001/* 002 * Units of Measurement Reference Implementation 003 * Copyright (c) 2005-2017, Jean-Marie Dautelle, Werner Keil, V2COM. 004 * 005 * All rights reserved. 006 * 007 * Redistribution and use in source and binary forms, with or without modification, 008 * are permitted provided that the following conditions are met: 009 * 010 * 1. Redistributions of source code must retain the above copyright notice, 011 * this list of conditions and the following disclaimer. 012 * 013 * 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions 014 * and the following disclaimer in the documentation and/or other materials provided with the distribution. 015 * 016 * 3. Neither the name of JSR-363 nor the names of its contributors may be used to endorse or promote products 017 * derived from this software without specific prior written permission. 018 * 019 * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" 020 * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, 021 * THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 022 * ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE 023 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES 024 * (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; 025 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED 026 * AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT 027 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, 028 * EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 029 */ 030package tec.units.ri.internal; 031 032/* 033 * Ported from the Sun Microsystems FDLIBM C-library. 034 * (Freely Distributable Library for Math) 035 * ==================================================== 036 * Portions Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. 037 * 038 * Permission to use, copy, modify, and distribute this 039 * software is freely granted, provided that this notice 040 * is preserved. 041 * ==================================================== 042 */ 043 044/** 045 * MathUtil for Java ME. This fills the gap in Java ME Math with a port of Sun's public FDLIBM C-library for IEEE-754. 046 * 047 * @author kmashint 048 * 049 * @see http://www.netlib.org/fdlibm/readme For the Freely Distributable C-library conforming to IEEE-754 floating point math. 050 * @see http://web.mit.edu/source/third/gcc/libjava/java/lang/ For the GNU C variant of the same IEEE-754 routines. 051 * @see http://www.dclausen.net/projects/microfloat/ Another take on the IEEE-754 routines. 052 * @see http://real-java.sourceforge.net/Real.html Yet another take on the IEEE-754 routines. 053 * @see http ://today.java.net/pub/a/today/2007/11/06/creating-java-me-math-pow-method .html For other approximations. 054 * @see http ://martin.ankerl.com/2007/10/04/optimized-pow-approximation-for-java- and-c-c/ For fast but rough approximations. 055 * @see http ://martin.ankerl.com/2007/02/11/optimized-exponential-functions-for-java / For more fast but rough approximations. 056 */ 057public abstract class MathUtil { 058 059 /* Common constants. */ 060 061 private static final double zero = 0.0, one = 1.0, two = 2.0, tiny = 1.0e-300, huge = 1.0e+300, two53 = 9007199254740992.0, /* 062 * 0x43400000 */ 063 two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */ 064 twom54 = 5.55111512312578270212e-17, /* 0x3C900000, 0x00000000 */ 065 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 066 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 067 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 068 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 069 P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ 070 071 // private static final double pio2_hi = 1.57079632679489655800e+00; 072 /* 073 * 0x3FF921FB, 074 * 0x54442D18 075 */ 076 // pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */ 077 // pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ 078 /* coefficient for R(x^2) */ 079 // pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */ 080 // pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */ 081 // pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */ 082 // pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */ 083 // pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */ 084 // pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */ 085 // qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */ 086 // qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */ 087 // qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */ 088 // qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */ 089 090 private static final double pi_o_4 = 7.8539816339744827900E-01, /* 091 * 0x3FE921FB, 092 * 0x54442D18 093 */ 094 pi_o_2 = 1.5707963267948965580E+00, /* 0x3FF921FB, 0x54442D18 */ 095 pi = 3.1415926535897931160E+00, /* 0x400921FB, 0x54442D18 */ 096 pi_lo = 1.2246467991473531772E-16; /* 0x3CA1A626, 0x33145C07 */ 097 098 private static final long HI_MASK = 0xffffffff00000000L, LO_MASK = 0x00000000ffffffffL; 099 100 private static final int HI_SHIFT = 32; 101 102 /** 103 * Return Math.E to the exponent a. This in turn uses ieee7854_exp(double). 104 */ 105 public static final double exp(double a) { 106 return ieee754_exp(a); 107 } 108 109 /** 110 * Return the natural logarithm, ln(a), as it relates to Math.E. This in turn uses ieee7854_log(double). 111 */ 112 public static final double log(double a) { 113 return ieee754_log(a); 114 } 115 116 /** 117 * Return a to the power of b, sometimes written as a ** b but not to be confused with the bitwise ^ operator. This in turn uses 118 * ieee7854_log(double). 119 */ 120 public static final double pow(double a, double b) { 121 return ieee754_pow(a, b); 122 } 123 124 /** 125 * Return the arcsine of a. 126 */ 127 /* public static final double asin(double a) { 128 return ieee754_asin(a); 129 } 130 */ 131 /** 132 * Return the arccosine of a. 133 */ 134 /*private static final double acos(double a) { 135 return ieee754_acos(a); 136 }*/ 137 138 /** 139 * Return the arctangent of a, call it b, where a = tan(b). 140 */ 141 public static final double atan(double a) { 142 return ieee754_atan(a); 143 } 144 145 /** 146 * For any real arguments x and y not both equal to zero, atan2(y, x) is the angle in radians between the positive x-axis of a plane and the point 147 * given by the coordinates (x, y) on it. The angle is positive for counter-clockwise angles (upper half-plane, y > 0), and negative for clockwise 148 * angles (lower half-plane, y < 0). This in turn uses ieee7854_arctan2(double). 149 */ 150 public static final double atan2(double b, double a) { 151 return ieee754_atan2(a, b); 152 } 153 154 /* 155 * __ieee754_exp(x) Returns the exponential of x. 156 * 157 * Method 1. Argument reduction: Reduce x to an r so that |r| <= 0.5*ln2 ~ 158 * 0.34658. Given x, find r and integer k such that 159 * 160 * x = k*ln2 + r, |r| <= 0.5*ln2. 161 * 162 * Here r will be represented as r = hi-lo for better accuracy. 163 * 164 * 2. Approximation of exp(r) by a special rational function on the interval 165 * [0,0.34658]: Write R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - 166 * r**4/360 + ... We use a special Remes algorithm on [0,0.34658] to 167 * generate a polynomial of degree 5 to approximate R. The maximum error of 168 * this polynomial approximation is bounded by 2**-59. In other words, R(z) 169 * ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 (where z=r*r, and 170 * the values of P1 to P5 are listed below) and | 5 | -59 | 171 * 2.0+P1*z+...+P5*z - R(z) | <= 2 | | The computation of exp(r) thus 172 * becomes 2*r exp(r) = 1 + ------- R - r r*R1(r) = 1 + r + ----------- (for 173 * better accuracy) 2 - R1(r) where 2 4 10 R1(r) = r - (P1*r + P2*r + ... + 174 * P5*r ). 175 * 176 * 3. Scale back to obtain exp(x): From step 1, we have exp(x) = 2^k * 177 * exp(r) 178 * 179 * Special cases: exp(INF) is INF, exp(NaN) is NaN; exp(-INF) is 0, and for 180 * finite argument, only exp(0)=1 is exact. 181 * 182 * Accuracy: according to an error analysis, the error is always less than 1 183 * ulp (unit in the last place). 184 * 185 * Misc. info. For IEEE double if x > 7.09782712893383973096e+02 then exp(x) 186 * overflow if x < -7.45133219101941108420e+02 then exp(x) underflow 187 * 188 * Constants: The hexadecimal values are the intended ones for the following 189 * constants. The decimal values may be used, provided that the compiler 190 * will convert from decimal to binary accurately enough to produce the 191 * hexadecimal values shown. 192 */ 193 194 private static final double twom1000 = 9.33263618503218878990e-302, /* 195 * 2**-1000 196 * = 197 * 0x01700000 198 * ,0 199 */ 200 o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ 201 u_threshold = -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ 202 invln2 = 1.44269504088896338700e+00; /* 0x3ff71547, 0x652b82fe */ 203 204 private static final double[] halF = new double[] { 0.5, -0.5 }, ln2HI = new double[] { 6.93147180369123816490e-01, /* 205 * 0x3fe62e42, 206 * 0xfee00000 207 */ 208 -6.93147180369123816490e-01 }, /* 0xbfe62e42, 0xfee00000 */ 209 ln2LO = new double[] { 1.90821492927058770002e-10, /* 210 * 0x3dea39ef, 211 * 0x35793c76 212 */ 213 -1.90821492927058770002e-10 }; /* 0xbdea39ef, 0x35793c76 */ 214 215 private static final double ieee754_exp(double x) { 216 double y, c, t; 217 double hi = 0, lo = 0; 218 int k = 0; 219 int xsb, hx, lx; 220 long yl; 221 long xl = Double.doubleToLongBits(x); 222 223 hx = (int) ((long) xl >>> HI_SHIFT); /* high word of x */ 224 xsb = (hx >> 31) & 1; /* sign bit of x */ 225 hx &= 0x7fffffff; /* high word of |x| */ 226 227 /* filter out non-finite argument */ 228 if (hx >= 0x40862E42) { /* if |x|>=709.78... */ 229 if (hx >= 0x7ff00000) { 230 lx = (int) ((long) xl & LO_MASK); /* low word of x */ 231 if (((hx & 0xfffff) | lx) != 0) 232 return x + x; /* NaN */ 233 else 234 return (xsb == 0) ? x : 0.0; /* exp(+-inf)={inf,0} */ 235 } 236 if (x > o_threshold) 237 return huge * huge; /* overflow */ 238 if (x < u_threshold) 239 return twom1000 * twom1000; /* underflow */ 240 } 241 242 /* argument reduction */ 243 if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ 244 if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ 245 hi = x - ln2HI[xsb]; 246 lo = ln2LO[xsb]; 247 k = 1 - xsb - xsb; 248 } else { 249 k = (int) (invln2 * x + halF[xsb]); 250 t = k; 251 hi = x - t * ln2HI[0]; /* t*ln2HI is exact here */ 252 lo = t * ln2LO[0]; 253 } 254 x = hi - lo; 255 } else if (hx < 0x3e300000) { /* when |x|<2**-28 */ 256 if (huge + x > one) 257 return one + x;/* trigger inexact */ 258 } 259 // else k = 0; // handled at declaration 260 261 /* x is now in primary range */ 262 t = x * x; 263 c = x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5)))); 264 if (k == 0) 265 return one - ((x * c) / (c - 2.0) - x); 266 else 267 y = one - ((lo - (x * c) / (2.0 - c)) - hi); 268 yl = Double.doubleToLongBits(y); 269 if (k >= -1021) { 270 yl += ((long) k << (20 + HI_SHIFT)); /* add k to y's exponent */ 271 return Double.longBitsToDouble(yl); 272 } else { 273 yl += ((long) (k + 1000) << (20 + HI_SHIFT));/* add k to y's exponent */ 274 return Double.longBitsToDouble(yl) * twom1000; 275 } 276 } 277 278 /* 279 * __ieee754_log(x) Return the logrithm of x 280 * 281 * Method : 1. Argument Reduction: find k and f such that x = 2^k * (1+f), 282 * where sqrt(2)/2 < 1+f < sqrt(2) . 283 * 284 * 2. Approximation of log(1+f). Let s = f/(2+f) ; based on log(1+f) = 285 * log(1+s) - log(1-s) = 2s + 2/3 s**3 + 2/5 s**5 + ....., = 2s + s*R We use 286 * a special Reme algorithm on [0,0.1716] to generate a polynomial of degree 287 * 14 to approximate R The maximum error of this polynomial approximation is 288 * bounded by 2**-58.45. In other words, 2 4 6 8 10 12 14 R(z) ~ Lg1*s 289 * +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s (the values of Lg1 to Lg7 are 290 * listed in the program) and | 2 14 | -58.45 | Lg1*s +...+Lg7*s - R(z) | <= 291 * 2 | | Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. In 292 * order to guarantee error in log below 1ulp, we compute log by log(1+f) = 293 * f - s*(f - R) (if f is not too large) log(1+f) = f - (hfsq - s*(hfsq+R)). 294 * (better accuracy) 295 * 296 * 3. Finally, log(x) = k*ln2 + log(1+f). = 297 * k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) Here ln2 is split into two 298 * floating point number: ln2_hi + ln2_lo, where n*ln2_hi is always exact 299 * for |n| < 2000. 300 * 301 * Special cases: log(x) is NaN with signal if x < 0 (including -INF) ; 302 * log(+INF) is +INF; log(0) is -INF with signal; log(NaN) is that NaN with 303 * no signal. 304 * 305 * Accuracy: according to an error analysis, the error is always less than 1 306 * ulp (unit in the last place). 307 * 308 * Constants: The hexadecimal values are the intended ones for the following 309 * constants. The decimal values may be used, provided that the compiler 310 * will convert from decimal to binary accurately enough to produce the 311 * hexadecimal values shown. 312 */ 313 314 private static final double ln2_hi = 6.93147180369123816490e-01, /* 315 * 3fe62e42 316 * fee00000 317 */ 318 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ 319 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ 320 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ 321 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ 322 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ 323 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ 324 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ 325 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ 326 327 private static final double ieee754_log(double x) { 328 double hfsq, f, s, z, R, w, t1, t2, dk; 329 int k, hx, lx, i, j; 330 long xl = Double.doubleToLongBits(x); 331 332 hx = (int) (xl >> HI_SHIFT); /* high word of x */ 333 lx = (int) (xl & LO_MASK); /* low word of x */ 334 335 k = 0; 336 if (hx < 0x00100000) { /* x < 2**-1022 */ 337 if (((hx & 0x7fffffff) | lx) == 0) 338 return -two54 / zero; /* log(+-0)=-inf */ 339 if (hx < 0) 340 return (x - x) / zero; /* log(-#) = NaN */ 341 k -= 54; 342 x *= two54; /* subnormal number, scale up x */ 343 hx = (int) (Double.doubleToLongBits(x) >>> HI_SHIFT); /* 344 * high word of 345 * x 346 */ 347 } 348 if (hx >= 0x7ff00000) 349 return x + x; 350 k += (hx >> 20) - 1023; 351 hx &= 0x000fffff; 352 i = (hx + 0x95f64) & 0x100000; 353 // __HI(x) = hx|(i^0x3ff00000); /* normalize x or x/2 */ 354 x = Double.longBitsToDouble(((long) (hx | (i ^ 0x3ff00000)) << HI_SHIFT) | (Double.doubleToLongBits(x) & LO_MASK)); 355 k += (i >> 20); 356 f = x - 1.0; 357 if ((0x000fffff & (2 + hx)) < 3) { /* |f| < 2**-20 */ 358 if (f == zero) 359 if (k == 0) 360 return zero; 361 else { 362 dk = (double) k; 363 return dk * ln2_hi + dk * ln2_lo; 364 } 365 R = f * f * (0.5 - 0.33333333333333333 * f); 366 if (k == 0) 367 return f - R; 368 else { 369 dk = (double) k; 370 return dk * ln2_hi - ((R - dk * ln2_lo) - f); 371 } 372 } 373 s = f / (2.0 + f); 374 dk = (double) k; 375 z = s * s; 376 i = hx - 0x6147a; 377 w = z * z; 378 j = 0x6b851 - hx; 379 t1 = w * (Lg2 + w * (Lg4 + w * Lg6)); 380 t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7))); 381 i |= j; 382 R = t2 + t1; 383 if (i > 0) { 384 hfsq = 0.5 * f * f; 385 if (k == 0) 386 return f - (hfsq - s * (hfsq + R)); 387 else 388 return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) - f); 389 } else { 390 if (k == 0) 391 return f - s * (f - R); 392 else 393 return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f); 394 } 395 } 396 397 /* 398 * __ieee754_pow(x,y) return x**y 399 * 400 * n Method: Let x = 2 * (1+f) 1. Compute and return log2(x) in two pieces: 401 * log2(x) = w1 + w2, where w1 has 53-24 = 29 bit trailing zeros. 2. Perform 402 * y*log2(x) = n+y' by simulating muti-precision arithmetic, where 403 * |y'|<=0.5. 3. Return x**y = 2**n*exp(y'*log2) 404 * 405 * Special cases: 1. (anything) ** 0 is 1 2. (anything) ** 1 is itself 3. 406 * (anything) ** NAN is NAN 4. NAN ** (anything except 0) is NAN 5. +-(|x| > 407 * 1) ** +INF is +INF 6. +-(|x| > 1) ** -INF is +0 7. +-(|x| < 1) ** +INF is 408 * +0 8. +-(|x| < 1) ** -INF is +INF 9. +-1 ** +-INF is NAN 10. +0 ** 409 * (+anything except 0, NAN) is +0 11. -0 ** (+anything except 0, NAN, odd 410 * integer) is +0 12. +0 ** (-anything except 0, NAN) is +INF 13. -0 ** 411 * (-anything except 0, NAN, odd integer) is +INF 14. -0 ** (odd integer) = 412 * -( +0 ** (odd integer) ) 15. +INF ** (+anything except 0,NAN) is +INF 16. 413 * +INF ** (-anything except 0,NAN) is +0 17. -INF ** (anything) = -0 ** 414 * (-anything) 18. (-anything) ** (integer) is 415 * (-1)**(integer)*(+anything**integer) 19. (-anything except 0 and inf) ** 416 * (non-integer) is NAN 417 * 418 * Accuracy: pow(x,y) returns x**y nearly rounded. In particular 419 * pow(integer,integer) always returns the correct integer provided it is 420 * representable. 421 * 422 * Constants : The hexadecimal values are the intended ones for the 423 * following constants. The decimal values may be used, provided that the 424 * compiler will convert from decimal to binary accurately enough to produce 425 * the hexadecimal values shown. 426 */ 427 428 private static final double bp[] = { 1.0, 1.5, }, dp_h[] = { 0.0, 5.84962487220764160156e-01, }, /* 0x3FE2B803, 0x40000000 */ 429 dp_l[] = { 0.0, 1.35003920212974897128e-08, }, /* 0x3E4CFDEB, 0x43CFD006 */ 430 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 431 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ 432 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ 433 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ 434 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ 435 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ 436 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ 437 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ 438 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ 439 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ 440 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ 441 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ 442 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ 443 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h */ 444 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ 445 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2 */ 446 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail */ 447 448 private static final double ieee754_pow(double x, double y) { 449 double z, ax, z_h, z_l, p_h, p_l; 450 double y1, t1, t2, r, s, t, u, v, w; 451 // int i0,i1; 452 int i, j, k, yisint, n; 453 int hx, hy, ix, iy; 454 int lx, ly; 455 456 // i0 = (int)((Double.doubleToLongBits(one)) >>> (29+HI_SHIFT))^1; 457 // i1 = 1-i0; 458 hx = (int) (Double.doubleToLongBits(x) >>> HI_SHIFT); 459 lx = (int) (Double.doubleToLongBits(x) & LO_MASK); 460 hy = (int) (Double.doubleToLongBits(y) >>> HI_SHIFT); 461 ly = (int) (Double.doubleToLongBits(y) & LO_MASK); 462 ix = hx & 0x7fffffff; 463 iy = hy & 0x7fffffff; 464 465 /* y==zero: x**0 = 1 */ 466 if ((iy | ly) == 0) 467 return one; 468 469 /* +-NaN return x+y */ 470 if (ix > 0x7ff00000 || ((ix == 0x7ff00000) && (lx != 0)) || iy > 0x7ff00000 || ((iy == 0x7ff00000) && (ly != 0))) 471 return x + y; 472 473 /* 474 * determine if y is an odd int when x < 0 yisint = 0 ... y is not an 475 * integer yisint = 1 ... y is an odd int yisint = 2 ... y is an even 476 * int 477 */ 478 yisint = 0; 479 if (hx < 0) { 480 if (iy >= 0x43400000) 481 yisint = 2; /* even integer y */ 482 else if (iy >= 0x3ff00000) { 483 k = (iy >> 20) - 0x3ff; /* exponent */ 484 if (k > 20) { 485 j = ly >> (52 - k); 486 if ((j << (52 - k)) == ly) 487 yisint = 2 - (j & 1); 488 } else if (ly == 0) { 489 j = iy >> (20 - k); 490 if ((j << (20 - k)) == iy) 491 yisint = 2 - (j & 1); 492 } 493 } 494 } 495 496 /* special value of y */ 497 if (ly == 0) { 498 if (iy == 0x7ff00000) { /* y is +-inf */ 499 if (((ix - 0x3ff00000) | lx) == 0) 500 return y - y; /* inf**+-1 is NaN */ 501 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ 502 return (hy >= 0) ? y : zero; 503 else 504 /* (|x|<1)**-,+inf = inf,0 */ 505 return (hy < 0) ? -y : zero; 506 } 507 if (iy == 0x3ff00000) { /* y is +-1 */ 508 if (hy < 0) 509 return one / x; 510 else 511 return x; 512 } 513 if (hy == 0x40000000) 514 return x * x; /* y is 2 */ 515 if (hy == 0x3fe00000) { /* y is 0.5 */ 516 if (hx >= 0) /* x >= +0 */ 517 return Math.sqrt(x); 518 } 519 } 520 521 ax = Math.abs(x); 522 /* special value of x */ 523 if (lx == 0) { 524 if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) { 525 z = ax; /* x is +-0,+-inf,+-1 */ 526 if (hy < 0) 527 z = one / z; /* z = (1/|x|) */ 528 if (hx < 0) { 529 if (((ix - 0x3ff00000) | yisint) == 0) { 530 z = (z - z) / (z - z); /* (-1)**non-int is NaN */ 531 } else if (yisint == 1) 532 z = -z; /* (x<0)**odd = -(|x|**odd) */ 533 } 534 return z; 535 } 536 } 537 538 n = (hx >>> 31) + 1; 539 540 /* (x<0)**(non-int) is NaN */ 541 if ((n | yisint) == 0) 542 return (x - x) / (x - x); 543 544 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 545 if ((n | (yisint - 1)) == 0) 546 s = -one;/* (-ve)**(odd int) */ 547 548 /* |y| is huge */ 549 if (iy > 0x41e00000) { /* if |y| > 2**31 */ 550 if (iy > 0x43f00000) { /* if |y| > 2**64, must o/uflow */ 551 if (ix <= 0x3fefffff) 552 return (hy < 0) ? huge * huge : tiny * tiny; 553 if (ix >= 0x3ff00000) 554 return (hy > 0) ? huge * huge : tiny * tiny; 555 } 556 /* over/underflow if x is not close to one */ 557 if (ix < 0x3fefffff) 558 return (hy < 0) ? s * huge * huge : s * tiny * tiny; 559 if (ix > 0x3ff00000) 560 return (hy > 0) ? s * huge * huge : s * tiny * tiny; 561 /* 562 * now |1-x| is tiny <= 2**-20, suffice to compute log(x) by 563 * x-x^2/2+x^3/3-x^4/4 564 */ 565 t = x - one; /* t has 20 trailing zeros */ 566 w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25)); 567 u = ivln2_h * t; /* ivln2_h has 21 sig. bits */ 568 v = t * ivln2_l - w * ivln2; 569 t1 = u + v; 570 // __LO(t1) = 0; // keep high word 571 t1 = Double.longBitsToDouble(Double.doubleToLongBits(t1) & HI_MASK); 572 t2 = v - (t1 - u); 573 } else { 574 double ss, s2, s_h, s_l, t_h, t_l; 575 n = 0; 576 /* take care subnormal number */ 577 if (ix < 0x00100000) { 578 ax *= two53; 579 n -= 53; 580 ix = (int) (Double.doubleToLongBits(ax) >>> HI_SHIFT); 581 } 582 n += ((ix) >> 20) - 0x3ff; 583 j = ix & 0x000fffff; 584 /* determine interval */ 585 ix = j | 0x3ff00000; /* normalize ix */ 586 if (j <= 0x3988E) 587 k = 0; /* |x|<sqrt(3/2) */ 588 else if (j < 0xBB67A) 589 k = 1; /* |x|<sqrt(3) */ 590 else { 591 k = 0; 592 n += 1; 593 ix -= 0x00100000; 594 } 595 // __HI(ax) = ix; 596 ax = Double.longBitsToDouble(((long) ix << HI_SHIFT) | (Double.doubleToLongBits(ax) & LO_MASK)); 597 598 /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 599 u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 600 v = one / (ax + bp[k]); 601 ss = u * v; 602 s_h = ss; 603 // __LO(s_h) = 0; // keep high word 604 s_h = Double.longBitsToDouble(Double.doubleToLongBits(s_h) & HI_MASK); 605 /* t_h=ax+bp[k] High */ 606 t_h = zero; 607 // __HI(t_h)=((ix>>1)|0x20000000)+0x00080000+(k<<18); 608 t_h = Double.longBitsToDouble(((long) ((int) ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18)) << HI_SHIFT) 609 | (Double.doubleToLongBits(t_h) & LO_MASK)); 610 t_l = ax - (t_h - bp[k]); 611 s_l = v * ((u - s_h * t_h) - s_h * t_l); 612 /* compute log(ax) */ 613 s2 = ss * ss; 614 r = s2 * s2 * (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6))))); 615 r += s_l * (s_h + ss); 616 s2 = s_h * s_h; 617 t_h = 3.0 + s2 + r; 618 // __LO(t_h) = 0; // keep high word 619 t_h = Double.longBitsToDouble(Double.doubleToLongBits(t_h) & HI_MASK); 620 t_l = r - ((t_h - 3.0) - s2); 621 /* u+v = ss*(1+...) */ 622 u = s_h * t_h; 623 v = s_l * t_h + t_l * ss; 624 /* 2/(3log2)*(ss+...) */ 625 p_h = u + v; 626 // __LO(p_h) = 0; // keep high word 627 p_h = Double.longBitsToDouble(Double.doubleToLongBits(p_h) & HI_MASK); 628 p_l = v - (p_h - u); 629 z_h = cp_h * p_h; /* cp_h+cp_l = 2/(3*log2) */ 630 z_l = cp_l * p_h + p_l * cp + dp_l[k]; 631 /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 632 t = (double) n; 633 t1 = (((z_h + z_l) + dp_h[k]) + t); 634 // __LO(t1) = 0; // keep high word 635 t1 = Double.longBitsToDouble(Double.doubleToLongBits(t1) & HI_MASK); 636 t2 = z_l - (((t1 - t) - dp_h[k]) - z_h); 637 } 638 639 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 640 y1 = y; 641 // __LO(y1) = 0; // keep high word 642 y1 = Double.longBitsToDouble(Double.doubleToLongBits(y1) & HI_MASK); 643 p_l = (y - y1) * t1 + y * t2; 644 p_h = y1 * t1; 645 z = p_l + p_h; 646 j = (int) (Double.doubleToLongBits(z) >>> HI_SHIFT); 647 i = (int) (Double.doubleToLongBits(z) & LO_MASK); 648 if (j >= 0x40900000) { /* z >= 1024 */ 649 if (((j - 0x40900000) | i) != 0) /* if z > 1024 */ 650 return s * huge * huge; /* overflow */ 651 else { 652 if (p_l + ovt > z - p_h) 653 return s * huge * huge; /* overflow */ 654 } 655 } else if ((j & 0x7fffffff) >= 0x4090cc00) { /* z <= -1075 */ 656 if (((j - 0xc090cc00) | i) != 0) /* z < -1075 */ 657 return s * tiny * tiny; /* underflow */ 658 else { 659 if (p_l <= z - p_h) 660 return s * tiny * tiny; /* underflow */ 661 } 662 } 663 /* 664 * compute 2**(p_h+p_l) 665 */ 666 i = j & 0x7fffffff; 667 k = (i >> 20) - 0x3ff; 668 n = 0; 669 if (i > 0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ 670 n = j + (0x00100000 >> (k + 1)); 671 k = ((n & 0x7fffffff) >> 20) - 0x3ff; /* new k for n */ 672 t = zero; 673 // __HI(t) = (n&~(0x000fffff>>k)); 674 t = Double.longBitsToDouble(((long) (n & ~(0x000fffff >> k)) << HI_SHIFT) | (Double.doubleToLongBits(t) & LO_MASK)); 675 n = ((n & 0x000fffff) | 0x00100000) >> (20 - k); 676 if (j < 0) 677 n = -n; 678 p_h -= t; 679 } 680 t = p_l + p_h; 681 // __LO(t) = 0; // keep high word 682 t = Double.longBitsToDouble(Double.doubleToLongBits(t) & HI_MASK); 683 u = t * lg2_h; 684 v = (p_l - (t - p_h)) * lg2 + t * lg2_l; 685 z = u + v; 686 w = v - (z - u); 687 t = z * z; 688 t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5)))); 689 r = (z * t1) / (t1 - two) - (w + z * w); 690 z = one - (r - z); 691 j = (int) ((long) Double.doubleToLongBits(z) >>> HI_SHIFT); 692 j += (n << 20); 693 if ((j >> 20) <= 0) 694 z = scalbn(z, n); /* subnormal output */ 695 else 696 // __HI(z) = j; 697 z = Double.longBitsToDouble(((long) j << HI_SHIFT) | (Double.doubleToLongBits(z) & LO_MASK)); 698 return s * z; 699 } 700 701 /* 702 * __ieee754_acos(x) Method : acos(x) = pi/2 - asin(x) acos(-x) = pi/2 + 703 * asin(x) For |x|<=0.5 acos(x) = pi/2 - (x + x*x^2*R(x^2)) (see asin.c) For 704 * x>0.5 acos(x) = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2))) = 705 * 2asin(sqrt((1-x)/2)) = 2s + 2s*z*R(z) ...z=(1-x)/2, s=sqrt(z) = 2f + (2c 706 * + 2s*z*R(z)) where f=hi part of s, and c = (z-f*f)/(s+f) is the 707 * correction term for f so that f+c ~ sqrt(z). For x<-0.5 acos(x) = pi - 708 * 2asin(sqrt((1-|x|)/2)) = pi - 0.5*(s+s*z*R(z)), where 709 * z=(1-|x|)/2,s=sqrt(z) 710 * 711 * Special cases: if x is NaN, return x itself; if |x|>1, return NaN with 712 * invalid signal. 713 * 714 * Function needed: sqrt 715 */ 716 717 // private static final double ieee754_acos(double x) { 718 // double z, p, q, r, w, s, c, df; 719 // int hx, ix; 720 // hx = (int) (Double.doubleToLongBits(x) >>> HI_SHIFT); 721 // ix = hx & 0x7fffffff; 722 // if (ix >= 0x3ff00000) { /* |x| >= 1 */ 723 // if (((ix - 0x3ff00000) | (int) (Double.doubleToLongBits(x) & LO_MASK)) == 0) { /* 724 // * | 725 // * x 726 // * |= 727 // * = 728 // * 1 729 // */ 730 // if (hx > 0) 731 // return 0.0; /* acos(1) = 0 */ 732 // else 733 // return pi + 2.0 * pio2_lo; /* acos(-1)= pi */ 734 // } 735 // return (x - x) / (x - x); /* acos(|x|>1) is NaN */ 736 // } 737 // if (ix < 0x3fe00000) { /* |x| < 0.5 */ 738 // if (ix <= 0x3c600000) 739 // return pio2_hi + pio2_lo;/* if|x|<2**-57 */ 740 // z = x * x; 741 // p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5))))); 742 // q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4))); 743 // r = p / q; 744 // return pio2_hi - (x - (pio2_lo - x * r)); 745 // } else if (hx < 0) { /* x < -0.5 */ 746 // z = (one + x) * 0.5; 747 // p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5))))); 748 // q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4))); 749 // s = Math.sqrt(z); 750 // r = p / q; 751 // w = r * s - pio2_lo; 752 // return pi - 2.0 * (s + w); 753 // } else { /* x > 0.5 */ 754 // z = (one - x) * 0.5; 755 // s = Math.sqrt(z); 756 // df = s; 757 // // __LO(df) = 0; // keep high word 758 // df = Double.longBitsToDouble(Double.doubleToLongBits(df) & HI_MASK); 759 // c = (z - df * df) / (s + df); 760 // p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5))))); 761 // q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4))); 762 // r = p / q; 763 // w = r * s + c; 764 // return 2.0 * (df + w); 765 // } 766 // } 767 768 /* 769 * __ieee754_asin(x) Method : Since asin(x) = x + x^3/6 + x^5*3/40 + 770 * x^7*15/336 + ... we approximate asin(x) on [0,0.5] by asin(x) = x + 771 * x*x^2*R(x^2) where R(x^2) is a rational approximation of (asin(x)-x)/x^3 772 * and its remez error is bounded by |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75) 773 * 774 * For x in [0.5,1] asin(x) = pi/2-2*asin(sqrt((1-x)/2)) Let y = (1-x), z = 775 * y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2; then for x>0.98 asin(x) = 776 * pi/2 - 2*(s+s*z*R(z)) = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo) For x<=0.98, 777 * let pio4_hi = pio2_hi/2, then f = hi part of s; c = sqrt(z) - f = 778 * (z-f*f)/(s+f) ...f+c=sqrt(z) and asin(x) = pi/2 - 2*(s+s*z*R(z)) = 779 * pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo) = 780 * pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c)) 781 * 782 * Special cases: if x is NaN, return x itself; if |x|>1, return NaN with 783 * invalid signal. 784 */ 785 786 // private static final double ieee754_asin(double x) { 787 // double t, w, p, q, c, r, s; 788 // int hx, ix; 789 // hx = (int) (Double.doubleToLongBits(x) >>> HI_SHIFT); 790 // ix = hx & 0x7fffffff; 791 // if (ix >= 0x3ff00000) { /* |x|>= 1 */ 792 // if (((ix - 0x3ff00000) | (int) (Double.doubleToLongBits(x) & LO_MASK)) == 0) 793 // /* asin(1)=+-pi/2 with inexact */ 794 // return x * pio2_hi + x * pio2_lo; 795 // return (x - x) / (x - x); /* asin(|x|>1) is NaN */ 796 // } else if (ix < 0x3fe00000) { /* |x|<0.5 */ 797 // if (ix < 0x3e400000) { /* if |x| < 2**-27 */ 798 // if (huge + x > one) 799 // return x;/* return x with inexact if x!=0 */ 800 // } else { 801 // t = x * x; 802 // p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5))))); 803 // q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4))); 804 // w = p / q; 805 // return x + x * w; 806 // } 807 // } 808 // /* 1> |x|>= 0.5 */ 809 // w = one - Math.abs(x); 810 // t = w * 0.5; 811 // p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5))))); 812 // q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4))); 813 // s = Math.sqrt(t); 814 // if (ix >= 0x3FEF3333) { /* if |x| > 0.975 */ 815 // w = p / q; 816 // t = pio2_hi - (2.0 * (s + s * w) - pio2_lo); 817 // } else { 818 // w = s; 819 // // __LO(w) = 0; // keep the high word 820 // w = Double.longBitsToDouble(Double.doubleToLongBits(w) & HI_MASK); 821 // c = (t - w * w) / (s + w); 822 // r = p / q; 823 // p = 2.0 * s * r - (pio2_lo - 2.0 * c); 824 // q = pio4_hi - 2.0 * w; 825 // t = pio4_hi - (p - q); 826 // } 827 // if (hx > 0) 828 // return t; 829 // else 830 // return -t; 831 // } 832 833 /* 834 * atan(x) Method 1. Reduce x to positive by atan(x) = -atan(-x). 2. 835 * According to the integer k=4t+0.25 chopped, t=x, the argument is further 836 * reduced to one of the following intervals and the arctangent of t is 837 * evaluated by the corresponding formula: 838 * 839 * [0,7/16] atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...) [7/16,11/16] 840 * atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) ) [11/16.19/16] atan(x) = 841 * atan( 1 ) + atan( (t-1)/(1+t) ) [19/16,39/16] atan(x) = atan(3/2) + atan( 842 * (t-1.5)/(1+1.5t) ) [39/16,INF] atan(x) = atan(INF) + atan( -1/t ) 843 * 844 * Constants: The hexadecimal values are the intended ones for the following 845 * constants. The decimal values may be used, provided that the compiler 846 * will convert from decimal to binary accurately enough to produce the 847 * hexadecimal values shown. 848 */ 849 850 private static final double atanhi[] = { 4.63647609000806093515e-01, /* 851 * atan(0.5 852 * )hi 853 * 0x3FDDAC67 854 * , 855 * 0x0561BB4F 856 */ 857 7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */ 858 9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */ 859 1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */ 860 }; 861 862 private static final double atanlo[] = { 2.26987774529616870924e-17, /* 863 * atan(0.5 864 * )lo 865 * 0x3C7A2B7F 866 * , 867 * 0x222F65E2 868 */ 869 3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */ 870 1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */ 871 6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */ 872 }; 873 874 private static final double aT[] = { 3.33333333333329318027e-01, /* 875 * 0x3FD55555, 876 * 0x5555550D 877 */ 878 -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */ 879 1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */ 880 -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */ 881 9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */ 882 -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */ 883 6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */ 884 -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */ 885 4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */ 886 -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */ 887 1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */ 888 }; 889 890 private static final double ieee754_atan(double x) { 891 double w, s1, s2, z; 892 int ix, hx, id; 893 894 hx = (int) (Double.doubleToLongBits(x) >>> HI_SHIFT); 895 ix = hx & 0x7fffffff; 896 if (ix >= 0x44100000) { /* if |x| >= 2^66 */ 897 if (ix > 0x7ff00000 || (ix == 0x7ff00000 && ((int) (Double.doubleToLongBits(x) & LO_MASK) != 0))) 898 return x + x; /* NaN */ 899 if (hx > 0) 900 return atanhi[3] + atanlo[3]; 901 else 902 return -atanhi[3] - atanlo[3]; 903 } 904 if (ix < 0x3fdc0000) { /* |x| < 0.4375 */ 905 if (ix < 0x3e200000) { /* |x| < 2^-29 */ 906 if (huge + x > one) 907 return x; /* raise inexact */ 908 } 909 id = -1; 910 } else { 911 x = Math.abs(x); 912 if (ix < 0x3ff30000) { /* |x| < 1.1875 */ 913 if (ix < 0x3fe60000) { /* 7/16 <=|x|<11/16 */ 914 id = 0; 915 x = (2.0 * x - one) / (2.0 + x); 916 } else { /* 11/16<=|x|< 19/16 */ 917 id = 1; 918 x = (x - one) / (x + one); 919 } 920 } else { 921 if (ix < 0x40038000) { /* |x| < 2.4375 */ 922 id = 2; 923 x = (x - 1.5) / (one + 1.5 * x); 924 } else { /* 2.4375 <= |x| < 2^66 */ 925 id = 3; 926 x = -1.0 / x; 927 } 928 } 929 } 930 /* end of argument reduction */ 931 z = x * x; 932 w = z * z; 933 /* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */ 934 s1 = z * (aT[0] + w * (aT[2] + w * (aT[4] + w * (aT[6] + w * (aT[8] + w * aT[10]))))); 935 s2 = w * (aT[1] + w * (aT[3] + w * (aT[5] + w * (aT[7] + w * aT[9])))); 936 if (id < 0) 937 return x - x * (s1 + s2); 938 else { 939 z = atanhi[id] - ((x * (s1 + s2) - atanlo[id]) - x); 940 return (hx < 0) ? -z : z; 941 } 942 } 943 944 /* 945 * __ieee754_atan2(y,x) Method : 1. Reduce y to positive by 946 * atan2(y,x)=-atan2(-y,x). 2. Reduce x to positive by (if x and y are 947 * unexceptional): ARG (x+iy) = arctan(y/x) ... if x > 0, ARG (x+iy) = pi - 948 * arctan[y/(-x)] ... if x < 0, 949 * 950 * Special cases: 951 * 952 * ATAN2((anything), NaN ) is NaN; ATAN2(NAN , (anything) ) is NaN; 953 * ATAN2(+-0, +(anything but NaN)) is +-0 ; ATAN2(+-0, -(anything but NaN)) 954 * is +-pi ; ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2; 955 * ATAN2(+-(anything but INF and NaN), +INF) is +-0 ; ATAN2(+-(anything but 956 * INF and NaN), -INF) is +-pi; ATAN2(+-INF,+INF ) is +-pi/4 ; 957 * ATAN2(+-INF,-INF ) is +-3pi/4; ATAN2(+-INF, (anything but,0,NaN, and 958 * INF)) is +-pi/2; 959 * 960 * Constants: The hexadecimal values are the intended ones for the following 961 * constants. The decimal values may be used, provided that the compiler 962 * will convert from decimal to binary accurately enough to produce the 963 * hexadecimal values shown. 964 */ 965 966 private static final double ieee754_atan2(double x, double y) { 967 double z; 968 int k, m; 969 int hx, hy, ix, iy; 970 int lx, ly; 971 972 // i0 = (int)((Double.doubleToLongBits(one)) >> (29+HI_SHIFT))^1; 973 // i1 = 1-i0; 974 hx = (int) (Double.doubleToLongBits(x) >>> HI_SHIFT); 975 lx = (int) (Double.doubleToLongBits(x) & LO_MASK); 976 hy = (int) (Double.doubleToLongBits(y) >>> HI_SHIFT); 977 ly = (int) (Double.doubleToLongBits(y) & LO_MASK); 978 ix = hx & 0x7fffffff; 979 iy = hy & 0x7fffffff; 980 981 if (((ix | ((lx | -lx) >> 31)) > 0x7ff00000) || ((iy | ((ly | -ly) >> 31)) > 0x7ff00000)) /* x or y is NaN */ 982 return x + y; 983 if ((hx - 0x3ff00000 | lx) == 0) 984 return ieee754_atan(y); /* x=1.0 */ 985 m = ((hy >> 31) & 1) | ((hx >> 30) & 2); /* 2*sign(x)+sign(y) */ 986 987 /* when y = 0 */ 988 if ((iy | ly) == 0) { 989 switch (m) { 990 case 0: 991 case 1: 992 return y; /* atan(+-0,+anything)=+-0 */ 993 case 2: 994 return pi + tiny;/* atan(+0,-anything) = pi */ 995 case 3: 996 return -pi - tiny;/* atan(-0,-anything) =-pi */ 997 } 998 } 999 /* when x = 0 */ 1000 if ((ix | lx) == 0) 1001 return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny; 1002 1003 /* when x is INF */ 1004 if (ix == 0x7ff00000) { 1005 if (iy == 0x7ff00000) { 1006 switch (m) { 1007 case 0: 1008 return pi_o_4 + tiny;/* atan(+INF,+INF) */ 1009 case 1: 1010 return -pi_o_4 - tiny;/* atan(-INF,+INF) */ 1011 case 2: 1012 return 3.0 * pi_o_4 + tiny;/* atan(+INF,-INF) */ 1013 case 3: 1014 return -3.0 * pi_o_4 - tiny;/* atan(-INF,-INF) */ 1015 } 1016 } else { 1017 switch (m) { 1018 case 0: 1019 return zero; /* atan(+...,+INF) */ 1020 case 1: 1021 return -zero; /* atan(-...,+INF) */ 1022 case 2: 1023 return pi + tiny; /* atan(+...,-INF) */ 1024 case 3: 1025 return -pi - tiny; /* atan(-...,-INF) */ 1026 } 1027 } 1028 } 1029 /* when y is INF */ 1030 if (iy == 0x7ff00000) 1031 return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny; 1032 1033 /* compute y/x */ 1034 k = (iy - ix) >> 20; 1035 if (k > 60) 1036 z = pi_o_2 + 0.5 * pi_lo; /* |y/x| > 2**60 */ 1037 else if (hx < 0 && k < -60) 1038 z = 0.0; /* |y|/x < -2**60 */ 1039 else 1040 z = ieee754_atan(Math.abs(y / x)); /* safe to do y/x */ 1041 switch (m) { 1042 case 0: 1043 return z; /* atan(+,+) */ 1044 case 1: 1045 z = Double.longBitsToDouble(Double.doubleToLongBits(z) ^ 0x80000000); // __HI(z) 1046 // ^= 1047 // 0x80000000; 1048 return z; /* atan(-,+) */ 1049 case 2: 1050 return pi - (z - pi_lo);/* atan(+,-) */ 1051 default: /* case 3 */ 1052 return (z - pi_lo) - pi;/* atan(-,-) */ 1053 } 1054 } 1055 1056 /** 1057 * scalbn (double x, int n) scalbn(x,n) returns x* 2**n computed by exponent manipulation rather than by actually performing an exponentiation or a 1058 * multiplication. 1059 */ 1060 public static final double scalbn(double x, int n) { 1061 int k, hx, lx; 1062 hx = (int) (Double.doubleToLongBits(x) >>> HI_SHIFT); 1063 lx = (int) (Double.doubleToLongBits(x) & LO_MASK); 1064 k = (hx & 0x7ff00000) >> 20; /* extract exponent */ 1065 if (k == 0) { /* 0 or subnormal x */ 1066 if ((lx | (hx & 0x7fffffff)) == 0) 1067 return x; /* +-0 */ 1068 x *= two54; 1069 hx = (int) (Double.doubleToLongBits(x) >>> HI_SHIFT); 1070 k = ((hx & 0x7ff00000) >> 20) - 54; 1071 if (n < -50000) 1072 return tiny * x; /* underflow */ 1073 } 1074 if (k == 0x7ff) 1075 return x + x; /* NaN or Inf */ 1076 k = k + n; 1077 if (k > 0x7fe) 1078 return huge * copysign(huge, x); /* overflow */ 1079 if (k > 0) /* normal result */ 1080 { 1081 // __HI(x) = (hx&0x800fffff)|(k<<20); 1082 x = Double.longBitsToDouble(((long) ((int) (hx & 0x800fffff) | (k << 20)) << HI_SHIFT) | (Double.doubleToLongBits(x) & LO_MASK)); 1083 return x; 1084 } 1085 if (k <= -54) 1086 if (n > 50000) /* in case integer overflow in n+k */ 1087 return huge * copysign(huge, x); /* overflow */ 1088 else 1089 return tiny * copysign(tiny, x); /* underflow */ 1090 k += 54; /* subnormal result */ 1091 // __HI(x) = (hx&0x800fffff)|(k<<20); 1092 x = Double.longBitsToDouble(((long) ((int) (hx & 0x800fffff) | (k << 20)) << HI_SHIFT) | (Double.doubleToLongBits(x) & LO_MASK)); 1093 return x * twom54; 1094 } 1095 1096 /* 1097 * copysign(double x, double y) copysign(x,y) returns a value with the 1098 * magnitude of x and with the sign bit of y. 1099 */ 1100 public static final double copysign(final double x, final double y) { 1101 // __HI(x) = (__HI(x)&0x7fffffff)|(__HI(y)&0x80000000); 1102 // The below is actually about 30% faster than doing greater/less 1103 // comparisons. 1104 return Double.longBitsToDouble((Double.doubleToLongBits(x) & 0x7fffffffffffffffL) | (Double.doubleToLongBits(y) & 0x8000000000000000L)); 1105 } 1106 1107 /* 1108 * fabs(x) returns the absolute value of x. This is already handled by Java 1109 * ME. public static final double fabs(double x) { //__HI(x) &= 0x7fffffff; 1110 * //return Double.longBitsToDouble(Double.doubleToLongBits(x) & 1111 * 0x7fffffffffffffffL); } 1112 */ 1113 1114 /** 1115 * Returns the negation of the argument, throwing an exception if the result exceeds a {@code double}. 1116 * 1117 * @param a 1118 * the value to negate 1119 * @return the result 1120 * @throws ArithmeticException 1121 * if the result overflows a double 1122 */ 1123 public static double negateExact(double a) { 1124 if (a == Double.MAX_VALUE || a == Double.MIN_VALUE) { 1125 throw new ArithmeticException("double overflow"); 1126 } 1127 1128 return -a; 1129 } 1130 1131 public static double gcd(double a, double b) { 1132 if (b == 0) 1133 return a; 1134 return gcd(b, a % b); 1135 } 1136 /* 1137 private static final double powSqrt(double x, double y) { 1138 int den = 1024, num = (int) (y * den), iterations = 10; 1139 double n = Double.MAX_VALUE; 1140 1141 while (n >= Double.MAX_VALUE && iterations > 1) { 1142 n = x; 1143 1144 for (int i = 1; i < num; i++) 1145 n *= x; 1146 1147 if (n >= Double.MAX_VALUE) { 1148 iterations--; 1149 den = (int) (den / 2); 1150 num = (int) (y * den); 1151 } 1152 } 1153 1154 for (int i = 0; i < iterations; i++) 1155 n = Math.sqrt(n); 1156 1157 return n; 1158 } 1159 */ 1160 1161 /* 1162 * 1163 * private static final double powDecay(double x, double y) { int num, den = 1164 * 1001, s = 0; double n = x, z = Double.MAX_VALUE; 1165 * 1166 * for( int i = 1; i < s; i++)n *= x; 1167 * 1168 * while( z >= Double.MAX_VALUE ) { den -=1; num = (int)(y*den); s = 1169 * (num/den)+1; 1170 * 1171 * z = x; for( int i = 1; i < num; i++ )z *= x; } 1172 * 1173 * while( n > 0 ) { double a = n; 1174 * 1175 * for( int i = 1; i < den; i++ )a *= n; 1176 * 1177 * if( (a-z) < .00001 || (z-a) > .00001 ) return n; 1178 * 1179 * n *= .9999; } 1180 * 1181 * return -1.0; } 1182 * 1183 * private static final double powTaylor(double a, double b) { boolean gt1 = 1184 * (Math.sqrt((a-1)*(a-1)) <= 1)? false:true; int oc = -1,iter = 30; double 1185 * p = a, x, x2, sumX, sumY; 1186 * 1187 * if( (b-Math.floor(b)) == 0 ) { for( int i = 1; i < b; i++ )p *= a; return 1188 * p; } 1189 * 1190 * x = (gt1)?(a /(a-1)):(a-1); sumX = (gt1)?(1/x):x; 1191 * 1192 * for( int i = 2; i < iter; i++ ) { p = x; for( int j = 1; j < i; j++)p *= 1193 * x; 1194 * 1195 * double xTemp = (gt1)?(1/(i*p)):(p/i); 1196 * 1197 * sumX = (gt1)?(sumX+xTemp):(sumX+(xTemp*oc)); 1198 * 1199 * oc *= -1; } 1200 * 1201 * x2 = b * sumX; sumY = 1+x2; 1202 * 1203 * for( int i = 2; i <= iter; i++ ) { p = x2; for( int j = 1; j < i; j++)p 1204 * *= x2; 1205 * 1206 * int yTemp = 2; for( int j = i; j > 2; j-- )yTemp *= j; 1207 * 1208 * sumY += p/yTemp; } 1209 * 1210 * return sumY; } 1211 */ 1212}