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 r, s, t=0.0, w;
+ int sign; // unsigned
+
+ hx = __HI(x); // high word of x
+ sign = hx & 0x80000000; // sign= sign(x)
+ hx ^= sign;
+ if (hx >= 0x7ff00000)
+ return (x+x); // cbrt(NaN,INF) is itself
+ if ((hx | __LO(x)) == 0)
+ return(x); // cbrt(0) is itself
+
+ x = __HI(x, hx); // x <- |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
+ 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;
+
+ // retore the sign bit
+ t = __HI(t, __HI(t) | sign);
+ return(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|