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trig.c
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trig.c
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/* Copyright JS Foundation and other contributors, http:https://js.foundation
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http:https://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*
* This file is based on work under the following copyright and permission
* notice:
*
* Copyright (C) 1993, 2004 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
*
* @(#)k_rem_pio2.c 1.3 95/01/18
* @(#)e_rem_pio2.c 1.4 95/01/18
* @(#)k_sin.c 1.3 95/01/18
* @(#)k_cos.c 1.3 95/01/18
* @(#)k_tan.c 1.5 04/04/22
* @(#)s_sin.c 1.3 95/01/18
* @(#)s_cos.c 1.3 95/01/18
* @(#)s_tan.c 1.3 95/01/18
*/
#include "jerry-math-internal.h"
#define zero 0.00000000000000000000e+00 /* 0x00000000, 0x00000000 */
#define half 5.00000000000000000000e-01 /* 0x3FE00000, 0x00000000 */
#define one 1.00000000000000000000e+00 /* 0x3FF00000, 0x00000000 */
#define two24 1.67772160000000000000e+07 /* 0x41700000, 0x00000000 */
#define twon24 5.96046447753906250000e-08 /* 0x3E700000, 0x00000000 */
/* __kernel_rem_pio2(x,y,e0,nx,prec)
* double x[],y[]; int e0,nx,prec;
*
* __kernel_rem_pio2 return the last three digits of N with
* y = x - N*pi/2
* so that |y| < pi/2.
*
* The method is to compute the integer (mod 8) and fraction parts of
* (2/pi)*x without doing the full multiplication. In general we
* skip the part of the product that are known to be a huge integer (
* more accurately, = 0 mod 8 ). Thus the number of operations are
* independent of the exponent of the input.
*
* (2/pi) is represented by an array of 24-bit integers in ipio2[].
*
* Input parameters:
* x[] The input value (must be positive) is broken into nx
* pieces of 24-bit integers in double precision format.
* x[i] will be the i-th 24 bit of x. The scaled exponent
* of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
* match x's up to 24 bits.
*
* Example of breaking a double positive z into x[0]+x[1]+x[2]:
* e0 = ilogb(z)-23
* z = scalbn(z,-e0)
* for i = 0,1,2
* x[i] = floor(z)
* z = (z-x[i])*2**24
*
* y[] ouput result in an array of double precision numbers.
* The dimension of y[] is:
* 24-bit precision 1
* 53-bit precision 2
* 64-bit precision 2
* 113-bit precision 3
* The actual value is the sum of them. Thus for 113-bit
* precison, one may have to do something like:
*
* long double t,w,r_head, r_tail;
* t = (long double)y[2] + (long double)y[1];
* w = (long double)y[0];
* r_head = t+w;
* r_tail = w - (r_head - t);
*
* e0 The exponent of x[0]
*
* nx dimension of x[]
*
* prec an integer indicating the precision:
* 0 24 bits (single)
* 1 53 bits (double)
* 2 64 bits (extended)
* 3 113 bits (quad)
*
* External function:
* double scalbn(), floor();
*
* Here is the description of some local variables:
*
* ipio2[] integer array, contains the (24*i)-th to (24*i+23)-th
* bit of 2/pi after binary point. The corresponding
* floating value is
*
* ipio2[i] * 2^(-24(i+1)).
*
* jk jk+1 is the initial number of terms of ipio2[] needed
* in the computation. The recommended value is 2,3,4,
* 6 for single, double, extended,and quad.
*
* jz local integer variable indicating the number of
* terms of ipio2[] used.
*
* jx nx - 1
*
* jv index for pointing to the suitable ipio2[] for the
* computation. In general, we want
* ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
* is an integer. Thus
* e0-3-24*jv >= 0 or (e0-3)/24 >= jv
* Hence jv = max(0,(e0-3)/24).
*
* jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
*
* q[] double array with integral value, representing the
* 24-bits chunk of the product of x and 2/pi.
*
* q0 the corresponding exponent of q[0]. Note that the
* exponent for q[i] would be q0-24*i.
*
* PIo2[] double precision array, obtained by cutting pi/2
* into 24 bits chunks.
*
* f[] ipio2[] in floating point
*
* iq[] integer array by breaking up q[] in 24-bits chunk.
*
* fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
*
* ih integer. If >0 it indicates q[] is >= 0.5, hence
* it also indicates the *sign* of the result.
*/
/*
* Constants:
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/
/* initial value for jk */
static const int init_jk[] = { 2, 3, 4, 6 };
static const double PIo2[] = {
1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
};
/*
* Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
*/
static const int ipio2[] = {
0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, 0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7,
0x246E3A, 0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, 0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C,
0x7026B4, 0x5F7E41, 0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8, 0x97FFDE, 0x05980F, 0xEF2F11,
0x8B5A0A, 0x6D1F6D, 0x367ECF, 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5, 0xF17B3D, 0x0739F7,
0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, 0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, 0x91615E,
0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880, 0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B,
};
static int
__kernel_rem_pio2 (double *x, double *y, int e0, int nx, int prec)
{
int jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
double z, fw, f[20], fq[20], q[20];
/* initialize jk */
jk = init_jk[prec];
jp = jk;
/* determine jx, jv, q0, note that 3 > q0 */
jx = nx - 1;
jv = (e0 - 3) / 24;
if (jv < 0)
{
jv = 0;
}
q0 = e0 - 24 * (jv + 1);
/* set up f[0] to f[jx + jk] where f[jx + jk] = ipio2[jv + jk] */
j = jv - jx;
m = jx + jk;
for (i = 0; i <= m; i++, j++)
{
f[i] = (j < 0) ? zero : (double) ipio2[j];
}
/* compute q[0], q[1], ... q[jk] */
for (i = 0; i <= jk; i++)
{
for (j = 0, fw = 0.0; j <= jx; j++)
{
fw += x[j] * f[jx + i - j];
}
q[i] = fw;
}
jz = jk;
recompute:
/* distill q[] into iq[] reversingly */
for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--)
{
fw = (double) ((int) (twon24 * z));
iq[i] = (int) (z - two24 * fw);
z = q[j - 1] + fw;
}
/* compute n */
z = scalbn (z, q0); /* actual value of z */
z -= 8.0 * floor (z * 0.125); /* trim off integer >= 8 */
n = (int) z;
z -= (double) n;
ih = 0;
if (q0 > 0) /* need iq[jz - 1] to determine n */
{
i = (iq[jz - 1] >> (24 - q0));
n += i;
iq[jz - 1] -= i << (24 - q0);
ih = iq[jz - 1] >> (23 - q0);
}
else if (q0 == 0)
{
ih = iq[jz - 1] >> 23;
}
else if (z >= 0.5)
{
ih = 2;
}
if (ih > 0) /* q > 0.5 */
{
n += 1;
carry = 0;
for (i = 0; i < jz; i++) /* compute 1 - q */
{
j = iq[i];
if (carry == 0)
{
if (j != 0)
{
carry = 1;
iq[i] = 0x1000000 - j;
}
}
else
{
iq[i] = 0xffffff - j;
}
}
if (q0 > 0) /* rare case: chance is 1 in 12 */
{
switch (q0)
{
case 1:
{
iq[jz - 1] &= 0x7fffff;
break;
}
case 2:
{
iq[jz - 1] &= 0x3fffff;
break;
}
}
}
if (ih == 2)
{
z = one - z;
if (carry != 0)
{
z -= scalbn (one, q0);
}
}
}
/* check if recomputation is needed */
if (z == zero)
{
j = 0;
for (i = jz - 1; i >= jk; i--)
{
j |= iq[i];
}
if (j == 0) /* need recomputation */
{
for (k = 1; iq[jk - k] == 0; k++) /* k = no. of terms needed */
{
}
for (i = jz + 1; i <= jz + k; i++) /* add q[jz + 1] to q[jz + k] */
{
f[jx + i] = (double) ipio2[jv + i];
for (j = 0, fw = 0.0; j <= jx; j++)
{
fw += x[j] * f[jx + i - j];
}
q[i] = fw;
}
jz += k;
goto recompute;
}
}
/* chop off zero terms */
if (z == 0.0)
{
jz -= 1;
q0 -= 24;
while (iq[jz] == 0)
{
jz--;
q0 -= 24;
}
}
else
{ /* break z into 24-bit if necessary */
z = scalbn (z, -q0);
if (z >= two24)
{
fw = (double) ((int) (twon24 * z));
iq[jz] = (int) (z - two24 * fw);
jz += 1;
q0 += 24;
iq[jz] = (int) fw;
}
else
{
iq[jz] = (int) z;
}
}
/* convert integer "bit" chunk to floating-point value */
fw = scalbn (one, q0);
for (i = jz; i >= 0; i--)
{
q[i] = fw * (double) iq[i];
fw *= twon24;
}
/* compute PIo2[0, ..., jp] * q[jz, ..., 0] */
for (i = jz; i >= 0; i--)
{
for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++)
{
fw += PIo2[k] * q[i + k];
}
fq[jz - i] = fw;
}
/* compress fq[] into y[] */
switch (prec)
{
case 0:
{
fw = 0.0;
for (i = jz; i >= 0; i--)
{
fw += fq[i];
}
y[0] = (ih == 0) ? fw : -fw;
break;
}
case 1:
case 2:
{
fw = 0.0;
for (i = jz; i >= 0; i--)
{
fw += fq[i];
}
y[0] = (ih == 0) ? fw : -fw;
fw = fq[0] - fw;
for (i = 1; i <= jz; i++)
{
fw += fq[i];
}
y[1] = (ih == 0) ? fw : -fw;
break;
}
case 3: /* painful */
{
for (i = jz; i > 0; i--)
{
fw = fq[i - 1] + fq[i];
fq[i] += fq[i - 1] - fw;
fq[i - 1] = fw;
}
for (i = jz; i > 1; i--)
{
fw = fq[i - 1] + fq[i];
fq[i] += fq[i - 1] - fw;
fq[i - 1] = fw;
}
for (fw = 0.0, i = jz; i >= 2; i--)
{
fw += fq[i];
}
if (ih == 0)
{
y[0] = fq[0];
y[1] = fq[1];
y[2] = fw;
}
else
{
y[0] = -fq[0];
y[1] = -fq[1];
y[2] = -fw;
}
}
}
return n & 7;
} /* __kernel_rem_pio2 */
/* __ieee754_rem_pio2(x,y)
* return the remainder of x rem pi/2 in y[0]+y[1]
* use __kernel_rem_pio2()
*/
static const int npio2_hw[] = {
0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C, 0x4025FDBB, 0x402921FB,
0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C, 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB,
0x403AB41B, 0x403C463A, 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB, 0x404858EB, 0x404921FB,
};
/*
* invpio2: 53 bits of 2/pi
* pio2_1: first 33 bit of pi/2
* pio2_1t: pi/2 - pio2_1
* pio2_2: second 33 bit of pi/2
* pio2_2t: pi/2 - (pio2_1 + pio2_2)
* pio2_3: third 33 bit of pi/2
* pio2_3t: pi/2 - (pio2_1 + pio2_2 + pio2_3)
*/
#define invpio2 6.36619772367581382433e-01 /* 0x3FE45F30, 0x6DC9C883 */
#define pio2_1 1.57079632673412561417e+00 /* 0x3FF921FB, 0x54400000 */
#define pio2_1t 6.07710050650619224932e-11 /* 0x3DD0B461, 0x1A626331 */
#define pio2_2 6.07710050630396597660e-11 /* 0x3DD0B461, 0x1A600000 */
#define pio2_2t 2.02226624879595063154e-21 /* 0x3BA3198A, 0x2E037073 */
#define pio2_3 2.02226624871116645580e-21 /* 0x3BA3198A, 0x2E000000 */
#define pio2_3t 8.47842766036889956997e-32 /* 0x397B839A, 0x252049C1 */
static int
__ieee754_rem_pio2 (double x, double *y)
{
double_accessor z;
double w, t, r, fn;
double tx[3];
int e0, i, j, nx, n, ix, hx;
hx = __HI (x); /* high word of x */
ix = hx & 0x7fffffff;
if (ix <= 0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */
{
y[0] = x;
y[1] = 0;
return 0;
}
if (ix < 0x4002d97c) /* |x| < 3pi/4, special case with n = +-1 */
{
if (hx > 0)
{
z.dbl = x - pio2_1;
if (ix != 0x3ff921fb) /* 33 + 53 bit pi is good enough */
{
y[0] = z.dbl - pio2_1t;
y[1] = (z.dbl - y[0]) - pio2_1t;
}
else /* near pi/2, use 33 + 33 + 53 bit pi */
{
z.dbl -= pio2_2;
y[0] = z.dbl - pio2_2t;
y[1] = (z.dbl - y[0]) - pio2_2t;
}
return 1;
}
else /* negative x */
{
z.dbl = x + pio2_1;
if (ix != 0x3ff921fb) /* 33 + 53 bit pi is good enough */
{
y[0] = z.dbl + pio2_1t;
y[1] = (z.dbl - y[0]) + pio2_1t;
}
else /* near pi/2, use 33 + 33 + 53 bit pi */
{
z.dbl += pio2_2;
y[0] = z.dbl + pio2_2t;
y[1] = (z.dbl - y[0]) + pio2_2t;
}
return -1;
}
}
if (ix <= 0x413921fb) /* |x| ~<= 2^19 * (pi/2), medium size */
{
t = fabs (x);
n = (int) (t * invpio2 + half);
fn = (double) n;
r = t - fn * pio2_1;
w = fn * pio2_1t; /* 1st round good to 85 bit */
if (n < 32 && ix != npio2_hw[n - 1])
{
y[0] = r - w; /* quick check no cancellation */
}
else
{
j = ix >> 20;
y[0] = r - w;
i = j - (((__HI (y[0])) >> 20) & 0x7ff);
if (i > 16) /* 2nd iteration needed, good to 118 */
{
t = r;
w = fn * pio2_2;
r = t - w;
w = fn * pio2_2t - ((t - r) - w);
y[0] = r - w;
i = j - (((__HI (y[0])) >> 20) & 0x7ff);
if (i > 49) /* 3rd iteration need, 151 bits acc, will cover all possible cases */
{
t = r;
w = fn * pio2_3;
r = t - w;
w = fn * pio2_3t - ((t - r) - w);
y[0] = r - w;
}
}
}
y[1] = (r - y[0]) - w;
if (hx < 0)
{
y[0] = -y[0];
y[1] = -y[1];
return -n;
}
else
{
return n;
}
}
/*
* all other (large) arguments
*/
if (ix >= 0x7ff00000) /* x is inf or NaN */
{
y[0] = y[1] = x - x;
return 0;
}
/* set z = scalbn(|x|, ilogb(x) - 23) */
z.as_int.lo = __LO (x);
e0 = (ix >> 20) - 1046; /* e0 = ilogb(z) - 23; */
z.as_int.hi = ix - (e0 << 20);
for (i = 0; i < 2; i++)
{
tx[i] = (double) ((int) (z.dbl));
z.dbl = (z.dbl - tx[i]) * two24;
}
tx[2] = z.dbl;
nx = 3;
while (tx[nx - 1] == zero) /* skip zero term */
{
nx--;
}
n = __kernel_rem_pio2 (tx, y, e0, nx, 2);
if (hx < 0)
{
y[0] = -y[0];
y[1] = -y[1];
return -n;
}
return n;
} /* __ieee754_rem_pio2 */
/* __kernel_sin( x, y, iy)
* kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854
* Input x is assumed to be bounded by ~pi/4 in magnitude.
* Input y is the tail of x.
* Input iy indicates whether y is 0. (if iy=0, y assume to be 0).
*
* Algorithm
* 1. Since sin(-x) = -sin(x), we need only to consider positive x.
* 2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0.
* 3. sin(x) is approximated by a polynomial of degree 13 on
* [0,pi/4]
* 3 13
* sin(x) ~ x + S1*x + ... + S6*x
* where
*
* |sin(x) 2 4 6 8 10 12 | -58
* |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2
* | x |
*
* 4. sin(x+y) = sin(x) + sin'(x')*y
* ~ sin(x) + (1-x*x/2)*y
* For better accuracy, let
* 3 2 2 2 2
* r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6))))
* then 3 2
* sin(x) = x + (S1*x + (x *(r-y/2)+y))
*/
#define S1 -1.66666666666666324348e-01 /* 0xBFC55555, 0x55555549 */
#define S2 8.33333333332248946124e-03 /* 0x3F811111, 0x1110F8A6 */
#define S3 -1.98412698298579493134e-04 /* 0xBF2A01A0, 0x19C161D5 */
#define S4 2.75573137070700676789e-06 /* 0x3EC71DE3, 0x57B1FE7D */
#define S5 -2.50507602534068634195e-08 /* 0xBE5AE5E6, 0x8A2B9CEB */
#define S6 1.58969099521155010221e-10 /* 0x3DE5D93A, 0x5ACFD57C */
static double
__kernel_sin (double x, double y, int iy)
{
double z, r, v;
int ix;
ix = __HI (x) & 0x7fffffff; /* high word of x */
if (ix < 0x3e400000) /* |x| < 2**-27 */
{
if ((int) x == 0)
{
return x; /* generate inexact */
}
}
z = x * x;
v = z * x;
r = S2 + z * (S3 + z * (S4 + z * (S5 + z * S6)));
if (iy == 0)
{
return x + v * (S1 + z * r);
}
else
{
return x - ((z * (half * y - v * r) - y) - v * S1);
}
} /* __kernel_sin */
/*
* __kernel_cos( x, y )
* kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
* Input x is assumed to be bounded by ~pi/4 in magnitude.
* Input y is the tail of x.
*
* Algorithm
* 1. Since cos(-x) = cos(x), we need only to consider positive x.
* 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
* 3. cos(x) is approximated by a polynomial of degree 14 on
* [0,pi/4]
* 4 14
* cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
* where the remez error is
*
* | 2 4 6 8 10 12 14 | -58
* |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
* | |
*
* 4 6 8 10 12 14
* 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
* cos(x) = 1 - x*x/2 + r
* since cos(x+y) ~ cos(x) - sin(x)*y
* ~ cos(x) - x*y,
* a correction term is necessary in cos(x) and hence
* cos(x+y) = 1 - (x*x/2 - (r - x*y))
* For better accuracy when x > 0.3, let qx = |x|/4 with
* the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
* Then
* cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
* Note that 1-qx and (x*x/2-qx) is EXACT here, and the
* magnitude of the latter is at least a quarter of x*x/2,
* thus, reducing the rounding error in the subtraction.
*/
#define C1 4.16666666666666019037e-02 /* 0x3FA55555, 0x5555554C */
#define C2 -1.38888888888741095749e-03 /* 0xBF56C16C, 0x16C15177 */
#define C3 2.48015872894767294178e-05 /* 0x3EFA01A0, 0x19CB1590 */
#define C4 -2.75573143513906633035e-07 /* 0xBE927E4F, 0x809C52AD */
#define C5 2.08757232129817482790e-09 /* 0x3E21EE9E, 0xBDB4B1C4 */
#define C6 -1.13596475577881948265e-11 /* 0xBDA8FAE9, 0xBE8838D4 */
static double
__kernel_cos (double x, double y)
{
double a, hz, z, r;
int ix;
ix = __HI (x) & 0x7fffffff; /* ix = |x|'s high word */
if (ix < 0x3e400000) /* if x < 2**27 */
{
if (((int) x) == 0)
{
return one; /* generate inexact */
}
}
z = x * x;
r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6)))));
if (ix < 0x3FD33333) /* if |x| < 0.3 */
{
return one - (0.5 * z - (z * r - x * y));
}
else
{
double_accessor qx;
if (ix > 0x3fe90000) /* x > 0.78125 */
{
qx.dbl = 0.28125;
}
else
{
qx.as_int.hi = ix - 0x00200000; /* x / 4 */
qx.as_int.lo = 0;
}
hz = 0.5 * z - qx.dbl;
a = one - qx.dbl;
return a - (hz - (z * r - x * y));
}
} /* __kernel_cos */
/* __kernel_tan( x, y, k )
* kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
* Input x is assumed to be bounded by ~pi/4 in magnitude.
* Input y is the tail of x.
* Input k indicates whether tan (if k = 1) or -1/tan (if k = -1) is returned.
*
* Algorithm
* 1. Since tan(-x) = -tan(x), we need only to consider positive x.
* 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
* 3. tan(x) is approximated by a odd polynomial of degree 27 on
* [0,0.67434]
* 3 27
* tan(x) ~ x + T1*x + ... + T13*x
* where
*
* |tan(x) 2 4 26 | -59.2
* |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
* | x |
*
* Note: tan(x+y) = tan(x) + tan'(x)*y
* ~ tan(x) + (1+x*x)*y
* Therefore, for better accuracy in computing tan(x+y), let
* 3 2 2 2 2
* r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
* then
* 3 2
* tan(x+y) = x + (T1*x + (x *(r+y)+y))
*
* 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
* tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
* = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
*/
#define T0 3.33333333333334091986e-01 /* 3FD55555, 55555563 */
#define T1 1.33333333333201242699e-01 /* 3FC11111, 1110FE7A */
#define T2 5.39682539762260521377e-02 /* 3FABA1BA, 1BB341FE */
#define T3 2.18694882948595424599e-02 /* 3F9664F4, 8406D637 */
#define T4 8.86323982359930005737e-03 /* 3F8226E3, E96E8493 */
#define T5 3.59207910759131235356e-03 /* 3F6D6D22, C9560328 */
#define T6 1.45620945432529025516e-03 /* 3F57DBC8, FEE08315 */
#define T7 5.88041240820264096874e-04 /* 3F4344D8, F2F26501 */
#define T8 2.46463134818469906812e-04 /* 3F3026F7, 1A8D1068 */
#define T9 7.81794442939557092300e-05 /* 3F147E88, A03792A6 */
#define T10 7.14072491382608190305e-05 /* 3F12B80F, 32F0A7E9 */
#define T11 -1.85586374855275456654e-05 /* BEF375CB, DB605373 */
#define T12 2.59073051863633712884e-05 /* 3EFB2A70, 74BF7AD4 */
#define pio4 7.85398163397448278999e-01 /* 3FE921FB, 54442D18 */
#define pio4lo 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */
static double
__kernel_tan (double x, double y, int iy)
{
double_accessor z;
double r, v, w, s;
int ix, hx;
hx = __HI (x); /* high word of x */
ix = hx & 0x7fffffff; /* high word of |x| */
if (ix < 0x3e300000) /* x < 2**-28 */
{
if ((int) x == 0) /* generate inexact */
{
if (((ix | __LO (x)) | (iy + 1)) == 0)
{
return one / fabs (x);
}
else
{
if (iy == 1)
{
return x;
}
else /* compute -1 / (x + y) carefully */
{
double a;
double_accessor t;
z.dbl = w = x + y;
z.as_int.lo = 0;
v = y - (z.dbl - x);
t.dbl = a = -one / w;
t.as_int.lo = 0;
s = one + t.dbl * z.dbl;
return t.dbl + a * (s + t.dbl * v);
}
}
}
}
if (ix >= 0x3FE59428) /* |x| >= 0.6744 */
{
if (hx < 0)
{
x = -x;
y = -y;
}
z.dbl = pio4 - x;
w = pio4lo - y;
x = z.dbl + w;
y = 0.0;
}
z.dbl = x * x;
w = z.dbl * z.dbl;
/*
* Break x^5 * (T[1] + x^2 * T[2] + ...) into
* x^5 (T[1] + x^4 * T[3] + ... + x^20 * T[11]) +
* x^5 (x^2 * (T[2] + x^4 * T[4] + ... + x^22 * [T12]))
*/
r = T1 + w * (T3 + w * (T5 + w * (T7 + w * (T9 + w * T11))));
v = z.dbl * (T2 + w * (T4 + w * (T6 + w * (T8 + w * (T10 + w * T12)))));
s = z.dbl * x;
r = y + z.dbl * (s * (r + v) + y);
r += T0 * s;
w = x + r;
if (ix >= 0x3FE59428)
{
v = (double) iy;
return (double) (1 - ((hx >> 30) & 2)) * (v - 2.0 * (x - (w * w / (w + v) - r)));
}
if (iy == 1)
{
return w;
}
else
{
/*
* if allow error up to 2 ulp, simply return
* -1.0 / (x + r) here
*/
/* compute -1.0 / (x + r) accurately */
double a;
double_accessor t;
z.dbl = w;
z.as_int.lo = 0;
v = r - (z.dbl - x); /* z + v = r + x */
t.dbl = a = -1.0 / w; /* a = -1.0 / w */
t.as_int.lo = 0;
s = 1.0 + t.dbl * z.dbl;
return t.dbl + a * (s + t.dbl * v);
}
} /* __kernel_tan */
/* Method:
* Let S,C and T denote the sin, cos and tan respectively on
* [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
* in [-pi/4 , +pi/4], and let n = k mod 4.
* We have
*
* n sin(x) cos(x) tan(x)
* ----------------------------------------------------------
* 0 S C T
* 1 C -S -1/T
* 2 -S -C T
* 3 -C S -1/T
* ----------------------------------------------------------
*
* Special cases:
* Let trig be any of sin, cos, or tan.
* trig(+-INF) is NaN, with signals;
* trig(NaN) is that NaN;
*
* Accuracy:
* TRIG(x) returns trig(x) nearly rounded
*/
/* sin(x)
* Return sine function of x.
*
* kernel function:
* __kernel_sin ... sine function on [-pi/4,pi/4]
* __kernel_cos ... cose function on [-pi/4,pi/4]
* __ieee754_rem_pio2 ... argument reduction routine
*/
double
sin (double x)
{
double y[2], z = 0.0;
int n, ix;
/* High word of x. */
ix = __HI (x);
/* |x| ~< pi/4 */
ix &= 0x7fffffff;
if (ix <= 0x3fe921fb)
{
return __kernel_sin (x, z, 0);
}
/* sin(Inf or NaN) is NaN */
else if (ix >= 0x7ff00000)
{
return x - x;
}
/* argument reduction needed */
else
{
n = __ieee754_rem_pio2 (x, y);
switch (n & 3)
{
case 0:
{
return __kernel_sin (y[0], y[1], 1);
}
case 1:
{
return __kernel_cos (y[0], y[1]);
}
case 2:
{
return -__kernel_sin (y[0], y[1], 1);
}
default:
{
return -__kernel_cos (y[0], y[1]);
}
}
}
} /* sin */
/* cos(x)
* Return cosine function of x.
*
* kernel function:
* __kernel_sin ... sine function on [-pi/4,pi/4]
* __kernel_cos ... cosine function on [-pi/4,pi/4]
* __ieee754_rem_pio2 ... argument reduction routine
*/
double
cos (double x)
{
double y[2], z = 0.0;
int n, ix;
/* High word of x. */
ix = __HI (x);
/* |x| ~< pi/4 */
ix &= 0x7fffffff;
if (ix <= 0x3fe921fb)
{
return __kernel_cos (x, z);
}
/* cos(Inf or NaN) is NaN */
else if (ix >= 0x7ff00000)
{
return x - x;
}
/* argument reduction needed */
else
{
n = __ieee754_rem_pio2 (x, y);
switch (n & 3)
{
case 0:
{
return __kernel_cos (y[0], y[1]);
}
case 1:
{
return -__kernel_sin (y[0], y[1], 1);
}
case 2: