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## src/java.base/share/classes/java/lang/FdLibm.java

 ``` 1 2 /* 3 * Copyright (c) 1998, 2004, 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_pow(x,y) return x**y 28 * 29 * n 30 * Method: Let x = 2 * (1+f) 31 * 1. Compute and return log2(x) in two pieces: 32 * log2(x) = w1 + w2, 33 * where w1 has 53-24 = 29 bit trailing zeros. 34 * 2. Perform y*log2(x) = n+y' by simulating muti-precision 35 * arithmetic, where |y'|<=0.5. 36 * 3. Return x**y = 2**n*exp(y'*log2) 37 * 38 * Special cases: 39 * 1. (anything) ** 0 is 1 40 * 2. (anything) ** 1 is itself 41 * 3. (anything) ** NAN is NAN 42 * 4. NAN ** (anything except 0) is NAN 43 * 5. +-(|x| > 1) ** +INF is +INF 44 * 6. +-(|x| > 1) ** -INF is +0 45 * 7. +-(|x| < 1) ** +INF is +0 46 * 8. +-(|x| < 1) ** -INF is +INF 47 * 9. +-1 ** +-INF is NAN 48 * 10. +0 ** (+anything except 0, NAN) is +0 49 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 50 * 12. +0 ** (-anything except 0, NAN) is +INF 51 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF 52 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) 53 * 15. +INF ** (+anything except 0,NAN) is +INF 54 * 16. +INF ** (-anything except 0,NAN) is +0 55 * 17. -INF ** (anything) = -0 ** (-anything) 56 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) 57 * 19. (-anything except 0 and inf) ** (non-integer) is NAN 58 * 59 * Accuracy: 60 * pow(x,y) returns x**y nearly rounded. In particular 61 * pow(integer,integer) 62 * always returns the correct integer provided it is 63 * representable. 64 * 65 * Constants : 66 * The hexadecimal values are the intended ones for the following 67 * constants. The decimal values may be used, provided that the 68 * compiler will convert from decimal to binary accurately enough 69 * to produce the hexadecimal values shown. 70 */ 71 72 #include "fdlibm.h" 73 74 #ifdef __STDC__ 75 static const double 76 #else 77 static double 78 #endif 79 bp[] = {1.0, 1.5,}, 80 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ 81 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ 82 zero = 0.0, 83 one = 1.0, 84 two = 2.0, 85 two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ 86 huge = 1.0e300, 87 tiny = 1.0e-300, 88 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 89 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ 90 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ 91 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ 92 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ 93 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ 94 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ 95 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 96 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 97 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 98 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 99 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ 100 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ 101 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ 102 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ 103 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ 104 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ 105 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ 106 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ 107 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ 108 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ 109 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ 110 111 #ifdef __STDC__ 112 double __ieee754_pow(double x, double y) 113 #else 114 double __ieee754_pow(x,y) 115 double x, y; 116 #endif 117 { 118 double z,ax,z_h,z_l,p_h,p_l; 119 double y1,t1,t2,r,s,t,u,v,w; 120 int i0,i1,i,j,k,yisint,n; 121 int hx,hy,ix,iy; 122 unsigned lx,ly; 123 124 i0 = ((*(int*)&one)>>29)^1; i1=1-i0; 125 hx = __HI(x); lx = __LO(x); 126 hy = __HI(y); ly = __LO(y); 127 ix = hx&0x7fffffff; iy = hy&0x7fffffff; 128 129 /* y==zero: x**0 = 1 */ 130 if((iy|ly)==0) return one; 131 132 /* +-NaN return x+y */ 133 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || 134 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) 135 return x+y; 136 137 /* determine if y is an odd int when x < 0 138 * yisint = 0 ... y is not an integer 139 * yisint = 1 ... y is an odd int 140 * yisint = 2 ... y is an even int 141 */ 142 yisint = 0; 143 if(hx<0) { 144 if(iy>=0x43400000) yisint = 2; /* even integer y */ 145 else if(iy>=0x3ff00000) { 146 k = (iy>>20)-0x3ff; /* exponent */ 147 if(k>20) { 148 j = ly>>(52-k); 149 if((j<<(52-k))==ly) yisint = 2-(j&1); 150 } else if(ly==0) { 151 j = iy>>(20-k); 152 if((j<<(20-k))==iy) yisint = 2-(j&1); 153 } 154 } 155 } 156 157 /* special value of y */ 158 if(ly==0) { 159 if (iy==0x7ff00000) { /* y is +-inf */ 160 if(((ix-0x3ff00000)|lx)==0) 161 return y - y; /* inf**+-1 is NaN */ 162 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ 163 return (hy>=0)? y: zero; 164 else /* (|x|<1)**-,+inf = inf,0 */ 165 return (hy<0)?-y: zero; 166 } 167 if(iy==0x3ff00000) { /* y is +-1 */ 168 if(hy<0) return one/x; else return x; 169 } 170 if(hy==0x40000000) return x*x; /* y is 2 */ 171 if(hy==0x3fe00000) { /* y is 0.5 */ 172 if(hx>=0) /* x >= +0 */ 173 return sqrt(x); 174 } 175 } 176 177 ax = fabs(x); 178 /* special value of x */ 179 if(lx==0) { 180 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ 181 z = ax; /*x is +-0,+-inf,+-1*/ 182 if(hy<0) z = one/z; /* z = (1/|x|) */ 183 if(hx<0) { 184 if(((ix-0x3ff00000)|yisint)==0) { 185 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 186 } else if(yisint==1) 187 z = -1.0*z; /* (x<0)**odd = -(|x|**odd) */ 188 } 189 return z; 190 } 191 } 192 193 n = (hx>>31)+1; 194 195 /* (x<0)**(non-int) is NaN */ 196 if((n|yisint)==0) return (x-x)/(x-x); 197 198 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 199 if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */ 200 201 /* |y| is huge */ 202 if(iy>0x41e00000) { /* if |y| > 2**31 */ 203 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ 204 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; 205 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; 206 } 207 /* over/underflow if x is not close to one */ 208 if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny; 209 if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny; 210 /* now |1-x| is tiny <= 2**-20, suffice to compute 211 log(x) by x-x^2/2+x^3/3-x^4/4 */ 212 t = ax-one; /* t has 20 trailing zeros */ 213 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); 214 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ 215 v = t*ivln2_l-w*ivln2; 216 t1 = u+v; 217 __LO(t1) = 0; 218 t2 = v-(t1-u); 219 } else { 220 double ss,s2,s_h,s_l,t_h,t_l; 221 n = 0; 222 /* take care subnormal number */ 223 if(ix<0x00100000) 224 {ax *= two53; n -= 53; ix = __HI(ax); } 225 n += ((ix)>>20)-0x3ff; 226 j = ix&0x000fffff; 227 /* determine interval */ 228 ix = j|0x3ff00000; /* normalize ix */ 229 if(j<=0x3988E) k=0; /* |x|>1)|0x20000000)+0x00080000+(k<<18); 243 t_l = ax - (t_h-bp[k]); 244 s_l = v*((u-s_h*t_h)-s_h*t_l); 245 /* compute log(ax) */ 246 s2 = ss*ss; 247 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 248 r += s_l*(s_h+ss); 249 s2 = s_h*s_h; 250 t_h = 3.0+s2+r; 251 __LO(t_h) = 0; 252 t_l = r-((t_h-3.0)-s2); 253 /* u+v = ss*(1+...) */ 254 u = s_h*t_h; 255 v = s_l*t_h+t_l*ss; 256 /* 2/(3log2)*(ss+...) */ 257 p_h = u+v; 258 __LO(p_h) = 0; 259 p_l = v-(p_h-u); 260 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 261 z_l = cp_l*p_h+p_l*cp+dp_l[k]; 262 /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 263 t = (double)n; 264 t1 = (((z_h+z_l)+dp_h[k])+t); 265 __LO(t1) = 0; 266 t2 = z_l-(((t1-t)-dp_h[k])-z_h); 267 } 268 269 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 270 y1 = y; 271 __LO(y1) = 0; 272 p_l = (y-y1)*t1+y*t2; 273 p_h = y1*t1; 274 z = p_l+p_h; 275 j = __HI(z); 276 i = __LO(z); 277 if (j>=0x40900000) { /* z >= 1024 */ 278 if(((j-0x40900000)|i)!=0) /* if z > 1024 */ 279 return s*huge*huge; /* overflow */ 280 else { 281 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ 282 } 283 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ 284 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ 285 return s*tiny*tiny; /* underflow */ 286 else { 287 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ 288 } 289 } 290 /* 291 * compute 2**(p_h+p_l) 292 */ 293 i = j&0x7fffffff; 294 k = (i>>20)-0x3ff; 295 n = 0; 296 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ 297 n = j+(0x00100000>>(k+1)); 298 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ 299 t = zero; 300 __HI(t) = (n&~(0x000fffff>>k)); 301 n = ((n&0x000fffff)|0x00100000)>>(20-k); 302 if(j<0) n = -n; 303 p_h -= t; 304 } 305 t = p_l+p_h; 306 __LO(t) = 0; 307 u = t*lg2_h; 308 v = (p_l-(t-p_h))*lg2+t*lg2_l; 309 z = u+v; 310 w = v-(z-u); 311 t = z*z; 312 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 313 r = (z*t1)/(t1-two)-(w+z*w); 314 z = one-(r-z); 315 j = __HI(z); 316 j += (n<<20); 317 if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */ 318 else __HI(z) += (n<<20); 319 return s*z; 320 } ``` ``` 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 *

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 *

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 class FdLibm { 52 // Constants used by multiple algorithms 53 private static final double INFINITY = Double.POSITIVE_INFINITY; 54 55 private FdLibm() { 56 throw new UnsupportedOperationException("No instances for you."); 57 } 58 59 /** 60 * Return the low-order 32 bits of the double argument as an int. 61 */ 62 private static int __LO(double x) { 63 long transducer = Double.doubleToRawLongBits(x); 64 return (int)transducer; 65 } 66 67 /** 68 * Return a double with its low-order bits of the second argument 69 * and the high-order bits of the first argument.. 70 */ 71 private static double __LO(double x, int low) { 72 long transX = Double.doubleToRawLongBits(x); 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 113 * 9. +-1 ** +-INF is NAN 114 * 10. +0 ** (+anything except 0, NAN) is +0 115 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 116 * 12. +0 ** (-anything except 0, NAN) is +INF 117 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF 118 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) 119 * 15. +INF ** (+anything except 0,NAN) is +INF 120 * 16. +INF ** (-anything except 0,NAN) is +0 121 * 17. -INF ** (anything) = -0 ** (-anything) 122 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) 123 * 19. (-anything except 0 and inf) ** (non-integer) is NAN 124 * 125 * Accuracy: 126 * pow(x,y) returns x**y nearly rounded. In particular 127 * pow(integer,integer) 128 * always returns the correct integer provided it is 129 * representable. 130 */ 131 public static class Pow { 132 public static strictfp double compute(final double x, final double y) { 133 double z; 134 double r, s, t, u, v, w; 135 int i, j, k, n; 136 137 // y == zero: x**0 = 1 138 if (y == 0.0) 139 return 1.0; 140 141 // +/-NaN return x + y to propagate NaN significands 142 if (Double.isNaN(x) || Double.isNaN(y)) 143 return x + y; 144 145 final double y_abs = Math.abs(y); 146 double x_abs = Math.abs(x); 147 // Special values of y 148 if (y == 2.0) { 149 return x * x; 150 } else if (y == 0.5) { 151 if (x >= -Double.MAX_VALUE) // Handle x == -infinity later 152 return Math.sqrt(x + 0.0); // Add 0.0 to properly handle x == -0.0 153 } else if (y_abs == 1.0) { // y is +/-1 154 return (y == 1.0) ? x : 1.0 / x; 155 } else if (y_abs == INFINITY) { // y is +/-infinity 156 if (x_abs == 1.0) 157 return y - y; // inf**+/-1 is NaN 158 else if (x_abs > 1.0) // (|x| > 1)**+/-inf = inf, 0 159 return (y >= 0) ? y : 0.0; 160 else // (|x| < 1)**-/+inf = inf, 0 161 return (y < 0) ? -y : 0.0; 162 } 163 164 final int hx = __HI(x); 165 int ix = hx & 0x7fffffff; 166 167 /* 168 * When x < 0, determine if y is an odd integer: 169 * y_is_int = 0 ... y is not an integer 170 * y_is_int = 1 ... y is an odd int 171 * y_is_int = 2 ... y is an even int 172 */ 173 int y_is_int = 0; 174 if (hx < 0) { 175 if (y_abs >= 0x1.0p53) // |y| >= 2^53 = 9.007199254740992E15 176 y_is_int = 2; // y is an even integer since ulp(2^53) = 2.0 177 else if (y_abs >= 1.0) { // |y| >= 1.0 178 long y_abs_as_long = (long) y_abs; 179 if ( ((double) y_abs_as_long) == y_abs) { 180 y_is_int = 2 - (int)(y_abs_as_long & 0x1L); 181 } 182 } 183 } 184 185 // Special value of x 186 if (x_abs == 0.0 || 187 x_abs == INFINITY || 188 x_abs == 1.0) { 189 z = x_abs; // x is +/-0, +/-inf, +/-1 190 if (y < 0.0) 191 z = 1.0/z; // z = (1/|x|) 192 if (hx < 0) { 193 if (((ix - 0x3ff00000) | y_is_int) == 0) { 194 z = (z-z)/(z-z); // (-1)**non-int is NaN 195 } else if (y_is_int == 1) 196 z = -1.0 * z; // (x < 0)**odd = -(|x|**odd) 197 } 198 return z; 199 } 200 201 n = (hx >> 31) + 1; 202 203 // (x < 0)**(non-int) is NaN 204 if ((n | y_is_int) == 0) 205 return (x-x)/(x-x); 206 207 s = 1.0; // s (sign of result -ve**odd) = -1 else = 1 208 if ( (n | (y_is_int - 1)) == 0) 209 s = -1.0; // (-ve)**(odd int) 210 211 double p_h, p_l, t1, t2; 212 // |y| is huge 213 if (y_abs > 0x1.0p31) { // if |y| > 2**31 214 final double INV_LN2 = 0x1.7154_7652_b82fep0; // 1.44269504088896338700e+00 = 1/ln2 215 final double INV_LN2_H = 0x1.715476p0; // 1.44269502162933349609e+00 = 24 bits of 1/ln2 216 final double INV_LN2_L = 0x1.4ae0_bf85_ddf44p-26; // 1.92596299112661746887e-08 = 1/ln2 tail 217 218 // Over/underflow if x is not close to one 219 if (x_abs < 0x1.fffffp-1) // |x| < 0.9999995231628418 220 return (y < 0.0) ? s * INFINITY : s * 0.0; 221 if (x_abs > 1.0) // |x| > 1.0 222 return (y > 0.0) ? s * INFINITY : s * 0.0; 223 /* 224 * now |1-x| is tiny <= 2**-20, sufficient to compute 225 * log(x) by x - x^2/2 + x^3/3 - x^4/4 226 */ 227 t = x_abs - 1.0; // t has 20 trailing zeros 228 w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25)); 229 u = INV_LN2_H * t; // INV_LN2_H has 21 sig. bits 230 v = t * INV_LN2_L - w * INV_LN2; 231 t1 = u + v; 232 t1 =__LO(t1, 0); 233 t2 = v - (t1 - u); 234 } else { 235 final double CP = 0x1.ec70_9dc3_a03fdp-1; // 9.61796693925975554329e-01 = 2/(3ln2) 236 final double CP_H = 0x1.ec709ep-1; // 9.61796700954437255859e-01 = (float)cp 237 final double CP_L = -0x1.e2fe_0145_b01f5p-28; // -7.02846165095275826516e-09 = tail of CP_H 238 239 double z_h, z_l, ss, s2, s_h, s_l, t_h, t_l; 240 n = 0; 241 // Take care of subnormal numbers 242 if (ix < 0x00100000) { 243 x_abs *= 0x1.0p53; // 2^53 = 9007199254740992.0 244 n -= 53; 245 ix = __HI(x_abs); 246 } 247 n += ((ix) >> 20) - 0x3ff; 248 j = ix & 0x000fffff; 249 // Determine interval 250 ix = j | 0x3ff00000; // Normalize ix 251 if (j <= 0x3988E) 252 k = 0; // |x| > 1) | 0x20000000) + 0x00080000 + (k << 18) ); 286 t_l = x_abs - (t_h - BP[k]); 287 s_l = v * ((u - s_h * t_h) - s_h * t_l); 288 // Compute log(x_abs) 289 s2 = ss * ss; 290 r = s2 * s2* (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6))))); 291 r += s_l * (s_h + ss); 292 s2 = s_h * s_h; 293 t_h = 3.0 + s2 + r; 294 t_h = __LO(t_h, 0); 295 t_l = r - ((t_h - 3.0) - s2); 296 // u+v = ss*(1+...) 297 u = s_h * t_h; 298 v = s_l * t_h + t_l * ss; 299 // 2/(3log2)*(ss + ...) 300 p_h = u + v; 301 p_h = __LO(p_h, 0); 302 p_l = v - (p_h - u); 303 z_h = CP_H * p_h; // CP_H + CP_L = 2/(3*log2) 304 z_l = CP_L * p_h + p_l * CP + DP_L[k]; 305 // log2(x_abs) = (ss + ..)*2/(3*log2) = n + DP_H + z_h + z_l 306 t = (double)n; 307 t1 = (((z_h + z_l) + DP_H[k]) + t); 308 t1 = __LO(t1, 0); 309 t2 = z_l - (((t1 - t) - DP_H[k]) - z_h); 310 } 311 312 // Split up y into (y1 + y2) and compute (y1 + y2) * (t1 + t2) 313 double y1 = y; 314 y1 = __LO(y1, 0); 315 p_l = (y - y1) * t1 + y * t2; 316 p_h = y1 * t1; 317 z = p_l + p_h; 318 j = __HI(z); 319 i = __LO(z); 320 if (j >= 0x40900000) { // z >= 1024 321 if (((j - 0x40900000) | i)!=0) // if z > 1024 322 return s * INFINITY; // Overflow 323 else { 324 final double OVT = 8.0085662595372944372e-0017; // -(1024-log2(ovfl+.5ulp)) 325 if (p_l + OVT > z - p_h) 326 return s * INFINITY; // Overflow 327 } 328 } else if ((j & 0x7fffffff) >= 0x4090cc00 ) { // z <= -1075 329 if (((j - 0xc090cc00) | i)!=0) // z < -1075 330 return s * 0.0; // Underflow 331 else { 332 if (p_l <= z - p_h) 333 return s * 0.0; // Underflow 334 } 335 } 336 /* 337 * Compute 2**(p_h+p_l) 338 */ 339 // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3 340 final double P1 = 0x1.5555_5555_5553ep-3; // 1.66666666666666019037e-01 341 final double P2 = -0x1.6c16_c16b_ebd93p-9; // -2.77777777770155933842e-03 342 final double P3 = 0x1.1566_aaf2_5de2cp-14; // 6.61375632143793436117e-05 343 final double P4 = -0x1.bbd4_1c5d_26bf1p-20; // -1.65339022054652515390e-06 344 final double P5 = 0x1.6376_972b_ea4d0p-25; // 4.13813679705723846039e-08 345 final double LG2 = 0x1.62e4_2fef_a39efp-1; // 6.93147180559945286227e-01 346 final double LG2_H = 0x1.62e43p-1; // 6.93147182464599609375e-01 347 final double LG2_L = -0x1.05c6_10ca_86c39p-29; // -1.90465429995776804525e-09 348 i = j & 0x7fffffff; 349 k = (i >> 20) - 0x3ff; 350 n = 0; 351 if (i > 0x3fe00000) { // if |z| > 0.5, set n = [z + 0.5] 352 n = j + (0x00100000 >> (k + 1)); 353 k = ((n & 0x7fffffff) >> 20) - 0x3ff; // new k for n 354 t = 0.0; 355 t = __HI(t, (n & ~(0x000fffff >> k)) ); 356 n = ((n & 0x000fffff) | 0x00100000) >> (20 - k); 357 if (j < 0) 358 n = -n; 359 p_h -= t; 360 } 361 t = p_l + p_h; 362 t = __LO(t, 0); 363 u = t * LG2_H; 364 v = (p_l - (t - p_h)) * LG2 + t * LG2_L; 365 z = u + v; 366 w = v - (z - u); 367 t = z * z; 368 t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5)))); 369 r = (z * t1)/(t1 - 2.0) - (w + z * w); 370 z = 1.0 - (r - z); 371 j = __HI(z); 372 j += (n << 20); 373 if ((j >> 20) <= 0) 374 z = Math.scalb(z, n); // subnormal output 375 else { 376 int z_hi = __HI(z); 377 z_hi += (n << 20); 378 z = __HI(z, z_hi); 379 } 380 return s * z; 381 } 382 } 383 } ```

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