2 * The implementations contained in this file are heavily based on the
3 * implementations found in the Berkeley SoftFloat library. As such, they are
4 * licensed under the same 3-clause BSD license:
6 * License for Berkeley SoftFloat Release 3e
11 * The following applies to the whole of SoftFloat Release 3e as well as to
12 * each source file individually.
14 * Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018 The Regents of the
15 * University of California. All rights reserved.
17 * Redistribution and use in source and binary forms, with or without
18 * modification, are permitted provided that the following conditions are met:
20 * 1. Redistributions of source code must retain the above copyright notice,
21 * this list of conditions, and the following disclaimer.
23 * 2. Redistributions in binary form must reproduce the above copyright
24 * notice, this list of conditions, and the following disclaimer in the
25 * documentation and/or other materials provided with the distribution.
27 * 3. Neither the name of the University nor the names of its contributors
28 * may be used to endorse or promote products derived from this software
29 * without specific prior written permission.
31 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS "AS IS", AND ANY
32 * EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
33 * WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE, ARE
34 * DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE FOR ANY
35 * DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
36 * (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
37 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
38 * ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
39 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
40 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
44 #extension GL_ARB_gpu_shader_int64 : enable
45 #extension GL_ARB_shader_bit_encoding : enable
46 #extension GL_EXT_shader_integer_mix : enable
47 #extension GL_MESA_shader_integer_functions : enable
51 /* Software IEEE floating-point rounding mode.
52 * GLSL spec section "4.7.1 Range and Precision":
53 * The rounding mode cannot be set and is undefined.
54 * But here, we are able to define the rounding mode at the compilation time.
56 #define FLOAT_ROUND_NEAREST_EVEN 0
57 #define FLOAT_ROUND_TO_ZERO 1
58 #define FLOAT_ROUND_DOWN 2
59 #define FLOAT_ROUND_UP 3
60 #define FLOAT_ROUNDING_MODE FLOAT_ROUND_NEAREST_EVEN
62 /* Relax propagation of NaN. Binary operations with a NaN source will still
63 * produce a NaN result, but it won't follow strict IEEE rules.
65 #define RELAXED_NAN_PROPAGATION
67 /* Absolute value of a Float64 :
71 __fabs64(uint64_t __a)
73 uvec2 a = unpackUint2x32(__a);
75 return packUint2x32(a);
78 /* Returns 1 if the double-precision floating-point value `a' is a NaN;
79 * otherwise returns 0.
82 __is_nan(uint64_t __a)
84 uvec2 a = unpackUint2x32(__a);
85 return (0xFFE00000u <= (a.y<<1)) &&
86 ((a.x != 0u) || ((a.y & 0x000FFFFFu) != 0u));
89 /* Negate value of a Float64 :
93 __fneg64(uint64_t __a)
95 uvec2 a = unpackUint2x32(__a);
97 return packUint2x32(a);
101 __fsign64(uint64_t __a)
103 uvec2 a = unpackUint2x32(__a);
106 retval.y = mix((a.y & 0x80000000u) | 0x3FF00000u, 0u, (a.y << 1 | a.x) == 0u);
107 return packUint2x32(retval);
110 /* Returns the fraction bits of the double-precision floating-point value `a'.*/
112 __extractFloat64FracLo(uint64_t a)
114 return unpackUint2x32(a).x;
118 __extractFloat64FracHi(uint64_t a)
120 return unpackUint2x32(a).y & 0x000FFFFFu;
123 /* Returns the exponent bits of the double-precision floating-point value `a'.*/
125 __extractFloat64Exp(uint64_t __a)
127 uvec2 a = unpackUint2x32(__a);
128 return int((a.y>>20) & 0x7FFu);
132 __feq64_nonnan(uint64_t __a, uint64_t __b)
134 uvec2 a = unpackUint2x32(__a);
135 uvec2 b = unpackUint2x32(__b);
136 return (a.x == b.x) &&
137 ((a.y == b.y) || ((a.x == 0u) && (((a.y | b.y)<<1) == 0u)));
140 /* Returns true if the double-precision floating-point value `a' is equal to the
141 * corresponding value `b', and false otherwise. The comparison is performed
142 * according to the IEEE Standard for Floating-Point Arithmetic.
145 __feq64(uint64_t a, uint64_t b)
147 if (__is_nan(a) || __is_nan(b))
150 return __feq64_nonnan(a, b);
153 /* Returns true if the double-precision floating-point value `a' is not equal
154 * to the corresponding value `b', and false otherwise. The comparison is
155 * performed according to the IEEE Standard for Floating-Point Arithmetic.
158 __fne64(uint64_t a, uint64_t b)
160 if (__is_nan(a) || __is_nan(b))
163 return !__feq64_nonnan(a, b);
166 /* Returns the sign bit of the double-precision floating-point value `a'.*/
168 __extractFloat64Sign(uint64_t a)
170 return unpackUint2x32(a).y & 0x80000000u;
173 /* Returns true if the signed 64-bit value formed by concatenating `a0' and
174 * `a1' is less than the signed 64-bit value formed by concatenating `b0' and
175 * `b1'. Otherwise, returns false.
178 ilt64(uint a0, uint a1, uint b0, uint b1)
180 return (int(a0) < int(b0)) || ((a0 == b0) && (a1 < b1));
184 __flt64_nonnan(uint64_t __a, uint64_t __b)
186 uvec2 a = unpackUint2x32(__a);
187 uvec2 b = unpackUint2x32(__b);
189 /* IEEE 754 floating point numbers are specifically designed so that, with
190 * two exceptions, values can be compared by bit-casting to signed integers
191 * with the same number of bits.
193 * From https://en.wikipedia.org/wiki/IEEE_754-1985#Comparing_floating-point_numbers:
195 * When comparing as 2's-complement integers: If the sign bits differ,
196 * the negative number precedes the positive number, so 2's complement
197 * gives the correct result (except that negative zero and positive zero
198 * should be considered equal). If both values are positive, the 2's
199 * complement comparison again gives the correct result. Otherwise (two
200 * negative numbers), the correct FP ordering is the opposite of the 2's
201 * complement ordering.
203 * The logic implied by the above quotation is:
205 * !both_are_zero(a, b) && (both_negative(a, b) ? a > b : a < b)
207 * This is equivalent to
209 * fne(a, b) && (both_negative(a, b) ? a >= b : a < b)
211 * fne(a, b) && (both_negative(a, b) ? !(a < b) : a < b)
213 * fne(a, b) && ((both_negative(a, b) && !(a < b)) ||
214 * (!both_negative(a, b) && (a < b)))
216 * (A!|B)&(A|!B) is (A xor B) which is implemented here using !=.
218 * fne(a, b) && (both_negative(a, b) != (a < b))
220 bool lt = ilt64(a.y, a.x, b.y, b.x);
221 bool both_negative = (a.y & b.y & 0x80000000u) != 0;
223 return !__feq64_nonnan(__a, __b) && (lt != both_negative);
226 /* Returns true if the double-precision floating-point value `a' is less than
227 * the corresponding value `b', and false otherwise. The comparison is performed
228 * according to the IEEE Standard for Floating-Point Arithmetic.
231 __flt64(uint64_t a, uint64_t b)
233 /* This weird layout matters. Doing the "obvious" thing results in extra
234 * flow control being inserted to implement the short-circuit evaluation
235 * rules. Flow control is bad!
237 bool x = !__is_nan(a);
238 bool y = !__is_nan(b);
239 bool z = __flt64_nonnan(a, b);
241 return (x && y && z);
244 /* Returns true if the double-precision floating-point value `a' is greater
245 * than or equal to * the corresponding value `b', and false otherwise. The
246 * comparison is performed * according to the IEEE Standard for Floating-Point
250 __fge64(uint64_t a, uint64_t b)
252 /* This weird layout matters. Doing the "obvious" thing results in extra
253 * flow control being inserted to implement the short-circuit evaluation
254 * rules. Flow control is bad!
256 bool x = !__is_nan(a);
257 bool y = !__is_nan(b);
258 bool z = !__flt64_nonnan(a, b);
260 return (x && y && z);
264 __fsat64(uint64_t __a)
266 uvec2 a = unpackUint2x32(__a);
268 /* fsat(NaN) should be zero. */
269 if (__is_nan(__a) || int(a.y) < 0)
272 /* IEEE 754 floating point numbers are specifically designed so that, with
273 * two exceptions, values can be compared by bit-casting to signed integers
274 * with the same number of bits.
276 * From https://en.wikipedia.org/wiki/IEEE_754-1985#Comparing_floating-point_numbers:
278 * When comparing as 2's-complement integers: If the sign bits differ,
279 * the negative number precedes the positive number, so 2's complement
280 * gives the correct result (except that negative zero and positive zero
281 * should be considered equal). If both values are positive, the 2's
282 * complement comparison again gives the correct result. Otherwise (two
283 * negative numbers), the correct FP ordering is the opposite of the 2's
284 * complement ordering.
286 * We know that both values are not negative, and we know that at least one
287 * value is not zero. Therefore, we can just use the 2's complement
288 * comparison ordering.
290 if (ilt64(0x3FF00000, 0x00000000, a.y, a.x))
291 return 0x3FF0000000000000ul;
296 /* Adds the 64-bit value formed by concatenating `a0' and `a1' to the 64-bit
297 * value formed by concatenating `b0' and `b1'. Addition is modulo 2^64, so
298 * any carry out is lost. The result is broken into two 32-bit pieces which
299 * are stored at the locations pointed to by `z0Ptr' and `z1Ptr'.
302 __add64(uint a0, uint a1, uint b0, uint b1,
308 z0Ptr = a0 + b0 + uint(z1 < a1);
312 /* Subtracts the 64-bit value formed by concatenating `b0' and `b1' from the
313 * 64-bit value formed by concatenating `a0' and `a1'. Subtraction is modulo
314 * 2^64, so any borrow out (carry out) is lost. The result is broken into two
315 * 32-bit pieces which are stored at the locations pointed to by `z0Ptr' and
319 __sub64(uint a0, uint a1, uint b0, uint b1,
324 z0Ptr = a0 - b0 - uint(a1 < b1);
327 /* Shifts the 64-bit value formed by concatenating `a0' and `a1' right by the
328 * number of bits given in `count'. If any nonzero bits are shifted off, they
329 * are "jammed" into the least significant bit of the result by setting the
330 * least significant bit to 1. The value of `count' can be arbitrarily large;
331 * in particular, if `count' is greater than 64, the result will be either 0
332 * or 1, depending on whether the concatenation of `a0' and `a1' is zero or
333 * nonzero. The result is broken into two 32-bit pieces which are stored at
334 * the locations pointed to by `z0Ptr' and `z1Ptr'.
337 __shift64RightJamming(uint a0,
345 int negCount = (-count) & 31;
347 z0 = mix(0u, a0, count == 0);
348 z0 = mix(z0, (a0 >> count), count < 32);
350 z1 = uint((a0 | a1) != 0u); /* count >= 64 */
351 uint z1_lt64 = (a0>>(count & 31)) | uint(((a0<<negCount) | a1) != 0u);
352 z1 = mix(z1, z1_lt64, count < 64);
353 z1 = mix(z1, (a0 | uint(a1 != 0u)), count == 32);
354 uint z1_lt32 = (a0<<negCount) | (a1>>count) | uint ((a1<<negCount) != 0u);
355 z1 = mix(z1, z1_lt32, count < 32);
356 z1 = mix(z1, a1, count == 0);
361 /* Shifts the 96-bit value formed by concatenating `a0', `a1', and `a2' right
362 * by 32 _plus_ the number of bits given in `count'. The shifted result is
363 * at most 64 nonzero bits; these are broken into two 32-bit pieces which are
364 * stored at the locations pointed to by `z0Ptr' and `z1Ptr'. The bits shifted
365 * off form a third 32-bit result as follows: The _last_ bit shifted off is
366 * the most-significant bit of the extra result, and the other 31 bits of the
367 * extra result are all zero if and only if _all_but_the_last_ bits shifted off
368 * were all zero. This extra result is stored in the location pointed to by
369 * `z2Ptr'. The value of `count' can be arbitrarily large.
370 * (This routine makes more sense if `a0', `a1', and `a2' are considered
371 * to form a fixed-point value with binary point between `a1' and `a2'. This
372 * fixed-point value is shifted right by the number of bits given in `count',
373 * and the integer part of the result is returned at the locations pointed to
374 * by `z0Ptr' and `z1Ptr'. The fractional part of the result may be slightly
375 * corrupted as described above, and is returned at the location pointed to by
379 __shift64ExtraRightJamming(uint a0, uint a1, uint a2,
388 int negCount = (-count) & 31;
390 z2 = mix(uint(a0 != 0u), a0, count == 64);
391 z2 = mix(z2, a0 << negCount, count < 64);
392 z2 = mix(z2, a1 << negCount, count < 32);
394 z1 = mix(0u, (a0 >> (count & 31)), count < 64);
395 z1 = mix(z1, (a0<<negCount) | (a1>>count), count < 32);
397 a2 = mix(a2 | a1, a2, count < 32);
398 z0 = mix(z0, a0 >> count, count < 32);
399 z2 |= uint(a2 != 0u);
401 z0 = mix(z0, 0u, (count == 32));
402 z1 = mix(z1, a0, (count == 32));
403 z2 = mix(z2, a1, (count == 32));
404 z0 = mix(z0, a0, (count == 0));
405 z1 = mix(z1, a1, (count == 0));
406 z2 = mix(z2, a2, (count == 0));
412 /* Shifts the 64-bit value formed by concatenating `a0' and `a1' left by the
413 * number of bits given in `count'. Any bits shifted off are lost. The value
414 * of `count' must be less than 32. The result is broken into two 32-bit
415 * pieces which are stored at the locations pointed to by `z0Ptr' and `z1Ptr'.
418 __shortShift64Left(uint a0, uint a1,
424 z0Ptr = mix((a0 << count | (a1 >> ((-count) & 31))), a0, count == 0);
427 /* Packs the sign `zSign', the exponent `zExp', and the significand formed by
428 * the concatenation of `zFrac0' and `zFrac1' into a double-precision floating-
429 * point value, returning the result. After being shifted into the proper
430 * positions, the three fields `zSign', `zExp', and `zFrac0' are simply added
431 * together to form the most significant 32 bits of the result. This means
432 * that any integer portion of `zFrac0' will be added into the exponent. Since
433 * a properly normalized significand will have an integer portion equal to 1,
434 * the `zExp' input should be 1 less than the desired result exponent whenever
435 * `zFrac0' and `zFrac1' concatenated form a complete, normalized significand.
438 __packFloat64(uint zSign, int zExp, uint zFrac0, uint zFrac1)
442 z.y = zSign + (uint(zExp) << 20) + zFrac0;
444 return packUint2x32(z);
447 /* Takes an abstract floating-point value having sign `zSign', exponent `zExp',
448 * and extended significand formed by the concatenation of `zFrac0', `zFrac1',
449 * and `zFrac2', and returns the proper double-precision floating-point value
450 * corresponding to the abstract input. Ordinarily, the abstract value is
451 * simply rounded and packed into the double-precision format, with the inexact
452 * exception raised if the abstract input cannot be represented exactly.
453 * However, if the abstract value is too large, the overflow and inexact
454 * exceptions are raised and an infinity or maximal finite value is returned.
455 * If the abstract value is too small, the input value is rounded to a
456 * subnormal number, and the underflow and inexact exceptions are raised if the
457 * abstract input cannot be represented exactly as a subnormal double-precision
458 * floating-point number.
459 * The input significand must be normalized or smaller. If the input
460 * significand is not normalized, `zExp' must be 0; in that case, the result
461 * returned is a subnormal number, and it must not require rounding. In the
462 * usual case that the input significand is normalized, `zExp' must be 1 less
463 * than the "true" floating-point exponent. The handling of underflow and
464 * overflow follows the IEEE Standard for Floating-Point Arithmetic.
467 __roundAndPackFloat64(uint zSign,
473 bool roundNearestEven;
476 roundNearestEven = FLOAT_ROUNDING_MODE == FLOAT_ROUND_NEAREST_EVEN;
477 increment = int(zFrac2) < 0;
478 if (!roundNearestEven) {
479 if (FLOAT_ROUNDING_MODE == FLOAT_ROUND_TO_ZERO) {
483 increment = (FLOAT_ROUNDING_MODE == FLOAT_ROUND_DOWN) &&
486 increment = (FLOAT_ROUNDING_MODE == FLOAT_ROUND_UP) &&
492 if ((0x7FD < zExp) ||
494 (0x001FFFFFu == zFrac0 && 0xFFFFFFFFu == zFrac1) &&
496 if ((FLOAT_ROUNDING_MODE == FLOAT_ROUND_TO_ZERO) ||
497 ((zSign != 0u) && (FLOAT_ROUNDING_MODE == FLOAT_ROUND_UP)) ||
498 ((zSign == 0u) && (FLOAT_ROUNDING_MODE == FLOAT_ROUND_DOWN))) {
499 return __packFloat64(zSign, 0x7FE, 0x000FFFFFu, 0xFFFFFFFFu);
501 return __packFloat64(zSign, 0x7FF, 0u, 0u);
504 __shift64ExtraRightJamming(
505 zFrac0, zFrac1, zFrac2, -zExp, zFrac0, zFrac1, zFrac2);
507 if (roundNearestEven) {
508 increment = zFrac2 < 0u;
511 increment = (FLOAT_ROUNDING_MODE == FLOAT_ROUND_DOWN) &&
514 increment = (FLOAT_ROUNDING_MODE == FLOAT_ROUND_UP) &&
521 __add64(zFrac0, zFrac1, 0u, 1u, zFrac0, zFrac1);
522 zFrac1 &= ~((zFrac2 + uint(zFrac2 == 0u)) & uint(roundNearestEven));
524 zExp = mix(zExp, 0, (zFrac0 | zFrac1) == 0u);
526 return __packFloat64(zSign, zExp, zFrac0, zFrac1);
530 __roundAndPackUInt64(uint zSign, uint zFrac0, uint zFrac1, uint zFrac2)
532 bool roundNearestEven;
534 uint64_t default_nan = 0xFFFFFFFFFFFFFFFFUL;
536 roundNearestEven = FLOAT_ROUNDING_MODE == FLOAT_ROUND_NEAREST_EVEN;
538 if (zFrac2 >= 0x80000000u)
541 if (!roundNearestEven) {
543 if ((FLOAT_ROUNDING_MODE == FLOAT_ROUND_DOWN) && (zFrac2 != 0u)) {
547 increment = (FLOAT_ROUNDING_MODE == FLOAT_ROUND_UP) &&
553 __add64(zFrac0, zFrac1, 0u, 1u, zFrac0, zFrac1);
554 if ((zFrac0 | zFrac1) != 0u)
555 zFrac1 &= ~(1u) + uint(zFrac2 == 0u) & uint(roundNearestEven);
557 return mix(packUint2x32(uvec2(zFrac1, zFrac0)), default_nan,
558 (zSign != 0u && (zFrac0 | zFrac1) != 0u));
562 __roundAndPackInt64(uint zSign, uint zFrac0, uint zFrac1, uint zFrac2)
564 bool roundNearestEven;
566 int64_t default_NegNaN = -0x7FFFFFFFFFFFFFFEL;
567 int64_t default_PosNaN = 0xFFFFFFFFFFFFFFFFL;
569 roundNearestEven = FLOAT_ROUNDING_MODE == FLOAT_ROUND_NEAREST_EVEN;
571 if (zFrac2 >= 0x80000000u)
574 if (!roundNearestEven) {
576 increment = ((FLOAT_ROUNDING_MODE == FLOAT_ROUND_DOWN) &&
579 increment = (FLOAT_ROUNDING_MODE == FLOAT_ROUND_UP) &&
585 __add64(zFrac0, zFrac1, 0u, 1u, zFrac0, zFrac1);
586 if ((zFrac0 | zFrac1) != 0u)
587 zFrac1 &= ~(1u) + uint(zFrac2 == 0u) & uint(roundNearestEven);
590 int64_t absZ = mix(int64_t(packUint2x32(uvec2(zFrac1, zFrac0))),
591 -int64_t(packUint2x32(uvec2(zFrac1, zFrac0))),
593 int64_t nan = mix(default_PosNaN, default_NegNaN, zSign != 0u);
594 return mix(absZ, nan, ((zSign != 0u) != (absZ < 0)) && bool(absZ));
597 /* Returns the number of leading 0 bits before the most-significant 1 bit of
598 * `a'. If `a' is zero, 32 is returned.
601 __countLeadingZeros32(uint a)
603 return 31 - findMSB(a);
606 /* Takes an abstract floating-point value having sign `zSign', exponent `zExp',
607 * and significand formed by the concatenation of `zSig0' and `zSig1', and
608 * returns the proper double-precision floating-point value corresponding
609 * to the abstract input. This routine is just like `__roundAndPackFloat64'
610 * except that the input significand has fewer bits and does not have to be
611 * normalized. In all cases, `zExp' must be 1 less than the "true" floating-
615 __normalizeRoundAndPackFloat64(uint zSign,
629 shiftCount = __countLeadingZeros32(zFrac0) - 11;
630 if (0 <= shiftCount) {
632 __shortShift64Left(zFrac0, zFrac1, shiftCount, zFrac0, zFrac1);
634 __shift64ExtraRightJamming(
635 zFrac0, zFrac1, 0u, -shiftCount, zFrac0, zFrac1, zFrac2);
638 return __roundAndPackFloat64(zSign, zExp, zFrac0, zFrac1, zFrac2);
641 /* Takes two double-precision floating-point values `a' and `b', one of which
642 * is a NaN, and returns the appropriate NaN result.
645 __propagateFloat64NaN(uint64_t __a, uint64_t __b)
647 #if defined RELAXED_NAN_PROPAGATION
648 uvec2 a = unpackUint2x32(__a);
649 uvec2 b = unpackUint2x32(__b);
651 return packUint2x32(uvec2(a.x | b.x, a.y | b.y));
653 bool aIsNaN = __is_nan(__a);
654 bool bIsNaN = __is_nan(__b);
655 uvec2 a = unpackUint2x32(__a);
656 uvec2 b = unpackUint2x32(__b);
660 return packUint2x32(mix(b, mix(a, b, bvec2(bIsNaN, bIsNaN)), bvec2(aIsNaN, aIsNaN)));
664 /* Returns the result of adding the double-precision floating-point values
665 * `a' and `b'. The operation is performed according to the IEEE Standard for
666 * Floating-Point Arithmetic.
669 __fadd64(uint64_t a, uint64_t b)
671 uint aSign = __extractFloat64Sign(a);
672 uint bSign = __extractFloat64Sign(b);
673 uint aFracLo = __extractFloat64FracLo(a);
674 uint aFracHi = __extractFloat64FracHi(a);
675 uint bFracLo = __extractFloat64FracLo(b);
676 uint bFracHi = __extractFloat64FracHi(b);
677 int aExp = __extractFloat64Exp(a);
678 int bExp = __extractFloat64Exp(b);
679 int expDiff = aExp - bExp;
680 if (aSign == bSign) {
685 bool orig_exp_diff_is_zero = (expDiff == 0);
687 if (orig_exp_diff_is_zero) {
689 bool propagate = ((aFracHi | bFracHi) | (aFracLo| bFracLo)) != 0u;
690 return mix(a, __propagateFloat64NaN(a, b), propagate);
692 __add64(aFracHi, aFracLo, bFracHi, bFracLo, zFrac0, zFrac1);
694 return __packFloat64(aSign, 0, zFrac0, zFrac1);
696 zFrac0 |= 0x00200000u;
698 __shift64ExtraRightJamming(
699 zFrac0, zFrac1, zFrac2, 1, zFrac0, zFrac1, zFrac2);
703 bool propagate = (aFracHi | aFracLo) != 0u;
704 return mix(a, __propagateFloat64NaN(a, b), propagate);
707 expDiff = mix(expDiff, expDiff - 1, bExp == 0);
708 bFracHi = mix(bFracHi | 0x00100000u, bFracHi, bExp == 0);
709 __shift64ExtraRightJamming(
710 bFracHi, bFracLo, 0u, expDiff, bFracHi, bFracLo, zFrac2);
714 bool propagate = (bFracHi | bFracLo) != 0u;
715 return mix(__packFloat64(aSign, 0x7ff, 0u, 0u), __propagateFloat64NaN(a, b), propagate);
717 expDiff = mix(expDiff, expDiff + 1, aExp == 0);
718 aFracHi = mix(aFracHi | 0x00100000u, aFracHi, aExp == 0);
719 __shift64ExtraRightJamming(
720 aFracHi, aFracLo, 0u, - expDiff, aFracHi, aFracLo, zFrac2);
724 if (!orig_exp_diff_is_zero) {
725 aFracHi |= 0x00100000u;
726 __add64(aFracHi, aFracLo, bFracHi, bFracLo, zFrac0, zFrac1);
728 if (!(zFrac0 < 0x00200000u)) {
729 __shift64ExtraRightJamming(zFrac0, zFrac1, zFrac2, 1, zFrac0, zFrac1, zFrac2);
733 return __roundAndPackFloat64(aSign, zExp, zFrac0, zFrac1, zFrac2);
738 __shortShift64Left(aFracHi, aFracLo, 10, aFracHi, aFracLo);
739 __shortShift64Left(bFracHi, bFracLo, 10, bFracHi, bFracLo);
745 bool propagate = (aFracHi | aFracLo) != 0u;
746 return mix(a, __propagateFloat64NaN(a, b), propagate);
748 expDiff = mix(expDiff, expDiff - 1, bExp == 0);
749 bFracHi = mix(bFracHi | 0x40000000u, bFracHi, bExp == 0);
750 __shift64RightJamming(bFracHi, bFracLo, expDiff, bFracHi, bFracLo);
751 aFracHi |= 0x40000000u;
752 __sub64(aFracHi, aFracLo, bFracHi, bFracLo, zFrac0, zFrac1);
755 return __normalizeRoundAndPackFloat64(aSign, zExp - 10, zFrac0, zFrac1);
762 bool propagate = (bFracHi | bFracLo) != 0u;
763 return mix(__packFloat64(aSign ^ 0x80000000u, 0x7ff, 0u, 0u), __propagateFloat64NaN(a, b), propagate);
765 expDiff = mix(expDiff, expDiff + 1, aExp == 0);
766 aFracHi = mix(aFracHi | 0x40000000u, aFracHi, aExp == 0);
767 __shift64RightJamming(aFracHi, aFracLo, - expDiff, aFracHi, aFracLo);
768 bFracHi |= 0x40000000u;
769 __sub64(bFracHi, bFracLo, aFracHi, aFracLo, zFrac0, zFrac1);
771 aSign ^= 0x80000000u;
773 return __normalizeRoundAndPackFloat64(aSign, zExp - 10, zFrac0, zFrac1);
776 bool propagate = ((aFracHi | bFracHi) | (aFracLo | bFracLo)) != 0u;
777 return mix(0xFFFFFFFFFFFFFFFFUL, __propagateFloat64NaN(a, b), propagate);
779 bExp = mix(bExp, 1, aExp == 0);
780 aExp = mix(aExp, 1, aExp == 0);
784 uint sign_of_difference = 0;
785 if (bFracHi < aFracHi) {
786 __sub64(aFracHi, aFracLo, bFracHi, bFracLo, zFrac0, zFrac1);
788 else if (aFracHi < bFracHi) {
789 __sub64(bFracHi, bFracLo, aFracHi, aFracLo, zFrac0, zFrac1);
790 sign_of_difference = 0x80000000;
792 else if (bFracLo <= aFracLo) {
793 /* It is possible that zFrac0 and zFrac1 may be zero after this. */
794 __sub64(aFracHi, aFracLo, bFracHi, bFracLo, zFrac0, zFrac1);
797 __sub64(bFracHi, bFracLo, aFracHi, aFracLo, zFrac0, zFrac1);
798 sign_of_difference = 0x80000000;
800 zExp = mix(bExp, aExp, sign_of_difference == 0u);
801 aSign ^= sign_of_difference;
802 uint64_t retval_0 = __packFloat64(uint(FLOAT_ROUNDING_MODE == FLOAT_ROUND_DOWN) << 31, 0, 0u, 0u);
803 uint64_t retval_1 = __normalizeRoundAndPackFloat64(aSign, zExp - 11, zFrac0, zFrac1);
804 return mix(retval_0, retval_1, zFrac0 != 0u || zFrac1 != 0u);
808 /* Multiplies the 64-bit value formed by concatenating `a0' and `a1' to the
809 * 64-bit value formed by concatenating `b0' and `b1' to obtain a 128-bit
810 * product. The product is broken into four 32-bit pieces which are stored at
811 * the locations pointed to by `z0Ptr', `z1Ptr', `z2Ptr', and `z3Ptr'.
814 __mul64To128(uint a0, uint a1, uint b0, uint b1,
827 umulExtended(a1, b1, z2, z3);
828 umulExtended(a1, b0, z1, more2);
829 __add64(z1, more2, 0u, z2, z1, z2);
830 umulExtended(a0, b0, z0, more1);
831 __add64(z0, more1, 0u, z1, z0, z1);
832 umulExtended(a0, b1, more1, more2);
833 __add64(more1, more2, 0u, z2, more1, z2);
834 __add64(z0, z1, 0u, more1, z0, z1);
841 /* Normalizes the subnormal double-precision floating-point value represented
842 * by the denormalized significand formed by the concatenation of `aFrac0' and
843 * `aFrac1'. The normalized exponent is stored at the location pointed to by
844 * `zExpPtr'. The most significant 21 bits of the normalized significand are
845 * stored at the location pointed to by `zFrac0Ptr', and the least significant
846 * 32 bits of the normalized significand are stored at the location pointed to
850 __normalizeFloat64Subnormal(uint aFrac0, uint aFrac1,
856 uint temp_zfrac0, temp_zfrac1;
857 shiftCount = __countLeadingZeros32(mix(aFrac0, aFrac1, aFrac0 == 0u)) - 11;
858 zExpPtr = mix(1 - shiftCount, -shiftCount - 31, aFrac0 == 0u);
860 temp_zfrac0 = mix(aFrac1<<shiftCount, aFrac1>>(-shiftCount), shiftCount < 0);
861 temp_zfrac1 = mix(0u, aFrac1<<(shiftCount & 31), shiftCount < 0);
863 __shortShift64Left(aFrac0, aFrac1, shiftCount, zFrac0Ptr, zFrac1Ptr);
865 zFrac0Ptr = mix(zFrac0Ptr, temp_zfrac0, aFrac0 == 0);
866 zFrac1Ptr = mix(zFrac1Ptr, temp_zfrac1, aFrac0 == 0);
869 /* Returns the result of multiplying the double-precision floating-point values
870 * `a' and `b'. The operation is performed according to the IEEE Standard for
871 * Floating-Point Arithmetic.
874 __fmul64(uint64_t a, uint64_t b)
882 uint aFracLo = __extractFloat64FracLo(a);
883 uint aFracHi = __extractFloat64FracHi(a);
884 uint bFracLo = __extractFloat64FracLo(b);
885 uint bFracHi = __extractFloat64FracHi(b);
886 int aExp = __extractFloat64Exp(a);
887 uint aSign = __extractFloat64Sign(a);
888 int bExp = __extractFloat64Exp(b);
889 uint bSign = __extractFloat64Sign(b);
890 uint zSign = aSign ^ bSign;
892 if (((aFracHi | aFracLo) != 0u) ||
893 ((bExp == 0x7FF) && ((bFracHi | bFracLo) != 0u))) {
894 return __propagateFloat64NaN(a, b);
896 if ((uint(bExp) | bFracHi | bFracLo) == 0u)
897 return 0xFFFFFFFFFFFFFFFFUL;
898 return __packFloat64(zSign, 0x7FF, 0u, 0u);
901 /* a cannot be NaN, but is b NaN? */
902 if ((bFracHi | bFracLo) != 0u)
903 #if defined RELAXED_NAN_PROPAGATION
906 return __propagateFloat64NaN(a, b);
908 if ((uint(aExp) | aFracHi | aFracLo) == 0u)
909 return 0xFFFFFFFFFFFFFFFFUL;
910 return __packFloat64(zSign, 0x7FF, 0u, 0u);
913 if ((aFracHi | aFracLo) == 0u)
914 return __packFloat64(zSign, 0, 0u, 0u);
915 __normalizeFloat64Subnormal(aFracHi, aFracLo, aExp, aFracHi, aFracLo);
918 if ((bFracHi | bFracLo) == 0u)
919 return __packFloat64(zSign, 0, 0u, 0u);
920 __normalizeFloat64Subnormal(bFracHi, bFracLo, bExp, bFracHi, bFracLo);
922 zExp = aExp + bExp - 0x400;
923 aFracHi |= 0x00100000u;
924 __shortShift64Left(bFracHi, bFracLo, 12, bFracHi, bFracLo);
926 aFracHi, aFracLo, bFracHi, bFracLo, zFrac0, zFrac1, zFrac2, zFrac3);
927 __add64(zFrac0, zFrac1, aFracHi, aFracLo, zFrac0, zFrac1);
928 zFrac2 |= uint(zFrac3 != 0u);
929 if (0x00200000u <= zFrac0) {
930 __shift64ExtraRightJamming(
931 zFrac0, zFrac1, zFrac2, 1, zFrac0, zFrac1, zFrac2);
934 return __roundAndPackFloat64(zSign, zExp, zFrac0, zFrac1, zFrac2);
938 __ffma64(uint64_t a, uint64_t b, uint64_t c)
940 return __fadd64(__fmul64(a, b), c);
943 /* Shifts the 64-bit value formed by concatenating `a0' and `a1' right by the
944 * number of bits given in `count'. Any bits shifted off are lost. The value
945 * of `count' can be arbitrarily large; in particular, if `count' is greater
946 * than 64, the result will be 0. The result is broken into two 32-bit pieces
947 * which are stored at the locations pointed to by `z0Ptr' and `z1Ptr'.
950 __shift64Right(uint a0, uint a1,
957 int negCount = (-count) & 31;
960 z0 = mix(z0, (a0 >> count), count < 32);
961 z0 = mix(z0, a0, count == 0);
963 z1 = mix(0u, (a0 >> (count & 31)), count < 64);
964 z1 = mix(z1, (a0<<negCount) | (a1>>count), count < 32);
965 z1 = mix(z1, a0, count == 0);
971 /* Returns the result of converting the double-precision floating-point value
972 * `a' to the unsigned integer format. The conversion is performed according
973 * to the IEEE Standard for Floating-Point Arithmetic.
976 __fp64_to_uint(uint64_t a)
978 uint aFracLo = __extractFloat64FracLo(a);
979 uint aFracHi = __extractFloat64FracHi(a);
980 int aExp = __extractFloat64Exp(a);
981 uint aSign = __extractFloat64Sign(a);
983 if ((aExp == 0x7FF) && ((aFracHi | aFracLo) != 0u))
986 aFracHi |= mix(0u, 0x00100000u, aExp != 0);
988 int shiftDist = 0x427 - aExp;
990 __shift64RightJamming(aFracHi, aFracLo, shiftDist, aFracHi, aFracLo);
992 if ((aFracHi & 0xFFFFF000u) != 0u)
993 return mix(~0u, 0u, aSign != 0u);
997 __shift64Right(aFracHi, aFracLo, 12, zero, z);
999 uint expt = mix(~0u, 0u, aSign != 0u);
1001 return mix(z, expt, (aSign != 0u) && (z != 0u));
1005 __uint_to_fp64(uint a)
1010 int shiftDist = __countLeadingZeros32(a) + 21;
1014 int negCount = (- shiftDist) & 31;
1016 aHigh = mix(0u, a<< shiftDist - 32, shiftDist < 64);
1018 aHigh = mix(aHigh, 0u, shiftDist == 0);
1019 aLow = mix(aLow, a, shiftDist ==0);
1020 aHigh = mix(aHigh, a >> negCount, shiftDist < 32);
1021 aLow = mix(aLow, a << shiftDist, shiftDist < 32);
1023 return __packFloat64(0u, 0x432 - shiftDist, aHigh, aLow);
1027 __uint64_to_fp64(uint64_t a)
1032 uvec2 aFrac = unpackUint2x32(a);
1033 uint aFracLo = __extractFloat64FracLo(a);
1034 uint aFracHi = __extractFloat64FracHi(a);
1036 if ((aFracHi & 0x80000000u) != 0u) {
1037 __shift64RightJamming(aFracHi, aFracLo, 1, aFracHi, aFracLo);
1038 return __roundAndPackFloat64(0, 0x433, aFracHi, aFracLo, 0u);
1040 return __normalizeRoundAndPackFloat64(0, 0x432, aFrac.y, aFrac.x);
1045 __fp64_to_uint64(uint64_t a)
1047 uint aFracLo = __extractFloat64FracLo(a);
1048 uint aFracHi = __extractFloat64FracHi(a);
1049 int aExp = __extractFloat64Exp(a);
1050 uint aSign = __extractFloat64Sign(a);
1052 uint64_t default_nan = 0xFFFFFFFFFFFFFFFFUL;
1054 aFracHi = mix(aFracHi, aFracHi | 0x00100000u, aExp != 0);
1055 int shiftCount = 0x433 - aExp;
1057 if ( shiftCount <= 0 ) {
1058 if (shiftCount < -11 && aExp == 0x7FF) {
1059 if ((aFracHi | aFracLo) != 0u)
1060 return __propagateFloat64NaN(a, a);
1061 return mix(default_nan, a, aSign == 0u);
1063 __shortShift64Left(aFracHi, aFracLo, -shiftCount, aFracHi, aFracLo);
1065 __shift64ExtraRightJamming(aFracHi, aFracLo, zFrac2, shiftCount,
1066 aFracHi, aFracLo, zFrac2);
1068 return __roundAndPackUInt64(aSign, aFracHi, aFracLo, zFrac2);
1072 __fp64_to_int64(uint64_t a)
1075 uint aFracLo = __extractFloat64FracLo(a);
1076 uint aFracHi = __extractFloat64FracHi(a);
1077 int aExp = __extractFloat64Exp(a);
1078 uint aSign = __extractFloat64Sign(a);
1079 int64_t default_NegNaN = -0x7FFFFFFFFFFFFFFEL;
1080 int64_t default_PosNaN = 0xFFFFFFFFFFFFFFFFL;
1082 aFracHi = mix(aFracHi, aFracHi | 0x00100000u, aExp != 0);
1083 int shiftCount = 0x433 - aExp;
1085 if (shiftCount <= 0) {
1086 if (shiftCount < -11 && aExp == 0x7FF) {
1087 if ((aFracHi | aFracLo) != 0u)
1088 return default_NegNaN;
1089 return mix(default_NegNaN, default_PosNaN, aSign == 0u);
1091 __shortShift64Left(aFracHi, aFracLo, -shiftCount, aFracHi, aFracLo);
1093 __shift64ExtraRightJamming(aFracHi, aFracLo, zFrac2, shiftCount,
1094 aFracHi, aFracLo, zFrac2);
1097 return __roundAndPackInt64(aSign, aFracHi, aFracLo, zFrac2);
1101 __fp32_to_uint64(float f)
1103 uint a = floatBitsToUint(f);
1104 uint aFrac = a & 0x007FFFFFu;
1105 int aExp = int((a>>23) & 0xFFu);
1106 uint aSign = a & 0x80000000u;
1110 uint64_t default_nan = 0xFFFFFFFFFFFFFFFFUL;
1111 int shiftCount = 0xBE - aExp;
1113 if (shiftCount <0) {
1118 aFrac = mix(aFrac, aFrac | 0x00800000u, aExp != 0);
1119 __shortShift64Left(aFrac, 0, 40, zFrac0, zFrac1);
1121 if (shiftCount != 0) {
1122 __shift64ExtraRightJamming(zFrac0, zFrac1, zFrac2, shiftCount,
1123 zFrac0, zFrac1, zFrac2);
1126 return __roundAndPackUInt64(aSign, zFrac0, zFrac1, zFrac2);
1130 __fp32_to_int64(float f)
1132 uint a = floatBitsToUint(f);
1133 uint aFrac = a & 0x007FFFFFu;
1134 int aExp = int((a>>23) & 0xFFu);
1135 uint aSign = a & 0x80000000u;
1139 int64_t default_NegNaN = -0x7FFFFFFFFFFFFFFEL;
1140 int64_t default_PosNaN = 0xFFFFFFFFFFFFFFFFL;
1141 int shiftCount = 0xBE - aExp;
1143 if (shiftCount <0) {
1144 if (aExp == 0xFF && aFrac != 0u)
1145 return default_NegNaN;
1146 return mix(default_NegNaN, default_PosNaN, aSign == 0u);
1149 aFrac = mix(aFrac, aFrac | 0x00800000u, aExp != 0);
1150 __shortShift64Left(aFrac, 0, 40, zFrac0, zFrac1);
1152 if (shiftCount != 0) {
1153 __shift64ExtraRightJamming(zFrac0, zFrac1, zFrac2, shiftCount,
1154 zFrac0, zFrac1, zFrac2);
1157 return __roundAndPackInt64(aSign, zFrac0, zFrac1, zFrac2);
1161 __int64_to_fp64(int64_t a)
1166 uint64_t absA = mix(uint64_t(a), uint64_t(-a), a < 0);
1167 uint aFracHi = __extractFloat64FracHi(absA);
1168 uvec2 aFrac = unpackUint2x32(absA);
1169 uint zSign = uint(unpackInt2x32(a).y) & 0x80000000u;
1171 if ((aFracHi & 0x80000000u) != 0u) {
1172 return mix(0ul, __packFloat64(0x80000000u, 0x434, 0u, 0u), a < 0);
1175 return __normalizeRoundAndPackFloat64(zSign, 0x432, aFrac.y, aFrac.x);
1178 /* Returns the result of converting the double-precision floating-point value
1179 * `a' to the 32-bit two's complement integer format. The conversion is
1180 * performed according to the IEEE Standard for Floating-Point Arithmetic---
1181 * which means in particular that the conversion is rounded according to the
1182 * current rounding mode. If `a' is a NaN, the largest positive integer is
1183 * returned. Otherwise, if the conversion overflows, the largest integer with
1184 * the same sign as `a' is returned.
1187 __fp64_to_int(uint64_t a)
1189 uint aFracLo = __extractFloat64FracLo(a);
1190 uint aFracHi = __extractFloat64FracHi(a);
1191 int aExp = __extractFloat64Exp(a);
1192 uint aSign = __extractFloat64Sign(a);
1195 uint aFracExtra = 0u;
1196 int shiftCount = aExp - 0x413;
1198 if (0 <= shiftCount) {
1200 if ((aExp == 0x7FF) && bool(aFracHi | aFracLo))
1202 return mix(0x7FFFFFFF, 0x80000000, aSign != 0u);
1204 __shortShift64Left(aFracHi | 0x00100000u, aFracLo, shiftCount, absZ, aFracExtra);
1209 aFracHi |= 0x00100000u;
1210 aFracExtra = ( aFracHi << (shiftCount & 31)) | aFracLo;
1211 absZ = aFracHi >> (- shiftCount);
1214 int z = mix(int(absZ), -int(absZ), aSign != 0u);
1215 int nan = mix(0x7FFFFFFF, 0x80000000, aSign != 0u);
1216 return mix(z, nan, ((aSign != 0u) != (z < 0)) && bool(z));
1219 /* Returns the result of converting the 32-bit two's complement integer `a'
1220 * to the double-precision floating-point format. The conversion is performed
1221 * according to the IEEE Standard for Floating-Point Arithmetic.
1224 __int_to_fp64(int a)
1229 return __packFloat64(0u, 0, 0u, 0u);
1230 uint zSign = uint(a) & 0x80000000u;
1231 uint absA = mix(uint(a), uint(-a), a < 0);
1232 int shiftCount = __countLeadingZeros32(absA) - 11;
1233 if (0 <= shiftCount) {
1234 zFrac0 = absA << shiftCount;
1237 __shift64Right(absA, 0u, -shiftCount, zFrac0, zFrac1);
1239 return __packFloat64(zSign, 0x412 - shiftCount, zFrac0, zFrac1);
1243 __fp64_to_bool(uint64_t a)
1245 return !__feq64_nonnan(__fabs64(a), 0ul);
1249 __bool_to_fp64(bool a)
1251 return packUint2x32(uvec2(0x00000000u, uint(-int(a) & 0x3ff00000)));
1254 /* Packs the sign `zSign', exponent `zExp', and significand `zFrac' into a
1255 * single-precision floating-point value, returning the result. After being
1256 * shifted into the proper positions, the three fields are simply added
1257 * together to form the result. This means that any integer portion of `zSig'
1258 * will be added into the exponent. Since a properly normalized significand
1259 * will have an integer portion equal to 1, the `zExp' input should be 1 less
1260 * than the desired result exponent whenever `zFrac' is a complete, normalized
1264 __packFloat32(uint zSign, int zExp, uint zFrac)
1266 return uintBitsToFloat(zSign + (uint(zExp)<<23) + zFrac);
1269 /* Takes an abstract floating-point value having sign `zSign', exponent `zExp',
1270 * and significand `zFrac', and returns the proper single-precision floating-
1271 * point value corresponding to the abstract input. Ordinarily, the abstract
1272 * value is simply rounded and packed into the single-precision format, with
1273 * the inexact exception raised if the abstract input cannot be represented
1274 * exactly. However, if the abstract value is too large, the overflow and
1275 * inexact exceptions are raised and an infinity or maximal finite value is
1276 * returned. If the abstract value is too small, the input value is rounded to
1277 * a subnormal number, and the underflow and inexact exceptions are raised if
1278 * the abstract input cannot be represented exactly as a subnormal single-
1279 * precision floating-point number.
1280 * The input significand `zFrac' has its binary point between bits 30
1281 * and 29, which is 7 bits to the left of the usual location. This shifted
1282 * significand must be normalized or smaller. If `zFrac' is not normalized,
1283 * `zExp' must be 0; in that case, the result returned is a subnormal number,
1284 * and it must not require rounding. In the usual case that `zFrac' is
1285 * normalized, `zExp' must be 1 less than the "true" floating-point exponent.
1286 * The handling of underflow and overflow follows the IEEE Standard for
1287 * Floating-Point Arithmetic.
1290 __roundAndPackFloat32(uint zSign, int zExp, uint zFrac)
1292 bool roundNearestEven;
1296 roundNearestEven = FLOAT_ROUNDING_MODE == FLOAT_ROUND_NEAREST_EVEN;
1297 roundIncrement = 0x40;
1298 if (!roundNearestEven) {
1299 if (FLOAT_ROUNDING_MODE == FLOAT_ROUND_TO_ZERO) {
1302 roundIncrement = 0x7F;
1304 if (FLOAT_ROUNDING_MODE == FLOAT_ROUND_UP)
1307 if (FLOAT_ROUNDING_MODE == FLOAT_ROUND_DOWN)
1312 roundBits = int(zFrac & 0x7Fu);
1313 if (0xFDu <= uint(zExp)) {
1314 if ((0xFD < zExp) || ((zExp == 0xFD) && (int(zFrac) + roundIncrement) < 0))
1315 return __packFloat32(zSign, 0xFF, 0u) - float(roundIncrement == 0);
1317 bool zexp_lt0 = zExp < 0;
1318 uint zFrac_lt0 = mix(uint(zFrac != 0u), (zFrac>>count) | uint((zFrac<<((-count) & 31)) != 0u), (-zExp) < 32);
1319 zFrac = mix(zFrac, zFrac_lt0, zexp_lt0);
1320 roundBits = mix(roundBits, int(zFrac) & 0x7f, zexp_lt0);
1321 zExp = mix(zExp, 0, zexp_lt0);
1323 zFrac = (zFrac + uint(roundIncrement))>>7;
1324 zFrac &= ~uint(((roundBits ^ 0x40) == 0) && roundNearestEven);
1326 return __packFloat32(zSign, mix(zExp, 0, zFrac == 0u), zFrac);
1329 /* Returns the result of converting the double-precision floating-point value
1330 * `a' to the single-precision floating-point format. The conversion is
1331 * performed according to the IEEE Standard for Floating-Point Arithmetic.
1334 __fp64_to_fp32(uint64_t __a)
1336 uvec2 a = unpackUint2x32(__a);
1340 uint aFracLo = __extractFloat64FracLo(__a);
1341 uint aFracHi = __extractFloat64FracHi(__a);
1342 int aExp = __extractFloat64Exp(__a);
1343 uint aSign = __extractFloat64Sign(__a);
1344 if (aExp == 0x7FF) {
1345 __shortShift64Left(a.y, a.x, 12, a.y, a.x);
1346 float rval = uintBitsToFloat(aSign | 0x7FC00000u | (a.y>>9));
1347 rval = mix(__packFloat32(aSign, 0xFF, 0u), rval, (aFracHi | aFracLo) != 0u);
1350 __shift64RightJamming(aFracHi, aFracLo, 22, allZero, zFrac);
1351 zFrac = mix(zFrac, zFrac | 0x40000000u, aExp != 0);
1352 return __roundAndPackFloat32(aSign, aExp - 0x381, zFrac);
1356 __uint64_to_fp32(uint64_t __a)
1358 uvec2 aFrac = unpackUint2x32(__a);
1359 int shiftCount = mix(__countLeadingZeros32(aFrac.y) - 33,
1360 __countLeadingZeros32(aFrac.x) - 1,
1363 if (0 <= shiftCount)
1364 __shortShift64Left(aFrac.y, aFrac.x, shiftCount, aFrac.y, aFrac.x);
1366 __shift64RightJamming(aFrac.y, aFrac.x, -shiftCount, aFrac.y, aFrac.x);
1368 return __roundAndPackFloat32(0u, 0x9C - shiftCount, aFrac.x);
1372 __int64_to_fp32(int64_t __a)
1374 uint aSign = uint(unpackInt2x32(__a).y) & 0x80000000u;
1375 uint64_t absA = mix(uint64_t(__a), uint64_t(-__a), __a < 0);
1376 uvec2 aFrac = unpackUint2x32(absA);
1377 int shiftCount = mix(__countLeadingZeros32(aFrac.y) - 33,
1378 __countLeadingZeros32(aFrac.x) - 1,
1381 if (0 <= shiftCount)
1382 __shortShift64Left(aFrac.y, aFrac.x, shiftCount, aFrac.y, aFrac.x);
1384 __shift64RightJamming(aFrac.y, aFrac.x, -shiftCount, aFrac.y, aFrac.x);
1386 return __roundAndPackFloat32(aSign, 0x9C - shiftCount, aFrac.x);
1389 /* Returns the result of converting the single-precision floating-point value
1390 * `a' to the double-precision floating-point format.
1393 __fp32_to_fp64(float f)
1395 uint a = floatBitsToUint(f);
1396 uint aFrac = a & 0x007FFFFFu;
1397 int aExp = int((a>>23) & 0xFFu);
1398 uint aSign = a & 0x80000000u;
1406 __shift64Right(nanHi, nanLo, 12, nanHi, nanLo);
1407 nanHi |= aSign | 0x7FF80000u;
1408 return packUint2x32(uvec2(nanLo, nanHi));
1410 return __packFloat64(aSign, 0x7FF, 0u, 0u);
1415 return __packFloat64(aSign, 0, 0u, 0u);
1416 /* Normalize subnormal */
1417 int shiftCount = __countLeadingZeros32(aFrac) - 8;
1418 aFrac <<= shiftCount;
1419 aExp = 1 - shiftCount;
1423 __shift64Right(aFrac, 0u, 3, zFrac0, zFrac1);
1424 return __packFloat64(aSign, aExp + 0x380, zFrac0, zFrac1);
1427 /* Adds the 96-bit value formed by concatenating `a0', `a1', and `a2' to the
1428 * 96-bit value formed by concatenating `b0', `b1', and `b2'. Addition is
1429 * modulo 2^96, so any carry out is lost. The result is broken into three
1430 * 32-bit pieces which are stored at the locations pointed to by `z0Ptr',
1431 * `z1Ptr', and `z2Ptr'.
1434 __add96(uint a0, uint a1, uint a2,
1435 uint b0, uint b1, uint b2,
1441 uint carry1 = uint(z2 < a2);
1443 uint carry0 = uint(z1 < a1);
1446 z0 += uint(z1 < carry1);
1453 /* Subtracts the 96-bit value formed by concatenating `b0', `b1', and `b2' from
1454 * the 96-bit value formed by concatenating `a0', `a1', and `a2'. Subtraction
1455 * is modulo 2^96, so any borrow out (carry out) is lost. The result is broken
1456 * into three 32-bit pieces which are stored at the locations pointed to by
1457 * `z0Ptr', `z1Ptr', and `z2Ptr'.
1460 __sub96(uint a0, uint a1, uint a2,
1461 uint b0, uint b1, uint b2,
1467 uint borrow1 = uint(a2 < b2);
1469 uint borrow0 = uint(a1 < b1);
1471 z0 -= uint(z1 < borrow1);
1479 /* Returns an approximation to the 32-bit integer quotient obtained by dividing
1480 * `b' into the 64-bit value formed by concatenating `a0' and `a1'. The
1481 * divisor `b' must be at least 2^31. If q is the exact quotient truncated
1482 * toward zero, the approximation returned lies between q and q + 2 inclusive.
1483 * If the exact quotient q is larger than 32 bits, the maximum positive 32-bit
1484 * unsigned integer is returned.
1487 __estimateDiv64To32(uint a0, uint a1, uint b)
1500 z = (b0<<16 <= a0) ? 0xFFFF0000u : (a0 / b0)<<16;
1501 umulExtended(b, z, term0, term1);
1502 __sub64(a0, a1, term0, term1, rem0, rem1);
1503 while (int(rem0) < 0) {
1506 __add64(rem0, rem1, b0, b1, rem0, rem1);
1508 rem0 = (rem0<<16) | (rem1>>16);
1509 z |= (b0<<16 <= rem0) ? 0xFFFFu : rem0 / b0;
1514 __sqrtOddAdjustments(int index)
1554 __sqrtEvenAdjustments(int index)
1593 /* Returns an approximation to the square root of the 32-bit significand given
1594 * by `a'. Considered as an integer, `a' must be at least 2^31. If bit 0 of
1595 * `aExp' (the least significant bit) is 1, the integer returned approximates
1596 * 2^31*sqrt(`a'/2^31), where `a' is considered an integer. If bit 0 of `aExp'
1597 * is 0, the integer returned approximates 2^31*sqrt(`a'/2^30). In either
1598 * case, the approximation returned lies strictly within +/-2 of the exact
1602 __estimateSqrt32(int aExp, uint a)
1606 int index = int(a>>27 & 15u);
1607 if ((aExp & 1) != 0) {
1608 z = 0x4000u + (a>>17) - __sqrtOddAdjustments(index);
1609 z = ((a / z)<<14) + (z<<15);
1612 z = 0x8000u + (a>>17) - __sqrtEvenAdjustments(index);
1614 z = (0x20000u <= z) ? 0xFFFF8000u : (z<<15);
1616 return uint(int(a)>>1);
1618 return ((__estimateDiv64To32(a, 0u, z))>>1) + (z>>1);
1621 /* Returns the square root of the double-precision floating-point value `a'.
1622 * The operation is performed according to the IEEE Standard for Floating-Point
1626 __fsqrt64(uint64_t a)
1631 uint doubleZFrac0 = 0u;
1640 uint64_t default_nan = 0xFFFFFFFFFFFFFFFFUL;
1642 uint aFracLo = __extractFloat64FracLo(a);
1643 uint aFracHi = __extractFloat64FracHi(a);
1644 int aExp = __extractFloat64Exp(a);
1645 uint aSign = __extractFloat64Sign(a);
1646 if (aExp == 0x7FF) {
1647 if ((aFracHi | aFracLo) != 0u)
1648 return __propagateFloat64NaN(a, a);
1654 if ((uint(aExp) | aFracHi | aFracLo) == 0u)
1659 if ((aFracHi | aFracLo) == 0u)
1660 return __packFloat64(0u, 0, 0u, 0u);
1661 __normalizeFloat64Subnormal(aFracHi, aFracLo, aExp, aFracHi, aFracLo);
1663 int zExp = ((aExp - 0x3FF)>>1) + 0x3FE;
1664 aFracHi |= 0x00100000u;
1665 __shortShift64Left(aFracHi, aFracLo, 11, term0, term1);
1666 zFrac0 = (__estimateSqrt32(aExp, term0)>>1) + 1u;
1668 zFrac0 = 0x7FFFFFFFu;
1669 doubleZFrac0 = zFrac0 + zFrac0;
1670 __shortShift64Left(aFracHi, aFracLo, 9 - (aExp & 1), aFracHi, aFracLo);
1671 umulExtended(zFrac0, zFrac0, term0, term1);
1672 __sub64(aFracHi, aFracLo, term0, term1, rem0, rem1);
1673 while (int(rem0) < 0) {
1676 __add64(rem0, rem1, 0u, doubleZFrac0 | 1u, rem0, rem1);
1678 zFrac1 = __estimateDiv64To32(rem1, 0u, doubleZFrac0);
1679 if ((zFrac1 & 0x1FFu) <= 5u) {
1682 umulExtended(doubleZFrac0, zFrac1, term1, term2);
1683 __sub64(rem1, 0u, term1, term2, rem1, rem2);
1684 umulExtended(zFrac1, zFrac1, term2, term3);
1685 __sub96(rem1, rem2, 0u, 0u, term2, term3, rem1, rem2, rem3);
1686 while (int(rem1) < 0) {
1688 __shortShift64Left(0u, zFrac1, 1, term2, term3);
1690 term2 |= doubleZFrac0;
1691 __add96(rem1, rem2, rem3, 0u, term2, term3, rem1, rem2, rem3);
1693 zFrac1 |= uint((rem1 | rem2 | rem3) != 0u);
1695 __shift64ExtraRightJamming(zFrac0, zFrac1, 0u, 10, zFrac0, zFrac1, zFrac2);
1696 return __roundAndPackFloat64(0u, zExp, zFrac0, zFrac1, zFrac2);
1700 __ftrunc64(uint64_t __a)
1702 uvec2 a = unpackUint2x32(__a);
1703 int aExp = __extractFloat64Exp(__a);
1707 int unbiasedExp = aExp - 1023;
1708 int fracBits = 52 - unbiasedExp;
1709 uint maskLo = mix(~0u << fracBits, 0u, fracBits >= 32);
1710 uint maskHi = mix(~0u << (fracBits - 32), ~0u, fracBits < 33);
1714 zLo = mix(zLo, 0u, unbiasedExp < 0);
1715 zHi = mix(zHi, 0u, unbiasedExp < 0);
1716 zLo = mix(zLo, a.x, unbiasedExp > 52);
1717 zHi = mix(zHi, a.y, unbiasedExp > 52);
1718 return packUint2x32(uvec2(zLo, zHi));
1722 __ffloor64(uint64_t a)
1724 /* The big assumtion is that when 'a' is NaN, __ftrunc(a) returns a. Based
1725 * on that assumption, NaN values that don't have the sign bit will safely
1726 * return NaN (identity). This is guarded by RELAXED_NAN_PROPAGATION
1727 * because otherwise the NaN should have the "signal" bit set. The
1728 * __fadd64 will ensure that occurs.
1731 #if defined RELAXED_NAN_PROPAGATION
1732 int(unpackUint2x32(a).y) >= 0
1737 uint64_t tr = __ftrunc64(a);
1739 if (is_positive || __feq64(tr, a)) {
1742 return __fadd64(tr, 0xbff0000000000000ul /* -1.0 */);
1747 __fround64(uint64_t __a)
1749 uvec2 a = unpackUint2x32(__a);
1750 int unbiasedExp = __extractFloat64Exp(__a) - 1023;
1754 if (unbiasedExp < 20) {
1755 if (unbiasedExp < 0) {
1756 if ((aHi & 0x80000000u) != 0u && aLo == 0u) {
1760 if ((a.y & 0x000FFFFFu) == 0u && a.x == 0u) {
1762 return packUint2x32(uvec2(aLo, aHi));
1764 aHi = mix(aHi, (aHi | 0x3FF00000u), unbiasedExp == -1);
1767 uint maskExp = 0x000FFFFFu >> unbiasedExp;
1768 uint lastBit = maskExp + 1;
1769 aHi += 0x00080000u >> unbiasedExp;
1770 if ((aHi & maskExp) == 0u)
1775 } else if (unbiasedExp > 51 || unbiasedExp == 1024) {
1778 uint maskExp = 0xFFFFFFFFu >> (unbiasedExp - 20);
1779 if ((aLo & maskExp) == 0u)
1781 uint tmp = aLo + (1u << (51 - unbiasedExp));
1788 return packUint2x32(uvec2(aLo, aHi));
1792 __fmin64(uint64_t a, uint64_t b)
1794 /* This weird layout matters. Doing the "obvious" thing results in extra
1795 * flow control being inserted to implement the short-circuit evaluation
1796 * rules. Flow control is bad!
1798 bool b_nan = __is_nan(b);
1799 bool a_lt_b = __flt64_nonnan(a, b);
1800 bool a_nan = __is_nan(a);
1802 return (b_nan || a_lt_b) && !a_nan ? a : b;
1806 __fmax64(uint64_t a, uint64_t b)
1808 /* This weird layout matters. Doing the "obvious" thing results in extra
1809 * flow control being inserted to implement the short-circuit evaluation
1810 * rules. Flow control is bad!
1812 bool b_nan = __is_nan(b);
1813 bool a_lt_b = __flt64_nonnan(a, b);
1814 bool a_nan = __is_nan(a);
1816 return (b_nan || a_lt_b) && !a_nan ? b : a;
1820 __ffract64(uint64_t a)
1822 return __fadd64(a, __fneg64(__ffloor64(a)));