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1 /*
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2 * Copyright 2005 Sun Microsystems, Inc. All Rights Reserved.
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3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
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4 *
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5 * This code is free software; you can redistribute it and/or modify it
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6 * under the terms of the GNU General Public License version 2 only, as
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7 * published by the Free Software Foundation.
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8 *
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9 * This code is distributed in the hope that it will be useful, but WITHOUT
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10 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
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11 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
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12 * version 2 for more details (a copy is included in the LICENSE file that
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13 * accompanied this code).
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14 *
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15 * You should have received a copy of the GNU General Public License version
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16 * 2 along with this work; if not, write to the Free Software Foundation,
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17 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
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18 *
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19 * Please contact Sun Microsystems, Inc., 4150 Network Circle, Santa Clara,
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20 * CA 95054 USA or visit www.sun.com if you need additional information or
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21 * have any questions.
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22 *
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23 */
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24
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25 #include "incls/_precompiled.incl"
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26 #include "incls/_sharedRuntimeTrans.cpp.incl"
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27
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28 // This file contains copies of the fdlibm routines used by
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29 // StrictMath. It turns out that it is almost always required to use
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30 // these runtime routines; the Intel CPU doesn't meet the Java
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31 // specification for sin/cos outside a certain limited argument range,
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32 // and the SPARC CPU doesn't appear to have sin/cos instructions. It
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33 // also turns out that avoiding the indirect call through function
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34 // pointer out to libjava.so in SharedRuntime speeds these routines up
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35 // by roughly 15% on both Win32/x86 and Solaris/SPARC.
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36
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37 // Enabling optimizations in this file causes incorrect code to be
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38 // generated; can not figure out how to turn down optimization for one
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39 // file in the IDE on Windows
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40 #ifdef WIN32
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41 # pragma optimize ( "", off )
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42 #endif
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43
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44 #include <math.h>
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45
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46 // VM_LITTLE_ENDIAN is #defined appropriately in the Makefiles
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47 // [jk] this is not 100% correct because the float word order may different
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48 // from the byte order (e.g. on ARM)
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49 #ifdef VM_LITTLE_ENDIAN
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50 # define __HI(x) *(1+(int*)&x)
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51 # define __LO(x) *(int*)&x
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52 #else
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53 # define __HI(x) *(int*)&x
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54 # define __LO(x) *(1+(int*)&x)
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55 #endif
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56
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57 double copysign(double x, double y) {
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58 __HI(x) = (__HI(x)&0x7fffffff)|(__HI(y)&0x80000000);
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59 return x;
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60 }
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61
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62 /*
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63 * ====================================================
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64 * Copyright (C) 1998 by Sun Microsystems, Inc. All rights reserved.
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65 *
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66 * Developed at SunSoft, a Sun Microsystems, Inc. business.
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67 * Permission to use, copy, modify, and distribute this
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68 * software is freely granted, provided that this notice
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69 * is preserved.
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70 * ====================================================
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71 */
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72
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73 /*
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74 * scalbn (double x, int n)
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75 * scalbn(x,n) returns x* 2**n computed by exponent
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76 * manipulation rather than by actually performing an
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77 * exponentiation or a multiplication.
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78 */
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79
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80 static const double
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81 two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
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82 twom54 = 5.55111512312578270212e-17, /* 0x3C900000, 0x00000000 */
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83 hugeX = 1.0e+300,
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84 tiny = 1.0e-300;
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85
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86 double scalbn (double x, int n) {
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87 int k,hx,lx;
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88 hx = __HI(x);
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89 lx = __LO(x);
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90 k = (hx&0x7ff00000)>>20; /* extract exponent */
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91 if (k==0) { /* 0 or subnormal x */
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92 if ((lx|(hx&0x7fffffff))==0) return x; /* +-0 */
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93 x *= two54;
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94 hx = __HI(x);
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95 k = ((hx&0x7ff00000)>>20) - 54;
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96 if (n< -50000) return tiny*x; /*underflow*/
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97 }
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98 if (k==0x7ff) return x+x; /* NaN or Inf */
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99 k = k+n;
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100 if (k > 0x7fe) return hugeX*copysign(hugeX,x); /* overflow */
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101 if (k > 0) /* normal result */
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102 {__HI(x) = (hx&0x800fffff)|(k<<20); return x;}
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103 if (k <= -54) {
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104 if (n > 50000) /* in case integer overflow in n+k */
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105 return hugeX*copysign(hugeX,x); /*overflow*/
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106 else return tiny*copysign(tiny,x); /*underflow*/
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107 }
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108 k += 54; /* subnormal result */
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109 __HI(x) = (hx&0x800fffff)|(k<<20);
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110 return x*twom54;
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111 }
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112
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113 /* __ieee754_log(x)
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114 * Return the logrithm of x
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115 *
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116 * Method :
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117 * 1. Argument Reduction: find k and f such that
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118 * x = 2^k * (1+f),
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119 * where sqrt(2)/2 < 1+f < sqrt(2) .
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120 *
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121 * 2. Approximation of log(1+f).
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122 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
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123 * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
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124 * = 2s + s*R
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125 * We use a special Reme algorithm on [0,0.1716] to generate
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126 * a polynomial of degree 14 to approximate R The maximum error
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127 * of this polynomial approximation is bounded by 2**-58.45. In
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128 * other words,
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129 * 2 4 6 8 10 12 14
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130 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
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131 * (the values of Lg1 to Lg7 are listed in the program)
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132 * and
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133 * | 2 14 | -58.45
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134 * | Lg1*s +...+Lg7*s - R(z) | <= 2
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135 * | |
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136 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
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137 * In order to guarantee error in log below 1ulp, we compute log
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138 * by
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139 * log(1+f) = f - s*(f - R) (if f is not too large)
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140 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
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141 *
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142 * 3. Finally, log(x) = k*ln2 + log(1+f).
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143 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
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144 * Here ln2 is split into two floating point number:
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145 * ln2_hi + ln2_lo,
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146 * where n*ln2_hi is always exact for |n| < 2000.
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147 *
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148 * Special cases:
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149 * log(x) is NaN with signal if x < 0 (including -INF) ;
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150 * log(+INF) is +INF; log(0) is -INF with signal;
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151 * log(NaN) is that NaN with no signal.
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152 *
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153 * Accuracy:
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154 * according to an error analysis, the error is always less than
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155 * 1 ulp (unit in the last place).
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156 *
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157 * Constants:
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158 * The hexadecimal values are the intended ones for the following
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159 * constants. The decimal values may be used, provided that the
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160 * compiler will convert from decimal to binary accurately enough
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161 * to produce the hexadecimal values shown.
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162 */
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163
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164 static const double
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165 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
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166 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
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167 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
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168 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
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169 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
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170 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
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171 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
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172 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
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173 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
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174
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175 static double zero = 0.0;
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176
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177 static double __ieee754_log(double x) {
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178 double hfsq,f,s,z,R,w,t1,t2,dk;
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179 int k,hx,i,j;
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180 unsigned lx;
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181
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182 hx = __HI(x); /* high word of x */
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183 lx = __LO(x); /* low word of x */
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184
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185 k=0;
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186 if (hx < 0x00100000) { /* x < 2**-1022 */
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187 if (((hx&0x7fffffff)|lx)==0)
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188 return -two54/zero; /* log(+-0)=-inf */
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189 if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
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190 k -= 54; x *= two54; /* subnormal number, scale up x */
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191 hx = __HI(x); /* high word of x */
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192 }
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193 if (hx >= 0x7ff00000) return x+x;
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194 k += (hx>>20)-1023;
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195 hx &= 0x000fffff;
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196 i = (hx+0x95f64)&0x100000;
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197 __HI(x) = hx|(i^0x3ff00000); /* normalize x or x/2 */
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198 k += (i>>20);
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199 f = x-1.0;
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200 if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */
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201 if(f==zero) {
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202 if (k==0) return zero;
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203 else {dk=(double)k; return dk*ln2_hi+dk*ln2_lo;}
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204 }
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205 R = f*f*(0.5-0.33333333333333333*f);
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206 if(k==0) return f-R; else {dk=(double)k;
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207 return dk*ln2_hi-((R-dk*ln2_lo)-f);}
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208 }
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209 s = f/(2.0+f);
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210 dk = (double)k;
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211 z = s*s;
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212 i = hx-0x6147a;
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213 w = z*z;
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214 j = 0x6b851-hx;
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215 t1= w*(Lg2+w*(Lg4+w*Lg6));
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216 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
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217 i |= j;
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218 R = t2+t1;
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219 if(i>0) {
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220 hfsq=0.5*f*f;
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221 if(k==0) return f-(hfsq-s*(hfsq+R)); else
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222 return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
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223 } else {
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224 if(k==0) return f-s*(f-R); else
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225 return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
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226 }
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227 }
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228
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229 JRT_LEAF(jdouble, SharedRuntime::dlog(jdouble x))
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230 return __ieee754_log(x);
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231 JRT_END
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232
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233 /* __ieee754_log10(x)
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234 * Return the base 10 logarithm of x
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235 *
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236 * Method :
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237 * Let log10_2hi = leading 40 bits of log10(2) and
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238 * log10_2lo = log10(2) - log10_2hi,
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239 * ivln10 = 1/log(10) rounded.
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240 * Then
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241 * n = ilogb(x),
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242 * if(n<0) n = n+1;
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243 * x = scalbn(x,-n);
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244 * log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x))
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245 *
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246 * Note 1:
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247 * To guarantee log10(10**n)=n, where 10**n is normal, the rounding
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248 * mode must set to Round-to-Nearest.
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249 * Note 2:
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250 * [1/log(10)] rounded to 53 bits has error .198 ulps;
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251 * log10 is monotonic at all binary break points.
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252 *
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253 * Special cases:
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254 * log10(x) is NaN with signal if x < 0;
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255 * log10(+INF) is +INF with no signal; log10(0) is -INF with signal;
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256 * log10(NaN) is that NaN with no signal;
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257 * log10(10**N) = N for N=0,1,...,22.
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258 *
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259 * Constants:
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260 * The hexadecimal values are the intended ones for the following constants.
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261 * The decimal values may be used, provided that the compiler will convert
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262 * from decimal to binary accurately enough to produce the hexadecimal values
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263 * shown.
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264 */
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265
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266 static const double
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267 ivln10 = 4.34294481903251816668e-01, /* 0x3FDBCB7B, 0x1526E50E */
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268 log10_2hi = 3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */
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269 log10_2lo = 3.69423907715893078616e-13; /* 0x3D59FEF3, 0x11F12B36 */
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270
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271 static double __ieee754_log10(double x) {
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272 double y,z;
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273 int i,k,hx;
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274 unsigned lx;
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275
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276 hx = __HI(x); /* high word of x */
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277 lx = __LO(x); /* low word of x */
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278
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279 k=0;
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280 if (hx < 0x00100000) { /* x < 2**-1022 */
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281 if (((hx&0x7fffffff)|lx)==0)
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282 return -two54/zero; /* log(+-0)=-inf */
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283 if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
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284 k -= 54; x *= two54; /* subnormal number, scale up x */
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285 hx = __HI(x); /* high word of x */
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286 }
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287 if (hx >= 0x7ff00000) return x+x;
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288 k += (hx>>20)-1023;
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289 i = ((unsigned)k&0x80000000)>>31;
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290 hx = (hx&0x000fffff)|((0x3ff-i)<<20);
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291 y = (double)(k+i);
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292 __HI(x) = hx;
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293 z = y*log10_2lo + ivln10*__ieee754_log(x);
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294 return z+y*log10_2hi;
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295 }
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296
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297 JRT_LEAF(jdouble, SharedRuntime::dlog10(jdouble x))
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298 return __ieee754_log10(x);
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299 JRT_END
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300
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301
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302 /* __ieee754_exp(x)
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303 * Returns the exponential of x.
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304 *
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305 * Method
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306 * 1. Argument reduction:
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307 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
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308 * Given x, find r and integer k such that
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309 *
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310 * x = k*ln2 + r, |r| <= 0.5*ln2.
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311 *
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312 * Here r will be represented as r = hi-lo for better
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313 * accuracy.
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314 *
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315 * 2. Approximation of exp(r) by a special rational function on
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316 * the interval [0,0.34658]:
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317 * Write
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318 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
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319 * We use a special Reme algorithm on [0,0.34658] to generate
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320 * a polynomial of degree 5 to approximate R. The maximum error
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321 * of this polynomial approximation is bounded by 2**-59. In
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322 * other words,
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323 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
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324 * (where z=r*r, and the values of P1 to P5 are listed below)
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325 * and
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326 * | 5 | -59
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327 * | 2.0+P1*z+...+P5*z - R(z) | <= 2
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328 * | |
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329 * The computation of exp(r) thus becomes
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330 * 2*r
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331 * exp(r) = 1 + -------
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332 * R - r
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333 * r*R1(r)
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334 * = 1 + r + ----------- (for better accuracy)
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335 * 2 - R1(r)
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336 * where
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337 * 2 4 10
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338 * R1(r) = r - (P1*r + P2*r + ... + P5*r ).
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339 *
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340 * 3. Scale back to obtain exp(x):
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341 * From step 1, we have
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342 * exp(x) = 2^k * exp(r)
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343 *
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344 * Special cases:
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345 * exp(INF) is INF, exp(NaN) is NaN;
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346 * exp(-INF) is 0, and
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347 * for finite argument, only exp(0)=1 is exact.
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348 *
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349 * Accuracy:
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350 * according to an error analysis, the error is always less than
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351 * 1 ulp (unit in the last place).
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352 *
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353 * Misc. info.
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354 * For IEEE double
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355 * if x > 7.09782712893383973096e+02 then exp(x) overflow
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356 * if x < -7.45133219101941108420e+02 then exp(x) underflow
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357 *
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358 * Constants:
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359 * The hexadecimal values are the intended ones for the following
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360 * constants. The decimal values may be used, provided that the
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361 * compiler will convert from decimal to binary accurately enough
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362 * to produce the hexadecimal values shown.
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363 */
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364
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365 static const double
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366 one = 1.0,
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367 halF[2] = {0.5,-0.5,},
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368 twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
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369 o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
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370 u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
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371 ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
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372 -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
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373 ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
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374 -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
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375 invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
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376 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
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377 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
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378 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
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379 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
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380 P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
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381
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382 static double __ieee754_exp(double x) {
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383 double y,hi=0,lo=0,c,t;
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384 int k=0,xsb;
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385 unsigned hx;
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386
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387 hx = __HI(x); /* high word of x */
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388 xsb = (hx>>31)&1; /* sign bit of x */
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389 hx &= 0x7fffffff; /* high word of |x| */
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390
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391 /* filter out non-finite argument */
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392 if(hx >= 0x40862E42) { /* if |x|>=709.78... */
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393 if(hx>=0x7ff00000) {
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394 if(((hx&0xfffff)|__LO(x))!=0)
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395 return x+x; /* NaN */
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396 else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
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397 }
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398 if(x > o_threshold) return hugeX*hugeX; /* overflow */
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399 if(x < u_threshold) return twom1000*twom1000; /* underflow */
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400 }
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401
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402 /* argument reduction */
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403 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
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404 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
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405 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
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406 } else {
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407 k = (int)(invln2*x+halF[xsb]);
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408 t = k;
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409 hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
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|
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410 lo = t*ln2LO[0];
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|
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411 }
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|
|
412 x = hi - lo;
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|
413 }
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414 else if(hx < 0x3e300000) { /* when |x|<2**-28 */
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415 if(hugeX+x>one) return one+x;/* trigger inexact */
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416 }
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417 else k = 0;
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418
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|
|
419 /* x is now in primary range */
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420 t = x*x;
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|
|
421 c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
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|
|
422 if(k==0) return one-((x*c)/(c-2.0)-x);
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|
423 else y = one-((lo-(x*c)/(2.0-c))-hi);
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|
|
424 if(k >= -1021) {
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|
425 __HI(y) += (k<<20); /* add k to y's exponent */
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|
426 return y;
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|
|
427 } else {
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|
428 __HI(y) += ((k+1000)<<20);/* add k to y's exponent */
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|
429 return y*twom1000;
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|
|
430 }
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|
|
431 }
|
|
|
432
|
|
|
433 JRT_LEAF(jdouble, SharedRuntime::dexp(jdouble x))
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|
|
434 return __ieee754_exp(x);
|
|
|
435 JRT_END
|
|
|
436
|
|
|
437 /* __ieee754_pow(x,y) return x**y
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|
|
438 *
|
|
|
439 * n
|
|
|
440 * Method: Let x = 2 * (1+f)
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|
|
441 * 1. Compute and return log2(x) in two pieces:
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|
|
442 * log2(x) = w1 + w2,
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|
|
443 * where w1 has 53-24 = 29 bit trailing zeros.
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|
|
444 * 2. Perform y*log2(x) = n+y' by simulating muti-precision
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|
|
445 * arithmetic, where |y'|<=0.5.
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|
|
446 * 3. Return x**y = 2**n*exp(y'*log2)
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|
|
447 *
|
|
|
448 * Special cases:
|
|
|
449 * 1. (anything) ** 0 is 1
|
|
|
450 * 2. (anything) ** 1 is itself
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|
|
451 * 3. (anything) ** NAN is NAN
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|
|
452 * 4. NAN ** (anything except 0) is NAN
|
|
|
453 * 5. +-(|x| > 1) ** +INF is +INF
|
|
|
454 * 6. +-(|x| > 1) ** -INF is +0
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|
|
455 * 7. +-(|x| < 1) ** +INF is +0
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|
|
456 * 8. +-(|x| < 1) ** -INF is +INF
|
|
|
457 * 9. +-1 ** +-INF is NAN
|
|
|
458 * 10. +0 ** (+anything except 0, NAN) is +0
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|
|
459 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
|
|
|
460 * 12. +0 ** (-anything except 0, NAN) is +INF
|
|
|
461 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
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|
|
462 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
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|
|
463 * 15. +INF ** (+anything except 0,NAN) is +INF
|
|
|
464 * 16. +INF ** (-anything except 0,NAN) is +0
|
|
|
465 * 17. -INF ** (anything) = -0 ** (-anything)
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|
|
466 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
|
|
|
467 * 19. (-anything except 0 and inf) ** (non-integer) is NAN
|
|
|
468 *
|
|
|
469 * Accuracy:
|
|
|
470 * pow(x,y) returns x**y nearly rounded. In particular
|
|
|
471 * pow(integer,integer)
|
|
|
472 * always returns the correct integer provided it is
|
|
|
473 * representable.
|
|
|
474 *
|
|
|
475 * Constants :
|
|
|
476 * The hexadecimal values are the intended ones for the following
|
|
|
477 * constants. The decimal values may be used, provided that the
|
|
|
478 * compiler will convert from decimal to binary accurately enough
|
|
|
479 * to produce the hexadecimal values shown.
|
|
|
480 */
|
|
|
481
|
|
|
482 static const double
|
|
|
483 bp[] = {1.0, 1.5,},
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|
|
484 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
|
|
|
485 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
|
|
|
486 zeroX = 0.0,
|
|
|
487 two = 2.0,
|
|
|
488 two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
|
|
|
489 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
|
|
|
490 L1X = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
|
|
|
491 L2X = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
|
|
|
492 L3X = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
|
|
|
493 L4X = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
|
|
|
494 L5X = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
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|
|
495 L6X = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
|
|
|
496 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
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|
|
497 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
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|
|
498 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
|
|
|
499 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
|
|
|
500 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
|
|
|
501 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
|
|
|
502 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
|
|
|
503 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
|
|
|
504 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
|
|
|
505 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
|
|
|
506
|
|
|
507 double __ieee754_pow(double x, double y) {
|
|
|
508 double z,ax,z_h,z_l,p_h,p_l;
|
|
|
509 double y1,t1,t2,r,s,t,u,v,w;
|
|
|
510 int i0,i1,i,j,k,yisint,n;
|
|
|
511 int hx,hy,ix,iy;
|
|
|
512 unsigned lx,ly;
|
|
|
513
|
|
|
514 i0 = ((*(int*)&one)>>29)^1; i1=1-i0;
|
|
|
515 hx = __HI(x); lx = __LO(x);
|
|
|
516 hy = __HI(y); ly = __LO(y);
|
|
|
517 ix = hx&0x7fffffff; iy = hy&0x7fffffff;
|
|
|
518
|
|
|
519 /* y==zero: x**0 = 1 */
|
|
|
520 if((iy|ly)==0) return one;
|
|
|
521
|
|
|
522 /* +-NaN return x+y */
|
|
|
523 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
|
|
|
524 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
|
|
|
525 return x+y;
|
|
|
526
|
|
|
527 /* determine if y is an odd int when x < 0
|
|
|
528 * yisint = 0 ... y is not an integer
|
|
|
529 * yisint = 1 ... y is an odd int
|
|
|
530 * yisint = 2 ... y is an even int
|
|
|
531 */
|
|
|
532 yisint = 0;
|
|
|
533 if(hx<0) {
|
|
|
534 if(iy>=0x43400000) yisint = 2; /* even integer y */
|
|
|
535 else if(iy>=0x3ff00000) {
|
|
|
536 k = (iy>>20)-0x3ff; /* exponent */
|
|
|
537 if(k>20) {
|
|
|
538 j = ly>>(52-k);
|
|
|
539 if((unsigned)(j<<(52-k))==ly) yisint = 2-(j&1);
|
|
|
540 } else if(ly==0) {
|
|
|
541 j = iy>>(20-k);
|
|
|
542 if((j<<(20-k))==iy) yisint = 2-(j&1);
|
|
|
543 }
|
|
|
544 }
|
|
|
545 }
|
|
|
546
|
|
|
547 /* special value of y */
|
|
|
548 if(ly==0) {
|
|
|
549 if (iy==0x7ff00000) { /* y is +-inf */
|
|
|
550 if(((ix-0x3ff00000)|lx)==0)
|
|
|
551 return y - y; /* inf**+-1 is NaN */
|
|
|
552 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
|
|
|
553 return (hy>=0)? y: zeroX;
|
|
|
554 else /* (|x|<1)**-,+inf = inf,0 */
|
|
|
555 return (hy<0)?-y: zeroX;
|
|
|
556 }
|
|
|
557 if(iy==0x3ff00000) { /* y is +-1 */
|
|
|
558 if(hy<0) return one/x; else return x;
|
|
|
559 }
|
|
|
560 if(hy==0x40000000) return x*x; /* y is 2 */
|
|
|
561 if(hy==0x3fe00000) { /* y is 0.5 */
|
|
|
562 if(hx>=0) /* x >= +0 */
|
|
|
563 return sqrt(x);
|
|
|
564 }
|
|
|
565 }
|
|
|
566
|
|
|
567 ax = fabsd(x);
|
|
|
568 /* special value of x */
|
|
|
569 if(lx==0) {
|
|
|
570 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
|
|
|
571 z = ax; /*x is +-0,+-inf,+-1*/
|
|
|
572 if(hy<0) z = one/z; /* z = (1/|x|) */
|
|
|
573 if(hx<0) {
|
|
|
574 if(((ix-0x3ff00000)|yisint)==0) {
|
|
|
575 z = (z-z)/(z-z); /* (-1)**non-int is NaN */
|
|
|
576 } else if(yisint==1)
|
|
|
577 z = -1.0*z; /* (x<0)**odd = -(|x|**odd) */
|
|
|
578 }
|
|
|
579 return z;
|
|
|
580 }
|
|
|
581 }
|
|
|
582
|
|
|
583 n = (hx>>31)+1;
|
|
|
584
|
|
|
585 /* (x<0)**(non-int) is NaN */
|
|
|
586 if((n|yisint)==0) return (x-x)/(x-x);
|
|
|
587
|
|
|
588 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
|
|
|
589 if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
|
|
|
590
|
|
|
591 /* |y| is huge */
|
|
|
592 if(iy>0x41e00000) { /* if |y| > 2**31 */
|
|
|
593 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
|
|
|
594 if(ix<=0x3fefffff) return (hy<0)? hugeX*hugeX:tiny*tiny;
|
|
|
595 if(ix>=0x3ff00000) return (hy>0)? hugeX*hugeX:tiny*tiny;
|
|
|
596 }
|
|
|
597 /* over/underflow if x is not close to one */
|
|
|
598 if(ix<0x3fefffff) return (hy<0)? s*hugeX*hugeX:s*tiny*tiny;
|
|
|
599 if(ix>0x3ff00000) return (hy>0)? s*hugeX*hugeX:s*tiny*tiny;
|
|
|
600 /* now |1-x| is tiny <= 2**-20, suffice to compute
|
|
|
601 log(x) by x-x^2/2+x^3/3-x^4/4 */
|
|
|
602 t = ax-one; /* t has 20 trailing zeros */
|
|
|
603 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
|
|
|
604 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
|
|
|
605 v = t*ivln2_l-w*ivln2;
|
|
|
606 t1 = u+v;
|
|
|
607 __LO(t1) = 0;
|
|
|
608 t2 = v-(t1-u);
|
|
|
609 } else {
|
|
|
610 double ss,s2,s_h,s_l,t_h,t_l;
|
|
|
611 n = 0;
|
|
|
612 /* take care subnormal number */
|
|
|
613 if(ix<0x00100000)
|
|
|
614 {ax *= two53; n -= 53; ix = __HI(ax); }
|
|
|
615 n += ((ix)>>20)-0x3ff;
|
|
|
616 j = ix&0x000fffff;
|
|
|
617 /* determine interval */
|
|
|
618 ix = j|0x3ff00000; /* normalize ix */
|
|
|
619 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
|
|
|
620 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
|
|
|
621 else {k=0;n+=1;ix -= 0x00100000;}
|
|
|
622 __HI(ax) = ix;
|
|
|
623
|
|
|
624 /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
|
|
|
625 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
|
|
|
626 v = one/(ax+bp[k]);
|
|
|
627 ss = u*v;
|
|
|
628 s_h = ss;
|
|
|
629 __LO(s_h) = 0;
|
|
|
630 /* t_h=ax+bp[k] High */
|
|
|
631 t_h = zeroX;
|
|
|
632 __HI(t_h)=((ix>>1)|0x20000000)+0x00080000+(k<<18);
|
|
|
633 t_l = ax - (t_h-bp[k]);
|
|
|
634 s_l = v*((u-s_h*t_h)-s_h*t_l);
|
|
|
635 /* compute log(ax) */
|
|
|
636 s2 = ss*ss;
|
|
|
637 r = s2*s2*(L1X+s2*(L2X+s2*(L3X+s2*(L4X+s2*(L5X+s2*L6X)))));
|
|
|
638 r += s_l*(s_h+ss);
|
|
|
639 s2 = s_h*s_h;
|
|
|
640 t_h = 3.0+s2+r;
|
|
|
641 __LO(t_h) = 0;
|
|
|
642 t_l = r-((t_h-3.0)-s2);
|
|
|
643 /* u+v = ss*(1+...) */
|
|
|
644 u = s_h*t_h;
|
|
|
645 v = s_l*t_h+t_l*ss;
|
|
|
646 /* 2/(3log2)*(ss+...) */
|
|
|
647 p_h = u+v;
|
|
|
648 __LO(p_h) = 0;
|
|
|
649 p_l = v-(p_h-u);
|
|
|
650 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
|
|
|
651 z_l = cp_l*p_h+p_l*cp+dp_l[k];
|
|
|
652 /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
|
|
|
653 t = (double)n;
|
|
|
654 t1 = (((z_h+z_l)+dp_h[k])+t);
|
|
|
655 __LO(t1) = 0;
|
|
|
656 t2 = z_l-(((t1-t)-dp_h[k])-z_h);
|
|
|
657 }
|
|
|
658
|
|
|
659 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
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660 y1 = y;
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661 __LO(y1) = 0;
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662 p_l = (y-y1)*t1+y*t2;
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663 p_h = y1*t1;
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664 z = p_l+p_h;
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665 j = __HI(z);
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666 i = __LO(z);
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667 if (j>=0x40900000) { /* z >= 1024 */
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668 if(((j-0x40900000)|i)!=0) /* if z > 1024 */
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669 return s*hugeX*hugeX; /* overflow */
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670 else {
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671 if(p_l+ovt>z-p_h) return s*hugeX*hugeX; /* overflow */
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672 }
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673 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
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674 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
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675 return s*tiny*tiny; /* underflow */
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676 else {
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677 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
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678 }
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679 }
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680 /*
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681 * compute 2**(p_h+p_l)
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682 */
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683 i = j&0x7fffffff;
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684 k = (i>>20)-0x3ff;
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685 n = 0;
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686 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
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687 n = j+(0x00100000>>(k+1));
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688 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
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689 t = zeroX;
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690 __HI(t) = (n&~(0x000fffff>>k));
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691 n = ((n&0x000fffff)|0x00100000)>>(20-k);
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692 if(j<0) n = -n;
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693 p_h -= t;
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694 }
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695 t = p_l+p_h;
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696 __LO(t) = 0;
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697 u = t*lg2_h;
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698 v = (p_l-(t-p_h))*lg2+t*lg2_l;
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699 z = u+v;
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700 w = v-(z-u);
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701 t = z*z;
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702 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
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703 r = (z*t1)/(t1-two)-(w+z*w);
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704 z = one-(r-z);
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705 j = __HI(z);
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706 j += (n<<20);
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707 if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
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708 else __HI(z) += (n<<20);
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709 return s*z;
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710 }
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711
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712
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713 JRT_LEAF(jdouble, SharedRuntime::dpow(jdouble x, jdouble y))
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714 return __ieee754_pow(x, y);
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715 JRT_END
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716
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717 #ifdef WIN32
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718 # pragma optimize ( "", on )
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719 #endif
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