56 */
57 private static int __HI(double x) {
58 long transducer = Double.doubleToRawLongBits(x);
59 return (int)(transducer >> 32);
60 }
61
62 /**
63 * Return a double with its high-order bits of the second argument
64 * and the low-order bits of the first argument..
65 */
66 private static double __HI(double x, int high) {
67 long transX = Double.doubleToRawLongBits(x);
68 return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL)|( ((long)high)) << 32 );
69 }
70
71 public static double hypot(double x, double y) {
72 return Hypot.compute(x, y);
73 }
74
75 /**
76 * hypot(x,y)
77 *
78 * Method :
79 * If (assume round-to-nearest) z = x*x + y*y
80 * has error less than sqrt(2)/2 ulp, than
81 * sqrt(z) has error less than 1 ulp (exercise).
82 *
83 * So, compute sqrt(x*x + y*y) with some care as
84 * follows to get the error below 1 ulp:
85 *
86 * Assume x > y > 0;
87 * (if possible, set rounding to round-to-nearest)
88 * 1. if x > 2y use
89 * x1*x1 + (y*y + (x2*(x + x1))) for x*x + y*y
90 * where x1 = x with lower 32 bits cleared, x2 = x - x1; else
91 * 2. if x <= 2y use
92 * t1*y1 + ((x-y) * (x-y) + (t1*y2 + t2*y))
93 * where t1 = 2x with lower 32 bits cleared, t2 = 2x - t1,
94 * y1= y with lower 32 bits chopped, y2 = y - y1.
95 *
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56 */
57 private static int __HI(double x) {
58 long transducer = Double.doubleToRawLongBits(x);
59 return (int)(transducer >> 32);
60 }
61
62 /**
63 * Return a double with its high-order bits of the second argument
64 * and the low-order bits of the first argument..
65 */
66 private static double __HI(double x, int high) {
67 long transX = Double.doubleToRawLongBits(x);
68 return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL)|( ((long)high)) << 32 );
69 }
70
71 public static double hypot(double x, double y) {
72 return Hypot.compute(x, y);
73 }
74
75 /**
76 * cbrt(x)
77 * Return cube root of x
78 */
79 public static class Cbrt {
80 // unsigned
81 private static final int B1 = 715094163; /* B1 = (682-0.03306235651)*2**20 */
82 private static final int B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */
83
84 private static final double C = 5.42857142857142815906e-01; /* 19/35 = 0x3FE15F15, 0xF15F15F1 */
85 private static final double D = -7.05306122448979611050e-01; /* -864/1225 = 0xBFE691DE, 0x2532C834 */
86 private static final double E = 1.41428571428571436819e+00; /* 99/70 = 0x3FF6A0EA, 0x0EA0EA0F */
87 private static final double F = 1.60714285714285720630e+00; /* 45/28 = 0x3FF9B6DB, 0x6DB6DB6E */
88 private static final double G = 3.57142857142857150787e-01; /* 5/14 = 0x3FD6DB6D, 0xB6DB6DB7 */
89
90 public static strictfp double compute(double x) {
91 int hx;
92 double r, s, t=0.0, w;
93 int sign; // unsigned
94
95 hx = __HI(x); // high word of x
96 sign = hx & 0x80000000; // sign= sign(x)
97 hx ^= sign;
98 if (hx >= 0x7ff00000)
99 return (x+x); // cbrt(NaN,INF) is itself
100 if ((hx | __LO(x)) == 0)
101 return(x); // cbrt(0) is itself
102
103 x = __HI(x, hx); // x <- |x|
104 // rough cbrt to 5 bits
105 if (hx < 0x00100000) { // subnormal number
106 t = __HI(t, 0x43500000); // set t= 2**54
107 t *= x;
108 t = __HI(t, __HI(t)/3+B2);
109 } else {
110 t = __HI(t, hx/3+B1);
111 }
112
113 // new cbrt to 23 bits, may be implemented in single precision
114 r = t * t/x;
115 s = C + r*t;
116 t *= G + F/(s + E + D/s);
117
118 // chopped to 20 bits and make it larger than cbrt(x)
119 t = __LO(t, 0);
120 t = __HI(t, __HI(t)+0x00000001);
121
122
123 // one step newton iteration to 53 bits with error less than 0.667 ulps
124 s = t * t; // t*t is exact
125 r = x / s;
126 w = t + t;
127 r= (r - t)/(w + r); // r-s is exact
128 t= t + t*r;
129
130 // retore the sign bit
131 t = __HI(t, __HI(t) | sign);
132 return(t);
133 }
134 }
135
136 /**
137 * hypot(x,y)
138 *
139 * Method :
140 * If (assume round-to-nearest) z = x*x + y*y
141 * has error less than sqrt(2)/2 ulp, than
142 * sqrt(z) has error less than 1 ulp (exercise).
143 *
144 * So, compute sqrt(x*x + y*y) with some care as
145 * follows to get the error below 1 ulp:
146 *
147 * Assume x > y > 0;
148 * (if possible, set rounding to round-to-nearest)
149 * 1. if x > 2y use
150 * x1*x1 + (y*y + (x2*(x + x1))) for x*x + y*y
151 * where x1 = x with lower 32 bits cleared, x2 = x - x1; else
152 * 2. if x <= 2y use
153 * t1*y1 + ((x-y) * (x-y) + (t1*y2 + t2*y))
154 * where t1 = 2x with lower 32 bits cleared, t2 = 2x - t1,
155 * y1= y with lower 32 bits chopped, y2 = y - y1.
156 *
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