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glibc/sysdeps/ieee754/dbl-64/s_sin.c
Siddhesh Poyarekar 09544cbcd6 Consolidate multiple precision sin/cos functions
2013-10-08 11:50:17 +05:30

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/*
* IBM Accurate Mathematical Library
* written by International Business Machines Corp.
* Copyright (C) 2001-2013 Free Software Foundation, Inc.
*
* 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>
#include <fenv.h>
/* Helper macros to compute sin of the input values. */
#define POLYNOMIAL2(xx) ((((s5.x * (xx) + s4.x) * (xx) + s3.x) * (xx) + s2.x) \
* (xx))
#define POLYNOMIAL(xx) (POLYNOMIAL2 (xx) + s1.x)
/* The computed polynomial is a variation of the Taylor series expansion for
sin(a):
a - a^3/3! + a^5/5! - a^7/7! + a^9/9! + (1 - a^2) * da / 2
The constants s1, s2, s3, etc. are pre-computed values of 1/3!, 1/5! and so
on. The result is returned to LHS and correction in COR. */
#define TAYLOR_SINCOS(xx, a, da, cor) \
({ \
double t = ((POLYNOMIAL (xx) * (a) - 0.5 * (da)) * (xx) + (da)); \
double res = (a) + t; \
(cor) = ((a) - res) + t; \
res; \
})
/* This is again a variation of the Taylor series expansion with the term
x^3/3! expanded into the following for better accuracy:
bb * x ^ 3 + 3 * aa * x * x1 * x2 + aa * x1 ^ 3 + aa * x2 ^ 3
The correction term is dx and bb + aa = -1/3!
*/
#define TAYLOR_SLOW(x0, dx, cor) \
({ \
static const double th2_36 = 206158430208.0; /* 1.5*2**37 */ \
double xx = (x0) * (x0); \
double x1 = ((x0) + th2_36) - th2_36; \
double y = aa.x * x1 * x1 * x1; \
double r = (x0) + y; \
double x2 = ((x0) - x1) + (dx); \
double t = (((POLYNOMIAL2 (xx) + bb.x) * xx + 3.0 * aa.x * x1 * x2) \
* (x0) + aa.x * x2 * x2 * x2 + (dx)); \
t = (((x0) - r) + y) + t; \
double res = r + t; \
(cor) = (r - res) + t; \
res; \
})
#define SINCOS_TABLE_LOOKUP(u, sn, ssn, cs, ccs) \
({ \
int4 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]; \
})
#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, bool reduce_range);
double __mpcos (double x, double dx, bool reduce_range);
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);
/* Reduce range of X and compute sin of a + da. K is the amount by which to
rotate the quadrants. This allows us to use the same routine to compute cos
by simply rotating the quadrants by 1. */
static inline double
__always_inline
reduce_and_compute (double x, double a, double da, unsigned int k)
{
double retval = 0;
unsigned int n = __branred (x, &a, &da);
k = (n + k) % 4;
switch (k)
{
case 0:
if (a * a < 0.01588)
retval = bsloww (a, da, x, n);
else
retval = bsloww1 (a, da, x, n);
break;
case 2:
if (a * a < 0.01588)
retval = bsloww (-a, -da, x, n);
else
retval = bsloww1 (-a, -da, x, n);
break;
case 1:
case 3:
retval = bsloww2 (a, da, x, n);
break;
}
return retval;
}
/*******************************************************************/
/* 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;
mynumber u, v;
int4 k, m, n;
double retval = 0;
SET_RESTORE_ROUND_53BIT (FE_TONEAREST);
u.x = x;
m = u.i[HIGH_HALF];
k = 0x7fffffff & m; /* no sign */
if (k < 0x3e500000) /* if x->0 =>sin(x)=x */
retval = x;
/*---------------------------- 2^-26 < |x|< 0.25 ----------------------*/
else if (k < 0x3fd00000)
{
xx = x * x;
/* Taylor series. */
t = POLYNOMIAL (xx) * (xx * x);
res = x + t;
cor = (x - res) + t;
retval = (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));
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
if (m <= 0)
{
sn = -sn;
ssn = -ssn;
}
cor = (ssn + s * ccs - sn * c) + cs * s;
res = sn + cor;
cor = (sn - res) + cor;
retval = (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));
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
cor = (ccs - s * ssn - cs * c) - sn * s;
res = cs + cor;
cor = (cs - res) + cor;
retval = (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. */
res = TAYLOR_SINCOS (xx, a, da, cor);
cor = (cor > 0) ? 1.02 * cor + eps : 1.02 * cor - eps;
retval = (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));
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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;
retval = ((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;
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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;
retval = ((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. */
res = TAYLOR_SINCOS (xx, a, da, cor);
cor = (cor > 0) ? 1.02 * cor + eps : 1.02 * cor - eps;
retval = (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));
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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;
retval = ((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;
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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;
retval = ((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)
retval = reduce_and_compute (x, a, da, 0);
/*--------------------- |x| > 2^1024 ----------------------------------*/
else
{
if (k == 0x7ff00000 && u.i[LOW_HALF] == 0)
__set_errno (EDOM);
retval = x / x;
}
return retval;
}
/*******************************************************************/
/* 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;
double retval = 0;
SET_RESTORE_ROUND_53BIT (FE_TONEAREST);
u.x = x;
m = u.i[HIGH_HALF];
k = 0x7fffffff & m;
/* |x|<2^-27 => cos(x)=1 */
if (k < 0x3e400000)
retval = 1.0;
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));
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
cor = (ccs - s * ssn - cs * c) - sn * s;
res = cs + cor;
cor = (cs - res) + cor;
retval = (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)
{
res = TAYLOR_SINCOS (xx, a, da, cor);
cor = (cor > 0) ? 1.02 * cor + 1.0e-31 : 1.02 * cor - 1.0e-31;
retval = (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));
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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;
retval = ((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)
{
res = TAYLOR_SINCOS (xx, a, da, cor);
cor = (cor > 0) ? 1.02 * cor + eps : 1.02 * cor - eps;
retval = (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));
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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;
retval = ((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;
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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;
retval = ((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)
{
res = TAYLOR_SINCOS (xx, a, da, cor);
cor = (cor > 0) ? 1.02 * cor + eps : 1.02 * cor - eps;
retval = (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));
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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;
retval = ((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;
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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;
retval = ((res == res + cor) ? ((n) ? -res : res)
: bsloww2 (a, da, x, n));
break;
}
} /* else if (k < 0x42F00000 ) */
/* 281474976710656 <|x| <2^1024 */
else if (k < 0x7ff00000)
retval = reduce_and_compute (x, a, da, 1);
else
{
if (k == 0x7ff00000 && u.i[LOW_HALF] == 0)
__set_errno (EDOM);
retval = x / x; /* |x| > 2^1024 */
}
return retval;
}
/************************************************************************/
/* 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)
{
double res, cor, w[2];
res = TAYLOR_SLOW (x, 0, cor);
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, false) : -__mpsin (-x, 0, false);
}
}
/*******************************************************************************/
/* 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;
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));
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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, false) : -__mpsin (-x, 0, false);
}
}
/**************************************************************************/
/* 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;
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));
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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, false) : -__mpsin (-x, 0, false);
}
}
/***************************************************************************/
/* 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)
{
double y, t, res, cor, w[2], a, da, xn;
union
{
int4 i[2];
double x;
} v;
int4 n;
res = TAYLOR_SLOW (x, dx, cor);
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);
if (w[1] > 0)
cor = 1.000000001 * w[1] + ABS (orig) * 1.1e-30;
else
cor = 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);
if (w[1] > 0)
cor = 1.000000001 * w[1] + ABS (orig) * 1.1e-40;
else
cor = 1.000000001 * w[1] - ABS (orig) * 1.1e-40;
if (w[0] == w[0] + cor)
return (a > 0) ? w[0] : -w[0];
else
return __mpsin (orig, 0, true);
}
}
}
/***************************************************************************/
/* 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;
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));
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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;
if (cor > 0)
cor = 1.0005 * cor + 3.1e-30 * ABS (orig);
else
cor = 1.0005 * cor - 3.1e-30 * ABS (orig);
if (res == res + cor)
return (x > 0) ? res : -res;
else
{
__dubsin (ABS (x), dx, w);
if (w[1] > 0)
cor = 1.000000005 * w[1] + 1.1e-30 * ABS (orig);
else
cor = 1.000000005 * w[1] - 1.1e-30 * ABS (orig);
if (w[0] == w[0] + cor)
return (x > 0) ? w[0] : -w[0];
else
return __mpsin (orig, 0, true);
}
}
/***************************************************************************/
/* 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;
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));
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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;
if (cor > 0)
cor = 1.0005 * cor + 3.1e-30 * ABS (orig);
else
cor = 1.0005 * cor - 3.1e-30 * ABS (orig);
if (res == res + cor)
return (n & 2) ? -res : res;
else
{
__docos (ABS (x), dx, w);
if (w[1] > 0)
cor = 1.000000005 * w[1] + 1.1e-30 * ABS (orig);
else
cor = 1.000000005 * w[1] - 1.1e-30 * ABS (orig);
if (w[0] == w[0] + cor)
return (n & 2) ? -w[0] : w[0];
else
return __mpsin (orig, 0, true);
}
}
/***************************************************************************/
/* 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)
{
double res, cor, w[2];
res = TAYLOR_SLOW (x, dx, cor);
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);
if (w[1] > 0)
cor = 1.000000001 * w[1] + 1.1e-24;
else
cor = 1.000000001 * w[1] - 1.1e-24;
if (w[0] == w[0] + cor)
return (x > 0) ? w[0] : -w[0];
else
return (n & 1) ? __mpcos (orig, 0, true) : __mpsin (orig, 0, true);
}
}
/***************************************************************************/
/* 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;
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));
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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);
if (w[1] > 0)
cor = 1.000000005 * w[1] + 1.1e-24;
else
cor = 1.000000005 * w[1] - 1.1e-24;
if (w[0] == w[0] + cor)
return (x > 0) ? w[0] : -w[0];
else
return (n & 1) ? __mpcos (orig, 0, true) : __mpsin (orig, 0, true);
}
}
/***************************************************************************/
/* 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;
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));
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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);
if (w[1] > 0)
cor = 1.000000005 * w[1] + 1.1e-24;
else
cor = 1.000000005 * w[1] - 1.1e-24;
if (w[0] == w[0] + cor)
return (n & 2) ? -w[0] : w[0];
else
return (n & 1) ? __mpsin (orig, 0, true) : __mpcos (orig, 0, true);
}
}
/************************************************************************/
/* 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;
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));
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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, false);
}
}
/***************************************************************************/
/* 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)
{
double y, t, res, cor, w[2], a, da, xn;
union
{
int4 i[2];
double x;
} v;
int4 n;
/* Taylor series */
t = TAYLOR_SLOW (x, dx, cor);
if (cor > 0)
cor = 1.0005 * cor + ABS (orig) * 3.1e-30;
else
cor = 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);
if (w[1] > 0)
cor = 1.000000001 * w[1] + ABS (orig) * 1.1e-30;
else
cor = 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);
if (w[1] > 0)
cor = 1.000000001 * w[1] + ABS (orig) * 1.1e-40;
else
cor = 1.000000001 * w[1] - ABS (orig) * 1.1e-40;
if (w[0] == w[0] + cor)
return (a > 0) ? w[0] : -w[0];
else
return __mpcos (orig, 0, true);
}
}
}
/***************************************************************************/
/* 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;
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));
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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;
if (cor > 0)
cor = 1.0005 * cor + 3.1e-30 * ABS (orig);
else
cor = 1.0005 * cor - 3.1e-30 * ABS (orig);
if (res == res + cor)
return (x > 0) ? res : -res;
else
{
__dubsin (ABS (x), dx, w);
if (w[1] > 0)
cor = 1.000000005 * w[1] + 1.1e-30 * ABS (orig);
else
cor = 1.000000005 * w[1] - 1.1e-30 * ABS (orig);
if (w[0] == w[0] + cor)
return (x > 0) ? w[0] : -w[0];
else
return __mpcos (orig, 0, true);
}
}
/***************************************************************************/
/* 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;
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));
SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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;
if (cor > 0)
cor = 1.0005 * cor + 3.1e-30 * ABS (orig);
else
cor = 1.0005 * cor - 3.1e-30 * ABS (orig);
if (res == res + cor)
return (n) ? -res : res;
else
{
__docos (ABS (x), dx, w);
if (w[1] > 0)
cor = 1.000000005 * w[1] + 1.1e-30 * ABS (orig);
else
cor = 1.000000005 * w[1] - 1.1e-30 * ABS (orig);
if (w[0] == w[0] + cor)
return (n) ? -w[0] : w[0];
else
return __mpcos (orig, 0, true);
}
}
#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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