1 2 /* 3 * Copyright (c) 1998, 2001, Oracle and/or its affiliates. All rights reserved. 4 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. 5 * 6 * This code is free software; you can redistribute it and/or modify it 7 * under the terms of the GNU General Public License version 2 only, as 8 * published by the Free Software Foundation. Oracle designates this 9 * particular file as subject to the "Classpath" exception as provided 10 * by Oracle in the LICENSE file that accompanied this code. 11 * 12 * This code is distributed in the hope that it will be useful, but WITHOUT 13 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or 14 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License 15 * version 2 for more details (a copy is included in the LICENSE file that 16 * accompanied this code). 17 * 18 * You should have received a copy of the GNU General Public License version 19 * 2 along with this work; if not, write to the Free Software Foundation, 20 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. 21 * 22 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA 23 * or visit www.oracle.com if you need additional information or have any 24 * questions. 25 */ 26 27 /* __ieee754_hypot(x,y) 28 * 29 * Method : 30 * If (assume round-to-nearest) z=x*x+y*y 31 * has error less than sqrt(2)/2 ulp, than 32 * sqrt(z) has error less than 1 ulp (exercise). 33 * 34 * So, compute sqrt(x*x+y*y) with some care as 35 * follows to get the error below 1 ulp: 36 * 37 * Assume x>y>0; 38 * (if possible, set rounding to round-to-nearest) 39 * 1. if x > 2y use 40 * x1*x1+(y*y+(x2*(x+x1))) for x*x+y*y 41 * where x1 = x with lower 32 bits cleared, x2 = x-x1; else 42 * 2. if x <= 2y use 43 * t1*y1+((x-y)*(x-y)+(t1*y2+t2*y)) 44 * where t1 = 2x with lower 32 bits cleared, t2 = 2x-t1, 45 * y1= y with lower 32 bits chopped, y2 = y-y1. 46 * 47 * NOTE: scaling may be necessary if some argument is too 48 * large or too tiny 49 * 50 * Special cases: 51 * hypot(x,y) is INF if x or y is +INF or -INF; else 52 * hypot(x,y) is NAN if x or y is NAN. 53 * 54 * Accuracy: 55 * hypot(x,y) returns sqrt(x^2+y^2) with error less 56 * than 1 ulps (units in the last place) 57 */ 58 59 #include "fdlibm.h" 60 61 #ifdef __STDC__ 62 double __ieee754_hypot(double x, double y) 63 #else 64 double __ieee754_hypot(x,y) 65 double x, y; 66 #endif 67 { 68 double a=x,b=y,t1,t2,y1,y2,w; 69 int j,k,ha,hb; 70 71 ha = __HI(x)&0x7fffffff; /* high word of x */ 72 hb = __HI(y)&0x7fffffff; /* high word of y */ 73 if(hb > ha) {a=y;b=x;j=ha; ha=hb;hb=j;} else {a=x;b=y;} 74 __HI(a) = ha; /* a <- |a| */ 75 __HI(b) = hb; /* b <- |b| */ 76 if((ha-hb)>0x3c00000) {return a+b;} /* x/y > 2**60 */ 77 k=0; 78 if(ha > 0x5f300000) { /* a>2**500 */ 79 if(ha >= 0x7ff00000) { /* Inf or NaN */ 80 w = a+b; /* for sNaN */ 81 if(((ha&0xfffff)|__LO(a))==0) w = a; 82 if(((hb^0x7ff00000)|__LO(b))==0) w = b; 83 return w; 84 } 85 /* scale a and b by 2**-600 */ 86 ha -= 0x25800000; hb -= 0x25800000; k += 600; 87 __HI(a) = ha; 88 __HI(b) = hb; 89 } 90 if(hb < 0x20b00000) { /* b < 2**-500 */ 91 if(hb <= 0x000fffff) { /* subnormal b or 0 */ 92 if((hb|(__LO(b)))==0) return a; 93 t1=0; 94 __HI(t1) = 0x7fd00000; /* t1=2^1022 */ 95 b *= t1; 96 a *= t1; 97 k -= 1022; 98 } else { /* scale a and b by 2^600 */ 99 ha += 0x25800000; /* a *= 2^600 */ 100 hb += 0x25800000; /* b *= 2^600 */ 101 k -= 600; 102 __HI(a) = ha; 103 __HI(b) = hb; 104 } 105 } 106 /* medium size a and b */ 107 w = a-b; 108 if (w>b) { 109 t1 = 0; 110 __HI(t1) = ha; 111 t2 = a-t1; 112 w = sqrt(t1*t1-(b*(-b)-t2*(a+t1))); 113 } else { 114 a = a+a; 115 y1 = 0; 116 __HI(y1) = hb; 117 y2 = b - y1; 118 t1 = 0; 119 __HI(t1) = ha+0x00100000; 120 t2 = a - t1; 121 w = sqrt(t1*y1-(w*(-w)-(t1*y2+t2*b))); 122 } 123 if(k!=0) { 124 t1 = 1.0; 125 __HI(t1) += (k<<20); 126 return t1*w; 127 } else return w; 128 } | 1 /* 2 * Copyright (c) 1998, 2015, Oracle and/or its affiliates. All rights reserved. 3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. 4 * 5 * This code is free software; you can redistribute it and/or modify it 6 * under the terms of the GNU General Public License version 2 only, as 7 * published by the Free Software Foundation. Oracle designates this 8 * particular file as subject to the "Classpath" exception as provided 9 * by Oracle in the LICENSE file that accompanied this code. 10 * 11 * This code is distributed in the hope that it will be useful, but WITHOUT 12 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or 13 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License 14 * version 2 for more details (a copy is included in the LICENSE file that 15 * accompanied this code). 16 * 17 * You should have received a copy of the GNU General Public License version 18 * 2 along with this work; if not, write to the Free Software Foundation, 19 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. 20 * 21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA 22 * or visit www.oracle.com if you need additional information or have any 23 * questions. 24 */ 25 26 package java.lang; 27 28 /** 29 * Port of the "Freely Distributable Math Library", version 5.3, from C to Java. 30 * 31 * <p>The C version of fdlibm relied on the idiom of pointer aliasing 32 * a 64-bit double floating-point value as a two-element array of 33 * 32-bit integers and reading and writing the two halves of the 34 * double independently. This coding pattern was problematic to C 35 * optimizers and not directly expressible in Java. Therefore, rather 36 * than a memory level overlay, if portions of a double need to be 37 * operated on as integer values, the standard library methods for 38 * bitwise floating-point to integer conversion, 39 * Double.longBitsToDouble and Double.doubleToRawLongBits, are directly 40 * or indirectly used . 41 * 42 * <p>The C version of fdlibm also took some pains to signal the 43 * correct IEEE 754 exceptional conditions divide by zero, invalid, 44 * overflow and underflow. For example, overflow would be signaled by 45 * {@code huge * huge} where {@code huge} was a large constant that 46 * would overflow when squared. Since IEEE floating-point exceptional 47 * handling is not supported natively in the JVM, such coding patterns 48 * have been omitted from this port. For example, rather than {@code 49 * return huge * huge}, this port will use {@code return INFINITY}. 50 * 51 * <p>Various comparison and arithmetic operations in fdlibm could be 52 * done either based on the integer view of a value or directly on the 53 * floating-point representation. Which idiom is faster may depend on 54 * platform specific factors. However,for code clarity if no other 55 * reason, this port will favor expressing those semantics operations 56 * in terms of floating-point operations when convenient to do so. 57 */ 58 class FdLibm { 59 // Constants used by multiple algorithms 60 private static final double INFINITY = Double.POSITIVE_INFINITY; 61 62 private FdLibm() { 63 throw new UnsupportedOperationException("No FdLibm instances for you."); 64 } 65 66 /** 67 * Return the low-order 32 bits of the double argument as an int. 68 */ 69 private static int __LO(double x) { 70 long transducer = Double.doubleToRawLongBits(x); 71 return (int)transducer; 72 } 73 74 /** 75 * Return a double with its low-order bits of the second argument 76 * and the high-order bits of the first argument.. 77 */ 78 private static double __LO(double x, int low) { 79 long transX = Double.doubleToRawLongBits(x); 80 return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L)|low ); 81 } 82 83 /** 84 * Return the high-order 32 bits of the double argument as an int. 85 */ 86 private static int __HI(double x) { 87 long transducer = Double.doubleToRawLongBits(x); 88 return (int)(transducer >> 32); 89 } 90 91 /** 92 * Return a double with its high-order bits of the second argument 93 * and the low-order bits of the first argument.. 94 */ 95 private static double __HI(double x, int high) { 96 long transX = Double.doubleToRawLongBits(x); 97 return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL)|( ((long)high)) << 32 ); 98 } 99 100 /** 101 * hypot(x,y) 102 * 103 * Method : 104 * If (assume round-to-nearest) z = x*x + y*y 105 * has error less than sqrt(2)/2 ulp, than 106 * sqrt(z) has error less than 1 ulp (exercise). 107 * 108 * So, compute sqrt(x*x + y*y) with some care as 109 * follows to get the error below 1 ulp: 110 * 111 * Assume x > y > 0; 112 * (if possible, set rounding to round-to-nearest) 113 * 1. if x > 2y use 114 * x1*x1 + (y*y + (x2*(x + x1))) for x*x + y*y 115 * where x1 = x with lower 32 bits cleared, x2 = x - x1; else 116 * 2. if x <= 2y use 117 * t1*y1 + ((x-y) * (x-y) + (t1*y2 + t2*y)) 118 * where t1 = 2x with lower 32 bits cleared, t2 = 2x - t1, 119 * y1= y with lower 32 bits chopped, y2 = y - y1. 120 * 121 * NOTE: scaling may be necessary if some argument is too 122 * large or too tiny 123 * 124 * Special cases: 125 * hypot(x,y) is INF if x or y is +INF or -INF; else 126 * hypot(x,y) is NAN if x or y is NAN. 127 * 128 * Accuracy: 129 * hypot(x,y) returns sqrt(x^2 + y^2) with error less 130 * than 1 ulps (units in the last place) 131 */ 132 public static class Hypot { 133 public static final double TWO_MINUS_600 = 0x1.0p-600; 134 public static final double TWO_PLUS_600 = 0x1.0p+600; 135 136 public static double compute(double x, double y) { 137 double a = Math.abs(x); 138 double b = Math.abs(y); 139 140 if (!Double.isFinite(a) || !Double.isFinite(b)) { 141 if (a == INFINITY || b == INFINITY) 142 return INFINITY; 143 else 144 return a + b; // Propagate NaN significand bits 145 } 146 147 if (b > a) { 148 double tmp = a; 149 a = b; 150 b = tmp; 151 } 152 assert a >= b; 153 154 // Doing bitwise conversion after screening for NaN allows 155 // the code to not worry about the possibility of 156 // "negative" NaN values. 157 158 // Note: the uses of ha and hb in hypot could be 159 // eliminated by changing the relative magnitude 160 // comparison below to either a floating-point divide or a 161 // comparison of getExponent results coupled initializing 162 // t1 and t2 using a split generated by floating-point 163 // operations. The range filtering and exponent 164 // adjustments already done by hypot implies should a 165 // split would not need to worry about overflow or 166 // underflow cases. 167 168 int ha = __HI(a); // high word of a 169 int hb = __HI(b); // high word of b 170 171 if ((ha - hb) > 0x3c00000) { 172 return a + b; // x / y > 2**60 173 } 174 175 int k = 0; 176 if (a > 0x1.0p500) { // a > 2**500 177 // scale a and b by 2**-600 178 ha -= 0x25800000; 179 hb -= 0x25800000; 180 a = a * TWO_MINUS_600; 181 b = b * TWO_MINUS_600; 182 k += 600; 183 } 184 double t1, t2; 185 if (b < 0x1.0p-500) { // b < 2**-500 186 if (hb <= Double.MIN_NORMAL) { // subnormal b or 0 */ 187 if (b == 0.0) 188 return a; 189 t1 = 0x1.0p1022; // t1 = 2^1022 190 b *= t1; 191 a *= t1; 192 k -= 1022; 193 } else { // scale a and b by 2^600 194 ha += 0x25800000; // a *= 2^600 195 hb += 0x25800000; // b *= 2^600 196 a = a * TWO_PLUS_600; 197 b = b * TWO_PLUS_600; 198 k -= 600; 199 } 200 } 201 // medium size a and b 202 double w = a - b; 203 if (w > b) { 204 t1 = 0; 205 t1 = __HI(t1, ha); 206 t2 = a - t1; 207 w = Math.sqrt(t1*t1 - (b*(-b) - t2 * (a + t1))); 208 } else { 209 double y1, y2; 210 a = a + a; 211 y1 = 0; 212 y1 = __HI(y1, hb); 213 y2 = b - y1; 214 t1 = 0; 215 t1 = __HI(t1, ha + 0x00100000); 216 t2 = a - t1; 217 w = Math.sqrt(t1*y1 - (w*(-w) - (t1*y2 + t2*b))); 218 } 219 if (k != 0) { 220 return Math.powerOfTwoD(k) * w; 221 } else 222 return w; 223 } 224 } 225 226 /** 227 * Compute x**y 228 * n 229 * Method: Let x = 2 * (1+f) 230 * 1. Compute and return log2(x) in two pieces: 231 * log2(x) = w1 + w2, 232 * where w1 has 53 - 24 = 29 bit trailing zeros. 233 * 2. Perform y*log2(x) = n+y' by simulating muti-precision 234 * arithmetic, where |y'| <= 0.5. 235 * 3. Return x**y = 2**n*exp(y'*log2) 236 * 237 * Special cases: 238 * 1. (anything) ** 0 is 1 239 * 2. (anything) ** 1 is itself 240 * 3. (anything) ** NAN is NAN 241 * 4. NAN ** (anything except 0) is NAN 242 * 5. +-(|x| > 1) ** +INF is +INF 243 * 6. +-(|x| > 1) ** -INF is +0 244 * 7. +-(|x| < 1) ** +INF is +0 245 * 8. +-(|x| < 1) ** -INF is +INF 246 * 9. +-1 ** +-INF is NAN 247 * 10. +0 ** (+anything except 0, NAN) is +0 248 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 249 * 12. +0 ** (-anything except 0, NAN) is +INF 250 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF 251 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) 252 * 15. +INF ** (+anything except 0,NAN) is +INF 253 * 16. +INF ** (-anything except 0,NAN) is +0 254 * 17. -INF ** (anything) = -0 ** (-anything) 255 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) 256 * 19. (-anything except 0 and inf) ** (non-integer) is NAN 257 * 258 * Accuracy: 259 * pow(x,y) returns x**y nearly rounded. In particular 260 * pow(integer,integer) 261 * always returns the correct integer provided it is 262 * representable. 263 */ 264 public static class Pow { 265 public static strictfp double compute(final double x, final double y) { 266 double z; 267 double r, s, t, u, v, w; 268 int i, j, k, n; 269 270 // y == zero: x**0 = 1 271 if (y == 0.0) 272 return 1.0; 273 274 // +/-NaN return x + y to propagate NaN significands 275 if (Double.isNaN(x) || Double.isNaN(y)) 276 return x + y; 277 278 final double y_abs = Math.abs(y); 279 double x_abs = Math.abs(x); 280 // Special values of y 281 if (y == 2.0) { 282 return x * x; 283 } else if (y == 0.5) { 284 if (x >= -Double.MAX_VALUE) // Handle x == -infinity later 285 return Math.sqrt(x + 0.0); // Add 0.0 to properly handle x == -0.0 286 } else if (y_abs == 1.0) { // y is +/-1 287 return (y == 1.0) ? x : 1.0 / x; 288 } else if (y_abs == INFINITY) { // y is +/-infinity 289 if (x_abs == 1.0) 290 return y - y; // inf**+/-1 is NaN 291 else if (x_abs > 1.0) // (|x| > 1)**+/-inf = inf, 0 292 return (y >= 0) ? y : 0.0; 293 else // (|x| < 1)**-/+inf = inf, 0 294 return (y < 0) ? -y : 0.0; 295 } 296 297 final int hx = __HI(x); 298 int ix = hx & 0x7fffffff; 299 300 /* 301 * When x < 0, determine if y is an odd integer: 302 * y_is_int = 0 ... y is not an integer 303 * y_is_int = 1 ... y is an odd int 304 * y_is_int = 2 ... y is an even int 305 */ 306 int y_is_int = 0; 307 if (hx < 0) { 308 if (y_abs >= 0x1.0p53) // |y| >= 2^53 = 9.007199254740992E15 309 y_is_int = 2; // y is an even integer since ulp(2^53) = 2.0 310 else if (y_abs >= 1.0) { // |y| >= 1.0 311 long y_abs_as_long = (long) y_abs; 312 if ( ((double) y_abs_as_long) == y_abs) { 313 y_is_int = 2 - (int)(y_abs_as_long & 0x1L); 314 } 315 } 316 } 317 318 // Special value of x 319 if (x_abs == 0.0 || 320 x_abs == INFINITY || 321 x_abs == 1.0) { 322 z = x_abs; // x is +/-0, +/-inf, +/-1 323 if (y < 0.0) 324 z = 1.0/z; // z = (1/|x|) 325 if (hx < 0) { 326 if (((ix - 0x3ff00000) | y_is_int) == 0) { 327 z = (z-z)/(z-z); // (-1)**non-int is NaN 328 } else if (y_is_int == 1) 329 z = -1.0 * z; // (x < 0)**odd = -(|x|**odd) 330 } 331 return z; 332 } 333 334 n = (hx >> 31) + 1; 335 336 // (x < 0)**(non-int) is NaN 337 if ((n | y_is_int) == 0) 338 return (x-x)/(x-x); 339 340 s = 1.0; // s (sign of result -ve**odd) = -1 else = 1 341 if ( (n | (y_is_int - 1)) == 0) 342 s = -1.0; // (-ve)**(odd int) 343 344 double p_h, p_l, t1, t2; 345 // |y| is huge 346 if (y_abs > 0x1.0p31) { // if |y| > 2**31 347 final double INV_LN2 = 0x1.7154_7652_b82fep0; // 1.44269504088896338700e+00 = 1/ln2 348 final double INV_LN2_H = 0x1.715476p0; // 1.44269502162933349609e+00 = 24 bits of 1/ln2 349 final double INV_LN2_L = 0x1.4ae0_bf85_ddf44p-26; // 1.92596299112661746887e-08 = 1/ln2 tail 350 351 // Over/underflow if x is not close to one 352 if (x_abs < 0x1.fffffp-1) // |x| < 0.9999995231628418 353 return (y < 0.0) ? s * INFINITY : s * 0.0; 354 if (x_abs > 1.0) // |x| > 1.0 355 return (y > 0.0) ? s * INFINITY : s * 0.0; 356 /* 357 * now |1-x| is tiny <= 2**-20, sufficient to compute 358 * log(x) by x - x^2/2 + x^3/3 - x^4/4 359 */ 360 t = x_abs - 1.0; // t has 20 trailing zeros 361 w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25)); 362 u = INV_LN2_H * t; // INV_LN2_H has 21 sig. bits 363 v = t * INV_LN2_L - w * INV_LN2; 364 t1 = u + v; 365 t1 =__LO(t1, 0); 366 t2 = v - (t1 - u); 367 } else { 368 final double CP = 0x1.ec70_9dc3_a03fdp-1; // 9.61796693925975554329e-01 = 2/(3ln2) 369 final double CP_H = 0x1.ec709ep-1; // 9.61796700954437255859e-01 = (float)cp 370 final double CP_L = -0x1.e2fe_0145_b01f5p-28; // -7.02846165095275826516e-09 = tail of CP_H 371 372 double z_h, z_l, ss, s2, s_h, s_l, t_h, t_l; 373 n = 0; 374 // Take care of subnormal numbers 375 if (ix < 0x00100000) { 376 x_abs *= 0x1.0p53; // 2^53 = 9007199254740992.0 377 n -= 53; 378 ix = __HI(x_abs); 379 } 380 n += ((ix) >> 20) - 0x3ff; 381 j = ix & 0x000fffff; 382 // Determine interval 383 ix = j | 0x3ff00000; // Normalize ix 384 if (j <= 0x3988E) 385 k = 0; // |x| <sqrt(3/2) 386 else if (j < 0xBB67A) 387 k = 1; // |x| <sqrt(3) 388 else { 389 k = 0; 390 n += 1; 391 ix -= 0x00100000; 392 } 393 x_abs = __HI(x_abs, ix); 394 395 // Compute ss = s_h + s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) 396 397 final double BP[] = {1.0, 398 1.5}; 399 final double DP_H[] = {0.0, 400 0x1.2b80_34p-1}; // 5.84962487220764160156e-01 401 final double DP_L[] = {0.0, 402 0x1.cfde_b43c_fd006p-27};// 1.35003920212974897128e-08 403 404 // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3 405 final double L1 = 0x1.3333_3333_33303p-1; // 5.99999999999994648725e-01 406 final double L2 = 0x1.b6db_6db6_fabffp-2; // 4.28571428578550184252e-01 407 final double L3 = 0x1.5555_5518_f264dp-2; // 3.33333329818377432918e-01 408 final double L4 = 0x1.1746_0a91_d4101p-2; // 2.72728123808534006489e-01 409 final double L5 = 0x1.d864_a93c_9db65p-3; // 2.30660745775561754067e-01 410 final double L6 = 0x1.a7e2_84a4_54eefp-3; // 2.06975017800338417784e-01 411 u = x_abs - BP[k]; // BP[0]=1.0, BP[1]=1.5 412 v = 1.0 / (x_abs + BP[k]); 413 ss = u * v; 414 s_h = ss; 415 s_h = __LO(s_h, 0); 416 // t_h=x_abs + BP[k] High 417 t_h = 0.0; 418 t_h = __HI(t_h, ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18) ); 419 t_l = x_abs - (t_h - BP[k]); 420 s_l = v * ((u - s_h * t_h) - s_h * t_l); 421 // Compute log(x_abs) 422 s2 = ss * ss; 423 r = s2 * s2* (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6))))); 424 r += s_l * (s_h + ss); 425 s2 = s_h * s_h; 426 t_h = 3.0 + s2 + r; 427 t_h = __LO(t_h, 0); 428 t_l = r - ((t_h - 3.0) - s2); 429 // u+v = ss*(1+...) 430 u = s_h * t_h; 431 v = s_l * t_h + t_l * ss; 432 // 2/(3log2)*(ss + ...) 433 p_h = u + v; 434 p_h = __LO(p_h, 0); 435 p_l = v - (p_h - u); 436 z_h = CP_H * p_h; // CP_H + CP_L = 2/(3*log2) 437 z_l = CP_L * p_h + p_l * CP + DP_L[k]; 438 // log2(x_abs) = (ss + ..)*2/(3*log2) = n + DP_H + z_h + z_l 439 t = (double)n; 440 t1 = (((z_h + z_l) + DP_H[k]) + t); 441 t1 = __LO(t1, 0); 442 t2 = z_l - (((t1 - t) - DP_H[k]) - z_h); 443 } 444 445 // Split up y into (y1 + y2) and compute (y1 + y2) * (t1 + t2) 446 double y1 = y; 447 y1 = __LO(y1, 0); 448 p_l = (y - y1) * t1 + y * t2; 449 p_h = y1 * t1; 450 z = p_l + p_h; 451 j = __HI(z); 452 i = __LO(z); 453 if (j >= 0x40900000) { // z >= 1024 454 if (((j - 0x40900000) | i)!=0) // if z > 1024 455 return s * INFINITY; // Overflow 456 else { 457 final double OVT = 8.0085662595372944372e-0017; // -(1024-log2(ovfl+.5ulp)) 458 if (p_l + OVT > z - p_h) 459 return s * INFINITY; // Overflow 460 } 461 } else if ((j & 0x7fffffff) >= 0x4090cc00 ) { // z <= -1075 462 if (((j - 0xc090cc00) | i)!=0) // z < -1075 463 return s * 0.0; // Underflow 464 else { 465 if (p_l <= z - p_h) 466 return s * 0.0; // Underflow 467 } 468 } 469 /* 470 * Compute 2**(p_h+p_l) 471 */ 472 // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3 473 final double P1 = 0x1.5555_5555_5553ep-3; // 1.66666666666666019037e-01 474 final double P2 = -0x1.6c16_c16b_ebd93p-9; // -2.77777777770155933842e-03 475 final double P3 = 0x1.1566_aaf2_5de2cp-14; // 6.61375632143793436117e-05 476 final double P4 = -0x1.bbd4_1c5d_26bf1p-20; // -1.65339022054652515390e-06 477 final double P5 = 0x1.6376_972b_ea4d0p-25; // 4.13813679705723846039e-08 478 final double LG2 = 0x1.62e4_2fef_a39efp-1; // 6.93147180559945286227e-01 479 final double LG2_H = 0x1.62e43p-1; // 6.93147182464599609375e-01 480 final double LG2_L = -0x1.05c6_10ca_86c39p-29; // -1.90465429995776804525e-09 481 i = j & 0x7fffffff; 482 k = (i >> 20) - 0x3ff; 483 n = 0; 484 if (i > 0x3fe00000) { // if |z| > 0.5, set n = [z + 0.5] 485 n = j + (0x00100000 >> (k + 1)); 486 k = ((n & 0x7fffffff) >> 20) - 0x3ff; // new k for n 487 t = 0.0; 488 t = __HI(t, (n & ~(0x000fffff >> k)) ); 489 n = ((n & 0x000fffff) | 0x00100000) >> (20 - k); 490 if (j < 0) 491 n = -n; 492 p_h -= t; 493 } 494 t = p_l + p_h; 495 t = __LO(t, 0); 496 u = t * LG2_H; 497 v = (p_l - (t - p_h)) * LG2 + t * LG2_L; 498 z = u + v; 499 w = v - (z - u); 500 t = z * z; 501 t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5)))); 502 r = (z * t1)/(t1 - 2.0) - (w + z * w); 503 z = 1.0 - (r - z); 504 j = __HI(z); 505 j += (n << 20); 506 if ((j >> 20) <= 0) 507 z = Math.scalb(z, n); // subnormal output 508 else { 509 int z_hi = __HI(z); 510 z_hi += (n << 20); 511 z = __HI(z, z_hi); 512 } 513 return s * z; 514 } 515 } 516 } |