4 ===============================================================================
6 This C source fragment is part of the SoftFloat IEC/IEEE Floating-point
7 Arithmetic Package, Release 2a.
9 Written by John R. Hauser. This work was made possible in part by the
10 International Computer Science Institute, located at Suite 600, 1947 Center
11 Street, Berkeley, California 94704. Funding was partially provided by the
12 National Science Foundation under grant MIP-9311980. The original version
13 of this code was written as part of a project to build a fixed-point vector
14 processor in collaboration with the University of California at Berkeley,
15 overseen by Profs. Nelson Morgan and John Wawrzynek. More information
16 is available through the Web page `http://HTTP.CS.Berkeley.EDU/~jhauser/
17 arithmetic/SoftFloat.html'.
19 THIS SOFTWARE IS DISTRIBUTED AS IS, FOR FREE. Although reasonable effort
20 has been made to avoid it, THIS SOFTWARE MAY CONTAIN FAULTS THAT WILL AT
21 TIMES RESULT IN INCORRECT BEHAVIOR. USE OF THIS SOFTWARE IS RESTRICTED TO
22 PERSONS AND ORGANIZATIONS WHO CAN AND WILL TAKE FULL RESPONSIBILITY FOR ANY
23 AND ALL LOSSES, COSTS, OR OTHER PROBLEMS ARISING FROM ITS USE.
25 Derivative works are acceptable, even for commercial purposes, so long as
26 (1) they include prominent notice that the work is derivative, and (2) they
27 include prominent notice akin to these four paragraphs for those parts of
28 this code that are retained.
30 ===============================================================================
34 -------------------------------------------------------------------------------
35 Shifts `a' right by the number of bits given in `count'. If any nonzero
36 bits are shifted off, they are ``jammed'' into the least significant bit of
37 the result by setting the least significant bit to 1. The value of `count'
38 can be arbitrarily large; in particular, if `count' is greater than 32, the
39 result will be either 0 or 1, depending on whether `a' is zero or nonzero.
40 The result is stored in the location pointed to by `zPtr'.
41 -------------------------------------------------------------------------------
43 INLINE void shift32RightJamming( bits32 a, int16 count, bits32 *zPtr )
50 else if ( count < 32 ) {
51 z = ( a>>count ) | ( ( a<<( ( - count ) & 31 ) ) != 0 );
61 -------------------------------------------------------------------------------
62 Shifts the 64-bit value formed by concatenating `a0' and `a1' right by the
63 number of bits given in `count'. Any bits shifted off are lost. The value
64 of `count' can be arbitrarily large; in particular, if `count' is greater
65 than 64, the result will be 0. The result is broken into two 32-bit pieces
66 which are stored at the locations pointed to by `z0Ptr' and `z1Ptr'.
67 -------------------------------------------------------------------------------
71 bits32 a0, bits32 a1, int16 count, bits32 *z0Ptr, bits32 *z1Ptr )
74 int8 negCount = ( - count ) & 31;
80 else if ( count < 32 ) {
81 z1 = ( a0<<negCount ) | ( a1>>count );
85 z1 = ( count < 64 ) ? ( a0>>( count & 31 ) ) : 0;
94 -------------------------------------------------------------------------------
95 Shifts the 64-bit value formed by concatenating `a0' and `a1' right by the
96 number of bits given in `count'. If any nonzero bits are shifted off, they
97 are ``jammed'' into the least significant bit of the result by setting the
98 least significant bit to 1. The value of `count' can be arbitrarily large;
99 in particular, if `count' is greater than 64, the result will be either 0
100 or 1, depending on whether the concatenation of `a0' and `a1' is zero or
101 nonzero. The result is broken into two 32-bit pieces which are stored at
102 the locations pointed to by `z0Ptr' and `z1Ptr'.
103 -------------------------------------------------------------------------------
107 bits32 a0, bits32 a1, int16 count, bits32 *z0Ptr, bits32 *z1Ptr )
110 int8 negCount = ( - count ) & 31;
116 else if ( count < 32 ) {
117 z1 = ( a0<<negCount ) | ( a1>>count ) | ( ( a1<<negCount ) != 0 );
122 z1 = a0 | ( a1 != 0 );
124 else if ( count < 64 ) {
125 z1 = ( a0>>( count & 31 ) ) | ( ( ( a0<<negCount ) | a1 ) != 0 );
128 z1 = ( ( a0 | a1 ) != 0 );
138 -------------------------------------------------------------------------------
139 Shifts the 96-bit value formed by concatenating `a0', `a1', and `a2' right
140 by 32 _plus_ the number of bits given in `count'. The shifted result is
141 at most 64 nonzero bits; these are broken into two 32-bit pieces which are
142 stored at the locations pointed to by `z0Ptr' and `z1Ptr'. The bits shifted
143 off form a third 32-bit result as follows: The _last_ bit shifted off is
144 the most-significant bit of the extra result, and the other 31 bits of the
145 extra result are all zero if and only if _all_but_the_last_ bits shifted off
146 were all zero. This extra result is stored in the location pointed to by
147 `z2Ptr'. The value of `count' can be arbitrarily large.
148 (This routine makes more sense if `a0', `a1', and `a2' are considered
149 to form a fixed-point value with binary point between `a1' and `a2'. This
150 fixed-point value is shifted right by the number of bits given in `count',
151 and the integer part of the result is returned at the locations pointed to
152 by `z0Ptr' and `z1Ptr'. The fractional part of the result may be slightly
153 corrupted as described above, and is returned at the location pointed to by
155 -------------------------------------------------------------------------------
158 shift64ExtraRightJamming(
169 int8 negCount = ( - count ) & 31;
179 z1 = ( a0<<negCount ) | ( a1>>count );
191 z1 = a0>>( count & 31 );
194 z2 = ( count == 64 ) ? a0 : ( a0 != 0 );
209 -------------------------------------------------------------------------------
210 Shifts the 64-bit value formed by concatenating `a0' and `a1' left by the
211 number of bits given in `count'. Any bits shifted off are lost. The value
212 of `count' must be less than 32. The result is broken into two 32-bit
213 pieces which are stored at the locations pointed to by `z0Ptr' and `z1Ptr'.
214 -------------------------------------------------------------------------------
218 bits32 a0, bits32 a1, int16 count, bits32 *z0Ptr, bits32 *z1Ptr )
223 ( count == 0 ) ? a0 : ( a0<<count ) | ( a1>>( ( - count ) & 31 ) );
228 -------------------------------------------------------------------------------
229 Shifts the 96-bit value formed by concatenating `a0', `a1', and `a2' left
230 by the number of bits given in `count'. Any bits shifted off are lost.
231 The value of `count' must be less than 32. The result is broken into three
232 32-bit pieces which are stored at the locations pointed to by `z0Ptr',
233 `z1Ptr', and `z2Ptr'.
234 -------------------------------------------------------------------------------
254 negCount = ( ( - count ) & 31 );
265 -------------------------------------------------------------------------------
266 Adds the 64-bit value formed by concatenating `a0' and `a1' to the 64-bit
267 value formed by concatenating `b0' and `b1'. Addition is modulo 2^64, so
268 any carry out is lost. The result is broken into two 32-bit pieces which
269 are stored at the locations pointed to by `z0Ptr' and `z1Ptr'.
270 -------------------------------------------------------------------------------
274 bits32 a0, bits32 a1, bits32 b0, bits32 b1, bits32 *z0Ptr, bits32 *z1Ptr )
280 *z0Ptr = a0 + b0 + ( z1 < a1 );
285 -------------------------------------------------------------------------------
286 Adds the 96-bit value formed by concatenating `a0', `a1', and `a2' to the
287 96-bit value formed by concatenating `b0', `b1', and `b2'. Addition is
288 modulo 2^96, so any carry out is lost. The result is broken into three
289 32-bit pieces which are stored at the locations pointed to by `z0Ptr',
290 `z1Ptr', and `z2Ptr'.
291 -------------------------------------------------------------------------------
310 carry1 = ( z2 < a2 );
312 carry0 = ( z1 < a1 );
315 z0 += ( z1 < carry1 );
324 -------------------------------------------------------------------------------
325 Subtracts the 64-bit value formed by concatenating `b0' and `b1' from the
326 64-bit value formed by concatenating `a0' and `a1'. Subtraction is modulo
327 2^64, so any borrow out (carry out) is lost. The result is broken into two
328 32-bit pieces which are stored at the locations pointed to by `z0Ptr' and
330 -------------------------------------------------------------------------------
334 bits32 a0, bits32 a1, bits32 b0, bits32 b1, bits32 *z0Ptr, bits32 *z1Ptr )
338 *z0Ptr = a0 - b0 - ( a1 < b1 );
343 -------------------------------------------------------------------------------
344 Subtracts the 96-bit value formed by concatenating `b0', `b1', and `b2' from
345 the 96-bit value formed by concatenating `a0', `a1', and `a2'. Subtraction
346 is modulo 2^96, so any borrow out (carry out) is lost. The result is broken
347 into three 32-bit pieces which are stored at the locations pointed to by
348 `z0Ptr', `z1Ptr', and `z2Ptr'.
349 -------------------------------------------------------------------------------
365 int8 borrow0, borrow1;
368 borrow1 = ( a2 < b2 );
370 borrow0 = ( a1 < b1 );
372 z0 -= ( z1 < borrow1 );
382 -------------------------------------------------------------------------------
383 Multiplies `a' by `b' to obtain a 64-bit product. The product is broken
384 into two 32-bit pieces which are stored at the locations pointed to by
386 -------------------------------------------------------------------------------
388 INLINE void mul32To64( bits32 a, bits32 b, bits32 *z0Ptr, bits32 *z1Ptr )
390 bits16 aHigh, aLow, bHigh, bLow;
391 bits32 z0, zMiddleA, zMiddleB, z1;
397 z1 = ( (bits32) aLow ) * bLow;
398 zMiddleA = ( (bits32) aLow ) * bHigh;
399 zMiddleB = ( (bits32) aHigh ) * bLow;
400 z0 = ( (bits32) aHigh ) * bHigh;
401 zMiddleA += zMiddleB;
402 z0 += ( ( (bits32) ( zMiddleA < zMiddleB ) )<<16 ) + ( zMiddleA>>16 );
405 z0 += ( z1 < zMiddleA );
412 -------------------------------------------------------------------------------
413 Multiplies the 64-bit value formed by concatenating `a0' and `a1' by `b'
414 to obtain a 96-bit product. The product is broken into three 32-bit pieces
415 which are stored at the locations pointed to by `z0Ptr', `z1Ptr', and
417 -------------------------------------------------------------------------------
429 bits32 z0, z1, z2, more1;
431 mul32To64( a1, b, &z1, &z2 );
432 mul32To64( a0, b, &z0, &more1 );
433 add64( z0, more1, 0, z1, &z0, &z1 );
441 -------------------------------------------------------------------------------
442 Multiplies the 64-bit value formed by concatenating `a0' and `a1' to the
443 64-bit value formed by concatenating `b0' and `b1' to obtain a 128-bit
444 product. The product is broken into four 32-bit pieces which are stored at
445 the locations pointed to by `z0Ptr', `z1Ptr', `z2Ptr', and `z3Ptr'.
446 -------------------------------------------------------------------------------
460 bits32 z0, z1, z2, z3;
463 mul32To64( a1, b1, &z2, &z3 );
464 mul32To64( a1, b0, &z1, &more2 );
465 add64( z1, more2, 0, z2, &z1, &z2 );
466 mul32To64( a0, b0, &z0, &more1 );
467 add64( z0, more1, 0, z1, &z0, &z1 );
468 mul32To64( a0, b1, &more1, &more2 );
469 add64( more1, more2, 0, z2, &more1, &z2 );
470 add64( z0, z1, 0, more1, &z0, &z1 );
479 -------------------------------------------------------------------------------
480 Returns an approximation to the 32-bit integer quotient obtained by dividing
481 `b' into the 64-bit value formed by concatenating `a0' and `a1'. The
482 divisor `b' must be at least 2^31. If q is the exact quotient truncated
483 toward zero, the approximation returned lies between q and q + 2 inclusive.
484 If the exact quotient q is larger than 32 bits, the maximum positive 32-bit
485 unsigned integer is returned.
486 -------------------------------------------------------------------------------
488 static bits32 estimateDiv64To32( bits32 a0, bits32 a1, bits32 b )
491 bits32 rem0, rem1, term0, term1;
494 if ( b <= a0 ) return 0xFFFFFFFF;
496 z = ( b0<<16 <= a0 ) ? 0xFFFF0000 : ( a0 / b0 )<<16;
497 mul32To64( b, z, &term0, &term1 );
498 sub64( a0, a1, term0, term1, &rem0, &rem1 );
499 while ( ( (sbits32) rem0 ) < 0 ) {
502 add64( rem0, rem1, b0, b1, &rem0, &rem1 );
504 rem0 = ( rem0<<16 ) | ( rem1>>16 );
505 z |= ( b0<<16 <= rem0 ) ? 0xFFFF : rem0 / b0;
510 #ifndef SOFTFLOAT_FOR_GCC
512 -------------------------------------------------------------------------------
513 Returns an approximation to the square root of the 32-bit significand given
514 by `a'. Considered as an integer, `a' must be at least 2^31. If bit 0 of
515 `aExp' (the least significant bit) is 1, the integer returned approximates
516 2^31*sqrt(`a'/2^31), where `a' is considered an integer. If bit 0 of `aExp'
517 is 0, the integer returned approximates 2^31*sqrt(`a'/2^30). In either
518 case, the approximation returned lies strictly within +/-2 of the exact
520 -------------------------------------------------------------------------------
522 static bits32 estimateSqrt32( int16 aExp, bits32 a )
524 static const bits16 sqrtOddAdjustments[] = {
525 0x0004, 0x0022, 0x005D, 0x00B1, 0x011D, 0x019F, 0x0236, 0x02E0,
526 0x039C, 0x0468, 0x0545, 0x0631, 0x072B, 0x0832, 0x0946, 0x0A67
528 static const bits16 sqrtEvenAdjustments[] = {
529 0x0A2D, 0x08AF, 0x075A, 0x0629, 0x051A, 0x0429, 0x0356, 0x029E,
530 0x0200, 0x0179, 0x0109, 0x00AF, 0x0068, 0x0034, 0x0012, 0x0002
535 index = ( a>>27 ) & 15;
537 z = 0x4000 + ( a>>17 ) - sqrtOddAdjustments[ index ];
538 z = ( ( a / z )<<14 ) + ( z<<15 );
542 z = 0x8000 + ( a>>17 ) - sqrtEvenAdjustments[ index ];
544 z = ( 0x20000 <= z ) ? 0xFFFF8000 : ( z<<15 );
545 if ( z <= a ) return (bits32) ( ( (sbits32) a )>>1 );
547 return ( ( estimateDiv64To32( a, 0, z ) )>>1 ) + ( z>>1 );
553 -------------------------------------------------------------------------------
554 Returns the number of leading 0 bits before the most-significant 1 bit of
555 `a'. If `a' is zero, 32 is returned.
556 -------------------------------------------------------------------------------
558 static int8 countLeadingZeros32( bits32 a )
560 static const int8 countLeadingZerosHigh[] = {
561 8, 7, 6, 6, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4,
562 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
563 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
564 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
565 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
566 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
567 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
568 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
569 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
570 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
571 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
572 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
573 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
574 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
575 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
576 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
585 if ( a < 0x1000000 ) {
589 shiftCount += countLeadingZerosHigh[ a>>24 ];
595 -------------------------------------------------------------------------------
596 Returns 1 if the 64-bit value formed by concatenating `a0' and `a1' is
597 equal to the 64-bit value formed by concatenating `b0' and `b1'. Otherwise,
599 -------------------------------------------------------------------------------
601 INLINE flag eq64( bits32 a0, bits32 a1, bits32 b0, bits32 b1 )
604 return ( a0 == b0 ) && ( a1 == b1 );
609 -------------------------------------------------------------------------------
610 Returns 1 if the 64-bit value formed by concatenating `a0' and `a1' is less
611 than or equal to the 64-bit value formed by concatenating `b0' and `b1'.
612 Otherwise, returns 0.
613 -------------------------------------------------------------------------------
615 INLINE flag le64( bits32 a0, bits32 a1, bits32 b0, bits32 b1 )
618 return ( a0 < b0 ) || ( ( a0 == b0 ) && ( a1 <= b1 ) );
623 -------------------------------------------------------------------------------
624 Returns 1 if the 64-bit value formed by concatenating `a0' and `a1' is less
625 than the 64-bit value formed by concatenating `b0' and `b1'. Otherwise,
627 -------------------------------------------------------------------------------
629 INLINE flag lt64( bits32 a0, bits32 a1, bits32 b0, bits32 b1 )
632 return ( a0 < b0 ) || ( ( a0 == b0 ) && ( a1 < b1 ) );
637 -------------------------------------------------------------------------------
638 Returns 1 if the 64-bit value formed by concatenating `a0' and `a1' is not
639 equal to the 64-bit value formed by concatenating `b0' and `b1'. Otherwise,
641 -------------------------------------------------------------------------------
643 INLINE flag ne64( bits32 a0, bits32 a1, bits32 b0, bits32 b1 )
646 return ( a0 != b0 ) || ( a1 != b1 );