endif
#ifndef __sin
weak_alias (__sin, sin)
# ifdef NO_LONG_DOUBLE
strong_alias (__sin, __sinl)
weak_alias (__sin, sinl)
# endif
#endif
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