1 /* 2 * Copyright (c) 1998, 2016, 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 /** 27 * A transliteration of the "Freely Distributable Math Library" 28 * algorithms from C into Java. That is, this port of the algorithms 29 * is as close to the C originals as possible while still being 30 * readable legal Java. 31 */ 32 public class FdlibmTranslit { 33 private FdlibmTranslit() { 34 throw new UnsupportedOperationException("No FdLibmTranslit instances for you."); 35 } 36 37 /** 38 * Return the low-order 32 bits of the double argument as an int. 39 */ 40 private static int __LO(double x) { 41 long transducer = Double.doubleToRawLongBits(x); 42 return (int)transducer; 43 } 44 45 /** 46 * Return a double with its low-order bits of the second argument 47 * and the high-order bits of the first argument.. 48 */ 49 private static double __LO(double x, int low) { 50 long transX = Double.doubleToRawLongBits(x); 51 return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L) | 52 (low & 0x0000_0000_FFFF_FFFFL)); 53 } 54 55 /** 56 * Return the high-order 32 bits of the double argument as an int. 57 */ 58 private static int __HI(double x) { 59 long transducer = Double.doubleToRawLongBits(x); 60 return (int)(transducer >> 32); 61 } 62 63 /** 64 * Return a double with its high-order bits of the second argument 65 * and the low-order bits of the first argument.. 66 */ 67 private static double __HI(double x, int high) { 68 long transX = Double.doubleToRawLongBits(x); 69 return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL) | 70 ( ((long)high)) << 32 ); 71 } 72 73 public static double hypot(double x, double y) { 74 return Hypot.compute(x, y); 75 } 76 77 /** 78 * cbrt(x) 79 * Return cube root of x 80 */ 81 public static class Cbrt { 82 // unsigned 83 private static final int B1 = 715094163; /* B1 = (682-0.03306235651)*2**20 */ 84 private static final int B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */ 85 86 private static final double C = 5.42857142857142815906e-01; /* 19/35 = 0x3FE15F15, 0xF15F15F1 */ 87 private static final double D = -7.05306122448979611050e-01; /* -864/1225 = 0xBFE691DE, 0x2532C834 */ 88 private static final double E = 1.41428571428571436819e+00; /* 99/70 = 0x3FF6A0EA, 0x0EA0EA0F */ 89 private static final double F = 1.60714285714285720630e+00; /* 45/28 = 0x3FF9B6DB, 0x6DB6DB6E */ 90 private static final double G = 3.57142857142857150787e-01; /* 5/14 = 0x3FD6DB6D, 0xB6DB6DB7 */ 91 92 public static strictfp double compute(double x) { 93 int hx; 94 double r, s, t=0.0, w; 95 int sign; // unsigned 96 97 hx = __HI(x); // high word of x 98 sign = hx & 0x80000000; // sign= sign(x) 99 hx ^= sign; 100 if (hx >= 0x7ff00000) 101 return (x+x); // cbrt(NaN,INF) is itself 102 if ((hx | __LO(x)) == 0) 103 return(x); // cbrt(0) is itself 104 105 x = __HI(x, hx); // x <- |x| 106 // rough cbrt to 5 bits 107 if (hx < 0x00100000) { // subnormal number 108 t = __HI(t, 0x43500000); // set t= 2**54 109 t *= x; 110 t = __HI(t, __HI(t)/3+B2); 111 } else { 112 t = __HI(t, hx/3+B1); 113 } 114 115 // new cbrt to 23 bits, may be implemented in single precision 116 r = t * t/x; 117 s = C + r*t; 118 t *= G + F/(s + E + D/s); 119 120 // chopped to 20 bits and make it larger than cbrt(x) 121 t = __LO(t, 0); 122 t = __HI(t, __HI(t)+0x00000001); 123 124 125 // one step newton iteration to 53 bits with error less than 0.667 ulps 126 s = t * t; // t*t is exact 127 r = x / s; 128 w = t + t; 129 r= (r - t)/(w + r); // r-s is exact 130 t= t + t*r; 131 132 // retore the sign bit 133 t = __HI(t, __HI(t) | sign); 134 return(t); 135 } 136 } 137 138 /** 139 * hypot(x,y) 140 * 141 * Method : 142 * If (assume round-to-nearest) z = x*x + y*y 143 * has error less than sqrt(2)/2 ulp, than 144 * sqrt(z) has error less than 1 ulp (exercise). 145 * 146 * So, compute sqrt(x*x + y*y) with some care as 147 * follows to get the error below 1 ulp: 148 * 149 * Assume x > y > 0; 150 * (if possible, set rounding to round-to-nearest) 151 * 1. if x > 2y use 152 * x1*x1 + (y*y + (x2*(x + x1))) for x*x + y*y 153 * where x1 = x with lower 32 bits cleared, x2 = x - x1; else 154 * 2. if x <= 2y use 155 * t1*y1 + ((x-y) * (x-y) + (t1*y2 + t2*y)) 156 * where t1 = 2x with lower 32 bits cleared, t2 = 2x - t1, 157 * y1= y with lower 32 bits chopped, y2 = y - y1. 158 * 159 * NOTE: scaling may be necessary if some argument is too 160 * large or too tiny 161 * 162 * Special cases: 163 * hypot(x,y) is INF if x or y is +INF or -INF; else 164 * hypot(x,y) is NAN if x or y is NAN. 165 * 166 * Accuracy: 167 * hypot(x,y) returns sqrt(x^2 + y^2) with error less 168 * than 1 ulps (units in the last place) 169 */ 170 static class Hypot { 171 public static double compute(double x, double y) { 172 double a = x; 173 double b = y; 174 double t1, t2, y1, y2, w; 175 int j, k, ha, hb; 176 177 ha = __HI(x) & 0x7fffffff; // high word of x 178 hb = __HI(y) & 0x7fffffff; // high word of y 179 if(hb > ha) { 180 a = y; 181 b = x; 182 j = ha; 183 ha = hb; 184 hb = j; 185 } else { 186 a = x; 187 b = y; 188 } 189 a = __HI(a, ha); // a <- |a| 190 b = __HI(b, hb); // b <- |b| 191 if ((ha - hb) > 0x3c00000) { 192 return a + b; // x / y > 2**60 193 } 194 k=0; 195 if (ha > 0x5f300000) { // a>2**500 196 if (ha >= 0x7ff00000) { // Inf or NaN 197 w = a + b; // for sNaN 198 if (((ha & 0xfffff) | __LO(a)) == 0) 199 w = a; 200 if (((hb ^ 0x7ff00000) | __LO(b)) == 0) 201 w = b; 202 return w; 203 } 204 // scale a and b by 2**-600 205 ha -= 0x25800000; 206 hb -= 0x25800000; 207 k += 600; 208 a = __HI(a, ha); 209 b = __HI(b, hb); 210 } 211 if (hb < 0x20b00000) { // b < 2**-500 212 if (hb <= 0x000fffff) { // subnormal b or 0 */ 213 if ((hb | (__LO(b))) == 0) 214 return a; 215 t1 = 0; 216 t1 = __HI(t1, 0x7fd00000); // t1=2^1022 217 b *= t1; 218 a *= t1; 219 k -= 1022; 220 } else { // scale a and b by 2^600 221 ha += 0x25800000; // a *= 2^600 222 hb += 0x25800000; // b *= 2^600 223 k -= 600; 224 a = __HI(a, ha); 225 b = __HI(b, hb); 226 } 227 } 228 // medium size a and b 229 w = a - b; 230 if (w > b) { 231 t1 = 0; 232 t1 = __HI(t1, ha); 233 t2 = a - t1; 234 w = Math.sqrt(t1*t1 - (b*(-b) - t2 * (a + t1))); 235 } else { 236 a = a + a; 237 y1 = 0; 238 y1 = __HI(y1, hb); 239 y2 = b - y1; 240 t1 = 0; 241 t1 = __HI(t1, ha + 0x00100000); 242 t2 = a - t1; 243 w = Math.sqrt(t1*y1 - (w*(-w) - (t1*y2 + t2*b))); 244 } 245 if (k != 0) { 246 t1 = 1.0; 247 int t1_hi = __HI(t1); 248 t1_hi += (k << 20); 249 t1 = __HI(t1, t1_hi); 250 return t1 * w; 251 } else 252 return w; 253 } 254 } 255 256 /** 257 * Returns the exponential of x. 258 * 259 * Method 260 * 1. Argument reduction: 261 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. 262 * Given x, find r and integer k such that 263 * 264 * x = k*ln2 + r, |r| <= 0.5*ln2. 265 * 266 * Here r will be represented as r = hi-lo for better 267 * accuracy. 268 * 269 * 2. Approximation of exp(r) by a special rational function on 270 * the interval [0,0.34658]: 271 * Write 272 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... 273 * We use a special Reme algorithm on [0,0.34658] to generate 274 * a polynomial of degree 5 to approximate R. The maximum error 275 * of this polynomial approximation is bounded by 2**-59. In 276 * other words, 277 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 278 * (where z=r*r, and the values of P1 to P5 are listed below) 279 * and 280 * | 5 | -59 281 * | 2.0+P1*z+...+P5*z - R(z) | <= 2 282 * | | 283 * The computation of exp(r) thus becomes 284 * 2*r 285 * exp(r) = 1 + ------- 286 * R - r 287 * r*R1(r) 288 * = 1 + r + ----------- (for better accuracy) 289 * 2 - R1(r) 290 * where 291 * 2 4 10 292 * R1(r) = r - (P1*r + P2*r + ... + P5*r ). 293 * 294 * 3. Scale back to obtain exp(x): 295 * From step 1, we have 296 * exp(x) = 2^k * exp(r) 297 * 298 * Special cases: 299 * exp(INF) is INF, exp(NaN) is NaN; 300 * exp(-INF) is 0, and 301 * for finite argument, only exp(0)=1 is exact. 302 * 303 * Accuracy: 304 * according to an error analysis, the error is always less than 305 * 1 ulp (unit in the last place). 306 * 307 * Misc. info. 308 * For IEEE double 309 * if x > 7.09782712893383973096e+02 then exp(x) overflow 310 * if x < -7.45133219101941108420e+02 then exp(x) underflow 311 * 312 * Constants: 313 * The hexadecimal values are the intended ones for the following 314 * constants. The decimal values may be used, provided that the 315 * compiler will convert from decimal to binary accurately enough 316 * to produce the hexadecimal values shown. 317 */ 318 static class Exp { 319 private static final double one = 1.0; 320 private static final double[] halF = {0.5,-0.5,}; 321 private static final double huge = 1.0e+300; 322 private static final double twom1000= 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0*/ 323 private static final double o_threshold= 7.09782712893383973096e+02; /* 0x40862E42, 0xFEFA39EF */ 324 private static final double u_threshold= -7.45133219101941108420e+02; /* 0xc0874910, 0xD52D3051 */ 325 private static final double[] ln2HI ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ 326 -6.93147180369123816490e-01}; /* 0xbfe62e42, 0xfee00000 */ 327 private static final double[] ln2LO ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ 328 -1.90821492927058770002e-10,}; /* 0xbdea39ef, 0x35793c76 */ 329 private static final double invln2 = 1.44269504088896338700e+00; /* 0x3ff71547, 0x652b82fe */ 330 private static final double P1 = 1.66666666666666019037e-01; /* 0x3FC55555, 0x5555553E */ 331 private static final double P2 = -2.77777777770155933842e-03; /* 0xBF66C16C, 0x16BEBD93 */ 332 private static final double P3 = 6.61375632143793436117e-05; /* 0x3F11566A, 0xAF25DE2C */ 333 private static final double P4 = -1.65339022054652515390e-06; /* 0xBEBBBD41, 0xC5D26BF1 */ 334 private static final double P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ 335 336 public static strictfp double compute(double x) { 337 double y,hi=0,lo=0,c,t; 338 int k=0,xsb; 339 /*unsigned*/ int hx; 340 341 hx = __HI(x); /* high word of x */ 342 xsb = (hx>>31)&1; /* sign bit of x */ 343 hx &= 0x7fffffff; /* high word of |x| */ 344 345 /* filter out non-finite argument */ 346 if(hx >= 0x40862E42) { /* if |x|>=709.78... */ 347 if(hx>=0x7ff00000) { 348 if(((hx&0xfffff)|__LO(x))!=0) 349 return x+x; /* NaN */ 350 else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */ 351 } 352 if(x > o_threshold) return huge*huge; /* overflow */ 353 if(x < u_threshold) return twom1000*twom1000; /* underflow */ 354 } 355 356 /* argument reduction */ 357 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 358 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 359 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb; 360 } else { 361 k = (int)(invln2*x+halF[xsb]); 362 t = k; 363 hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */ 364 lo = t*ln2LO[0]; 365 } 366 x = hi - lo; 367 } 368 else if(hx < 0x3e300000) { /* when |x|<2**-28 */ 369 if(huge+x>one) return one+x;/* trigger inexact */ 370 } 371 else k = 0; 372 373 /* x is now in primary range */ 374 t = x*x; 375 c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 376 if(k==0) return one-((x*c)/(c-2.0)-x); 377 else y = one-((lo-(x*c)/(2.0-c))-hi); 378 if(k >= -1021) { 379 y = __HI(y, __HI(y) + (k<<20)); /* add k to y's exponent */ 380 return y; 381 } else { 382 y = __HI(y, __HI(y) + ((k+1000)<<20));/* add k to y's exponent */ 383 return y*twom1000; 384 } 385 } 386 } 387 }