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8302027: Port fdlibm trig functions (sin, cos, tan) to Java

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Reviewed-by: bpb
This commit is contained in:
Joe Darcy 2023-03-04 23:52:03 +00:00
parent 9fdbf3cfc4
commit 1bb39a95eb
7 changed files with 2293 additions and 22 deletions
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905
src/java.base/share/classes/java/lang/FdLibm.java
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@ -60,6 +60,7 @@ package java.lang;
class FdLibm {
// Constants used by multiple algorithms
private static final double INFINITY = Double.POSITIVE_INFINITY;
private static final double TWO24 = 0x1.0p24; // 1.67772160000000000000e+07
private static final double TWO54 = 0x1.0p54; // 1.80143985094819840000e+16
private static final double HUGE = 1.0e+300;
@ -113,6 +114,910 @@ class FdLibm {
(low & 0xffff_ffffL));
}
/** 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
*
* 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
*/
static class Sin {
private Sin() {throw new UnsupportedOperationException();}
static double compute(double x) {
double[] y = new double[2];
double z = 0.0;
int n, ix;
// High word of x.
ix = __HI(x);
// |x| ~< pi/4
ix &= 0x7fff_ffff;
if (ix <= 0x3fe9_21fb) {
return __kernel_sin(x, z, 0);
} else if (ix>=0x7ff0_0000) { // sin(Inf or NaN) is NaN
return x - x;
} else { // argument reduction needed
n = RemPio2.__ieee754_rem_pio2(x, y);
switch(n & 3) {
case 0: return Sin.__kernel_sin(y[0], y[1], 1);
case 1: return Cos.__kernel_cos(y[0], y[1]);
case 2: return -Sin.__kernel_sin(y[0], y[1], 1);
default:
return -Cos.__kernel_cos(y[0], y[1]);
}
}
}
/** __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))
*/
private static final double
S1 = -0x1.5555555555549p-3, // -1.66666666666666324348e-01
S2 = 0x1.111111110f8a6p-7, // 8.33333333332248946124e-03
S3 = -0x1.a01a019c161d5p-13, // -1.98412698298579493134e-04
S4 = 0x1.71de357b1fe7dp-19, // 2.75573137070700676789e-06
S5 = -0x1.ae5e68a2b9cebp-26, // -2.50507602534068634195e-08
S6 = 0x1.5d93a5acfd57cp-33; // 1.58969099521155010221e-10
static double __kernel_sin(double x, double y, int iy) {
double z, r, v;
int ix;
ix = __HI(x) & 0x7fff_ffff; // high word of x
if (ix < 0x3e40_0000) { // |x| < 2**-27
if ((int)x == 0) // generate inexact
return x;
}
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*(0.5*y - v*r) - y) - v*S1);
}
}
}
/** 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
*
* 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
*/
static class Cos {
private Cos() {throw new UnsupportedOperationException();}
static double compute(double x) {
double[] y = new double[2];
double z = 0.0;
int n, ix;
// High word of x.
ix = __HI(x);
// |x| ~< pi/4
ix &= 0x7fff_ffff;
if (ix <= 0x3fe9_21fb) {
return __kernel_cos(x, z);
} else if (ix >= 0x7ff0_0000) { // cos(Inf or NaN) is NaN
return x-x;
} else { // argument reduction needed
n = RemPio2.__ieee754_rem_pio2(x,y);
switch (n & 3) {
case 0: return Cos.__kernel_cos(y[0], y[1]);
case 1: return -Sin.__kernel_sin(y[0], y[1],1);
case 2: return -Cos.__kernel_cos(y[0], y[1]);
default:
return Sin.__kernel_sin(y[0], y[1], 1);
}
}
}
/**
* __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 < 0x3e4000000), 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.
*/
private static final double
C1 = 0x1.555555555554cp-5, // 4.16666666666666019037e-02
C2 = -0x1.6c16c16c15177p-10, // -1.38888888888741095749e-03
C3 = 0x1.a01a019cb159p-16, // 2.48015872894767294178e-05
C4 = -0x1.27e4f809c52adp-22, // -2.75573143513906633035e-07
C5 = 0x1.1ee9ebdb4b1c4p-29, // 2.08757232129817482790e-09
C6 = -0x1.8fae9be8838d4p-37; // -1.13596475577881948265e-11
static double __kernel_cos(double x, double y) {
double a, hz, z, r, qx = 0.0;
int ix;
ix = __HI(x) & 0x7fff_ffff; // ix = |x|'s high word
if (ix < 0x3e40_0000) { // if x < 2**27
if (((int)x) == 0) { // generate inexact
return 1.0;
}
}
z = x*x;
r = z*(C1 + z*(C2 + z*(C3 + z*(C4 + z*(C5 + z*C6)))));
if (ix < 0x3FD3_3333) { // if |x| < 0.3
return 1.0 - (0.5*z - (z*r - x*y));
} else {
if (ix > 0x3fe9_0000) { // x > 0.78125
qx = 0.28125;
} else {
qx = __HI_LO(ix - 0x0020_0000, 0);
}
hz = 0.5*z - qx;
a = 1.0 - qx;
return a - (hz - (z*r - x*y));
}
}
}
/** tan(x)
* Return tangent function of x.
*
* kernel function:
* __kernel_tan ... tangent function on [-pi/4,pi/4]
* __ieee754_rem_pio2 ... argument reduction routine
*
* 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
*/
static class Tan {
private Tan() {throw new UnsupportedOperationException();}
static double compute(double x) {
double[] y = new double[2];
double z = 0.0;
int n, ix;
// High word of x.
ix = __HI(x);
// |x| ~< pi/4
ix &= 0x7fff_ffff;
if (ix <= 0x3fe9_21fb) {
return __kernel_tan(x, z, 1);
} else if (ix >= 0x7ff0_0000) { // tan(Inf or NaN) is NaN
return x-x; // NaN
} else { // argument reduction needed
n = RemPio2.__ieee754_rem_pio2(x, y);
return __kernel_tan(y[0], y[1], 1 - ((n & 1) << 1)); // 1 -- n even; -1 -- n odd
}
}
/** __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)))
*/
private static final double
pio4 = 0x1.921fb54442d18p-1, // 7.85398163397448278999e-01
pio4lo= 0x1.1a62633145c07p-55, // 3.06161699786838301793e-17
T[] = {
0x1.5555555555563p-2, // 3.33333333333334091986e-01
0x1.111111110fe7ap-3, // 1.33333333333201242699e-01
0x1.ba1ba1bb341fep-5, // 5.39682539762260521377e-02
0x1.664f48406d637p-6, // 2.18694882948595424599e-02
0x1.226e3e96e8493p-7, // 8.86323982359930005737e-03
0x1.d6d22c9560328p-9, // 3.59207910759131235356e-03
0x1.7dbc8fee08315p-10, // 1.45620945432529025516e-03
0x1.344d8f2f26501p-11, // 5.88041240820264096874e-04
0x1.026f71a8d1068p-12, // 2.46463134818469906812e-04
0x1.47e88a03792a6p-14, // 7.81794442939557092300e-05
0x1.2b80f32f0a7e9p-14, // 7.14072491382608190305e-05
-0x1.375cbdb605373p-16, // -1.85586374855275456654e-05
0x1.b2a7074bf7ad4p-16, // 2.59073051863633712884e-05
};
static double __kernel_tan(double x, double y, int iy) {
double z, r, v, w, s;
int ix, hx;
hx = __HI(x); // high word of x
ix = hx&0x7fff_ffff; // high word of |x|
if (ix < 0x3e30_0000) { // x < 2**-28
if ((int)x == 0) { // generate inexact
if (((ix | __LO(x)) | (iy + 1)) == 0) {
return 1.0 / Math.abs(x);
} else {
if (iy == 1) {
return x;
} else { // compute -1 / (x+y) carefully
double a, t;
z = w = x + y;
z= __LO(z, 0);
v = y - (z - x);
t = a = -1.0 / w;
t = __LO(t, 0);
s = 1.0 + t * z;
return t + a * (s + t * v);
}
}
}
}
if (ix >= 0x3FE5_9428) { // |x| >= 0.6744
if ( hx < 0) {
x = -x;
y = -y;
}
z = pio4 - x;
w = pio4lo - y;
x = z + w;
y = 0.0;
}
z = x*x;
w = z*z;
/* 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 = T[1] + w*(T[3] + w*(T[5] + w*(T[7] + w*(T[9] + w*T[11]))));
v = z*(T[2] + w*(T[4] + w*(T[6] + w*(T[8] + w*(T[10] + w*T[12])))));
s = z*x;
r = y + z*(s*(r + v) + y);
r += T[0]*s;
w = x + r;
if (ix >= 0x3FE5_9428) {
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 were to allow error up to 2 ulp,
could simply return -1.0/(x + r) here */
// compute -1.0/(x + r) accurately
double a,t;
z = w;
z = __LO(z, 0);
v = r - (z - x); // z + v = r + x
t = a = -1.0/w; // a = -1.0/w
t = __LO(t, 0);
s = 1.0 + t*z;
return t + a*(s + t*v);
}
}
}
/** __ieee754_rem_pio2(x,y)
*
* return the remainder of x rem pi/2 in y[0]+y[1]
* use __kernel_rem_pio2()
*/
static class RemPio2 {
/*
* Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
*/
private static final int[] two_over_pi = {
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,
};
private static final 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)
*/
private static final double
invpio2 = 0x1.45f306dc9c883p-1, // 6.36619772367581382433e-01
pio2_1 = 0x1.921fb544p0, // 1.57079632673412561417e+00
pio2_1t = 0x1.0b4611a626331p-34, // 6.07710050650619224932e-11
pio2_2 = 0x1.0b4611a6p-34, // 6.07710050630396597660e-11
pio2_2t = 0x1.3198a2e037073p-69, // 2.02226624879595063154e-21
pio2_3 = 0x1.3198a2ep-69, // 2.02226624871116645580e-21
pio2_3t = 0x1.b839a252049c1p-104; // 8.47842766036889956997e-32
static int __ieee754_rem_pio2(double x, double[] y) {
double z = 0.0, w, t, r, fn;
double[] tx = new double[3];
int e0, i, j, nx, n, ix, hx;
hx = __HI(x); // high word of x
ix = hx & 0x7fff_ffff;
if (ix <= 0x3fe9_21fb) { // |x| ~<= pi/4 , no need for reduction
y[0] = x;
y[1] = 0;
return 0;
}
if (ix < 0x4002_d97c) { // |x| < 3pi/4, special case with n=+-1
if (hx > 0) {
z = x - pio2_1;
if (ix != 0x3ff9_21fb) { // 33+53 bit pi is good enough
y[0] = z - pio2_1t;
y[1] = (z - y[0]) - pio2_1t;
} else { // near pi/2, use 33+33+53 bit pi
z -= pio2_2;
y[0] = z - pio2_2t;
y[1] = (z-y[0])-pio2_2t;
}
return 1;
} else { // negative x
z = x + pio2_1;
if (ix != 0x3ff_921fb) { // 33+53 bit pi is good enough
y[0] = z + pio2_1t;
y[1] = (z - y[0]) + pio2_1t;
} else { // near pi/2, use 33+33+53 bit pi
z += pio2_2;
y[0] = z + pio2_2t;
y[1] = (z - y[0]) + pio2_2t;
}
return -1;
}
}
if (ix <= 0x4139_21fb) { // |x| ~<= 2^19*(pi/2), medium size
t = Math.abs(x);
n = (int) (t*invpio2 + 0.5);
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
t = r; // will cover all possible cases
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 >= 0x7ff0_0000) { // x is inf or NaN
y[0] = y[1] = x - x;
return 0;
}
// set z = scalbn(|x|,ilogb(x)-23)
z = __LO(z, __LO(x));
e0 = (ix >> 20) - 1046; /* e0 = ilogb(z)-23; */
z = __HI(z, ix - (e0 << 20));
for (i=0; i < 2; i++) {
tx[i] = (double)((int)(z));
z = (z - tx[i])*TWO24;
}
tx[2] = z;
nx = 3;
while (tx[nx - 1] == 0.0) { // skip zero term
nx--;
}
n = KernelRemPio2.__kernel_rem_pio2(tx, y, e0, nx, 2, two_over_pi);
if (hx < 0) {
y[0] = -y[0];
y[1] = -y[1];
return -n;
}
return n;
}
}
/**
* __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
* double x[],y[]; int e0,nx,prec; int ipio2[];
*
* __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[] output 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
* precision, 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)
*
* 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)).
*
* External function:
* double scalbn(), floor();
*
*
* Here is the description of some local variables:
*
* 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.
*
*/
static class KernelRemPio2 {
/*
* 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.
*/
private static final int init_jk[] = {2, 3, 4, 6}; // initial value for jk
private static final double PIo2[] = {
0x1.921fb4p0, // 1.57079625129699707031e+00
0x1.4442dp-24, // 7.54978941586159635335e-08
0x1.846988p-48, // 5.39030252995776476554e-15
0x1.8cc516p-72, // 3.28200341580791294123e-22
0x1.01b838p-96, // 1.27065575308067607349e-29
0x1.a25204p-120, // 1.22933308981111328932e-36
0x1.382228p-145, // 2.73370053816464559624e-44
0x1.9f31dp-169, // 2.16741683877804819444e-51
};
static final double
twon24 = 0x1.0p-24; // 5.96046447753906250000e-08
static int __kernel_rem_pio2(double[] x, double[] y, int e0, int nx, int prec, final int[] ipio2) {
int jz, jx, jv, jp, jk, carry, n, i, j, k, m, q0, ih;
int[] iq = new int[20];
double z,fw;
double [] f = new double[20];
double [] fq= new double[20];
double [] q = new double[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) ? 0.0 : (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;
while(true) {
// 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 = Math.scalb(z, q0); // actual value of z
z -= 8.0*Math.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] = 0x100_0000 - j;
}
} else {
iq[i] = 0xff_ffff - j;
}
}
if (q0 > 0) { // rare case: chance is 1 in 12
switch(q0) {
case 1:
iq[jz-1] &= 0x7f_ffff;
break;
case 2:
iq[jz-1] &= 0x3f_ffff;
break;
}
}
if (ih == 2) {
z = 1.0 - z;
if (carry != 0) {
z -= Math.scalb(1.0, q0);
}
}
}
// check if recomputation is needed
if (z == 0.0) {
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;
continue;
} else {
break;
}
} else {
break;
}
}
// 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 = Math.scalb(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 = Math.scalb(1.0, 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;
}
}
/** Returns the arcsine of x.
*
* Method :

12
src/java.base/share/classes/java/lang/StrictMath.java
View File

@ -127,7 +127,9 @@ public final class StrictMath {
* @param a an angle, in radians.
* @return the sine of the argument.
*/
public static native double sin(double a);
public static double sin(double a) {
return FdLibm.Sin.compute(a);
}
/**
* Returns the trigonometric cosine of an angle. Special cases:
@ -139,7 +141,9 @@ public final class StrictMath {
* @param a an angle, in radians.
* @return the cosine of the argument.
*/
public static native double cos(double a);
public static double cos(double a) {
return FdLibm.Cos.compute(a);
}
/**
* Returns the trigonometric tangent of an angle. Special cases:
@ -151,7 +155,9 @@ public final class StrictMath {
* @param a an angle, in radians.
* @return the tangent of the argument.
*/
public static native double tan(double a);
public static double tan(double a) {
return FdLibm.Tan.compute(a);
}
/**
* Returns the arc sine of a value; the returned angle is in the

129
test/jdk/java/lang/Math/SinCosTests.java Normal file
View File

@ -0,0 +1,129 @@
/*
* Copyright (c) 2003, 2023, Oracle and/or its affiliates. All rights reserved.
* DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
*
* This code is free software; you can redistribute it and/or modify it
* under the terms of the GNU General Public License version 2 only, as
* published by the Free Software Foundation.
*
* This code is distributed in the hope that it will be useful, but WITHOUT
* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
* FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
* version 2 for more details (a copy is included in the LICENSE file that
* accompanied this code).
*
* You should have received a copy of the GNU General Public License version
* 2 along with this work; if not, write to the Free Software Foundation,
* Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
*
* Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
* or visit www.oracle.com if you need additional information or have any
* questions.
*/
/*
* @test
* @library /test/lib
* @build Tests
* @run main SinCosTests
* @bug 8302040
* @summary Tests for {Math, StrictMath}.sqrt
*/
public class SinCosTests {
private SinCosTests(){}
public static void main(String... argv) {
int failures = 0;
failures += testSin();
failures += testCos();
if (failures > 0) {
System.err.println("Testing sin and cos incurred "
+ failures + " failures.");
throw new RuntimeException();
}
}
private static final double InfinityD = Double.POSITIVE_INFINITY;
private static final double NaNd = Double.NaN;
/**
* "Special cases:
*
* If the argument is NaN or an infinity, then the result is NaN.
*
* If the argument is zero, then the result is a zero with the
* same sign as the argument."
*/
private static int testSin() {
int failures = 0;
for(double nan : Tests.NaNs) {
failures += testSinCase(nan, NaNd);
}
double [][] testCases = {
{+InfinityD, NaNd},
{-InfinityD, NaNd},
{+0.0, +0.0},
{-0.0, -0.0},
};
for(int i = 0; i < testCases.length; i++) {
failures += testSinCase(testCases[i][0], testCases[i][1]);
}
return failures;
}
/**
* "Special cases:
*
* If the argument is NaN or an infinity, then the result is NaN.
* If the argument is zero, then the result is 1.0."
*/
private static int testCos() {
int failures = 0;
for(double nan : Tests.NaNs) {
failures += testCosCase(nan, NaNd);
}
double [][] testCases = {
{+InfinityD, NaNd},
{-InfinityD, NaNd},
{+0.0, +1.0},
{-0.0, +1.0},
};
for(int i = 0; i < testCases.length; i++) {
failures += testCosCase(testCases[i][0], testCases[i][1]);
}
return failures;
}
private static int testSinCase(double input, double expected) {
int failures=0;
failures+=Tests.test("Math.sin", input, Math::sin, expected);
failures+=Tests.test("StrictMath.sin", input, StrictMath::sin, expected);
return failures;
}
private static int testCosCase(double input, double expected) {
int failures=0;
failures+=Tests.test("Math.cos", input, Math::cos, expected);
failures+=Tests.test("StrictMath.cos", input, StrictMath::cos, expected);
return failures;
}
}

71
test/jdk/java/lang/Math/TanTests.java
View File

@ -23,21 +23,74 @@
/*
* @test
* @bug 5033578
* @bug 5033578 8302027
* @build Tests
* @run main TanTests
* @summary Tests for {Math, StrictMath}.tan
*/
public class TanTests {
private TanTests(){}
static int testTanCase(double input, double expected, double ulps) {
private static final double NaNd = Double.NaN;
private static final double InfinityD = Double.POSITIVE_INFINITY;
public static void main(String... argv) {
int failures = 0;
failures += testTanNaN();
failures += testTanCardinal();
failures += testTan();
if (failures > 0) {
System.err.println("Testing tan incurred "
+ failures + " failures.");
throw new RuntimeException();
}
}
private static int testTanCase(double input, double expected, double ulps) {
int failures = 0;
failures += Tests.testUlpDiff("StrictMath.tan", input, StrictMath::tan, expected, ulps);
failures += Tests.testUlpDiff("Math.tan", input, Math::tan, expected, ulps);
return failures;
}
static int testTan() {
private static int testTanNaN() {
int failures = 0;
// "If the argument is NaN or an infinity, then the result is NaN."
for(double nan : Tests.NaNs) {
failures += Tests.test("StrictMath.tan", nan, StrictMath::tan, NaNd);
failures += Tests.test("Math.tan", nan, Math::tan, NaNd);
}
return failures;
}
private static int testTanCardinal() {
int failures = 0;
double [][] testCases = {
// "If the argument is NaN or an infinity, then the result is NaN."
{ InfinityD, NaNd},
{-InfinityD, NaNd},
// "If the argument is zero, then the result is a zero
// with the same sign as the argument."
{-0.0, -0.0},
{+0.0, +0.0},
};
for(double[] testCase : testCases) {
failures += Tests.test("StrictMath.tan", testCase[0], StrictMath::tan, testCase[1]);
failures += Tests.test("Math.tan", testCase[0], Math::tan, testCase[1]);
}
return failures;
}
private static int testTan() {
int failures = 0;
double [][] testCases = {
@ -169,16 +222,4 @@ public class TanTests {
return failures;
}
public static void main(String... argv) {
int failures = 0;
failures += testTan();
if (failures > 0) {
System.err.println("Testing tan incurred "
+ failures + " failures.");
throw new RuntimeException();
}
}
}

8
test/jdk/java/lang/StrictMath/ExhaustingTests.java
View File

@ -23,7 +23,7 @@
/*
* @test
* @bug 8301833 8302026 8301444 8302028 8302040
* @bug 8301833 8302026 8301444 8302028 8302040 8302027
* @build Tests
* @build FdlibmTranslit
* @build ExhaustingTests
@ -84,9 +84,9 @@ public class ExhaustingTests {
new UnaryTestCase("cosh", FdlibmTranslit::cosh, StrictMath::cosh, DEFAULT_SHIFT),
new UnaryTestCase("tanh", FdlibmTranslit::tanh, StrictMath::tanh, DEFAULT_SHIFT),
// new UnaryTestCase("sin", FdlibmTranslit::sin, StrictMath::sin, DEFAULT_SHIFT),
// new UnaryTestCase("cos", FdlibmTranslit::cos, StrictMath::cos, DEFAULT_SHIFT),
// new UnaryTestCase("tan", FdlibmTranslit::tan, StrictMath::tan, DEFAULT_SHIFT),
new UnaryTestCase("sin", FdlibmTranslit::sin, StrictMath::sin, DEFAULT_SHIFT),
new UnaryTestCase("cos", FdlibmTranslit::cos, StrictMath::cos, DEFAULT_SHIFT),
new UnaryTestCase("tan", FdlibmTranslit::tan, StrictMath::tan, DEFAULT_SHIFT),
new UnaryTestCase("asin", FdlibmTranslit::asin, StrictMath::asin, DEFAULT_SHIFT),
new UnaryTestCase("acos", FdlibmTranslit::acos, StrictMath::acos, DEFAULT_SHIFT),

872
test/jdk/java/lang/StrictMath/FdlibmTranslit.java
View File

@ -70,6 +70,18 @@ public class FdlibmTranslit {
( ((long)high)) << 32 );
}
public static double sin(double x) {
return Sin.compute(x);
}
public static double cos(double x) {
return Cos.compute(x);
}
public static double tan(double x) {
return Tan.compute(x);
}
public static double asin(double x) {
return Asin.compute(x);
}
@ -130,6 +142,866 @@ public class FdlibmTranslit {
return Tanh.compute(x);
}
// -----------------------------------------------------------------------------------------
/** 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
*
* 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
*/
static class Sin {
static double compute(double x) {
double[] y = new double[2];
double 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 = RemPio2.__ieee754_rem_pio2(x,y);
switch(n&3) {
case 0: return Sin.__kernel_sin(y[0],y[1],1);
case 1: return Cos.__kernel_cos(y[0],y[1]);
case 2: return -Sin.__kernel_sin(y[0],y[1],1);
default:
return -Cos.__kernel_cos(y[0],y[1]);
}
}
}
/** __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))
*/
private static final double
half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
S1 = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */
S2 = 8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */
S3 = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */
S4 = 2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */
S5 = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */
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);
}
}
/** 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
*
* 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
*/
static class Cos {
static double compute(double x) {
double[] y = new double[2];
double 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 = RemPio2.__ieee754_rem_pio2(x,y);
switch(n&3) {
case 0: return Cos.__kernel_cos(y[0],y[1]);
case 1: return -Sin.__kernel_sin(y[0],y[1],1);
case 2: return -Cos.__kernel_cos(y[0],y[1]);
default:
return Sin.__kernel_sin(y[0],y[1],1);
}
}
}
/**
* __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.
*/
private static final double
one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
static double __kernel_cos(double x, double y) {
double a,hz,z,r,qx = 0.0;
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 {
if(ix > 0x3fe90000) { /* x > 0.78125 */
qx = 0.28125;
} else {
//__HI(qx) = ix-0x00200000; /* x/4 */
qx = __HI(qx, ix-0x00200000);
// __LO(qx) = 0;
qx = __LO(qx, 0);
}
hz = 0.5*z-qx;
a = one-qx;
return a - (hz - (z*r-x*y));
}
}
}
/** tan(x)
* Return tangent function of x.
*
* kernel function:
* __kernel_tan ... tangent function on [-pi/4,pi/4]
* __ieee754_rem_pio2 ... argument reduction routine
*
* 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
*/
static class Tan {
static double compute(double x) {
double[] y= new double[2];
double z=0.0;
int n, ix;
/* High word of x. */
ix = __HI(x);
/* |x| ~< pi/4 */
ix &= 0x7fffffff;
if(ix <= 0x3fe921fb) return __kernel_tan(x,z,1);
/* tan(Inf or NaN) is NaN */
else if (ix>=0x7ff00000) return x-x; /* NaN */
/* argument reduction needed */
else {
n = RemPio2.__ieee754_rem_pio2(x,y);
return __kernel_tan(y[0],y[1],1-((n&1)<<1)); /* 1 -- n even
-1 -- n odd */
}
}
/** __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)))
*/
private static final double
one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
T[] = {
3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
-1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
};
static double __kernel_tan(double x, double y, int iy) {
double z,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 / Math.abs(x);
else {
if (iy == 1)
return x;
else { /* compute -1 / (x+y) carefully */
double a, t;
z = w = x + y;
// __LO(z) = 0;
z= __LO(z, 0);
v = y - (z - x);
t = a = -one / w;
//__LO(t) = 0;
t = __LO(t, 0);
s = one + t * z;
return t + a * (s + t * v);
}
}
}
}
if(ix>=0x3FE59428) { /* |x|>=0.6744 */
if(hx<0) {x = -x; y = -y;}
z = pio4-x;
w = pio4lo-y;
x = z+w; y = 0.0;
}
z = x*x;
w = z*z;
/* 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 = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
s = z*x;
r = y + z*(s*(r+v)+y);
r += T[0]*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,t;
z = w;
// __LO(z) = 0;
z = __LO(z, 0);
v = r-(z - x); /* z+v = r+x */
t = a = -1.0/w; /* a = -1.0/w */
// __LO(t) = 0;
t = __LO(t, 0);
s = 1.0+t*z;
return t+a*(s+t*v);
}
}
}
/** __ieee754_rem_pio2(x,y)
*
* return the remainder of x rem pi/2 in y[0]+y[1]
* use __kernel_rem_pio2()
*/
static class RemPio2 {
/*
* Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
*/
private static final int[] two_over_pi = {
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,
};
private static final 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)
*/
private static final double
zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
static int __ieee754_rem_pio2(double x, double[] y) {
double z = 0.0,w,t,r,fn;
double[] tx = new double[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 = x - pio2_1;
if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
y[0] = z - pio2_1t;
y[1] = (z-y[0])-pio2_1t;
} else { /* near pi/2, use 33+33+53 bit pi */
z -= pio2_2;
y[0] = z - pio2_2t;
y[1] = (z-y[0])-pio2_2t;
}
return 1;
} else { /* negative x */
z = x + pio2_1;
if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
y[0] = z + pio2_1t;
y[1] = (z-y[0])+pio2_1t;
} else { /* near pi/2, use 33+33+53 bit pi */
z += pio2_2;
y[0] = z + pio2_2t;
y[1] = (z-y[0])+pio2_2t;
}
return -1;
}
}
if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
t = Math.abs(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 */
t = r; /* will cover all possible cases */
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) */
// __LO(z) = __LO(x);
z = __LO(z, __LO(x));
e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */
// __HI(z) = ix - (e0<<20);
z = __HI(z, ix - (e0<<20));
for(i=0;i<2;i++) {
tx[i] = (double)((int)(z));
z = (z-tx[i])*two24;
}
tx[2] = z;
nx = 3;
while(tx[nx-1]==zero) nx--; /* skip zero term */
n = KernelRemPio2.__kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi);
if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
return n;
}
}
/**
* __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
* double x[],y[]; int e0,nx,prec; int ipio2[];
*
* __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[] output 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
* precision, 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)
*
* 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)).
*
* External function:
* double scalbn(), floor();
*
*
* Here is the description of some local variables:
*
* 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.
*
*/
static class KernelRemPio2 {
/*
* 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.
*/
private static final int[] init_jk = {2,3,4,6}; /* initial value for jk */
private static final 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 */
};
static final double
zero = 0.0,
one = 1.0,
two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
static int __kernel_rem_pio2(double[] x, double[] y, int e0, int nx, int prec, final int[] ipio2) {
int jz,jx,jv,jp,jk,carry,n,i,j,k,m,q0,ih;
int[] iq = new int[20];
double z,fw;
double [] f = new double[20];
double [] fq= new double[20];
double [] q = new double[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;
/*
* Transliteration note: the FDLIBM C sources have a
* "recompute:" label at this point and a "goto
* recompute;" later on at the indicated point. This
* structure was replaced by wrapping the code in the
* while(true){...} loop below, replacing the goto with
* the continue to re-execute the loop and by adding
* breaks to exit the loop on the other control flow
* paths.
*/
while(true) {
/* 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 = Math.scalb(z,q0); /* actual value of z */
z -= 8.0*Math.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 -= Math.scalb(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;
// At this point "goto recompute;" in the original C sources.
continue;
} else { break;}
} else {break;}
}
/* 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 = Math.scalb(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 = Math.scalb(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;
}
}
/** Returns the arcsine of x.
*

318
test/jdk/java/lang/StrictMath/TrigTests.java Normal file
View File

@ -0,0 +1,318 @@
/*
* Copyright (c) 2003, 2023, Oracle and/or its affiliates. All rights reserved.
* DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
*
* This code is free software; you can redistribute it and/or modify it
* under the terms of the GNU General Public License version 2 only, as
* published by the Free Software Foundation.
*
* This code is distributed in the hope that it will be useful, but WITHOUT
* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
* FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
* version 2 for more details (a copy is included in the LICENSE file that
* accompanied this code).
*
* You should have received a copy of the GNU General Public License version
* 2 along with this work; if not, write to the Free Software Foundation,
* Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
*
* Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
* or visit www.oracle.com if you need additional information or have any
* questions.
*/
import jdk.test.lib.RandomFactory;
import java.util.function.DoubleUnaryOperator;
/*
* @test
* @bug 8302027
* @key randomness
* @library /test/lib
* @build jdk.test.lib.RandomFactory
* @build Tests
* @build FdlibmTranslit
* @build TrigTests
* @run main TrigTests
* @summary Tests for StrictMath.{sin, cos, tan}
*/
/**
* The tests in ../Math/{TanTests.java, SinCosTests.java} test
* properties that should hold for any implementation of the trig
* functions sin, cos, and tan, including the FDLIBM-based ones
* required by the StrictMath class. Therefore, the test cases in
* ../Math/{TanTests.java, SinCosTests.java} are run against both the
* Math and StrictMath versions of the trig methods. The role of this
* test is to verify that the FDLIBM algorithms are being used by
* running golden file tests on values that may vary from one
* conforming implementation of the trig functions to another.
*/
public class TrigTests {
private TrigTests(){}
public static void main(String... args) {
int failures = 0;
failures += testAgainstTranslitCommon();
failures += testAgainstTranslitSin();
failures += testAgainstTranslitCos();
failures += testAgainstTranslitTan();
if (failures > 0) {
System.err.println("Testing the trig functions incurred "
+ failures + " failures.");
throw new RuntimeException();
}
}
/**
* Bundle together groups of testing methods.
*/
private static enum TrigTest {
SIN(TrigTests::testSinCase, FdlibmTranslit::sin),
COS(TrigTests::testCosCase, FdlibmTranslit::cos),
TAN(TrigTests::testTanCase, FdlibmTranslit::tan);
private DoubleDoubleToInt testCase;
private DoubleUnaryOperator transliteration;
TrigTest(DoubleDoubleToInt testCase, DoubleUnaryOperator transliteration) {
this.testCase = testCase;
this.transliteration = transliteration;
}
public DoubleDoubleToInt testCase() {return testCase;}
public DoubleUnaryOperator transliteration() {return transliteration;}
}
// Initialize shared random number generator
private static java.util.Random random = RandomFactory.getRandom();
/**
* Test against shared points of interest.
*/
private static int testAgainstTranslitCommon() {
int failures = 0;
double[] pointsOfInterest = {
Math.PI/4.0,
-Math.PI/4.0,
Math.PI/2.0,
-Math.PI/2.0,
3.0*Math.PI/2.0,
-3.0*Math.PI/2.0,
Math.PI,
-Math.PI,
2.0*Math.PI,
-2.0*Math.PI,
Double.MIN_NORMAL,
1.0,
Tests.createRandomDouble(random),
};
for (var testMethods : TrigTest.values()) {
for (double testPoint : pointsOfInterest) {
failures += testRangeMidpoint(testPoint, Math.ulp(testPoint), 1000, testMethods);
}
}
return failures;
}
/**
* Test StrictMath.sin against transliteration port of sin.
*/
private static int testAgainstTranslitSin() {
int failures = 0;
// Probe near decision points in the FDLIBM algorithm.
double[] decisionPoints = {
0x1.0p-27,
-0x1.0p-27,
};
for (double testPoint : decisionPoints) {
failures += testRangeMidpoint(testPoint, Math.ulp(testPoint), 1000, TrigTest.SIN);
}
// Inputs where Math.sin and StrictMath.sin differ for at least
// one Math.sin implementation.
double [][] testCases = {
{0x1.00000006eeeefp-12, 0x1.ffffffb888889p-13},
{0x1.00000006eeefp-12, 0x1.ffffffb88888bp-13},
{0x1.00000006eeef1p-12, 0x1.ffffffb88888dp-13},
{0x1.000000001bba2p-9, 0x1.ffffeaaae2633p-10},
{0x1.000000000013p-1, 0x1.eaee8744b0806p-2},
{0x1.0000000000012p0, 0x1.aed548f090d02p-1},
{0x1.00000000004e1p9, 0x1.45b52f29ac36p-4},
{0x1.00000000000cp10, -0x1.44ad26136ce5fp-3},
{0x1.000000000020bp11, -0x1.4092047afcd2p-2},
{0x1.0000000000003p12, -0x1.3074ea23314dep-1},
{0x1.0000000000174p50, -0x1.54cd5e7e9e3d2p-1},
{0x1.0000000000005p51, -0x1.8c35b0d728faep-2},
{0x1.0000000000101p113, -0x1.69e9ed300b1dcp-1},
{0x1.0000000000017p114, 0x1.f6b44aa2a1c9cp-1},
{0x1.00000000001abp128, -0x1.ecaddc1136bb2p-1},
{0x1.000000000001bp129, -0x1.682ccb977e4dp-1},
{0x1.0p233, 0x1.7c54e75ed6077p-1},
{0x1.00000000000fcp299, 0x1.78ad2fd7aef78p-1},
{0x1.0000000000002p300, -0x1.1adaf3550facp-1},
{0x1.00000000001afp1023, 0x1.d1c804ef2eeccp-1},
};
for (double[] testCase: testCases) {
failures+=testSinCase(testCase[0], testCase[1]);
}
return failures;
}
/**
* Test StrictMath.cos against transliteration port of cos.
*/
private static int testAgainstTranslitCos() {
int failures = 0;
// Probe near decision points in the FDLIBM algorithm.
double[] decisionPoints = {
0x1.0p27,
-0x1.0p27,
0.78125,
-0.78125,
};
for (double testPoint : decisionPoints) {
failures += testRangeMidpoint(testPoint, Math.ulp(testPoint), 1000, TrigTest.COS);
}
// Inputs where Math.cos and StrictMath.cos differ for at least
// one Math.cos implementation.
double [][] testCases = {
{0x1.000000076aaa6p-10, 0x1.fffff00000147p-1},
{0x1.000000002e4fbp-8, 0x1.ffff00001554fp-1},
{0x1.0000000000318p-2, 0x1.f01549f7dee4p-1},
{0x1.000000000011ep-1, 0x1.c1528065b7cc6p-1},
{0x1.0000000000174p0, 0x1.14a280fb50419p-1},
{0x1.0000000000019p1, -0x1.aa226575372bbp-2},
{0x1.00000000018c9p9, -0x1.fe60f23b0016ap-1},
{0x1.0000000000022p10, 0x1.f98669d7b18d6p-1},
{0x1.0000000000281p11, 0x1.e6439428b217p-1},
{0x1.0000000000001p12, 0x1.9ba4a85e6e173p-1},
{0x1.0000000000211p20, 0x1.e33ad93554beep-1},
{0x1.0000000000006p21, 0x1.9027223f77694p-1},
{0x1.00000000000b8p95, 0x1.8315138968a66p-1},
{0x1.0000000000043p96, 0x1.5b302d1c86cbcp-4},
{0x1.000000000013ap127, -0x1.740d46d7821f4p-1},
{0x1.0000000000002p128, -0x1.e050345cf2161p-1},
{0x1.000000000014p299, 0x1.6c5f3c84352fep-1},
{0x1.0000000000007p300, -0x1.55109bfdf1c5cp-1},
{0x1.000000000010ep400, 0x1.e725637029938p-2},
{0x1.0000000000007p401, 0x1.1f89e14e29ccep-1},
{0x1.0p402, 0x1.be2d53c4560dcp-1},
{0x1.000000000015fp1023, -0x1.2f2596c42735cp-1},
};
for (double[] testCase: testCases) {
failures+=testCosCase(testCase[0], testCase[1]);
}
return failures;
}
/**
* Test StrictMath.tan against transliteration port of tan
*/
private static int testAgainstTranslitTan() {
int failures = 0;
// Probe near decision points in the FDLIBM algorithm.
double[] decisionPoints = {
0x1.0p-28,
-0x1.0p-28,
0.6744,
-0.6744,
};
for (double testPoint : decisionPoints) {
failures += testRangeMidpoint(testPoint, Math.ulp(testPoint), 1000, TrigTest.TAN);
}
// Inputs where Math.tan and StrictMath.tan differ for at least
// one Math.tan implementation.
double [][] testCases = {
{0x1.00000002221fep-13, 0x1.0000001777753p-13},
{0x1.0000000088859p-12, 0x1.00000055dddbp-12},
{0x1.0000000008787p-10, 0x1.000005555defep-10},
{0x1.0000000001423p-9, 0x1.0000155558b9ap-9},
{0x1.00000000005d9p-2, 0x1.05785a43c529p-2},
{0x1.000000000001fp-1, 0x1.17b4f5bf34772p-1},
{0x1.000000000006ep0, 0x1.8eb245cbee51ep0},
{0x1.0000000000032p1, -0x1.17af62e094fd7p1},
{0x1.00000000006a7p9, -0x1.46be0efd0f8cp-4},
{0x1.0p10, -0x1.48d5be43ada01p-3},
{0x1.00000000000c3p32, 0x1.0ad3757181cbap-1},
{0x1.0000000000005p33, 0x1.6e07fbf43d47p0},
{0x1.0000000000124p127, -0x1.3baa73a93958p0},
{0x1.000000000002p128, -0x1.bf05a77a8df0cp-1},
{0x1.000000000011cp299, 0x1.8a6f42eaa3d1fp0},
{0x1.000000000001cp300, -0x1.b30fc9f73002cp-1},
{0x1.0000000000013p500, -0x1.c4e46751be12cp-1},
{0x1.00000000000ep1023, -0x1.d52c4ec04f108p-2}
};
for (double[] testCase: testCases) {
failures+=testTanCase(testCase[0], testCase[1]);
}
return failures;
}
private interface DoubleDoubleToInt {
int apply(double x, double y);
}
private static int testRange(double start, double increment, int count,
TrigTest testMethods) {
int failures = 0;
double x = start;
for (int i = 0; i < count; i++, x += increment) {
failures +=
testMethods.testCase().apply(x, testMethods.transliteration().applyAsDouble(x));
}
return failures;
}
private static int testRangeMidpoint(double midpoint, double increment, int count,
TrigTest testMethods) {
int failures = 0;
double x = midpoint - increment*(count / 2) ;
for (int i = 0; i < count; i++, x += increment) {
failures +=
testMethods.testCase().apply(x, testMethods.transliteration().applyAsDouble(x));
}
return failures;
}
private static int testSinCase(double input, double expected) {
return Tests.test("StrictMath.sin(double)", input,
StrictMath::sin, expected);
}
private static int testCosCase(double input, double expected) {
return Tests.test("StrictMath.cos(double)", input,
StrictMath::cos, expected);
}
private static int testTanCase(double input, double expected) {
return Tests.test("StrictMath.tan(double)", input,
StrictMath::tan, expected);
}
}