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.
*/
class FdLibm {
// Constants used by multiple algorithms
private static final double INFINITY = Double.POSITIVE_INFINITY;
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 );
}
/**
* 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 );
}
/**
* cbrt(x)
* Return cube root of x
*/
public static class Cbrt {
// unsigned
private static final int B1 = 715094163; /* B1 = (682-0.03306235651)*2**20 */
private static final int B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */
private static final double C = 5.42857142857142815906e-01; /* 19/35 = 0x3FE15F15, 0xF15F15F1 */
private static final double D = -7.05306122448979611050e-01; /* -864/1225 = 0xBFE691DE, 0x2532C834 */
private static final double E = 1.41428571428571436819e+00; /* 99/70 = 0x3FF6A0EA, 0x0EA0EA0F */
private static final double F = 1.60714285714285720630e+00; /* 45/28 = 0x3FF9B6DB, 0x6DB6DB6E */
private static final double G = 3.57142857142857150787e-01; /* 5/14 = 0x3FD6DB6D, 0xB6DB6DB7 */
public static strictfp double compute(double x) {
int hx;
double t = 0.0;
double sign;
if (!Double.isFinite(x) || x == 0.0)
return x; // Handles signed zeros properly
sign = (x < 0.0) ? -1.0: 1.0;
x = Math.abs(x); // x <- |x|
hx = __HI(x); // high word of x
// Rough cbrt to 5 bits
if (hx < 0x00100000) { // subnormal number
t = __HI(t, 0x43500000); // set t= 2**54
t *= x;
t = __HI(t, __HI(t)/3 + B2);
} else {
t = __HI(t, hx/3 + B1);
}
// New cbrt to 23 bits, may be implemented in single precision
double r, s, w;
r = t * t/x;
s = C + r*t;
t *= G + F/(s + E + D/s);
// Chopped to 20 bits and make it larger than cbrt(x)
t = __LO(t, 0);
t = __HI(t, __HI(t) + 0x00000001);
// One step newton iteration to 53 bits with error less than 0.667 ulps
s = t * t; // t*t is exact
r = x / s;
w = t + t;
r = (r - t)/(w + r); // r-s is exact
t = t + t*r;
// Restore the original sign bit
return sign * t;
}
}
/**
* hypot(x,y)
*
* Method :
* If (assume round-to-nearest) z = x*x + y*y
* has error less than sqrt(2)/2 ulp, than
* sqrt(z) has error less than 1 ulp (exercise).
*
* So, compute sqrt(x*x + y*y) with some care as
* follows to get the error below 1 ulp:
*
* Assume x > y > 0;
* (if possible, set rounding to round-to-nearest)
* 1. if x > 2y use
* x1*x1 + (y*y + (x2*(x + x1))) for x*x + y*y
* where x1 = x with lower 32 bits cleared, x2 = x - x1; else
* 2. if x <= 2y use
* t1*y1 + ((x-y) * (x-y) + (t1*y2 + t2*y))
* where t1 = 2x with lower 32 bits cleared, t2 = 2x - t1,
* y1= y with lower 32 bits chopped, y2 = y - y1.
*
* NOTE: scaling may be necessary if some argument is too
* large or too tiny
*
* Special cases:
* hypot(x,y) is INF if x or y is +INF or -INF; else
* hypot(x,y) is NAN if x or y is NAN.
*
* Accuracy:
* hypot(x,y) returns sqrt(x^2 + y^2) with error less
* than 1 ulp (unit in the last place)
*/
public static class Hypot {
public static final double TWO_MINUS_600 = 0x1.0p-600;
public static final double TWO_PLUS_600 = 0x1.0p+600;
public static strictfp double compute(double x, double y) {
double a = Math.abs(x);
double b = Math.abs(y);
if (!Double.isFinite(a) || !Double.isFinite(b)) {
if (a == INFINITY || b == INFINITY)
return INFINITY;
else
return a + b; // Propagate NaN significand bits
}
if (b > a) {
double tmp = a;
a = b;
b = tmp;
}
assert a >= b;
// Doing bitwise conversion after screening for NaN allows
// the code to not worry about the possibility of
// "negative" NaN values.
// Note: the ha and hb variables are the high-order
// 32-bits of a and b stored as integer values. The ha and
// hb values are used first for a rough magnitude
// comparison of a and b and second for simulating higher
// precision by allowing a and b, respectively, to be
// decomposed into non-overlapping portions. Both of these
// uses could be eliminated. The magnitude comparison
// could be eliminated by extracting and comparing the
// exponents of a and b or just be performing a
// floating-point divide. Splitting a floating-point
// number into non-overlapping portions can be
// accomplished by judicious use of multiplies and
// additions. For details see T. J. Dekker, A Floating
// Point Technique for Extending the Available Precision ,
// Numerische Mathematik, vol. 18, 1971, pp.224-242 and
// subsequent work.
int ha = __HI(a); // high word of a
int hb = __HI(b); // high word of b
if ((ha - hb) > 0x3c00000) {
return a + b; // x / y > 2**60
}
int k = 0;
if (a > 0x1.00000_ffff_ffffp500) { // a > ~2**500
// scale a and b by 2**-600
ha -= 0x25800000;
hb -= 0x25800000;
a = a * TWO_MINUS_600;
b = b * TWO_MINUS_600;
k += 600;
}
double t1, t2;
if (b < 0x1.0p-500) { // b < 2**-500
if (b < Double.MIN_NORMAL) { // subnormal b or 0 */
if (b == 0.0)
return a;
t1 = 0x1.0p1022; // t1 = 2^1022
b *= t1;
a *= t1;
k -= 1022;
} else { // scale a and b by 2^600
ha += 0x25800000; // a *= 2^600
hb += 0x25800000; // b *= 2^600
a = a * TWO_PLUS_600;
b = b * TWO_PLUS_600;
k -= 600;
}
}
// medium size a and b
double w = a - b;
if (w > b) {
t1 = 0;
t1 = __HI(t1, ha);
t2 = a - t1;
w = Math.sqrt(t1*t1 - (b*(-b) - t2 * (a + t1)));
} else {
double y1, y2;
a = a + a;
y1 = 0;
y1 = __HI(y1, hb);
y2 = b - y1;
t1 = 0;
t1 = __HI(t1, ha + 0x00100000);
t2 = a - t1;
w = Math.sqrt(t1*y1 - (w*(-w) - (t1*y2 + t2*b)));
}
if (k != 0) {
return Math.powerOfTwoD(k) * w;
} else
return w;
}
}
/**
* Compute x**y
* n
* Method: Let x = 2 * (1+f)
* 1. Compute and return log2(x) in two pieces:
* log2(x) = w1 + w2,
* where w1 has 53 - 24 = 29 bit trailing zeros.
* 2. Perform y*log2(x) = n+y' by simulating multi-precision
* arithmetic, where |y'| <= 0.5.
* 3. Return x**y = 2**n*exp(y'*log2)
*
* Special cases:
* 1. (anything) ** 0 is 1
* 2. (anything) ** 1 is itself
* 3. (anything) ** NAN is NAN
* 4. NAN ** (anything except 0) is NAN
* 5. +-(|x| > 1) ** +INF is +INF
* 6. +-(|x| > 1) ** -INF is +0
* 7. +-(|x| < 1) ** +INF is +0
* 8. +-(|x| < 1) ** -INF is +INF
* 9. +-1 ** +-INF is NAN
* 10. +0 ** (+anything except 0, NAN) is +0
* 11. -0 ** (+anything except 0, NAN, odd integer) is +0
* 12. +0 ** (-anything except 0, NAN) is +INF
* 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
* 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
* 15. +INF ** (+anything except 0,NAN) is +INF
* 16. +INF ** (-anything except 0,NAN) is +0
* 17. -INF ** (anything) = -0 ** (-anything)
* 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
* 19. (-anything except 0 and inf) ** (non-integer) is NAN
*
* Accuracy:
* pow(x,y) returns x**y nearly rounded. In particular
* pow(integer,integer)
* always returns the correct integer provided it is
* representable.
*/
public static class Pow {
public static strictfp double compute(final double x, final double y) {
double z;
double r, s, t, u, v, w;
int i, j, k, n;
// y == zero: x**0 = 1
if (y == 0.0)
return 1.0;
// +/-NaN return x + y to propagate NaN significands
if (Double.isNaN(x) || Double.isNaN(y))
return x + y;
final double y_abs = Math.abs(y);
double x_abs = Math.abs(x);
// Special values of y
if (y == 2.0) {
return x * x;
} else if (y == 0.5) {
if (x >= -Double.MAX_VALUE) // Handle x == -infinity later
return Math.sqrt(x + 0.0); // Add 0.0 to properly handle x == -0.0
} else if (y_abs == 1.0) { // y is +/-1
return (y == 1.0) ? x : 1.0 / x;
} else if (y_abs == INFINITY) { // y is +/-infinity
if (x_abs == 1.0)
return y - y; // inf**+/-1 is NaN
else if (x_abs > 1.0) // (|x| > 1)**+/-inf = inf, 0
return (y >= 0) ? y : 0.0;
else // (|x| < 1)**-/+inf = inf, 0
return (y < 0) ? -y : 0.0;
}
final int hx = __HI(x);
int ix = hx & 0x7fffffff;
/*
* When x < 0, determine if y is an odd integer:
* y_is_int = 0 ... y is not an integer
* y_is_int = 1 ... y is an odd int
* y_is_int = 2 ... y is an even int
*/
int y_is_int = 0;
if (hx < 0) {
if (y_abs >= 0x1.0p53) // |y| >= 2^53 = 9.007199254740992E15
y_is_int = 2; // y is an even integer since ulp(2^53) = 2.0
else if (y_abs >= 1.0) { // |y| >= 1.0
long y_abs_as_long = (long) y_abs;
if ( ((double) y_abs_as_long) == y_abs) {
y_is_int = 2 - (int)(y_abs_as_long & 0x1L);
}
}
}
// Special value of x
if (x_abs == 0.0 ||
x_abs == INFINITY ||
x_abs == 1.0) {
z = x_abs; // x is +/-0, +/-inf, +/-1
if (y < 0.0)
z = 1.0/z; // z = (1/|x|)
if (hx < 0) {
if (((ix - 0x3ff00000) | y_is_int) == 0) {
z = (z-z)/(z-z); // (-1)**non-int is NaN
} else if (y_is_int == 1)
z = -1.0 * z; // (x < 0)**odd = -(|x|**odd)
}
return z;
}
n = (hx >> 31) + 1;
// (x < 0)**(non-int) is NaN
if ((n | y_is_int) == 0)
return (x-x)/(x-x);
s = 1.0; // s (sign of result -ve**odd) = -1 else = 1
if ( (n | (y_is_int - 1)) == 0)
s = -1.0; // (-ve)**(odd int)
double p_h, p_l, t1, t2;
// |y| is huge
if (y_abs > 0x1.00000_ffff_ffffp31) { // if |y| > ~2**31
final double INV_LN2 = 0x1.7154_7652_b82fep0; // 1.44269504088896338700e+00 = 1/ln2
final double INV_LN2_H = 0x1.715476p0; // 1.44269502162933349609e+00 = 24 bits of 1/ln2
final double INV_LN2_L = 0x1.4ae0_bf85_ddf44p-26; // 1.92596299112661746887e-08 = 1/ln2 tail
// Over/underflow if x is not close to one
if (x_abs < 0x1.fffff_0000_0000p-1) // |x| < ~0.9999995231628418
return (y < 0.0) ? s * INFINITY : s * 0.0;
if (x_abs > 0x1.00000_ffff_ffffp0) // |x| > ~1.0
return (y > 0.0) ? s * INFINITY : s * 0.0;
/*
* now |1-x| is tiny <= 2**-20, sufficient to compute
* log(x) by x - x^2/2 + x^3/3 - x^4/4
*/
t = x_abs - 1.0; // t has 20 trailing zeros
w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
u = INV_LN2_H * t; // INV_LN2_H has 21 sig. bits
v = t * INV_LN2_L - w * INV_LN2;
t1 = u + v;
t1 =__LO(t1, 0);
t2 = v - (t1 - u);
} else {
final double CP = 0x1.ec70_9dc3_a03fdp-1; // 9.61796693925975554329e-01 = 2/(3ln2)
final double CP_H = 0x1.ec709ep-1; // 9.61796700954437255859e-01 = (float)cp
final double CP_L = -0x1.e2fe_0145_b01f5p-28; // -7.02846165095275826516e-09 = tail of CP_H
double z_h, z_l, ss, s2, s_h, s_l, t_h, t_l;
n = 0;
// Take care of subnormal numbers
if (ix < 0x00100000) {
x_abs *= 0x1.0p53; // 2^53 = 9007199254740992.0
n -= 53;
ix = __HI(x_abs);
}
n += ((ix) >> 20) - 0x3ff;
j = ix & 0x000fffff;
// Determine interval
ix = j | 0x3ff00000; // Normalize ix
if (j <= 0x3988E)
k = 0; // |x|