/* Invert *X and store in *Y. Relative error bound:
- For P = 2: 1.001 * R ^ (1 - P)
- For P = 3: 1.063 * R ^ (1 - P)
- For P > 3: 2.001 * R ^ (1 - P)
*X = 0 is not permissible. */
static void
SECTION
__inv (const mp_no *x, mp_no *y, int p)
{
long i;
double t;
mp_no z, w;
static const int np1[] =
{ 0, 0, 0, 0, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3,
4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4
};
__cpy (x, &z, p);
z.e = 0;
__mp_dbl (&z, &t, p);
t = 1 / t;
__dbl_mp (t, y, p);
EY -= EX;
for (i = 0; i < np1[p]; i++)
{
__cpy (y, &w, p);
__mul (x, &w, y, p);
__sub (&__mptwo, y, &z, p);
__mul (&w, &z, y, p);
}
}
static void
SECTION
cc32(mp_no *x, mp_no *y, int p) {
int i;
double a;
#if 0
double b;
static const mp_no mpone = {1,{1.0,1.0}};
#endif
mp_no mpt1,x2,gor,sum ,mpk={1,{1.0}};
#if 0
mp_no mpt2;
#endif
for (i=1;i<=p;i++) mpk.d[i]=0;
__mul(x,x,&x2,p);
mpk.d[1]=27.0;
__mul(&oofac27,&mpk,&gor,p);
__cpy(&gor,&sum,p);
for (a=26.0;a>2.0;a-=2.0) {
mpk.d[1]=a*(a-1.0);
__mul(&gor,&mpk,&mpt1,p);
__cpy(&mpt1,&gor,p);
__mul(&x2,&sum,&mpt1,p);
__sub(&gor,&mpt1,&sum,p);
}
__mul(&x2,&sum,y,p);
}
void
SECTION
__mpsqrt (mp_no *x, mp_no *y, int p)
{
int i, m, ey;
double dx, dy;
static const mp_no mphalf = {0, {1.0, HALFRAD}};
static const mp_no mp3halfs = {1, {1.0, 1.0, HALFRAD}};
mp_no mpxn, mpz, mpu, mpt1, mpt2;
ey = EX / 2;
__cpy (x, &mpxn, p);
mpxn.e -= (ey + ey);
__mp_dbl (&mpxn, &dx, p);
dy = fastiroot (dx);
__dbl_mp (dy, &mpu, p);
__mul (&mpxn, &mphalf, &mpz, p);
m = __mpsqrt_mp[p];
for (i = 0; i < m; i++)
{
__sqr (&mpu, &mpt1, p);
__mul (&mpt1, &mpz, &mpt2, p);
__sub (&mp3halfs, &mpt2, &mpt1, p);
__mul (&mpu, &mpt1, &mpt2, p);
__cpy (&mpt2, &mpu, p);
}
__mul (&mpxn, &mpu, y, p);
EY += ey;
}
void __inv(const mp_no *x, mp_no *y, int p) {
int i;
#if 0
int l;
#endif
double t;
mp_no z,w;
static const int np1[] = {0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,
4,4,4,4,4,4,4,4,4,4,4,4,4,4,4};
const mp_no mptwo = {1,{1.0,2.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0}};
__cpy(x,&z,p); z.e=0; __mp_dbl(&z,&t,p);
t=ONE/t; __dbl_mp(t,y,p); EY -= EX;
for (i=0; i<np1[p]; i++) {
__cpy(y,&w,p);
__mul(x,&w,y,p);
__sub(&mptwo,y,&z,p);
__mul(&w,&z,y,p);
}
return;
}
double __slowpow(double x, double y, double z) {
double res,res1;
mp_no mpx, mpy, mpz,mpw,mpp,mpr,mpr1;
static const mp_no eps = {-3,{1.0,4.0}};
int p;
res = __halfulp(x,y); /* halfulp() returns -10 or x^y */
if (res >= 0) return res; /* if result was really computed by halfulp */
/* else, if result was not really computed by halfulp */
p = 10; /* p=precision */
__dbl_mp(x,&mpx,p);
__dbl_mp(y,&mpy,p);
__dbl_mp(z,&mpz,p);
__mplog(&mpx, &mpz, p); /* log(x) = z */
__mul(&mpy,&mpz,&mpw,p); /* y * z =w */
__mpexp(&mpw, &mpp, p); /* e^w =pp */
__add(&mpp,&eps,&mpr,p); /* pp+eps =r */
__mp_dbl(&mpr, &res, p);
__sub(&mpp,&eps,&mpr1,p); /* pp -eps =r1 */
__mp_dbl(&mpr1, &res1, p); /* converting into double precision */
if (res == res1) return res;
p = 32; /* if we get here result wasn't calculated exactly, continue */
__dbl_mp(x,&mpx,p); /* for more exact calculation */
__dbl_mp(y,&mpy,p);
__dbl_mp(z,&mpz,p);
__mplog(&mpx, &mpz, p); /* log(c)=z */
__mul(&mpy,&mpz,&mpw,p); /* y*z =w */
__mpexp(&mpw, &mpp, p); /* e^w=pp */
__mp_dbl(&mpp, &res, p); /* converting into double precision */
return res;
}
/* Perform range reduction of a double number x into multi precision number y,
such that y = x - n * pi / 2, abs (y) < pi / 4, n = 0, +-1, +-2, ...
Return int which indicates in which quarter of circle x is. */
int
SECTION
__mpranred (double x, mp_no *y, int p)
{
number v;
double t, xn;
int i, k, n;
mp_no a, b, c;
if (ABS (x) < 2.8e14)
{
t = (x * hpinv.d + toint.d);
xn = t - toint.d;
v.d = t;
n = v.i[LOW_HALF] & 3;
__dbl_mp (xn, &a, p);
__mul (&a, &hp, &b, p);
__dbl_mp (x, &c, p);
__sub (&c, &b, y, p);
return n;
}
else
{
/* If x is very big more precision required. */
__dbl_mp (x, &a, p);
a.d[0] = 1.0;
k = a.e - 5;
if (k < 0)
k = 0;
b.e = -k;
b.d[0] = 1.0;
for (i = 0; i < p; i++)
b.d[i + 1] = toverp[i + k];
__mul (&a, &b, &c, p);
t = c.d[c.e];
for (i = 1; i <= p - c.e; i++)
c.d[i] = c.d[i + c.e];
for (i = p + 1 - c.e; i <= p; i++)
c.d[i] = 0;
c.e = 0;
if (c.d[1] >= HALFRAD)
{
t += 1.0;
__sub (&c, &__mpone, &b, p);
__mul (&b, &hp, y, p);
}
else
__mul (&c, &hp, y, p);
n = (int) t;
if (x < 0)
{
y->d[0] = -y->d[0];
n = -n;
}
return (n & 3);
}
}
/* Stage 3: Perform a multi-Precision computation */
static double
SECTION
atan2Mp (double x, double y, const int pr[])
{
double z1, z2;
int i, p;
mp_no mpx, mpy, mpz, mpz1, mpz2, mperr, mpt1;
for (i = 0; i < MM; i++)
{
p = pr[i];
__dbl_mp (x, &mpx, p);
__dbl_mp (y, &mpy, p);
__mpatan2 (&mpy, &mpx, &mpz, p);
__dbl_mp (ud[i].d, &mpt1, p);
__mul (&mpz, &mpt1, &mperr, p);
__add (&mpz, &mperr, &mpz1, p);
__sub (&mpz, &mperr, &mpz2, p);
__mp_dbl (&mpz1, &z1, p);
__mp_dbl (&mpz2, &z2, p);
if (z1 == z2)
{
LIBC_PROBE (slowatan2, 4, &p, &x, &y, &z1);
return z1;
}
}
LIBC_PROBE (slowatan2_inexact, 4, &p, &x, &y, &z1);
return z1; /*if impossible to do exact computing */
}
/*Converting from double precision to Multi-precision and calculating e^x */
double
SECTION
__slowexp (double x)
{
#ifndef USE_LONG_DOUBLE_FOR_MP
double w, z, res, eps = 3.0e-26;
int p;
mp_no mpx, mpy, mpz, mpw, mpeps, mpcor;
/* Use the multiple precision __MPEXP function to compute the exponential
First at 144 bits and if it is not accurate enough, at 768 bits. */
p = 6;
__dbl_mp (x, &mpx, p);
__mpexp (&mpx, &mpy, p);
__dbl_mp (eps, &mpeps, p);
__mul (&mpeps, &mpy, &mpcor, p);
__add (&mpy, &mpcor, &mpw, p);
__sub (&mpy, &mpcor, &mpz, p);
__mp_dbl (&mpw, &w, p);
__mp_dbl (&mpz, &z, p);
if (w == z)
return w;
else
{
p = 32;
__dbl_mp (x, &mpx, p);
__mpexp (&mpx, &mpy, p);
__mp_dbl (&mpy, &res, p);
return res;
}
#else
return (double) __ieee754_expl((long double)x);
#endif
}
/*Converting from double precision to Multi-precision and calculating e^x */
double
SECTION
__slowexp(double x) {
double w,z,res,eps=3.0e-26;
#if 0
double y;
#endif
int p;
#if 0
int orig,i;
#endif
mp_no mpx, mpy, mpz,mpw,mpeps,mpcor;
p=6;
__dbl_mp(x,&mpx,p); /* Convert a double precision number x */
/* into a multiple precision number mpx with prec. p. */
__mpexp(&mpx, &mpy, p); /* Multi-Precision exponential function */
__dbl_mp(eps,&mpeps,p);
__mul(&mpeps,&mpy,&mpcor,p);
__add(&mpy,&mpcor,&mpw,p);
__sub(&mpy,&mpcor,&mpz,p);
__mp_dbl(&mpw, &w, p);
__mp_dbl(&mpz, &z, p);
if (w == z) return w;
else { /* if calculating is not exactly */
p = 32;
__dbl_mp(x,&mpx,p);
__mpexp(&mpx, &mpy, p);
__mp_dbl(&mpy, &res, p);
return res;
}
}
void __mplog(mp_no *x, mp_no *y, int p) {
#include "mplog.h"
int i,m;
#if 0
int j,k,m1,m2,n;
double a,b;
#endif
static const int mp[33] = {0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,
4,4,4,4,4,4,4,4,4,4,4,4,4,4};
mp_no mpone = {0,{0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0}};
mp_no mpt1,mpt2;
/* Choose m and initiate mpone */
m = mp[p]; mpone.e = 1; mpone.d[0]=mpone.d[1]=ONE;
/* Perform m newton iterations to solve for y: exp(y)-x=0. */
/* The iterations formula is: y(n+1)=y(n)+(x*exp(-y(n))-1). */
__cpy(y,&mpt1,p);
for (i=0; i<m; i++) {
mpt1.d[0]=-mpt1.d[0];
__mpexp(&mpt1,&mpt2,p);
__mul(x,&mpt2,&mpt1,p);
__sub(&mpt1,&mpone,&mpt2,p);
__add(y,&mpt2,&mpt1,p);
__cpy(&mpt1,y,p);
}
return;
}
/*Converting from double precision to Multi-precision and calculating e^x */
double __slowexp(double x) {
#ifdef NO_LONG_DOUBLE
double w,z,res,eps=3.0e-26;
int p;
mp_no mpx, mpy, mpz,mpw,mpeps,mpcor;
p=6;
__dbl_mp(x,&mpx,p); /* Convert a double precision number x */
/* into a multiple precision number mpx with prec. p. */
__mpexp(&mpx, &mpy, p); /* Multi-Precision exponential function */
__dbl_mp(eps,&mpeps,p);
__mul(&mpeps,&mpy,&mpcor,p);
__add(&mpy,&mpcor,&mpw,p);
__sub(&mpy,&mpcor,&mpz,p);
__mp_dbl(&mpw, &w, p);
__mp_dbl(&mpz, &z, p);
if (w == z) return w;
else { /* if calculating is not exactly */
p = 32;
__dbl_mp(x,&mpx,p);
__mpexp(&mpx, &mpy, p);
__mp_dbl(&mpy, &res, p);
return res;
}
#else
return (double) __ieee754_expl((long double)x);
#endif
}
/* y=0 is not permitted if x<=0. No error messages are given. */
void __mpatan2(mp_no *y, mp_no *x, mp_no *z, int p) {
static const double ZERO = 0.0, ONE = 1.0;
mp_no mpone = {0,{0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0}};
mp_no mpt1,mpt2,mpt3;
if (X[0] <= ZERO) {
mpone.e = 1; mpone.d[0] = mpone.d[1] = ONE;
__dvd(x,y,&mpt1,p); __mul(&mpt1,&mpt1,&mpt2,p);
if (mpt1.d[0] != ZERO) mpt1.d[0] = ONE;
__add(&mpt2,&mpone,&mpt3,p); __mpsqrt(&mpt3,&mpt2,p);
__add(&mpt1,&mpt2,&mpt3,p); mpt3.d[0]=Y[0];
__mpatan(&mpt3,&mpt1,p); __add(&mpt1,&mpt1,z,p);
}
else
{ __dvd(y,x,&mpt1,p);
__mpatan(&mpt1,z,p);
}
return;
}
/* Final stages. Compute atan(x) by multiple precision arithmetic */
static double
atanMp (double x, const int pr[])
{
mp_no mpx, mpy, mpy2, mperr, mpt1, mpy1;
double y1, y2;
int i, p;
for (i = 0; i < M; i++)
{
p = pr[i];
__dbl_mp (x, &mpx, p);
__mpatan (&mpx, &mpy, p);
__dbl_mp (u9[i].d, &mpt1, p);
__mul (&mpy, &mpt1, &mperr, p);
__add (&mpy, &mperr, &mpy1, p);
__sub (&mpy, &mperr, &mpy2, p);
__mp_dbl (&mpy1, &y1, p);
__mp_dbl (&mpy2, &y2, p);
if (y1 == y2)
{
LIBC_PROBE (slowatan, 3, &p, &x, &y1);
return y1;
}
}
LIBC_PROBE (slowatan_inexact, 3, &p, &x, &y1);
return y1; /*if impossible to do exact computing */
}
void __dvd(const mp_no *x, const mp_no *y, mp_no *z, int p) {
mp_no w;
if (X[0] == ZERO) Z[0] = ZERO;
else {__inv(y,&w,p); __mul(x,&w,z,p);}
return;
}
void __c32(mp_no *x, mp_no *y, mp_no *z, int p) {
static const mp_no mpt={1,{1.0,2.0}}, one={1,{1.0,1.0}};
mp_no u,t,t1,t2,c,s;
int i;
__cpy(x,&u,p);
u.e=u.e-1;
cc32(&u,&c,p);
ss32(&u,&s,p);
for (i=0;i<24;i++) {
__mul(&c,&s,&t,p);
__sub(&s,&t,&t1,p);
__add(&t1,&t1,&s,p);
__sub(&mpt,&c,&t1,p);
__mul(&t1,&c,&t2,p);
__add(&t2,&t2,&c,p);
}
__sub(&one,&c,y,p);
__cpy(&s,z,p);
}
static void
SECTION
ss32(mp_no *x, mp_no *y, int p) {
int i;
double a;
mp_no mpt1,x2,gor,sum ,mpk={1,{1.0}};
for (i=1;i<=p;i++) mpk.d[i]=0;
__sqr(x,&x2,p);
__cpy(&oofac27,&gor,p);
__cpy(&gor,&sum,p);
for (a=27.0;a>1.0;a-=2.0) {
mpk.d[1]=a*(a-1.0);
__mul(&gor,&mpk,&mpt1,p);
__cpy(&mpt1,&gor,p);
__mul(&x2,&sum,&mpt1,p);
__sub(&gor,&mpt1,&sum,p);
}
__mul(x,&sum,y,p);
}
void
SECTION
__c32(mp_no *x, mp_no *y, mp_no *z, int p) {
mp_no u,t,t1,t2,c,s;
int i;
__cpy(x,&u,p);
u.e=u.e-1;
cc32(&u,&c,p);
ss32(&u,&s,p);
for (i=0;i<24;i++) {
__mul(&c,&s,&t,p);
__sub(&s,&t,&t1,p);
__add(&t1,&t1,&s,p);
__sub(&mptwo,&c,&t1,p);
__mul(&t1,&c,&t2,p);
__add(&t2,&t2,&c,p);
}
__sub(&mpone,&c,y,p);
__cpy(&s,z,p);
}
/* Divide *X by *Y and store result in *Z. X and Y may overlap but not X and Z
or Y and Z. Relative error bound:
- For P = 2: 2.001 * R ^ (1 - P)
- For P = 3: 2.063 * R ^ (1 - P)
- For P > 3: 3.001 * R ^ (1 - P)
*X = 0 is not permissible. */
void
SECTION
__dvd (const mp_no *x, const mp_no *y, mp_no *z, int p)
{
mp_no w;
if (X[0] == 0)
Z[0] = 0;
else
{
__inv (y, &w, p);
__mul (x, &w, z, p);
}
}
/* Treat the Denormalized case */
static double
SECTION
normalized (double ax, double ay, double y, double z)
{
int p;
mp_no mpx, mpy, mpz, mperr, mpz2, mpt1;
p = 6;
__dbl_mp (ax, &mpx, p);
__dbl_mp (ay, &mpy, p);
__dvd (&mpy, &mpx, &mpz, p);
__dbl_mp (ue.d, &mpt1, p);
__mul (&mpz, &mpt1, &mperr, p);
__sub (&mpz, &mperr, &mpz2, p);
__mp_dbl (&mpz2, &z, p);
return signArctan2 (y, z);
}
/*Converting from double precision to Multi-precision and calculating e^x */
double
SECTION
__slowexp (double x)
{
#ifndef USE_LONG_DOUBLE_FOR_MP
double w, z, res, eps = 3.0e-26;
int p;
mp_no mpx, mpy, mpz, mpw, mpeps, mpcor;
/* Use the multiple precision __MPEXP function to compute the exponential
First at 144 bits and if it is not accurate enough, at 768 bits. */
p = 6;
__dbl_mp (x, &mpx, p);
__mpexp (&mpx, &mpy, p);
__dbl_mp (eps, &mpeps, p);
__mul (&mpeps, &mpy, &mpcor, p);
__add (&mpy, &mpcor, &mpw, p);
__sub (&mpy, &mpcor, &mpz, p);
__mp_dbl (&mpw, &w, p);
__mp_dbl (&mpz, &z, p);
if (w == z)
{
/* Track how often we get to the slow exp code plus
its input/output values. */
LIBC_PROBE (slowexp_p6, 2, &x, &w);
return w;
}
else
{
p = 32;
__dbl_mp (x, &mpx, p);
__mpexp (&mpx, &mpy, p);
__mp_dbl (&mpy, &res, p);
/* Track how often we get to the uber-slow exp code plus
its input/output values. */
LIBC_PROBE (slowexp_p32, 2, &x, &res);
return res;
}
#else
return (double) __ieee754_expl((long double)x);
#endif
}
/*Converting from double precision to Multi-precision and calculating e^x */
double __slowexp(double x) {
double w,z,res,eps=3.0e-26;
#if 0
double y;
#endif
int p;
#if 0
int orig,i;
#endif
mp_no mpx, mpy, mpz,mpw,mpeps,mpcor;
p=6;
__dbl_mp(x,&mpx,p); /* Convert a double precision number x */
/* into a multiple precision number mpx with prec. p. */
__mpexp(&mpx, &mpy, p); /* Multi-Precision exponential function */
__dbl_mp(eps,&mpeps,p);
__mul(&mpeps,&mpy,&mpcor,p);
__add(&mpy,&mpcor,&mpw,p);
__sub(&mpy,&mpcor,&mpz,p);
__mp_dbl(&mpw, &w, p);
__mp_dbl(&mpz, &z, p);
if (w == z) {
/* Track how often we get to the slow exp code plus
its input/output values. */
LIBC_PROBE (slowexp_p6, 2, &x, &w);
return w;
}
else { /* if calculating is not exactly */
p = 32;
__dbl_mp(x,&mpx,p);
__mpexp(&mpx, &mpy, p);
__mp_dbl(&mpy, &res, p);
/* Track how often we get to the uber-slow exp code plus
its input/output values. */
LIBC_PROBE (slowexp_p32, 2, &x, &res);
return res;
}
}
void __mpexp(mp_no *x, mp_no *y, int p) {
int i,j,k,m,m1,m2,n;
Double a,b;
static const int np[33] = {0,0,0,0,3,3,4,4,5,4,4,5,5,5,6,6,6,6,6,6,
6,6,6,6,7,7,7,7,8,8,8,8,8};
static const int m1p[33]= {0,0,0,0,17,23,23,28,27,38,42,39,43,47,43,47,50,54,
57,60,64,67,71,74,68,71,74,77,70,73,76,78,81};
static const int m1np[7][18] = {
{ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{ 0, 0, 0, 0,36,48,60,72, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{ 0, 0, 0, 0,24,32,40,48,56,64,72, 0, 0, 0, 0, 0, 0, 0},
{ 0, 0, 0, 0,17,23,29,35,41,47,53,59,65, 0, 0, 0, 0, 0},
{ 0, 0, 0, 0, 0, 0,23,28,33,38,42,47,52,57,62,66, 0, 0},
{ 0, 0, 0, 0, 0, 0, 0, 0,27, 0, 0,39,43,47,51,55,59,63},
{ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,43,47,50,54}};
mp_no mpone = {0,{0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0}};
mp_no mpk = {0,{0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0}};
mp_no mps,mpak,mpt1,mpt2;
/* Choose m,n and compute a=2**(-m) */
n = np[p]; m1 = m1p[p]; a = twomm1[p].d();
for (i=0; i<EX; i++) a *= RADIXI;
for ( ; i>EX; i--) a *= RADIX;
b = X[1]*RADIXI; m2 = 24*EX;
for (; b<HALF; m2--) { a *= TWO; b *= TWO; }
if (b == HALF) {
for (i=2; i<=p; i++) { if (X[i]!=ZERO) break; }
if (i==p+1) { m2--; a *= TWO; }
}
if ((m=m1+m2) <= 0) {
m=0; a=ONE;
for (i=n-1; i>0; i--,n--) { if (m1np[i][p]+m2>0) break; }
}
/* Compute s=x*2**(-m). Put result in mps */
__dbl_mp(a,&mpt1,p);
__mul(x,&mpt1,&mps,p);
/* Evaluate the polynomial. Put result in mpt2 */
mpone.e=1; mpone.d(0)=ONE; mpone.d(1)=ONE;
mpk.e = 1; mpk.d(0) = ONE; mpk.d(1)=nn[n].d();
__dvd(&mps,&mpk,&mpt1,p);
__add(&mpone,&mpt1,&mpak,p);
for (k=n-1; k>1; k--) {
__mul(&mps,&mpak,&mpt1,p);
mpk.d(1)=nn[k].d();
__dvd(&mpt1,&mpk,&mpt2,p);
__add(&mpone,&mpt2,&mpak,p);
}
__mul(&mps,&mpak,&mpt1,p);
__add(&mpone,&mpt1,&mpt2,p);
/* Raise polynomial value to the power of 2**m. Put result in y */
for (k=0,j=0; k<m; ) {
__mul(&mpt2,&mpt2,&mpt1,p); k++;
if (k==m) { j=1; break; }
__mul(&mpt1,&mpt1,&mpt2,p); k++;
}
if (j) __cpy(&mpt1,y,p);
else __cpy(&mpt2,y,p);
return;
}
void
SECTION
__mpatan(mp_no *x, mp_no *y, int p) {
int i,m,n;
double dx;
mp_no
mpone = {0,{0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0}},
mptwo = {0,{0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0}},
mptwoim1 = {0,{0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0}};
mp_no mps,mpsm,mpt,mpt1,mpt2,mpt3;
/* Choose m and initiate mpone, mptwo & mptwoim1 */
if (EX>0) m=7;
else if (EX<0) m=0;
else {
__mp_dbl(x,&dx,p); dx=ABS(dx);
for (m=6; m>0; m--)
{if (dx>__atan_xm[m].d) break;}
}
mpone.e = mptwo.e = mptwoim1.e = 1;
mpone.d[0] = mpone.d[1] = mptwo.d[0] = mptwoim1.d[0] = ONE;
mptwo.d[1] = TWO;
/* Reduce x m times */
__mul(x,x,&mpsm,p);
if (m==0) __cpy(x,&mps,p);
else {
for (i=0; i<m; i++) {
__add(&mpone,&mpsm,&mpt1,p);
__mpsqrt(&mpt1,&mpt2,p);
__add(&mpt2,&mpt2,&mpt1,p);
__add(&mptwo,&mpsm,&mpt2,p);
__add(&mpt1,&mpt2,&mpt3,p);
__dvd(&mpsm,&mpt3,&mpt1,p);
__cpy(&mpt1,&mpsm,p);
}
__mpsqrt(&mpsm,&mps,p); mps.d[0] = X[0];
}
/* Evaluate a truncated power series for Atan(s) */
n=__atan_np[p]; mptwoim1.d[1] = __atan_twonm1[p].d;
__dvd(&mpsm,&mptwoim1,&mpt,p);
for (i=n-1; i>1; i--) {
mptwoim1.d[1] -= TWO;
__dvd(&mpsm,&mptwoim1,&mpt1,p);
__mul(&mpsm,&mpt,&mpt2,p);
__sub(&mpt1,&mpt2,&mpt,p);
}
__mul(&mps,&mpt,&mpt1,p);
__sub(&mps,&mpt1,&mpt,p);
/* Compute Atan(x) */
mptwoim1.d[1] = __atan_twom[m].d;
__mul(&mptwoim1,&mpt,y,p);
return;
}
slong _nmod_poly_xgcd_hgcd(mp_ptr G, mp_ptr S, mp_ptr T,
mp_srcptr A, slong lenA, mp_srcptr B, slong lenB,
nmod_t mod)
{
const slong cutoff = FLINT_BIT_COUNT(mod.n) <= 8 ?
NMOD_POLY_SMALL_GCD_CUTOFF : NMOD_POLY_GCD_CUTOFF;
slong lenG, lenS, lenT;
if (lenB == 1)
{
G[0] = B[0];
T[0] = 1;
lenG = 1;
lenS = 0;
lenT = 1;
}
else
{
mp_ptr q = _nmod_vec_init(lenA + lenB);
mp_ptr r = q + lenA;
slong lenq, lenr;
__divrem(q, lenq, r, lenr, A, lenA, B, lenB);
if (lenr == 0)
{
__set(G, lenG, B, lenB);
T[0] = 1;
lenS = 0;
lenT = 1;
}
else
{
mp_ptr h, j, v, w, R[4], X;
slong lenh, lenj, lenv, lenw, lenR[4];
int sgnR;
lenh = lenj = lenB;
lenv = lenw = lenA + lenB - 2;
lenR[0] = lenR[1] = lenR[2] = lenR[3] = (lenB + 1) / 2;
X = _nmod_vec_init(2 * lenh + 2 * lenv + 4 * lenR[0]);
h = X;
j = h + lenh;
v = j + lenj;
w = v + lenv;
R[0] = w + lenw;
R[1] = R[0] + lenR[0];
R[2] = R[1] + lenR[1];
R[3] = R[2] + lenR[2];
sgnR = _nmod_poly_hgcd(R, lenR, h, &lenh, j, &lenj, B, lenB, r, lenr, mod);
if (sgnR > 0)
{
_nmod_vec_neg(S, R[1], lenR[1], mod);
_nmod_vec_set(T, R[0], lenR[0]);
}
else
{
_nmod_vec_set(S, R[1], lenR[1]);
_nmod_vec_neg(T, R[0], lenR[0], mod);
}
lenS = lenR[1];
lenT = lenR[0];
while (lenj != 0)
{
__divrem(q, lenq, r, lenr, h, lenh, j, lenj);
__mul(v, lenv, q, lenq, T, lenT);
{
slong l;
_nmod_vec_swap(S, T, FLINT_MAX(lenS, lenT));
l = lenS; lenS = lenT; lenT = l;
}
__sub(T, lenT, T, lenT, v, lenv);
if (lenr == 0)
{
__set(G, lenG, j, lenj);
goto cofactor;
}
if (lenj < cutoff)
{
mp_ptr u0 = R[0], u1 = R[1];
slong lenu0 = lenr - 1, lenu1 = lenj - 1;
lenG = _nmod_poly_xgcd_euclidean(G, u0, u1, j, lenj, r, lenr, mod);
MPN_NORM(u0, lenu0);
MPN_NORM(u1, lenu1);
__mul(v, lenv, S, lenS, u0, lenu0);
__mul(w, lenw, T, lenT, u1, lenu1);
__add(S, lenS, v, lenv, w, lenw);
goto cofactor;
}
sgnR = _nmod_poly_hgcd(R, lenR, h, &lenh, j, &lenj, j,lenj, r, lenr, mod);
__mul(v, lenv, R[1], lenR[1], T, lenT);
__mul(w, lenw, R[2], lenR[2], S, lenS);
__mul(q, lenq, S, lenS, R[3], lenR[3]);
if (sgnR > 0)
__sub(S, lenS, q, lenq, v, lenv);
else
__sub(S, lenS, v, lenv, q, lenq);
__mul(q, lenq, T, lenT, R[0], lenR[0]);
if (sgnR > WORD(0))
__sub(T, lenT, q, lenq, w, lenw);
else
__sub(T, lenT, w, lenw, q, lenq);
}
__set(G, lenG, h, lenh);
cofactor:
__mul(v, lenv, S, lenS, A, lenA);
__sub(w, lenw, G, lenG, v, lenv);
__div(T, lenT, w, lenw, B, lenB);
_nmod_vec_clear(X);
}
_nmod_vec_clear(q);
}
flint_mpn_zero(S + lenS, lenB - 1 - lenS);
flint_mpn_zero(T + lenT, lenA - 1 - lenT);
return lenG;
}
/* Multi-Precision exponential function subroutine (for p >= 4,
2**(-55) <= abs(x) <= 1024). */
void
SECTION
__mpexp (mp_no *x, mp_no *y, int p)
{
int i, j, k, m, m1, m2, n;
mantissa_t b;
static const int np[33] =
{
0, 0, 0, 0, 3, 3, 4, 4, 5, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 6,
6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8
};
static const int m1p[33] =
{
0, 0, 0, 0,
17, 23, 23, 28,
27, 38, 42, 39,
43, 47, 43, 47,
50, 54, 57, 60,
64, 67, 71, 74,
68, 71, 74, 77,
70, 73, 76, 78,
81
};
static const int m1np[7][18] =
{
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{0, 0, 0, 0, 36, 48, 60, 72, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{0, 0, 0, 0, 24, 32, 40, 48, 56, 64, 72, 0, 0, 0, 0, 0, 0, 0},
{0, 0, 0, 0, 17, 23, 29, 35, 41, 47, 53, 59, 65, 0, 0, 0, 0, 0},
{0, 0, 0, 0, 0, 0, 23, 28, 33, 38, 42, 47, 52, 57, 62, 66, 0, 0},
{0, 0, 0, 0, 0, 0, 0, 0, 27, 0, 0, 39, 43, 47, 51, 55, 59, 63},
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 43, 47, 50, 54}
};
mp_no mps, mpk, mpt1, mpt2;
/* Choose m,n and compute a=2**(-m). */
n = np[p];
m1 = m1p[p];
b = X[1];
m2 = 24 * EX;
for (; b < HALFRAD; m2--)
b *= 2;
if (b == HALFRAD)
{
for (i = 2; i <= p; i++)
{
if (X[i] != 0)
break;
}
if (i == p + 1)
m2--;
}
m = m1 + m2;
if (__glibc_unlikely (m <= 0))
{
/* The m1np array which is used to determine if we can reduce the
polynomial expansion iterations, has only 18 elements. Besides,
numbers smaller than those required by p >= 18 should not come here
at all since the fast phase of exp returns 1.0 for anything less
than 2^-55. */
assert (p < 18);
m = 0;
for (i = n - 1; i > 0; i--, n--)
if (m1np[i][p] + m2 > 0)
break;
}
/* Compute s=x*2**(-m). Put result in mps. This is the range-reduced input
that we will use to compute e^s. For the final result, simply raise it
to 2^m. */
__pow_mp (-m, &mpt1, p);
__mul (x, &mpt1, &mps, p);
/* Compute the Taylor series for e^s:
1 + x/1! + x^2/2! + x^3/3! ...
for N iterations. We compute this as:
e^x = 1 + (x * n!/1! + x^2 * n!/2! + x^3 * n!/3!) / n!
= 1 + (x * (n!/1! + x * (n!/2! + x * (n!/3! + x ...)))) / n!
k! is computed on the fly as KF and at the end of the polynomial loop, KF
is n!, which can be used directly. */
__cpy (&mps, &mpt2, p);
double kf = 1.0;
/* Evaluate the rest. The result will be in mpt2. */
for (k = n - 1; k > 0; k--)
{
/* n! / k! = n * (n - 1) ... * (n - k + 1) */
kf *= k + 1;
__dbl_mp (kf, &mpk, p);
__add (&mpt2, &mpk, &mpt1, p);
__mul (&mps, &mpt1, &mpt2, p);
}
__dbl_mp (kf, &mpk, p);
__dvd (&mpt2, &mpk, &mpt1, p);
__add (&__mpone, &mpt1, &mpt2, p);
/* Raise polynomial value to the power of 2**m. Put result in y. */
for (k = 0, j = 0; k < m;)
{
__sqr (&mpt2, &mpt1, p);
k++;
if (k == m)
{
j = 1;
break;
}
__sqr (&mpt1, &mpt2, p);
k++;
}
if (j)
__cpy (&mpt1, y, p);
else
__cpy (&mpt2, y, p);
return;
}
