fpu/softfloat-macros.h

   1
   2 /*============================================================================
   3
   4 This C source fragment is part of the SoftFloat IEC/IEEE Floating-point
   5 Arithmetic Package, Release 2b.
   6
   7 Written by John R. Hauser.  This work was made possible in part by the
   8 International Computer Science Institute, located at Suite 600, 1947 Center
   9 Street, Berkeley, California 94704.  Funding was partially provided by the
  10 National Science Foundation under grant MIP-9311980.  The original version
  11 of this code was written as part of a project to build a fixed-point vector
  12 processor in collaboration with the University of California at Berkeley,
  13 overseen by Profs. Nelson Morgan and John Wawrzynek.  More information
  14 is available through the Web page `http://www.cs.berkeley.edu/~jhauser/
  15 arithmetic/SoftFloat.html'.
  16
  17 THIS SOFTWARE IS DISTRIBUTED AS IS, FOR FREE.  Although reasonable effort has
  18 been made to avoid it, THIS SOFTWARE MAY CONTAIN FAULTS THAT WILL AT TIMES
  19 RESULT IN INCORRECT BEHAVIOR.  USE OF THIS SOFTWARE IS RESTRICTED TO PERSONS
  20 AND ORGANIZATIONS WHO CAN AND WILL TAKE FULL RESPONSIBILITY FOR ALL LOSSES,
  21 COSTS, OR OTHER PROBLEMS THEY INCUR DUE TO THE SOFTWARE, AND WHO FURTHERMORE
  22 EFFECTIVELY INDEMNIFY JOHN HAUSER AND THE INTERNATIONAL COMPUTER SCIENCE
  23 INSTITUTE (possibly via similar legal notice) AGAINST ALL LOSSES, COSTS, OR
  24 OTHER PROBLEMS INCURRED BY THEIR CUSTOMERS AND CLIENTS DUE TO THE SOFTWARE.
  25
  26 Derivative works are acceptable, even for commercial purposes, so long as
  27 (1) the source code for the derivative work includes prominent notice that
  28 the work is derivative, and (2) the source code includes prominent notice with
  29 these four paragraphs for those parts of this code that are retained.
  30
  31 =============================================================================*/
  32
  33 /*----------------------------------------------------------------------------
  34 | Shifts `a' right by the number of bits given in `count'.  If any nonzero
  35 | bits are shifted off, they are ``jammed'' into the least significant bit of
  36 | the result by setting the least significant bit to 1.  The value of `count'
  37 | can be arbitrarily large; in particular, if `count' is greater than 32, the
  38 | result will be either 0 or 1, depending on whether `a' is zero or nonzero.
  39 | The result is stored in the location pointed to by `zPtr'.
  40 *----------------------------------------------------------------------------*/
  41
  42 INLINE void shift32RightJamming( bits32 a, int16 count, bits32 *zPtr )
  43 {
  44     bits32 z;
  45
  46     if ( count == 0 ) {
  47         z = a;
  48     }
  49     else if ( count < 32 ) {
  50         z = ( a>>count ) | ( ( a<<( ( - count ) & 31 ) ) != 0 );
  51     }
  52     else {
  53         z = ( a != 0 );
  54     }
  55     *zPtr = z;
  56
  57 }
  58
  59 /*----------------------------------------------------------------------------
  60 | Shifts `a' right by the number of bits given in `count'.  If any nonzero
  61 | bits are shifted off, they are ``jammed'' into the least significant bit of
  62 | the result by setting the least significant bit to 1.  The value of `count'
  63 | can be arbitrarily large; in particular, if `count' is greater than 64, the
  64 | result will be either 0 or 1, depending on whether `a' is zero or nonzero.
  65 | The result is stored in the location pointed to by `zPtr'.
  66 *----------------------------------------------------------------------------*/
  67
  68 INLINE void shift64RightJamming( bits64 a, int16 count, bits64 *zPtr )
  69 {
  70     bits64 z;
  71
  72     if ( count == 0 ) {
  73         z = a;
  74     }
  75     else if ( count < 64 ) {
  76         z = ( a>>count ) | ( ( a<<( ( - count ) & 63 ) ) != 0 );
  77     }
  78     else {
  79         z = ( a != 0 );
  80     }
  81     *zPtr = z;
  82
  83 }
  84
  85 /*----------------------------------------------------------------------------
  86 | Shifts the 128-bit value formed by concatenating `a0' and `a1' right by 64
  87 | _plus_ the number of bits given in `count'.  The shifted result is at most
  88 | 64 nonzero bits; this is stored at the location pointed to by `z0Ptr'.  The
  89 | bits shifted off form a second 64-bit result as follows:  The _last_ bit
  90 | shifted off is the most-significant bit of the extra result, and the other
  91 | 63 bits of the extra result are all zero if and only if _all_but_the_last_
  92 | bits shifted off were all zero.  This extra result is stored in the location
  93 | pointed to by `z1Ptr'.  The value of `count' can be arbitrarily large.
  94 |     (This routine makes more sense if `a0' and `a1' are considered to form
  95 | a fixed-point value with binary point between `a0' and `a1'.  This fixed-
  96 | point value is shifted right by the number of bits given in `count', and
  97 | the integer part of the result is returned at the location pointed to by
  98 | `z0Ptr'.  The fractional part of the result may be slightly corrupted as
  99 | described above, and is returned at the location pointed to by `z1Ptr'.)
 100 *----------------------------------------------------------------------------*/
 101
 102 INLINE void
 103  shift64ExtraRightJamming(
 104      bits64 a0, bits64 a1, int16 count, bits64 *z0Ptr, bits64 *z1Ptr )
 105 {
 106     bits64 z0, z1;
 107     int8 negCount = ( - count ) & 63;
 108
 109     if ( count == 0 ) {
 110         z1 = a1;
 111         z0 = a0;
 112     }
 113     else if ( count < 64 ) {
 114         z1 = ( a0<<negCount ) | ( a1 != 0 );
 115         z0 = a0>>count;
 116     }
 117     else {
 118         if ( count == 64 ) {
 119             z1 = a0 | ( a1 != 0 );
 120         }
 121         else {
 122             z1 = ( ( a0 | a1 ) != 0 );
 123         }
 124         z0 = 0;
 125     }
 126     *z1Ptr = z1;
 127     *z0Ptr = z0;
 128
 129 }
 130
 131 /*----------------------------------------------------------------------------
 132 | Shifts the 128-bit value formed by concatenating `a0' and `a1' right by the
 133 | number of bits given in `count'.  Any bits shifted off are lost.  The value
 134 | of `count' can be arbitrarily large; in particular, if `count' is greater
 135 | than 128, the result will be 0.  The result is broken into two 64-bit pieces
 136 | which are stored at the locations pointed to by `z0Ptr' and `z1Ptr'.
 137 *----------------------------------------------------------------------------*/
 138
 139 INLINE void
 140  shift128Right(
 141      bits64 a0, bits64 a1, int16 count, bits64 *z0Ptr, bits64 *z1Ptr )
 142 {
 143     bits64 z0, z1;
 144     int8 negCount = ( - count ) & 63;
 145
 146     if ( count == 0 ) {
 147         z1 = a1;
 148         z0 = a0;
 149     }
 150     else if ( count < 64 ) {
 151         z1 = ( a0<<negCount ) | ( a1>>count );
 152         z0 = a0>>count;
 153     }
 154     else {
 155         z1 = ( count < 64 ) ? ( a0>>( count & 63 ) ) : 0;
 156         z0 = 0;
 157     }
 158     *z1Ptr = z1;
 159     *z0Ptr = z0;
 160
 161 }
 162
 163 /*----------------------------------------------------------------------------
 164 | Shifts the 128-bit value formed by concatenating `a0' and `a1' right by the
 165 | number of bits given in `count'.  If any nonzero bits are shifted off, they
 166 | are ``jammed'' into the least significant bit of the result by setting the
 167 | least significant bit to 1.  The value of `count' can be arbitrarily large;
 168 | in particular, if `count' is greater than 128, the result will be either
 169 | 0 or 1, depending on whether the concatenation of `a0' and `a1' is zero or
 170 | nonzero.  The result is broken into two 64-bit pieces which are stored at
 171 | the locations pointed to by `z0Ptr' and `z1Ptr'.
 172 *----------------------------------------------------------------------------*/
 173
 174 INLINE void
 175  shift128RightJamming(
 176      bits64 a0, bits64 a1, int16 count, bits64 *z0Ptr, bits64 *z1Ptr )
 177 {
 178     bits64 z0, z1;
 179     int8 negCount = ( - count ) & 63;
 180
 181     if ( count == 0 ) {
 182         z1 = a1;
 183         z0 = a0;
 184     }
 185     else if ( count < 64 ) {
 186         z1 = ( a0<<negCount ) | ( a1>>count ) | ( ( a1<<negCount ) != 0 );
 187         z0 = a0>>count;
 188     }
 189     else {
 190         if ( count == 64 ) {
 191             z1 = a0 | ( a1 != 0 );
 192         }
 193         else if ( count < 128 ) {
 194             z1 = ( a0>>( count & 63 ) ) | ( ( ( a0<<negCount ) | a1 ) != 0 );
 195         }
 196         else {
 197             z1 = ( ( a0 | a1 ) != 0 );
 198         }
 199         z0 = 0;
 200     }
 201     *z1Ptr = z1;
 202     *z0Ptr = z0;
 203
 204 }
 205
 206 /*----------------------------------------------------------------------------
 207 | Shifts the 192-bit value formed by concatenating `a0', `a1', and `a2' right
 208 | by 64 _plus_ the number of bits given in `count'.  The shifted result is
 209 | at most 128 nonzero bits; these are broken into two 64-bit pieces which are
 210 | stored at the locations pointed to by `z0Ptr' and `z1Ptr'.  The bits shifted
 211 | off form a third 64-bit result as follows:  The _last_ bit shifted off is
 212 | the most-significant bit of the extra result, and the other 63 bits of the
 213 | extra result are all zero if and only if _all_but_the_last_ bits shifted off
 214 | were all zero.  This extra result is stored in the location pointed to by
 215 | `z2Ptr'.  The value of `count' can be arbitrarily large.
 216 |     (This routine makes more sense if `a0', `a1', and `a2' are considered
 217 | to form a fixed-point value with binary point between `a1' and `a2'.  This
 218 | fixed-point value is shifted right by the number of bits given in `count',
 219 | and the integer part of the result is returned at the locations pointed to
 220 | by `z0Ptr' and `z1Ptr'.  The fractional part of the result may be slightly
 221 | corrupted as described above, and is returned at the location pointed to by
 222 | `z2Ptr'.)
 223 *----------------------------------------------------------------------------*/
 224
 225 INLINE void
 226  shift128ExtraRightJamming(
 227      bits64 a0,
 228      bits64 a1,
 229      bits64 a2,
 230      int16 count,
 231      bits64 *z0Ptr,
 232      bits64 *z1Ptr,
 233      bits64 *z2Ptr
 234  )
 235 {
 236     bits64 z0, z1, z2;
 237     int8 negCount = ( - count ) & 63;
 238
 239     if ( count == 0 ) {
 240         z2 = a2;
 241         z1 = a1;
 242         z0 = a0;
 243     }
 244     else {
 245         if ( count < 64 ) {
 246             z2 = a1<<negCount;
 247             z1 = ( a0<<negCount ) | ( a1>>count );
 248             z0 = a0>>count;
 249         }
 250         else {
 251             if ( count == 64 ) {
 252                 z2 = a1;
 253                 z1 = a0;
 254             }
 255             else {
 256                 a2 |= a1;
 257                 if ( count < 128 ) {
 258                     z2 = a0<<negCount;
 259                     z1 = a0>>( count & 63 );
 260                 }
 261                 else {
 262                     z2 = ( count == 128 ) ? a0 : ( a0 != 0 );
 263                     z1 = 0;
 264                 }
 265             }
 266             z0 = 0;
 267         }
 268         z2 |= ( a2 != 0 );
 269     }
 270     *z2Ptr = z2;
 271     *z1Ptr = z1;
 272     *z0Ptr = z0;
 273
 274 }
 275
 276 /*----------------------------------------------------------------------------
 277 | Shifts the 128-bit value formed by concatenating `a0' and `a1' left by the
 278 | number of bits given in `count'.  Any bits shifted off are lost.  The value
 279 | of `count' must be less than 64.  The result is broken into two 64-bit
 280 | pieces which are stored at the locations pointed to by `z0Ptr' and `z1Ptr'.
 281 *----------------------------------------------------------------------------*/
 282
 283 INLINE void
 284  shortShift128Left(
 285      bits64 a0, bits64 a1, int16 count, bits64 *z0Ptr, bits64 *z1Ptr )
 286 {
 287
 288     *z1Ptr = a1<<count;
 289     *z0Ptr =
 290         ( count == 0 ) ? a0 : ( a0<<count ) | ( a1>>( ( - count ) & 63 ) );
 291
 292 }
 293
 294 /*----------------------------------------------------------------------------
 295 | Shifts the 192-bit value formed by concatenating `a0', `a1', and `a2' left
 296 | by the number of bits given in `count'.  Any bits shifted off are lost.
 297 | The value of `count' must be less than 64.  The result is broken into three
 298 | 64-bit pieces which are stored at the locations pointed to by `z0Ptr',
 299 | `z1Ptr', and `z2Ptr'.
 300 *----------------------------------------------------------------------------*/
 301
 302 INLINE void
 303  shortShift192Left(
 304      bits64 a0,
 305      bits64 a1,
 306      bits64 a2,
 307      int16 count,
 308      bits64 *z0Ptr,
 309      bits64 *z1Ptr,
 310      bits64 *z2Ptr
 311  )
 312 {
 313     bits64 z0, z1, z2;
 314     int8 negCount;
 315
 316     z2 = a2<<count;
 317     z1 = a1<<count;
 318     z0 = a0<<count;
 319     if ( 0 < count ) {
 320         negCount = ( ( - count ) & 63 );
 321         z1 |= a2>>negCount;
 322         z0 |= a1>>negCount;
 323     }
 324     *z2Ptr = z2;
 325     *z1Ptr = z1;
 326     *z0Ptr = z0;
 327
 328 }
 329
 330 /*----------------------------------------------------------------------------
 331 | Adds the 128-bit value formed by concatenating `a0' and `a1' to the 128-bit
 332 | value formed by concatenating `b0' and `b1'.  Addition is modulo 2^128, so
 333 | any carry out is lost.  The result is broken into two 64-bit pieces which
 334 | are stored at the locations pointed to by `z0Ptr' and `z1Ptr'.
 335 *----------------------------------------------------------------------------*/
 336
 337 INLINE void
 338  add128(
 339      bits64 a0, bits64 a1, bits64 b0, bits64 b1, bits64 *z0Ptr, bits64 *z1Ptr )
 340 {
 341     bits64 z1;
 342
 343     z1 = a1 + b1;
 344     *z1Ptr = z1;
 345     *z0Ptr = a0 + b0 + ( z1 < a1 );
 346
 347 }
 348
 349 /*----------------------------------------------------------------------------
 350 | Adds the 192-bit value formed by concatenating `a0', `a1', and `a2' to the
 351 | 192-bit value formed by concatenating `b0', `b1', and `b2'.  Addition is
 352 | modulo 2^192, so any carry out is lost.  The result is broken into three
 353 | 64-bit pieces which are stored at the locations pointed to by `z0Ptr',
 354 | `z1Ptr', and `z2Ptr'.
 355 *----------------------------------------------------------------------------*/
 356
 357 INLINE void
 358  add192(
 359      bits64 a0,
 360      bits64 a1,
 361      bits64 a2,
 362      bits64 b0,
 363      bits64 b1,
 364      bits64 b2,
 365      bits64 *z0Ptr,
 366      bits64 *z1Ptr,
 367      bits64 *z2Ptr
 368  )
 369 {
 370     bits64 z0, z1, z2;
 371     int8 carry0, carry1;
 372
 373     z2 = a2 + b2;
 374     carry1 = ( z2 < a2 );
 375     z1 = a1 + b1;
 376     carry0 = ( z1 < a1 );
 377     z0 = a0 + b0;
 378     z1 += carry1;
 379     z0 += ( z1 < carry1 );
 380     z0 += carry0;
 381     *z2Ptr = z2;
 382     *z1Ptr = z1;
 383     *z0Ptr = z0;
 384
 385 }
 386
 387 /*----------------------------------------------------------------------------
 388 | Subtracts the 128-bit value formed by concatenating `b0' and `b1' from the
 389 | 128-bit value formed by concatenating `a0' and `a1'.  Subtraction is modulo
 390 | 2^128, so any borrow out (carry out) is lost.  The result is broken into two
 391 | 64-bit pieces which are stored at the locations pointed to by `z0Ptr' and
 392 | `z1Ptr'.
 393 *----------------------------------------------------------------------------*/
 394
 395 INLINE void
 396  sub128(
 397      bits64 a0, bits64 a1, bits64 b0, bits64 b1, bits64 *z0Ptr, bits64 *z1Ptr )
 398 {
 399
 400     *z1Ptr = a1 - b1;
 401     *z0Ptr = a0 - b0 - ( a1 < b1 );
 402
 403 }
 404
 405 /*----------------------------------------------------------------------------
 406 | Subtracts the 192-bit value formed by concatenating `b0', `b1', and `b2'
 407 | from the 192-bit value formed by concatenating `a0', `a1', and `a2'.
 408 | Subtraction is modulo 2^192, so any borrow out (carry out) is lost.  The
 409 | result is broken into three 64-bit pieces which are stored at the locations
 410 | pointed to by `z0Ptr', `z1Ptr', and `z2Ptr'.
 411 *----------------------------------------------------------------------------*/
 412
 413 INLINE void
 414  sub192(
 415      bits64 a0,
 416      bits64 a1,
 417      bits64 a2,
 418      bits64 b0,
 419      bits64 b1,
 420      bits64 b2,
 421      bits64 *z0Ptr,
 422      bits64 *z1Ptr,
 423      bits64 *z2Ptr
 424  )
 425 {
 426     bits64 z0, z1, z2;
 427     int8 borrow0, borrow1;
 428
 429     z2 = a2 - b2;
 430     borrow1 = ( a2 < b2 );
 431     z1 = a1 - b1;
 432     borrow0 = ( a1 < b1 );
 433     z0 = a0 - b0;
 434     z0 -= ( z1 < borrow1 );
 435     z1 -= borrow1;
 436     z0 -= borrow0;
 437     *z2Ptr = z2;
 438     *z1Ptr = z1;
 439     *z0Ptr = z0;
 440
 441 }
 442
 443 /*----------------------------------------------------------------------------
 444 | Multiplies `a' by `b' to obtain a 128-bit product.  The product is broken
 445 | into two 64-bit pieces which are stored at the locations pointed to by
 446 | `z0Ptr' and `z1Ptr'.
 447 *----------------------------------------------------------------------------*/
 448
 449 INLINE void mul64To128( bits64 a, bits64 b, bits64 *z0Ptr, bits64 *z1Ptr )
 450 {
 451     bits32 aHigh, aLow, bHigh, bLow;
 452     bits64 z0, zMiddleA, zMiddleB, z1;
 453
 454     aLow = a;
 455     aHigh = a>>32;
 456     bLow = b;
 457     bHigh = b>>32;
 458     z1 = ( (bits64) aLow ) * bLow;
 459     zMiddleA = ( (bits64) aLow ) * bHigh;
 460     zMiddleB = ( (bits64) aHigh ) * bLow;
 461     z0 = ( (bits64) aHigh ) * bHigh;
 462     zMiddleA += zMiddleB;
 463     z0 += ( ( (bits64) ( zMiddleA < zMiddleB ) )<<32 ) + ( zMiddleA>>32 );
 464     zMiddleA <<= 32;
 465     z1 += zMiddleA;
 466     z0 += ( z1 < zMiddleA );
 467     *z1Ptr = z1;
 468     *z0Ptr = z0;
 469
 470 }
 471
 472 /*----------------------------------------------------------------------------
 473 | Multiplies the 128-bit value formed by concatenating `a0' and `a1' by
 474 | `b' to obtain a 192-bit product.  The product is broken into three 64-bit
 475 | pieces which are stored at the locations pointed to by `z0Ptr', `z1Ptr', and
 476 | `z2Ptr'.
 477 *----------------------------------------------------------------------------*/
 478
 479 INLINE void
 480  mul128By64To192(
 481      bits64 a0,
 482      bits64 a1,
 483      bits64 b,
 484      bits64 *z0Ptr,
 485      bits64 *z1Ptr,
 486      bits64 *z2Ptr
 487  )
 488 {
 489     bits64 z0, z1, z2, more1;
 490
 491     mul64To128( a1, b, &z1, &z2 );
 492     mul64To128( a0, b, &z0, &more1 );
 493     add128( z0, more1, 0, z1, &z0, &z1 );
 494     *z2Ptr = z2;
 495     *z1Ptr = z1;
 496     *z0Ptr = z0;
 497
 498 }
 499
 500 /*----------------------------------------------------------------------------
 501 | Multiplies the 128-bit value formed by concatenating `a0' and `a1' to the
 502 | 128-bit value formed by concatenating `b0' and `b1' to obtain a 256-bit
 503 | product.  The product is broken into four 64-bit pieces which are stored at
 504 | the locations pointed to by `z0Ptr', `z1Ptr', `z2Ptr', and `z3Ptr'.
 505 *----------------------------------------------------------------------------*/
 506
 507 INLINE void
 508  mul128To256(
 509      bits64 a0,
 510      bits64 a1,
 511      bits64 b0,
 512      bits64 b1,
 513      bits64 *z0Ptr,
 514      bits64 *z1Ptr,
 515      bits64 *z2Ptr,
 516      bits64 *z3Ptr
 517  )
 518 {
 519     bits64 z0, z1, z2, z3;
 520     bits64 more1, more2;
 521
 522     mul64To128( a1, b1, &z2, &z3 );
 523     mul64To128( a1, b0, &z1, &more2 );
 524     add128( z1, more2, 0, z2, &z1, &z2 );
 525     mul64To128( a0, b0, &z0, &more1 );
 526     add128( z0, more1, 0, z1, &z0, &z1 );
 527     mul64To128( a0, b1, &more1, &more2 );
 528     add128( more1, more2, 0, z2, &more1, &z2 );
 529     add128( z0, z1, 0, more1, &z0, &z1 );
 530     *z3Ptr = z3;
 531     *z2Ptr = z2;
 532     *z1Ptr = z1;
 533     *z0Ptr = z0;
 534
 535 }
 536
 537 /*----------------------------------------------------------------------------
 538 | Returns an approximation to the 64-bit integer quotient obtained by dividing
 539 | `b' into the 128-bit value formed by concatenating `a0' and `a1'.  The
 540 | divisor `b' must be at least 2^63.  If q is the exact quotient truncated
 541 | toward zero, the approximation returned lies between q and q + 2 inclusive.
 542 | If the exact quotient q is larger than 64 bits, the maximum positive 64-bit
 543 | unsigned integer is returned.
 544 *----------------------------------------------------------------------------*/
 545
 546 static bits64 estimateDiv128To64( bits64 a0, bits64 a1, bits64 b )
 547 {
 548     bits64 b0, b1;
 549     bits64 rem0, rem1, term0, term1;
 550     bits64 z;
 551
 552     if ( b <= a0 ) return LIT64( 0xFFFFFFFFFFFFFFFF );
 553     b0 = b>>32;
 554     z = ( b0<<32 <= a0 ) ? LIT64( 0xFFFFFFFF00000000 ) : ( a0 / b0 )<<32;
 555     mul64To128( b, z, &term0, &term1 );
 556     sub128( a0, a1, term0, term1, &rem0, &rem1 );
 557     while ( ( (sbits64) rem0 ) < 0 ) {
 558         z -= LIT64( 0x100000000 );
 559         b1 = b<<32;
 560         add128( rem0, rem1, b0, b1, &rem0, &rem1 );
 561     }
 562     rem0 = ( rem0<<32 ) | ( rem1>>32 );
 563     z |= ( b0<<32 <= rem0 ) ? 0xFFFFFFFF : rem0 / b0;
 564     return z;
 565
 566 }
 567
 568 /*----------------------------------------------------------------------------
 569 | Returns an approximation to the square root of the 32-bit significand given
 570 | by `a'.  Considered as an integer, `a' must be at least 2^31.  If bit 0 of
 571 | `aExp' (the least significant bit) is 1, the integer returned approximates
 572 | 2^31*sqrt(`a'/2^31), where `a' is considered an integer.  If bit 0 of `aExp'
 573 | is 0, the integer returned approximates 2^31*sqrt(`a'/2^30).  In either
 574 | case, the approximation returned lies strictly within +/-2 of the exact
 575 | value.
 576 *----------------------------------------------------------------------------*/
 577
 578 static bits32 estimateSqrt32( int16 aExp, bits32 a )
 579 {
 580     static const bits16 sqrtOddAdjustments[] = {
 581         0x0004, 0x0022, 0x005D, 0x00B1, 0x011D, 0x019F, 0x0236, 0x02E0,
 582         0x039C, 0x0468, 0x0545, 0x0631, 0x072B, 0x0832, 0x0946, 0x0A67
 583     };
 584     static const bits16 sqrtEvenAdjustments[] = {
 585         0x0A2D, 0x08AF, 0x075A, 0x0629, 0x051A, 0x0429, 0x0356, 0x029E,
 586         0x0200, 0x0179, 0x0109, 0x00AF, 0x0068, 0x0034, 0x0012, 0x0002
 587     };
 588     int8 index;
 589     bits32 z;
 590
 591     index = ( a>>27 ) & 15;
 592     if ( aExp & 1 ) {
 593         z = 0x4000 + ( a>>17 ) - sqrtOddAdjustments[ index ];
 594         z = ( ( a / z )<<14 ) + ( z<<15 );
 595         a >>= 1;
 596     }
 597     else {
 598         z = 0x8000 + ( a>>17 ) - sqrtEvenAdjustments[ index ];
 599         z = a / z + z;
 600         z = ( 0x20000 <= z ) ? 0xFFFF8000 : ( z<<15 );
 601         if ( z <= a ) return (bits32) ( ( (sbits32) a )>>1 );
 602     }
 603     return ( (bits32) ( ( ( (bits64) a )<<31 ) / z ) ) + ( z>>1 );
 604
 605 }
 606
 607 /*----------------------------------------------------------------------------
 608 | Returns the number of leading 0 bits before the most-significant 1 bit of
 609 | `a'.  If `a' is zero, 32 is returned.
 610 *----------------------------------------------------------------------------*/
 611
 612 static int8 countLeadingZeros32( bits32 a )
 613 {
 614     static const int8 countLeadingZerosHigh[] = {
 615         8, 7, 6, 6, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4,
 616         3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
 617         2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
 618         2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
 619         1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
 620         1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
 621         1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
 622         1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
 623         0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
 624         0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
 625         0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
 626         0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
 627         0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
 628         0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
 629         0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
 630         0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
 631     };
 632     int8 shiftCount;
 633
 634     shiftCount = 0;
 635     if ( a < 0x10000 ) {
 636         shiftCount += 16;
 637         a <<= 16;
 638     }
 639     if ( a < 0x1000000 ) {
 640         shiftCount += 8;
 641         a <<= 8;
 642     }
 643     shiftCount += countLeadingZerosHigh[ a>>24 ];
 644     return shiftCount;
 645
 646 }
 647
 648 /*----------------------------------------------------------------------------
 649 | Returns the number of leading 0 bits before the most-significant 1 bit of
 650 | `a'.  If `a' is zero, 64 is returned.
 651 *----------------------------------------------------------------------------*/
 652
 653 static int8 countLeadingZeros64( bits64 a )
 654 {
 655     int8 shiftCount;
 656
 657     shiftCount = 0;
 658     if ( a < ( (bits64) 1 )<<32 ) {
 659         shiftCount += 32;
 660     }
 661     else {
 662         a >>= 32;
 663     }
 664     shiftCount += countLeadingZeros32( a );
 665     return shiftCount;
 666
 667 }
 668
 669 /*----------------------------------------------------------------------------
 670 | Returns 1 if the 128-bit value formed by concatenating `a0' and `a1'
 671 | is equal to the 128-bit value formed by concatenating `b0' and `b1'.
 672 | Otherwise, returns 0.
 673 *----------------------------------------------------------------------------*/
 674
 675 INLINE flag eq128( bits64 a0, bits64 a1, bits64 b0, bits64 b1 )
 676 {
 677
 678     return ( a0 == b0 ) && ( a1 == b1 );
 679
 680 }
 681
 682 /*----------------------------------------------------------------------------
 683 | Returns 1 if the 128-bit value formed by concatenating `a0' and `a1' is less
 684 | than or equal to the 128-bit value formed by concatenating `b0' and `b1'.
 685 | Otherwise, returns 0.
 686 *----------------------------------------------------------------------------*/
 687
 688 INLINE flag le128( bits64 a0, bits64 a1, bits64 b0, bits64 b1 )
 689 {
 690
 691     return ( a0 < b0 ) || ( ( a0 == b0 ) && ( a1 <= b1 ) );
 692
 693 }
 694
 695 /*----------------------------------------------------------------------------
 696 | Returns 1 if the 128-bit value formed by concatenating `a0' and `a1' is less
 697 | than the 128-bit value formed by concatenating `b0' and `b1'.  Otherwise,
 698 | returns 0.
 699 *----------------------------------------------------------------------------*/
 700
 701 INLINE flag lt128( bits64 a0, bits64 a1, bits64 b0, bits64 b1 )
 702 {
 703
 704     return ( a0 < b0 ) || ( ( a0 == b0 ) && ( a1 < b1 ) );
 705
 706 }
 707
 708 /*----------------------------------------------------------------------------
 709 | Returns 1 if the 128-bit value formed by concatenating `a0' and `a1' is
 710 | not equal to the 128-bit value formed by concatenating `b0' and `b1'.
 711 | Otherwise, returns 0.
 712 *----------------------------------------------------------------------------*/
 713
 714 INLINE flag ne128( bits64 a0, bits64 a1, bits64 b0, bits64 b1 )
 715 {
 716
 717     return ( a0 != b0 ) || ( a1 != b1 );
 718
 719 }
 720