The C version of fdlibm relied on the idiom of pointer aliasing * a 64-bit double floating-point value as a two-element array of * 32-bit integers and reading and writing the two halves of the * double independently. This coding pattern was problematic to C * optimizers and not directly expressible in Java. Therefore, rather * than a memory level overlay, if portions of a double need to be * operated on as integer values, the standard library methods for * bitwise floating-point to integer conversion, * Double.longBitsToDouble and Double.doubleToRawLongBits, are directly * or indirectly used. * *
The C version of fdlibm also took some pains to signal the * correct IEEE 754 exceptional conditions divide by zero, invalid, * overflow and underflow. For example, overflow would be signaled by * {@code huge * huge} where {@code huge} was a large constant that * would overflow when squared. Since IEEE floating-point exceptional * handling is not supported natively in the JVM, such coding patterns * have been omitted from this port. For example, rather than {@code * return huge * huge}, this port will use {@code return INFINITY}. * *
Various comparison and arithmetic operations in fdlibm could be
* done either based on the integer view of a value or directly on the
* floating-point representation. Which idiom is faster may depend on
* platform specific factors. However, for code clarity if no other
* reason, this port will favor expressing the semantics of those
* operations in terms of floating-point operations when convenient to
* do so.
*/
final 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;
/*
* Constants for bit-wise manipulation of IEEE 754 double
* values. These constants are for the high-order 32-bits of a
* 64-bit double value: 1 sign bit as the most significant bit,
* followed by 11 exponent bits, and then the remaining bits as
* the significand.
*/
private static final int SIGN_BIT = 0x8000_0000;
private static final int EXP_BITS = 0x7ff0_0000;
private static final int EXP_SIGNIF_BITS = 0x7fff_ffff;
private FdLibm() {
throw new UnsupportedOperationException("No FdLibm instances for you.");
}
/**
* Return the low-order 32 bits of the double argument as an int.
*/
private static int __LO(double x) {
long transducer = Double.doubleToRawLongBits(x);
return (int)transducer;
}
/**
* Return a double with its low-order bits of the second argument
* and the high-order bits of the first argument..
*/
private static double __LO(double x, int low) {
long transX = Double.doubleToRawLongBits(x);
return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L) |
(low & 0x0000_0000_FFFF_FFFFL));
}
/**
* Return the high-order 32 bits of the double argument as an int.
*/
private static int __HI(double x) {
long transducer = Double.doubleToRawLongBits(x);
return (int)(transducer >> 32);
}
/**
* Return a double with its high-order bits of the second argument
* and the low-order bits of the first argument..
*/
private static double __HI(double x, int high) {
long transX = Double.doubleToRawLongBits(x);
return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL) |
( ((long)high)) << 32 );
}
/**
* Return a double with its high-order bits of the first argument
* and the low-order bits of the second argument..
*/
private static double __HI_LO(int high, int low) {
return Double.longBitsToDouble(((long)high << 32) |
(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 final 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 &= EXP_SIGNIF_BITS;
if (ix <= 0x3fe9_21fb) {
return __kernel_sin(x, z, 0);
} else if (ix >= EXP_BITS) { // 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) & EXP_SIGNIF_BITS; // 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 final 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 &= EXP_SIGNIF_BITS;
if (ix <= 0x3fe9_21fb) {
return __kernel_cos(x, z);
} else if (ix >= EXP_BITS) { // 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) & EXP_SIGNIF_BITS; // 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 final 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 &= EXP_SIGNIF_BITS;
if (ix <= 0x3fe9_21fb) {
return __kernel_tan(x, z, 1);
} else if (ix >= EXP_BITS) { // 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 & EXP_SIGNIF_BITS; // 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 final 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 & EXP_SIGNIF_BITS;
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 >= EXP_BITS) { // 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 final 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 :
* Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
* we approximate asin(x) on [0,0.5] by
* asin(x) = x + x*x^2*R(x^2)
* where
* R(x^2) is a rational approximation of (asin(x)-x)/x^3
* and its remez error is bounded by
* |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
*
* For x in [0.5,1]
* asin(x) = pi/2-2*asin(sqrt((1-x)/2))
* Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
* then for x>0.98
* asin(x) = pi/2 - 2*(s+s*z*R(z))
* = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
* For x<=0.98, let pio4_hi = pio2_hi/2, then
* f = hi part of s;
* c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
* and
* asin(x) = pi/2 - 2*(s+s*z*R(z))
* = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
* = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
*
* Special cases:
* if x is NaN, return x itself;
* if |x|>1, return NaN with invalid signal.
*
*/
static final class Asin {
private Asin() {throw new UnsupportedOperationException();}
private static final double
pio2_hi = 0x1.921fb54442d18p0, // 1.57079632679489655800e+00
pio2_lo = 0x1.1a62633145c07p-54, // 6.12323399573676603587e-17
pio4_hi = 0x1.921fb54442d18p-1, // 7.85398163397448278999e-01
// coefficient for R(x^2)
pS0 = 0x1.5555555555555p-3, // 1.66666666666666657415e-01
pS1 = -0x1.4d61203eb6f7dp-2, // -3.25565818622400915405e-01
pS2 = 0x1.9c1550e884455p-3, // 2.01212532134862925881e-01
pS3 = -0x1.48228b5688f3bp-5, // -4.00555345006794114027e-02
pS4 = 0x1.9efe07501b288p-11, // 7.91534994289814532176e-04
pS5 = 0x1.23de10dfdf709p-15, // 3.47933107596021167570e-05
qS1 = -0x1.33a271c8a2d4bp1, // -2.40339491173441421878e+00
qS2 = 0x1.02ae59c598ac8p1, // 2.02094576023350569471e+00
qS3 = -0x1.6066c1b8d0159p-1, // -6.88283971605453293030e-01
qS4 = 0x1.3b8c5b12e9282p-4; // 7.70381505559019352791e-02
static double compute(double x) {
double t = 0, w, p, q, c, r, s;
int hx, ix;
hx = __HI(x);
ix = hx & EXP_SIGNIF_BITS;
if (ix >= 0x3ff0_0000) { // |x| >= 1
if(((ix - 0x3ff0_0000) | __LO(x)) == 0) {
// asin(1) = +-pi/2 with inexact
return x*pio2_hi + x*pio2_lo;
}
return (x - x)/(x - x); // asin(|x| > 1) is NaN
} else if (ix < 0x3fe0_0000) { // |x| < 0.5
if (ix < 0x3e40_0000) { // if |x| < 2**-27
if (HUGE + x > 1.0) {// return x with inexact if x != 0
return x;
}
} else {
t = x*x;
}
p = t*(pS0 + t*(pS1 + t*(pS2 + t*(pS3 + t*(pS4 + t*pS5)))));
q = 1.0 + t*(qS1 + t*(qS2 + t*(qS3 + t*qS4)));
w = p/q;
return x + x*w;
}
// 1 > |x| >= 0.5
w = 1.0 - Math.abs(x);
t = w*0.5;
p = t*(pS0 + t*(pS1 + t*(pS2 + t*(pS3 + t*(pS4 + t*pS5)))));
q = 1.0 + t*(qS1 + t*(qS2 + t*(qS3 + t*qS4)));
s = Math.sqrt(t);
if (ix >= 0x3FEF_3333) { // if |x| > 0.975
w = p/q;
t = pio2_hi - (2.0*(s + s*w) - pio2_lo);
} else {
w = s;
w = __LO(w, 0);
c = (t - w*w)/(s + w);
r = p/q;
p = 2.0*s*r - (pio2_lo - 2.0*c);
q = pio4_hi - 2.0*w;
t = pio4_hi - (p - q);
}
return (hx > 0) ? t : -t;
}
}
/** Returns the arccosine of x.
* Method :
* acos(x) = pi/2 - asin(x)
* acos(-x) = pi/2 + asin(x)
* For |x| <= 0.5
* acos(x) = pi/2 - (x + x*x^2*R(x^2)) (see asin.c)
* For x > 0.5
* acos(x) = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2)))
* = 2asin(sqrt((1-x)/2))
* = 2s + 2s*z*R(z) ...z=(1-x)/2, s=sqrt(z)
* = 2f + (2c + 2s*z*R(z))
* where f=hi part of s, and c = (z-f*f)/(s+f) is the correction term
* for f so that f+c ~ sqrt(z).
* For x <- 0.5
* acos(x) = pi - 2asin(sqrt((1-|x|)/2))
* = pi - 0.5*(s+s*z*R(z)), where z=(1-|x|)/2,s=sqrt(z)
*
* Special cases:
* if x is NaN, return x itself;
* if |x|>1, return NaN with invalid signal.
*
* Function needed: sqrt
*/
static final class Acos {
private Acos() {throw new UnsupportedOperationException();}
private static final double
pio2_hi = 0x1.921fb54442d18p0, // 1.57079632679489655800e+00
pio2_lo = 0x1.1a62633145c07p-54, // 6.12323399573676603587e-17
pS0 = 0x1.5555555555555p-3, // 1.66666666666666657415e-01
pS1 = -0x1.4d61203eb6f7dp-2, // -3.25565818622400915405e-01
pS2 = 0x1.9c1550e884455p-3, // 2.01212532134862925881e-01
pS3 = -0x1.48228b5688f3bp-5, // -4.00555345006794114027e-02
pS4 = 0x1.9efe07501b288p-11, // 7.91534994289814532176e-04
pS5 = 0x1.23de10dfdf709p-15, // 3.47933107596021167570e-05
qS1 = -0x1.33a271c8a2d4bp1, // -2.40339491173441421878e+00
qS2 = 0x1.02ae59c598ac8p1, // 2.02094576023350569471e+00
qS3 = -0x1.6066c1b8d0159p-1, // -6.88283971605453293030e-01
qS4 = 0x1.3b8c5b12e9282p-4; // 7.70381505559019352791e-02
static double compute(double x) {
double z, p, q, r, w, s, c, df;
int hx, ix;
hx = __HI(x);
ix = hx & EXP_SIGNIF_BITS;
if (ix >= 0x3ff0_0000) { // |x| >= 1
if (((ix - 0x3ff0_0000) | __LO(x)) == 0) { // |x| == 1
if (hx > 0) {// acos(1) = 0
return 0.0;
}else { // acos(-1)= pi
return Math.PI + 2.0*pio2_lo;
}
}
return (x - x)/(x - x); // acos(|x| > 1) is NaN
}
if (ix < 0x3fe0_0000) { // |x| < 0.5
if (ix <= 0x3c60_0000) { // if |x| < 2**-57
return pio2_hi + pio2_lo;
}
z = x*x;
p = z*(pS0 + z*(pS1 + z*(pS2 + z*(pS3 + z*(pS4 + z*pS5)))));
q = 1.0 + z*(qS1 + z*(qS2 + z*(qS3 + z*qS4)));
r = p/q;
return pio2_hi - (x - (pio2_lo - x*r));
} else if (hx < 0) { // x < -0.5
z = (1.0 + x)*0.5;
p = z*(pS0 + z*(pS1 + z*(pS2 + z*(pS3 + z*(pS4 + z*pS5)))));
q = 1.0 + z*(qS1 + z*(qS2 + z*(qS3 + z*qS4)));
s = Math.sqrt(z);
r = p/q;
w = r*s - pio2_lo;
return Math.PI - 2.0*(s+w);
} else { // x > 0.5
z = (1.0 - x)*0.5;
s = Math.sqrt(z);
df = s;
df = __LO(df, 0);
c = (z - df*df)/(s + df);
p = z*(pS0 + z*(pS1 + z*(pS2 + z*(pS3 + z*(pS4 + z*pS5)))));
q = 1.0 + z*(qS1 + z*(qS2 + z*(qS3 + z*qS4)));
r = p/q;
w = r*s + c;
return 2.0*(df + w);
}
}
}
/* Returns the arctangent of x.
* Method
* 1. Reduce x to positive by atan(x) = -atan(-x).
* 2. According to the integer k=4t+0.25 chopped, t=x, the argument
* is further reduced to one of the following intervals and the
* arctangent of t is evaluated by the corresponding formula:
*
* [0,7/16] atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
* [7/16,11/16] atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
* [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
* [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
* [39/16,INF] atan(x) = atan(INF) + atan( -1/t )
*
* 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.
*/
static final class Atan {
private Atan() {throw new UnsupportedOperationException();}
private static final double atanhi[] = {
0x1.dac670561bb4fp-2, // atan(0.5)hi 4.63647609000806093515e-01
0x1.921fb54442d18p-1, // atan(1.0)hi 7.85398163397448278999e-01
0x1.f730bd281f69bp-1, // atan(1.5)hi 9.82793723247329054082e-01
0x1.921fb54442d18p0, // atan(inf)hi 1.57079632679489655800e+00
};
private static final double atanlo[] = {
0x1.a2b7f222f65e2p-56, // atan(0.5)lo 2.26987774529616870924e-17
0x1.1a62633145c07p-55, // atan(1.0)lo 3.06161699786838301793e-17
0x1.007887af0cbbdp-56, // atan(1.5)lo 1.39033110312309984516e-17
0x1.1a62633145c07p-54, // atan(inf)lo 6.12323399573676603587e-17
};
private static final double aT[] = {
0x1.555555555550dp-2, // 3.33333333333329318027e-01
-0x1.999999998ebc4p-3, // -1.99999999998764832476e-01
0x1.24924920083ffp-3, // 1.42857142725034663711e-01
-0x1.c71c6fe231671p-4, // -1.11111104054623557880e-01
0x1.745cdc54c206ep-4, // 9.09088713343650656196e-02
-0x1.3b0f2af749a6dp-4, // -7.69187620504482999495e-02
0x1.10d66a0d03d51p-4, // 6.66107313738753120669e-02
-0x1.dde2d52defd9ap-5, // -5.83357013379057348645e-02
0x1.97b4b24760debp-5, // 4.97687799461593236017e-02
-0x1.2b4442c6a6c2fp-5, // -3.65315727442169155270e-02
0x1.0ad3ae322da11p-6, // 1.62858201153657823623e-02
};
static double compute(double x) {
double w, s1, s2, z;
int ix, hx, id;
hx = __HI(x);
ix = hx & EXP_SIGNIF_BITS;
if (ix >= 0x4410_0000) { // if |x| >= 2^66
if (ix > EXP_BITS ||
(ix == EXP_BITS && (__LO(x) != 0))) {
return x+x; // NaN
}
if (hx > 0) {
return atanhi[3] + atanlo[3];
} else {
return -atanhi[3] - atanlo[3];
}
} if (ix < 0x3fdc_0000) { // |x| < 0.4375
if (ix < 0x3e20_0000) { // |x| < 2^-29
if (HUGE + x > 1.0) { // raise inexact
return x;
}
}
id = -1;
} else {
x = Math.abs(x);
if (ix < 0x3ff3_0000) { // |x| < 1.1875
if (ix < 0x3fe60000) { // 7/16 <= |x| < 11/16
id = 0;
x = (2.0*x - 1.0)/(2.0 + x);
} else { // 11/16 <= |x| < 19/16
id = 1;
x = (x - 1.0)/(x + 1.0);
}
} else {
if (ix < 0x4003_8000) { // |x| < 2.4375
id = 2;
x = (x - 1.5)/(1.0 + 1.5*x);
} else { // 2.4375 <= |x| < 2^66
id = 3;
x = -1.0/x;
}
}
}
// end of argument reduction
z = x*x;
w = z*z;
// break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly
s1 = z*(aT[0] + w*(aT[2] + w*(aT[4] + w*(aT[6] + w*(aT[8] + w*aT[10])))));
s2 = w*(aT[1] + w*(aT[3] + w*(aT[5] + w*(aT[7] + w*aT[9]))));
if (id < 0) {
return x - x*(s1 + s2);
} else {
z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x);
return (hx < 0) ? -z: z;
}
}
}
/**
* Returns the angle theta from the conversion of rectangular
* coordinates (x, y) to polar coordinates (r, theta).
*
* Method :
* 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x).
* 2. Reduce x to positive by (if x and y are unexceptional):
* ARG (x+iy) = arctan(y/x) ... if x > 0,
* ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0,
*
* Special cases:
*
* ATAN2((anything), NaN ) is NaN;
* ATAN2(NAN , (anything) ) is NaN;
* ATAN2(+-0, +(anything but NaN)) is +-0 ;
* ATAN2(+-0, -(anything but NaN)) is +-pi ;
* ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2;
* ATAN2(+-(anything but INF and NaN), +INF) is +-0 ;
* ATAN2(+-(anything but INF and NaN), -INF) is +-pi;
* ATAN2(+-INF,+INF ) is +-pi/4 ;
* ATAN2(+-INF,-INF ) is +-3pi/4;
* ATAN2(+-INF, (anything but,0,NaN, and INF)) is +-pi/2;
*
* 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.
*/
static final class Atan2 {
private Atan2() {throw new UnsupportedOperationException();}
private static final double
tiny = 1.0e-300,
pi_o_4 = 0x1.921fb54442d18p-1, // 7.8539816339744827900E-01
pi_o_2 = 0x1.921fb54442d18p0, // 1.5707963267948965580E+00
pi_lo = 0x1.1a62633145c07p-53; // 1.2246467991473531772E-16
static double compute(double y, double x) {
double z;
int k, m, hx, hy, ix, iy;
/*unsigned*/ int lx, ly;
hx = __HI(x);
ix = hx & EXP_SIGNIF_BITS;
lx = __LO(x);
hy = __HI(y);
iy = hy & EXP_SIGNIF_BITS;
ly = __LO(y);
if (Double.isNaN(x) || Double.isNaN(y))
return x + y;
if (((hx - 0x3ff0_0000) | lx) == 0) // x = 1.0
return StrictMath.atan(y);
m = ((hy >> 31) & 1)|((hx >> 30) & 2); // 2*sign(x) + sign(y)
// when y = 0
if ((iy | ly) == 0) {
switch(m) {
case 0:
case 1: return y; // atan(+/-0, +anything) = +/-0
case 2: return Math.PI + tiny; // atan(+0, -anything) = pi
case 3: return -Math.PI - tiny; // atan(-0, -anything) = -pi
}
}
// when x = 0
if ((ix | lx) == 0) {
return (hy < 0)? -pi_o_2 - tiny : pi_o_2 + tiny;
}
// when x is INF
if (ix == EXP_BITS) {
if (iy == EXP_BITS) {
switch(m) {
case 0: return pi_o_4 + tiny; // atan(+INF, +INF)
case 1: return -pi_o_4 - tiny; // atan(-INF, +INF)
case 2: return 3.0*pi_o_4 + tiny; // atan(+INF, -INF)
case 3: return -3.0*pi_o_4 - tiny; // atan(-INF, -INF)
}
} else {
switch(m) {
case 0: return 0.0; // atan(+..., +INF)
case 1: return -0.0; // atan(-..., +INF)
case 2: return Math.PI + tiny; // atan(+..., -INF)
case 3: return -Math.PI - tiny; // atan(-..., -INF)
}
}
}
// when y is INF
if (iy == EXP_BITS) {
return (hy < 0)? -pi_o_2 - tiny : pi_o_2 + tiny;
}
// compute y/x
k = (iy - ix) >> 20;
if (k > 60) { // |y/x| > 2**60
z = pi_o_2+0.5*pi_lo;
} else if (hx < 0 && k < -60) { // |y|/x < -2**60
z = 0.0;
} else { // safe to do y/x
z = StrictMath.atan(Math.abs(y/x));
}
switch (m) {
case 0: return z; // atan(+, +)
case 1: return -z; // atan(-, +)
case 2: return Math.PI - (z - pi_lo); // atan(+, -)
default: return (z - pi_lo) - Math.PI; // atan(-, -), case 3
}
}
}
/**
* Return correctly rounded sqrt.
* ------------------------------------------
* | Use the hardware sqrt if you have one |
* ------------------------------------------
* Method:
* Bit by bit method using integer arithmetic. (Slow, but portable)
* 1. Normalization
* Scale x to y in [1,4) with even powers of 2:
* find an integer k such that 1 <= (y=x*2^(2k)) < 4, then
* sqrt(x) = 2^k * sqrt(y)
* 2. Bit by bit computation
* Let q = sqrt(y) truncated to i bit after binary point (q = 1),
* i 0
* i+1 2
* s = 2*q , and y = 2 * ( y - q ). (1)
* i i i i
*
* To compute q from q , one checks whether
* i+1 i
*
* -(i+1) 2
* (q + 2 ) <= y. (2)
* i
* -(i+1)
* If (2) is false, then q = q ; otherwise q = q + 2 .
* i+1 i i+1 i
*
* With some algebraic manipulation, it is not difficult to see
* that (2) is equivalent to
* -(i+1)
* s + 2 <= y (3)
* i i
*
* The advantage of (3) is that s and y can be computed by
* i i
* the following recurrence formula:
* if (3) is false
*
* s = s , y = y ; (4)
* i+1 i i+1 i
*
* otherwise,
* -i -(i+1)
* s = s + 2 , y = y - s - 2 (5)
* i+1 i i+1 i i
*
* One may easily use induction to prove (4) and (5).
* Note. Since the left hand side of (3) contain only i+2 bits,
* it does not necessary to do a full (53-bit) comparison
* in (3).
* 3. Final rounding
* After generating the 53 bits result, we compute one more bit.
* Together with the remainder, we can decide whether the
* result is exact, bigger than 1/2ulp, or less than 1/2ulp
* (it will never equal to 1/2ulp).
* The rounding mode can be detected by checking whether
* huge + tiny is equal to huge, and whether huge - tiny is
* equal to huge for some floating point number "huge" and "tiny".
*
* Special cases:
* sqrt(+-0) = +-0 ... exact
* sqrt(inf) = inf
* sqrt(-ve) = NaN ... with invalid signal
* sqrt(NaN) = NaN ... with invalid signal for signaling NaN
*
* Other methods : see the appended file at the end of the program below.
*---------------
*/
static final class Sqrt {
private Sqrt() {throw new UnsupportedOperationException();}
private static final double tiny = 1.0e-300;
static double compute(double x) {
double z = 0.0;
int sign = SIGN_BIT;
/*unsigned*/ int r, t1, s1, ix1, q1;
int ix0, s0, q, m, t, i;
ix0 = __HI(x); // high word of x
ix1 = __LO(x); // low word of x
// take care of Inf and NaN
if ((ix0 & EXP_BITS) == EXP_BITS) {
return x*x + x; // sqrt(NaN)=NaN, sqrt(+inf)=+inf, sqrt(-inf)=sNaN
}
// take care of zero
if (ix0 <= 0) {
if (((ix0 & (~sign)) | ix1) == 0)
return x; // sqrt(+-0) = +-0
else if (ix0 < 0)
return (x - x)/(x - x); // sqrt(-ve) = sNaN
}
// normalize x
m = (ix0 >> 20);
if (m == 0) { // subnormal x
while (ix0 == 0) {
m -= 21;
ix0 |= (ix1 >>> 11); // unsigned shift
ix1 <<= 21;
}
for(i = 0; (ix0 & 0x0010_0000) == 0; i++) {
ix0 <<= 1;
}
m -= i-1;
ix0 |= (ix1 >>> (32 - i)); // unsigned shift
ix1 <<= i;
}
m -= 1023; // unbias exponent */
ix0 = (ix0 & 0x000f_ffff) | 0x0010_0000;
if ((m & 1) != 0){ // odd m, double x to make it even
ix0 += ix0 + ((ix1 & sign) >>> 31); // unsigned shift
ix1 += ix1;
}
m >>= 1; // m = [m/2]
// generate sqrt(x) bit by bit
ix0 += ix0 + ((ix1 & sign) >>> 31); // unsigned shift
ix1 += ix1;
q = q1 = s0 = s1 = 0; // [q,q1] = sqrt(x)
r = 0x0020_0000; // r = moving bit from right to left
while (r != 0) {
t = s0 + r;
if (t <= ix0) {
s0 = t + r;
ix0 -= t;
q += r;
}
ix0 += ix0 + ((ix1 & sign) >>> 31); // unsigned shift
ix1 += ix1;
r >>>= 1; // unsigned shift
}
r = sign;
while (r != 0) {
t1 = s1 + r;
t = s0;
if ((t < ix0) ||
((t == ix0) && (Integer.compareUnsigned(t1, ix1) <= 0 ))) { // t1 <= ix1
s1 = t1 + r;
if (((t1 & sign) == sign) && (s1 & sign) == 0) {
s0 += 1;
}
ix0 -= t;
if (Integer.compareUnsigned(ix1, t1) < 0) { // ix1 < t1
ix0 -= 1;
}
ix1 -= t1;
q1 += r;
}
ix0 += ix0 + ((ix1 & sign) >>> 31); // unsigned shift
ix1 += ix1;
r >>>= 1; // unsigned shift
}
// use floating add to find out rounding direction
if ((ix0 | ix1) != 0) {
z = 1.0 - tiny; // trigger inexact flag
if (z >= 1.0) {
z = 1.0 + tiny;
if (q1 == 0xffff_ffff) {
q1 = 0;
q += 1;
} else if (z > 1.0) {
if (q1 == 0xffff_fffe) {
q += 1;
}
q1 += 2;
} else {
q1 += (q1 & 1);
}
}
}
ix0 = (q >> 1) + 0x3fe0_0000;
ix1 = q1 >>> 1; // unsigned shift
if ((q & 1) == 1) {
ix1 |= sign;
}
ix0 += (m << 20);
return __HI_LO(ix0, ix1);
}
}
// The following comment is supplementary information from the FDLIBM sources.
/*
* Other methods (use floating-point arithmetic)
* -------------
* (This is a copy of a drafted paper by Prof W. Kahan
* and K.C. Ng, written in May, 1986)
*
* Two algorithms are given here to implement sqrt(x)
* (IEEE double precision arithmetic) in software.
* Both supply sqrt(x) correctly rounded. The first algorithm (in
* Section A) uses newton iterations and involves four divisions.
* The second one uses reciproot iterations to avoid division, but
* requires more multiplications. Both algorithms need the ability
* to chop results of arithmetic operations instead of round them,
* and the INEXACT flag to indicate when an arithmetic operation
* is executed exactly with no roundoff error, all part of the
* standard (IEEE 754-1985). The ability to perform shift, add,
* subtract and logical AND operations upon 32-bit words is needed
* too, though not part of the standard.
*
* A. sqrt(x) by Newton Iteration
*
* (1) Initial approximation
*
* Let x0 and x1 be the leading and the trailing 32-bit words of
* a floating point number x (in IEEE double format) respectively
*
* 1 11 52 ...widths
* ------------------------------------------------------
* x: |s| e | f |
* ------------------------------------------------------
* msb lsb msb lsb ...order
*
*
* ------------------------ ------------------------
* x0: |s| e | f1 | x1: | f2 |
* ------------------------ ------------------------
*
* By performing shifts and subtracts on x0 and x1 (both regarded
* as integers), we obtain an 8-bit approximation of sqrt(x) as
* follows.
*
* k := (x0>>1) + 0x1ff80000;
* y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits
* Here k is a 32-bit integer and T1[] is an integer array containing
* correction terms. Now magically the floating value of y (y's
* leading 32-bit word is y0, the value of its trailing word is 0)
* approximates sqrt(x) to almost 8-bit.
*
* Value of T1:
* static int T1[32]= {
* 0, 1024, 3062, 5746, 9193, 13348, 18162, 23592,
* 29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215,
* 83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581,
* 16499, 12183, 8588, 5674, 3403, 1742, 661, 130,};
*
* (2) Iterative refinement
*
* Apply Heron's rule three times to y, we have y approximates
* sqrt(x) to within 1 ulp (Unit in the Last Place):
*
* y := (y+x/y)/2 ... almost 17 sig. bits
* y := (y+x/y)/2 ... almost 35 sig. bits
* y := y-(y-x/y)/2 ... within 1 ulp
*
*
* Remark 1.
* Another way to improve y to within 1 ulp is:
*
* y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x)
* y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x)
*
* 2
* (x-y )*y
* y := y + 2* ---------- ...within 1 ulp
* 2
* 3y + x
*
*
* This formula has one division fewer than the one above; however,
* it requires more multiplications and additions. Also x must be
* scaled in advance to avoid spurious overflow in evaluating the
* expression 3y*y+x. Hence it is not recommended unless division
* is slow. If division is very slow, then one should use the
* reciproot algorithm given in section B.
*
* (3) Final adjustment
*
* By twiddling y's last bit it is possible to force y to be
* correctly rounded according to the prevailing rounding mode
* as follows. Let r and i be copies of the rounding mode and
* inexact flag before entering the square root program. Also we
* use the expression y+-ulp for the next representable floating
* numbers (up and down) of y. Note that y+-ulp = either fixed
* point y+-1, or multiply y by nextafter(1,+-inf) in chopped
* mode.
*
* I := FALSE; ... reset INEXACT flag I
* R := RZ; ... set rounding mode to round-toward-zero
* z := x/y; ... chopped quotient, possibly inexact
* If(not I) then { ... if the quotient is exact
* if(z=y) {
* I := i; ... restore inexact flag
* R := r; ... restore rounded mode
* return sqrt(x):=y.
* } else {
* z := z - ulp; ... special rounding
* }
* }
* i := TRUE; ... sqrt(x) is inexact
* If (r=RN) then z=z+ulp ... rounded-to-nearest
* If (r=RP) then { ... round-toward-+inf
* y = y+ulp; z=z+ulp;
* }
* y := y+z; ... chopped sum
* y0:=y0-0x00100000; ... y := y/2 is correctly rounded.
* I := i; ... restore inexact flag
* R := r; ... restore rounded mode
* return sqrt(x):=y.
*
* (4) Special cases
*
* Square root of +inf, +-0, or NaN is itself;
* Square root of a negative number is NaN with invalid signal.
*
*
* B. sqrt(x) by Reciproot Iteration
*
* (1) Initial approximation
*
* Let x0 and x1 be the leading and the trailing 32-bit words of
* a floating point number x (in IEEE double format) respectively
* (see section A). By performing shifs and subtracts on x0 and y0,
* we obtain a 7.8-bit approximation of 1/sqrt(x) as follows.
*
* k := 0x5fe80000 - (x0>>1);
* y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits
*
* Here k is a 32-bit integer and T2[] is an integer array
* containing correction terms. Now magically the floating
* value of y (y's leading 32-bit word is y0, the value of
* its trailing word y1 is set to zero) approximates 1/sqrt(x)
* to almost 7.8-bit.
*
* Value of T2:
* static int T2[64]= {
* 0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866,
* 0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f,
* 0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d,
* 0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0,
* 0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989,
* 0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd,
* 0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e,
* 0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,};
*
* (2) Iterative refinement
*
* Apply Reciproot iteration three times to y and multiply the
* result by x to get an approximation z that matches sqrt(x)
* to about 1 ulp. To be exact, we will have
* -1ulp < sqrt(x)-z<1.0625ulp.
*
* ... set rounding mode to Round-to-nearest
* y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x)
* y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x)
* ... special arrangement for better accuracy
* z := x*y ... 29 bits to sqrt(x), with z*y<1
* z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x)
*
* Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that
* (a) the term z*y in the final iteration is always less than 1;
* (b) the error in the final result is biased upward so that
* -1 ulp < sqrt(x) - z < 1.0625 ulp
* instead of |sqrt(x)-z|<1.03125ulp.
*
* (3) Final adjustment
*
* By twiddling y's last bit it is possible to force y to be
* correctly rounded according to the prevailing rounding mode
* as follows. Let r and i be copies of the rounding mode and
* inexact flag before entering the square root program. Also we
* use the expression y+-ulp for the next representable floating
* numbers (up and down) of y. Note that y+-ulp = either fixed
* point y+-1, or multiply y by nextafter(1,+-inf) in chopped
* mode.
*
* R := RZ; ... set rounding mode to round-toward-zero
* switch(r) {
* case RN: ... round-to-nearest
* if(x<= z*(z-ulp)...chopped) z = z - ulp; else
* if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp;
* break;
* case RZ:case RM: ... round-to-zero or round-to--inf
* R:=RP; ... reset rounding mod to round-to-+inf
* if(x