73 return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L)|low );
74 }
75
76 /**
77 * Return the high-order 32 bits of the double argument as an int.
78 */
79 private static int __HI(double x) {
80 long transducer = Double.doubleToRawLongBits(x);
81 return (int)(transducer >> 32);
82 }
83
84 /**
85 * Return a double with its high-order bits of the second argument
86 * and the low-order bits of the first argument..
87 */
88 private static double __HI(double x, int high) {
89 long transX = Double.doubleToRawLongBits(x);
90 return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL)|( ((long)high)) << 32 );
91 }
92
93 /**
94 * Compute x**y
95 * n
96 * Method: Let x = 2 * (1+f)
97 * 1. Compute and return log2(x) in two pieces:
98 * log2(x) = w1 + w2,
99 * where w1 has 53 - 24 = 29 bit trailing zeros.
100 * 2. Perform y*log2(x) = n+y' by simulating muti-precision
101 * arithmetic, where |y'| <= 0.5.
102 * 3. Return x**y = 2**n*exp(y'*log2)
103 *
104 * Special cases:
105 * 1. (anything) ** 0 is 1
106 * 2. (anything) ** 1 is itself
107 * 3. (anything) ** NAN is NAN
108 * 4. NAN ** (anything except 0) is NAN
109 * 5. +-(|x| > 1) ** +INF is +INF
110 * 6. +-(|x| > 1) ** -INF is +0
111 * 7. +-(|x| < 1) ** +INF is +0
112 * 8. +-(|x| < 1) ** -INF is +INF
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73 return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L)|low );
74 }
75
76 /**
77 * Return the high-order 32 bits of the double argument as an int.
78 */
79 private static int __HI(double x) {
80 long transducer = Double.doubleToRawLongBits(x);
81 return (int)(transducer >> 32);
82 }
83
84 /**
85 * Return a double with its high-order bits of the second argument
86 * and the low-order bits of the first argument..
87 */
88 private static double __HI(double x, int high) {
89 long transX = Double.doubleToRawLongBits(x);
90 return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL)|( ((long)high)) << 32 );
91 }
92
93 /**
94 * hypot(x,y)
95 *
96 * Method :
97 * If (assume round-to-nearest) z=x*x+y*y
98 * has error less than sqrt(2)/2 ulp, than
99 * sqrt(z) has error less than 1 ulp (exercise).
100 *
101 * So, compute sqrt(x*x+y*y) with some care as
102 * follows to get the error below 1 ulp:
103 *
104 * Assume x>y>0;
105 * (if possible, set rounding to round-to-nearest)
106 * 1. if x > 2y use
107 * x1*x1+(y*y+(x2*(x+x1))) for x*x+y*y
108 * where x1 = x with lower 32 bits cleared, x2 = x-x1; else
109 * 2. if x <= 2y use
110 * t1*y1+((x-y)*(x-y)+(t1*y2+t2*y))
111 * where t1 = 2x with lower 32 bits cleared, t2 = 2x-t1,
112 * y1= y with lower 32 bits chopped, y2 = y-y1.
113 *
114 * NOTE: scaling may be necessary if some argument is too
115 * large or too tiny
116 *
117 * Special cases:
118 * hypot(x,y) is INF if x or y is +INF or -INF; else
119 * hypot(x,y) is NAN if x or y is NAN.
120 *
121 * Accuracy:
122 * hypot(x,y) returns sqrt(x^2+y^2) with error less
123 * than 1 ulps (units in the last place)
124 */
125 public static class Hypot {
126 public static double compute(double x, double y) {
127 double a=x,b=y,t1,t2,y1,y2,w;
128 int j,k,ha,hb;
129
130 ha = __HI(x)&0x7fffffff; /* high word of x */
131 hb = __HI(y)&0x7fffffff; /* high word of y */
132 if(hb > ha) {a=y;b=x;j=ha; ha=hb;hb=j;} else {a=x;b=y;}
133 a = __HI(a, ha); /* a <- |a| */
134 b = __HI(b, hb); /* b <- |b| */
135 if((ha-hb)>0x3c00000) {return a+b;} /* x/y > 2**60 */
136 k=0;
137 if(ha > 0x5f300000) { /* a>2**500 */
138 if(ha >= 0x7ff00000) { /* Inf or NaN */
139 w = a+b; /* for sNaN */
140 if(((ha&0xfffff)|__LO(a))==0) w = a;
141 if(((hb^0x7ff00000)|__LO(b))==0) w = b;
142 return w;
143 }
144 /* scale a and b by 2**-600 */
145 ha -= 0x25800000; hb -= 0x25800000; k += 600;
146 a = __HI(a, ha);
147 b = __HI(b, hb);
148 }
149 if(hb < 0x20b00000) { /* b < 2**-500 */
150 if(hb <= 0x000fffff) { /* subnormal b or 0 */
151 if((hb|(__LO(b)))==0) return a;
152 t1=0;
153 t1 = __HI(t1, 0x7fd00000); /* t1=2^1022 */
154 b *= t1;
155 a *= t1;
156 k -= 1022;
157 } else { /* scale a and b by 2^600 */
158 ha += 0x25800000; /* a *= 2^600 */
159 hb += 0x25800000; /* b *= 2^600 */
160 k -= 600;
161 a = __HI(a, ha);
162 b = __HI(b, hb);
163 }
164 }
165 /* medium size a and b */
166 w = a-b;
167 if (w>b) {
168 t1 = 0;
169 t1 = __HI(t1, ha);
170 t2 = a-t1;
171 w = Math.sqrt(t1*t1-(b*(-b)-t2*(a+t1)));
172 } else {
173 a = a+a;
174 y1 = 0;
175 y1 = __HI(y1, hb);
176 y2 = b - y1;
177 t1 = 0;
178 t1 = __HI(t1, ha+0x00100000);
179 t2 = a - t1;
180 w = Math.sqrt(t1*y1-(w*(-w)-(t1*y2+t2*b)));
181 }
182 if(k!=0) {
183 t1 = 1.0;
184 int t1_hi = __HI(t1);
185 t1_hi += (k<<20);
186 t1 = __HI(t1, t1_hi);
187 return t1*w;
188 } else return w;
189 }
190 }
191
192 /**
193 * Compute x**y
194 * n
195 * Method: Let x = 2 * (1+f)
196 * 1. Compute and return log2(x) in two pieces:
197 * log2(x) = w1 + w2,
198 * where w1 has 53 - 24 = 29 bit trailing zeros.
199 * 2. Perform y*log2(x) = n+y' by simulating muti-precision
200 * arithmetic, where |y'| <= 0.5.
201 * 3. Return x**y = 2**n*exp(y'*log2)
202 *
203 * Special cases:
204 * 1. (anything) ** 0 is 1
205 * 2. (anything) ** 1 is itself
206 * 3. (anything) ** NAN is NAN
207 * 4. NAN ** (anything except 0) is NAN
208 * 5. +-(|x| > 1) ** +INF is +INF
209 * 6. +-(|x| > 1) ** -INF is +0
210 * 7. +-(|x| < 1) ** +INF is +0
211 * 8. +-(|x| < 1) ** -INF is +INF
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