4 // Copyright (c) 2002 - 2003, Intel Corporation
5 // All rights reserved.
7 // Contributed 2002 by the Intel Numerics Group, Intel Corporation
9 // Redistribution and use in source and binary forms, with or without
10 // modification, are permitted provided that the following conditions are
13 // * Redistributions of source code must retain the above copyright
14 // notice, this list of conditions and the following disclaimer.
16 // * Redistributions in binary form must reproduce the above copyright
17 // notice, this list of conditions and the following disclaimer in the
18 // documentation and/or other materials provided with the distribution.
20 // * The name of Intel Corporation may not be used to endorse or promote
21 // products derived from this software without specific prior written
24 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
25 // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES,INCLUDING,BUT NOT
26 // LIMITED TO,THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
27 // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INTEL OR ITS
28 // CONTRIBUTORS BE LIABLE FOR ANY DIRECT,INDIRECT,INCIDENTAL,SPECIAL,
29 // EXEMPLARY,OR CONSEQUENTIAL DAMAGES (INCLUDING,BUT NOT LIMITED TO,
30 // PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,DATA,OR
31 // PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY
32 // OF LIABILITY,WHETHER IN CONTRACT,STRICT LIABILITY OR TORT (INCLUDING
33 // NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
34 // SOFTWARE,EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
36 // Intel Corporation is the author of this code,and requests that all
37 // problem reports or change requests be submitted to it directly at
38 // http://www.intel.com/software/products/opensource/libraries/num.htm.
40 //*********************************************************************
43 // 01/10/02 Initial version
44 // 01/25/02 Corrected parameter store, load, and tag for __libm_error_support
45 // 02/01/02 Added support of SIGN(GAMMA(x)) calculation
46 // 05/20/02 Cleaned up namespace and sf0 syntax
47 // 09/16/02 Improved accuracy on intervals reduced to [1;1.25]
48 // 10/21/02 Now it returns SIGN(GAMMA(x))=-1 for negative zero
49 // 02/10/03 Reordered header: .section, .global, .proc, .align
51 //*********************************************************************
53 //*********************************************************************
55 // Function: __libm_lgammaf(float x, int* signgam, int szsigngam)
56 // computes the principle value of the logarithm of the GAMMA function
57 // of x. Signum of GAMMA(x) is stored to memory starting at the address
58 // specified by the signgam.
60 //*********************************************************************
64 // Floating-Point Registers: f6-f15
67 // General Purpose Registers:
71 // r37-r40 (Used to pass arguments to error handling routine)
73 // Predicate Registers: p6-p15
75 //*********************************************************************
77 // IEEE Special Conditions:
79 // lgamma(+inf) = +inf
80 // lgamma(-inf) = +inf
81 // lgamma(+/-0) = +inf
82 // lgamma(x<0, x - integer) = +inf
83 // lgamma(SNaN) = QNaN
84 // lgamma(QNaN) = QNaN
86 //*********************************************************************
90 // The method consists of three cases.
92 // If 2^13 <= x < OVERFLOW_BOUNDARY use case lgammaf_pstirling;
93 // else if 1 < x < 2^13 use case lgammaf_regular;
94 // else if -9 < x < 1 use case lgammaf_negrecursion;
95 // else if -2^13 < x < -9 use case lgammaf_negpoly;
96 // else if x < -2^13 use case lgammaf_negstirling;
97 // else if x is close to negative
98 // roots of ln(GAMMA(x)) use case lgammaf_negroots;
101 // Case 2^13 <= x < OVERFLOW_BOUNDARY
102 // ----------------------------------
103 // Here we use algorithm based on the Stirling formula:
104 // ln(GAMMA(x)) = ln(sqrt(2*Pi)) + (x-0.5)*ln(x) - x
108 // To calculate ln(GAMMA(x)) for such arguments we use polynomial
109 // approximation on following intervals: [1.0; 1.25), [1.25; 1.5),
110 // [1.5, 1.75), [1.75; 2), [2; 4), [2^i; 2^(i+1)), i=1..8
112 // Following variants of approximation and argument reduction are used:
114 // ln(GAMMA(x)) ~ (x-1.0)*P7(x)
117 // ln(GAMMA(x)) ~ ln(GAMMA(x0))+(x-x0)*P8(x-x0),
118 // where x0 - point of local minimum on [1;2] rounded to nearest double
122 // ln(GAMMA(x)) ~ P8(x)
125 // ln(GAMMA(x)) ~ (x-2)*P7(x)
128 // ln(GAMMA(x)) ~ (x-2)*P10(x)
130 // 6. [2^i; 2^(i+1)), i=2..8
131 // ln(GAMMA(x)) ~ P10((x-2^i)/2^i)
135 // Here we use the recursive formula:
136 // ln(GAMMA(x)) = ln(GAMMA(x+1)) - ln(x)
138 // Using this formula we reduce argument to base interval [1.0; 2.0]
140 // Case -2^13 < x < -9
141 // --------------------
142 // Here we use the formula:
143 // ln(GAMMA(x)) = ln(Pi/(|x|*GAMMA(|x|)*sin(Pi*|x|))) =
144 // = -ln(|x|) - ln((GAMMA(|x|)) - ln(sin(Pi*r)/(Pi*r)) - ln(|r|)
145 // where r = x - rounded_to_nearest(x), i.e |r| <= 0.5 and
146 // ln(sin(Pi*r)/(Pi*r)) is approximated by 8-degree polynomial of r^2
150 // Here we use algorithm based on the Stirling formula:
151 // ln(GAMMA(x)) = -ln(sqrt(2*Pi)) + (|x|-0.5)ln(x) - |x| -
152 // - ln(sin(Pi*r)/(Pi*r)) - ln(|r|)
153 // where r = x - rounded_to_nearest(x).
155 // Neighbourhoods of negative roots
156 // --------------------------------
157 // Here we use polynomial approximation
158 // ln(GAMMA(x-x0)) = ln(GAMMA(x0)) + (x-x0)*P14(x-x0),
159 // where x0 is a root of ln(GAMMA(x)) rounded to nearest double
163 // Claculation of logarithm
164 // ------------------------
165 // Consider x = 2^N * xf so
166 // ln(x) = ln(frcpa(x)*x/frcpa(x))
167 // = ln(1/frcpa(x)) + ln(frcpa(x)*x)
169 // frcpa(x) = 2^(-N) * frcpa(xf)
171 // ln(1/frcpa(x)) = -ln(2^(-N)) - ln(frcpa(xf))
172 // = N*ln(2) - ln(frcpa(xf))
173 // = N*ln(2) + ln(1/frcpa(xf))
175 // ln(x) = ln(1/frcpa(x)) + ln(frcpa(x)*x) =
176 // = N*ln(2) + ln(1/frcpa(xf)) + ln(frcpa(x)*x)
177 // = N*ln(2) + T + ln(frcpa(x)*x)
179 // Let r = 1 - frcpa(x)*x, note that r is quite small by
182 // ln(x) = N*ln(2) + T + ln(1+r) ~ N*ln(2) + T + Series(r),
183 // where T - is precomputed tabular value,
184 // Series(r) = (P3*r + P2)*r^2 + (P1*r + 1)
186 //*********************************************************************
236 GR_Parameter_RESULT = r39
237 GR_Parameter_TAG = r40
239 //*********************************************************************
242 FR_Y = f1 // lgammaf is single argument function
331 FR_OverflowBound = f93
338 //*********************************************************************
342 LOCAL_OBJECT_START(lgammaf_data)
344 data8 0xbfd0001008f39d59 // P3
345 data8 0x3fd5556073e0c45a // P2
346 data8 0x3fe62e42fefa39ef // ln(2)
347 data8 0x3fe0000000000000 // 0.5
349 data8 0x3F60040155D5889E //ln(1/frcpa(1+ 0/256)
350 data8 0x3F78121214586B54 //ln(1/frcpa(1+ 1/256)
351 data8 0x3F841929F96832F0 //ln(1/frcpa(1+ 2/256)
352 data8 0x3F8C317384C75F06 //ln(1/frcpa(1+ 3/256)
353 data8 0x3F91A6B91AC73386 //ln(1/frcpa(1+ 4/256)
354 data8 0x3F95BA9A5D9AC039 //ln(1/frcpa(1+ 5/256)
355 data8 0x3F99D2A8074325F4 //ln(1/frcpa(1+ 6/256)
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358 data8 0x3FA2954C78CBCE1B //ln(1/frcpa(1+ 9/256)
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361 data8 0x3FA85188B630F068 //ln(1/frcpa(1+ 12/256)
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366 data8 0x3FB0EC139C5DA601 //ln(1/frcpa(1+ 17/256)
367 data8 0x3FB1DBD2643D190B //ln(1/frcpa(1+ 18/256)
368 data8 0x3FB2CC7284FE5F1C //ln(1/frcpa(1+ 19/256)
369 data8 0x3FB3BDF5A7D1EE64 //ln(1/frcpa(1+ 20/256)
370 data8 0x3FB4B05D7AA012E0 //ln(1/frcpa(1+ 21/256)
371 data8 0x3FB580DB7CEB5702 //ln(1/frcpa(1+ 22/256)
372 data8 0x3FB674F089365A7A //ln(1/frcpa(1+ 23/256)
373 data8 0x3FB769EF2C6B568D //ln(1/frcpa(1+ 24/256)
374 data8 0x3FB85FD927506A48 //ln(1/frcpa(1+ 25/256)
375 data8 0x3FB9335E5D594989 //ln(1/frcpa(1+ 26/256)
376 data8 0x3FBA2B0220C8E5F5 //ln(1/frcpa(1+ 27/256)
377 data8 0x3FBB0004AC1A86AC //ln(1/frcpa(1+ 28/256)
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379 data8 0x3FBCCFEDBFEE13A8 //ln(1/frcpa(1+ 30/256)
380 data8 0x3FBDA727638446A2 //ln(1/frcpa(1+ 31/256)
381 data8 0x3FBEA3257FE10F7A //ln(1/frcpa(1+ 32/256)
382 data8 0x3FBF7BE9FEDBFDE6 //ln(1/frcpa(1+ 33/256)
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385 data8 0x3FC1178E8227E47C //ln(1/frcpa(1+ 36/256)
386 data8 0x3FC185747DBECF34 //ln(1/frcpa(1+ 37/256)
387 data8 0x3FC1F3B925F25D41 //ln(1/frcpa(1+ 38/256)
388 data8 0x3FC2625D1E6DDF57 //ln(1/frcpa(1+ 39/256)
389 data8 0x3FC2D1610C86813A //ln(1/frcpa(1+ 40/256)
390 data8 0x3FC340C59741142E //ln(1/frcpa(1+ 41/256)
391 data8 0x3FC3B08B6757F2A9 //ln(1/frcpa(1+ 42/256)
392 data8 0x3FC40DFB08378003 //ln(1/frcpa(1+ 43/256)
393 data8 0x3FC47E74E8CA5F7C //ln(1/frcpa(1+ 44/256)
394 data8 0x3FC4EF51F6466DE4 //ln(1/frcpa(1+ 45/256)
395 data8 0x3FC56092E02BA516 //ln(1/frcpa(1+ 46/256)
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398 data8 0x3FC6A399DABBD383 //ln(1/frcpa(1+ 49/256)
399 data8 0x3FC70337DD3CE41B //ln(1/frcpa(1+ 50/256)
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401 data8 0x3FC7E9D82A0B022D //ln(1/frcpa(1+ 52/256)
402 data8 0x3FC84A6B759F512F //ln(1/frcpa(1+ 53/256)
403 data8 0x3FC8AB47D5F5A310 //ln(1/frcpa(1+ 54/256)
404 data8 0x3FC91FE49096581B //ln(1/frcpa(1+ 55/256)
405 data8 0x3FC981634011AA75 //ln(1/frcpa(1+ 56/256)
406 data8 0x3FC9F6C407089664 //ln(1/frcpa(1+ 57/256)
407 data8 0x3FCA58E729348F43 //ln(1/frcpa(1+ 58/256)
408 data8 0x3FCABB55C31693AD //ln(1/frcpa(1+ 59/256)
409 data8 0x3FCB1E104919EFD0 //ln(1/frcpa(1+ 60/256)
410 data8 0x3FCB94EE93E367CB //ln(1/frcpa(1+ 61/256)
411 data8 0x3FCBF851C067555F //ln(1/frcpa(1+ 62/256)
412 data8 0x3FCC5C0254BF23A6 //ln(1/frcpa(1+ 63/256)
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531 data8 0x3FE1365252BF0865 //ln(1/frcpa(1+ 182/256)
532 data8 0x3FE14AE558B4A92D //ln(1/frcpa(1+ 183/256)
533 data8 0x3FE15F85A19C765B //ln(1/frcpa(1+ 184/256)
534 data8 0x3FE16D4D38C119FA //ln(1/frcpa(1+ 185/256)
535 data8 0x3FE18203C20DD133 //ln(1/frcpa(1+ 186/256)
536 data8 0x3FE196C7BC4B1F3B //ln(1/frcpa(1+ 187/256)
537 data8 0x3FE1A4A738B7A33C //ln(1/frcpa(1+ 188/256)
538 data8 0x3FE1B981C0C9653D //ln(1/frcpa(1+ 189/256)
539 data8 0x3FE1CE69E8BB106B //ln(1/frcpa(1+ 190/256)
540 data8 0x3FE1DC619DE06944 //ln(1/frcpa(1+ 191/256)
541 data8 0x3FE1F160A2AD0DA4 //ln(1/frcpa(1+ 192/256)
542 data8 0x3FE2066D7740737E //ln(1/frcpa(1+ 193/256)
543 data8 0x3FE2147DBA47A394 //ln(1/frcpa(1+ 194/256)
544 data8 0x3FE229A1BC5EBAC3 //ln(1/frcpa(1+ 195/256)
545 data8 0x3FE237C1841A502E //ln(1/frcpa(1+ 196/256)
546 data8 0x3FE24CFCE6F80D9A //ln(1/frcpa(1+ 197/256)
547 data8 0x3FE25B2C55CD5762 //ln(1/frcpa(1+ 198/256)
548 data8 0x3FE2707F4D5F7C41 //ln(1/frcpa(1+ 199/256)
549 data8 0x3FE285E0842CA384 //ln(1/frcpa(1+ 200/256)
550 data8 0x3FE294294708B773 //ln(1/frcpa(1+ 201/256)
551 data8 0x3FE2A9A2670AFF0C //ln(1/frcpa(1+ 202/256)
552 data8 0x3FE2B7FB2C8D1CC1 //ln(1/frcpa(1+ 203/256)
553 data8 0x3FE2C65A6395F5F5 //ln(1/frcpa(1+ 204/256)
554 data8 0x3FE2DBF557B0DF43 //ln(1/frcpa(1+ 205/256)
555 data8 0x3FE2EA64C3F97655 //ln(1/frcpa(1+ 206/256)
556 data8 0x3FE3001823684D73 //ln(1/frcpa(1+ 207/256)
557 data8 0x3FE30E97E9A8B5CD //ln(1/frcpa(1+ 208/256)
558 data8 0x3FE32463EBDD34EA //ln(1/frcpa(1+ 209/256)
559 data8 0x3FE332F4314AD796 //ln(1/frcpa(1+ 210/256)
560 data8 0x3FE348D90E7464D0 //ln(1/frcpa(1+ 211/256)
561 data8 0x3FE35779F8C43D6E //ln(1/frcpa(1+ 212/256)
562 data8 0x3FE36621961A6A99 //ln(1/frcpa(1+ 213/256)
563 data8 0x3FE37C299F3C366A //ln(1/frcpa(1+ 214/256)
564 data8 0x3FE38AE2171976E7 //ln(1/frcpa(1+ 215/256)
565 data8 0x3FE399A157A603E7 //ln(1/frcpa(1+ 216/256)
566 data8 0x3FE3AFCCFE77B9D1 //ln(1/frcpa(1+ 217/256)
567 data8 0x3FE3BE9D503533B5 //ln(1/frcpa(1+ 218/256)
568 data8 0x3FE3CD7480B4A8A3 //ln(1/frcpa(1+ 219/256)
569 data8 0x3FE3E3C43918F76C //ln(1/frcpa(1+ 220/256)
570 data8 0x3FE3F2ACB27ED6C7 //ln(1/frcpa(1+ 221/256)
571 data8 0x3FE4019C2125CA93 //ln(1/frcpa(1+ 222/256)
572 data8 0x3FE4181061389722 //ln(1/frcpa(1+ 223/256)
573 data8 0x3FE42711518DF545 //ln(1/frcpa(1+ 224/256)
574 data8 0x3FE436194E12B6BF //ln(1/frcpa(1+ 225/256)
575 data8 0x3FE445285D68EA69 //ln(1/frcpa(1+ 226/256)
576 data8 0x3FE45BCC464C893A //ln(1/frcpa(1+ 227/256)
577 data8 0x3FE46AED21F117FC //ln(1/frcpa(1+ 228/256)
578 data8 0x3FE47A1527E8A2D3 //ln(1/frcpa(1+ 229/256)
579 data8 0x3FE489445EFFFCCC //ln(1/frcpa(1+ 230/256)
580 data8 0x3FE4A018BCB69835 //ln(1/frcpa(1+ 231/256)
581 data8 0x3FE4AF5A0C9D65D7 //ln(1/frcpa(1+ 232/256)
582 data8 0x3FE4BEA2A5BDBE87 //ln(1/frcpa(1+ 233/256)
583 data8 0x3FE4CDF28F10AC46 //ln(1/frcpa(1+ 234/256)
584 data8 0x3FE4DD49CF994058 //ln(1/frcpa(1+ 235/256)
585 data8 0x3FE4ECA86E64A684 //ln(1/frcpa(1+ 236/256)
586 data8 0x3FE503C43CD8EB68 //ln(1/frcpa(1+ 237/256)
587 data8 0x3FE513356667FC57 //ln(1/frcpa(1+ 238/256)
588 data8 0x3FE522AE0738A3D8 //ln(1/frcpa(1+ 239/256)
589 data8 0x3FE5322E26867857 //ln(1/frcpa(1+ 240/256)
590 data8 0x3FE541B5CB979809 //ln(1/frcpa(1+ 241/256)
591 data8 0x3FE55144FDBCBD62 //ln(1/frcpa(1+ 242/256)
592 data8 0x3FE560DBC45153C7 //ln(1/frcpa(1+ 243/256)
593 data8 0x3FE5707A26BB8C66 //ln(1/frcpa(1+ 244/256)
594 data8 0x3FE587F60ED5B900 //ln(1/frcpa(1+ 245/256)
595 data8 0x3FE597A7977C8F31 //ln(1/frcpa(1+ 246/256)
596 data8 0x3FE5A760D634BB8B //ln(1/frcpa(1+ 247/256)
597 data8 0x3FE5B721D295F10F //ln(1/frcpa(1+ 248/256)
598 data8 0x3FE5C6EA94431EF9 //ln(1/frcpa(1+ 249/256)
599 data8 0x3FE5D6BB22EA86F6 //ln(1/frcpa(1+ 250/256)
600 data8 0x3FE5E6938645D390 //ln(1/frcpa(1+ 251/256)
601 data8 0x3FE5F673C61A2ED2 //ln(1/frcpa(1+ 252/256)
602 data8 0x3FE6065BEA385926 //ln(1/frcpa(1+ 253/256)
603 data8 0x3FE6164BFA7CC06B //ln(1/frcpa(1+ 254/256)
604 data8 0x3FE62643FECF9743 //ln(1/frcpa(1+ 255/256)
607 data8 0xBEB2CC7A38B9355F,0x3F035F2D1833BF4C // A10,A9
608 data8 0xBFF51BAA7FD27785,0x3FFC9D5D5B6CDEFF // A2,A1
609 data8 0xBF421676F9CB46C7,0x3F7437F2FA1436C6 // A8,A7
610 data8 0xBFD7A7041DE592FE,0x3FE9F107FEE8BD29 // A4,A3
612 data8 0x3F6BBBD68451C0CD,0xBF966EC3272A16F7 // A10,A9
613 data8 0x40022A24A39AD769,0x4014190EDF49C8C5 // A2,A1
614 data8 0x3FB130FD016EE241,0xBFC151B46E635248 // A8,A7
615 data8 0x3FDE8F611965B5FE,0xBFEB5110EB265E3D // A4,A3
617 data8 0x3F736EF93508626A,0xBF9FE5DBADF58AF1 // A10,A9
618 data8 0x40110A9FC5192058,0x40302008A6F96B29 // A2,A1
619 data8 0x3FB8E74E0CE1E4B5,0xBFC9B5DA78873656 // A8,A7
620 data8 0x3FE99D0DF10022DC,0xBFF829C0388F9484 // A4,A3
622 data8 0x3F7FFF9D6D7E9269,0xBFAA780A249AEDB1 // A10,A9
623 data8 0x402082A807AEA080,0x4045ED9868408013 // A2,A1
624 data8 0x3FC4E1E54C2F99B7,0xBFD5DE2D6FFF1490 // A8,A7
625 data8 0x3FF75FC89584AE87,0xC006B4BADD886CAE // A4,A3
627 data8 0x3F8CE54375841A5F,0xBFB801ABCFFA1BE2 // A10,A9
628 data8 0x403040A8B1815BDA,0x405B99A917D24B7A // A2,A1
629 data8 0x3FD30CAB81BFFA03,0xBFE41AEF61ECF48B // A8,A7
630 data8 0x400650CC136BEC43,0xC016022046E8292B // A4,A3
632 data8 0x3F9B69BD22CAA8B8,0xBFC6D48875B7A213 // A10,A9
633 data8 0x40402028CCAA2F6D,0x40709AACEB3CBE0F // A2,A1
634 data8 0x3FE22C6A5924761E,0xBFF342F5F224523D // A8,A7
635 data8 0x4015CD405CCA331F,0xC025AAD10482C769 // A4,A3
637 data8 0x3FAAAD9CD0E40D06,0xBFD63FC8505D80CB // A10,A9
638 data8 0x40501008D56C2648,0x408364794B0F4376 // A2,A1
639 data8 0x3FF1BE0126E00284,0xC002D8E3F6F7F7CA // A8,A7
640 data8 0x40258C757E95D860,0xC0357FA8FD398011 // A4,A3
642 data8 0x3FBA4DAC59D49FEB,0xBFE5F476D1C43A77 // A10,A9
643 data8 0x40600800D890C7C6,0x40962C42AAEC8EF0 // A2,A1
644 data8 0x40018680ECF19B89,0xC012A3EB96FB7BA4 // A8,A7
645 data8 0x40356C4CDD3B60F9,0xC0456A34BF18F440 // A4,A3
647 data8 0x3FCA1B54F6225A5A,0xBFF5CD67BA10E048 // A10,A9
648 data8 0x407003FED94C58C2,0x40A8F30B4ACBCD22 // A2,A1
649 data8 0x40116A135EB66D8C,0xC022891B1CED527E // A8,A7
650 data8 0x40455C4617FDD8BC,0xC0555F82729E59C4 // A4,A3
652 data8 0x3FD9FFF9095C6EC9,0xC005B88CB25D76C9 // A10,A9
653 data8 0x408001FE58FA734D,0x40BBB953BAABB0F3 // A2,A1
654 data8 0x40215B2F9FEB5D87,0xC0327B539DEA5058 // A8,A7
655 data8 0x40555444B3E8D64D,0xC0655A2B26F9FC8A // A4,A3
657 data8 0x3FE9F065A1C3D6B1,0xC015ACF6FAE8D78D // A10,A9
658 data8 0x409000FE383DD2B7,0x40CE7F5C1E8BCB8B // A2,A1
659 data8 0x40315324E5DB2EBE,0xC04274194EF70D18 // A8,A7
660 data8 0x4065504353FF2207,0xC075577FE1BFE7B6 // A4,A3
662 data8 0x3FF9E6FBC6B1C70D,0xC025A62DAF76F85D // A10,A9
663 data8 0x40A0007E2F61EBE8,0x40E0A2A23FB5F6C3 // A2,A1
664 data8 0x40414E9BC0A0141A,0xC0527030F2B69D43 // A8,A7
665 data8 0x40754E417717B45B,0xC085562A447258E5 // A4,A3
667 data8 0xbfdffffffffaea15 // P1
668 data8 0x3FDD8B618D5AF8FE // point of local minimum on [1;2]
669 data8 0x3FED67F1C864BEB5 // ln(sqrt(2*Pi))
670 data8 0x4008000000000000 // 3.0
672 data8 0xBF9E1C289FB224AB,0x3FBF7422445C9460 // A6,A5
673 data8 0xBFF01E76D66F8D8A // A0
674 data8 0xBFE2788CFC6F91DA // A1 [1.0;1.25)
675 data8 0x3FCB8CC69000EB5C,0xBFD41997A0C2C641 // A6,A5
676 data8 0x3FFCAB0BFA0EA462 // A0
677 data8 0xBFBF19B9BCC38A42 // A0 [1.25;1.5)
678 data8 0x3FD51EE4DE0A364C,0xBFE00D7F98A16E4B // A6,A5
679 data8 0x40210CE1F327E9E4 // A0
680 data8 0x4001DB08F9DFA0CC // A0 [1.5;1.75)
681 data8 0x3FE24F606742D252,0xBFEC81D7D12574EC // A6,A5
682 data8 0x403BE636A63A9C27 // A0
683 data8 0x4000A0CB38D6CF0A // A0 [1.75;2.0)
684 data8 0x3FF1029A9DD542B4,0xBFFAD37C209D3B25 // A6,A5
685 data8 0x405385E6FD9BE7EA // A0
686 data8 0x478895F1C0000000 // Overflow boundary
687 data8 0x400062D97D26B523,0xC00A03E1529FF023 // A6,A5
688 data8 0x4069204C51E566CE,0 // A0
689 data8 0x40101476B38FD501,0xC0199DE7B387C0FC // A6,A5
690 data8 0x407EB8DAEC83D759,0 // A0
691 data8 0x401FDB008D65125A,0xC0296B506E665581 // A6,A5
692 data8 0x409226D93107EF66,0 // A0
693 data8 0x402FB3EAAF3E7B2D,0xC039521142AD8E0D // A6,A5
694 data8 0x40A4EFA4F072792E,0 // A0
695 data8 0x403FA024C66B2563,0xC0494569F250E691 // A6,A5
696 data8 0x40B7B747C9235BB8,0 // A0
697 data8 0x404F9607D6DA512C,0xC0593F0B2EDDB4BC // A6,A5
698 data8 0x40CA7E29C5F16DE2,0 // A0
699 data8 0x405F90C5F613D98D,0xC0693BD130E50AAF // A6,A5
700 data8 0x40DD4495238B190C,0 // A0
702 // polynomial approximation of ln(sin(Pi*x)/(Pi*x)), |x| <= 0.5
703 data8 0xBFD58731A486E820,0xBFA4452CC28E15A9 // S16,S14
704 data8 0xBFD013F6E1B86C4F,0xBFD5B3F19F7A341F // S8,S6
705 data8 0xBFC86A0D5252E778,0xBFC93E08C9EE284B // S12,S10
706 data8 0xBFE15132555C9EDD,0xBFFA51A662480E35 // S4,S2
709 data8 0xBFA697D6775F48EA,0x3FB9894B682A98E7 // A9,A8
710 data8 0xBFCA8969253CFF55,0x3FD15124EFB35D9D // A5,A4
711 data8 0xBFC1B00158AB719D,0x3FC5997D04E7F1C1 // A7,A6
712 data8 0xBFD9A4D50BAFF989,0x3FEA51A661F5176A // A3,A2
714 data8 0x3F838E0D35A6171A,0xBF831BBBD61313B7 // A8,A7
715 data8 0x3FB08B40196425D0,0xBFC2E427A53EB830 // A4,A3
716 data8 0x3F9285DDDC20D6C3,0xBFA0C90C9C223044 // A6,A5
717 data8 0x3FDEF72BC8F5287C,0x3D890B3DAEBC1DFC // A2,A1
719 data8 0x3F65D5A7EB31047F,0xBFA44EAC9BFA7FDE // A8,A7
720 data8 0x40051FEFE7A663D8,0xC012A5CFE00A2522 // A4,A3
721 data8 0x3FD0E1583AB00E08,0xBFF084AF95883BA5 // A6,A5
722 data8 0x40185982877AE0A2,0xC015F83DB73B57B7 // A2,A1
724 data8 0x3F4A9222032EB39A,0xBF8CBC9587EEA5A3 // A8,A7
725 data8 0x3FF795400783BE49,0xC00851BC418B8A25 // A4,A3
726 data8 0x3FBBC992783E8C5B,0xBFDFA67E65E89B29 // A6,A5
727 data8 0x4012B408F02FAF88,0xC013284CE7CB0C39 // A2,A1
730 data8 0xC003A7FC9600F86C // -2.4570247382208005860
731 data8 0xC009260DBC9E59AF // -3.1435808883499798405
732 data8 0xC005FB410A1BD901 // -2.7476826467274126919
733 data8 0xC00FA471547C2FE5 // -3.9552942848585979085
735 // polynomial approximation of ln(GAMMA(x)) near roots
736 // near -2.4570247382208005860
737 data8 0x3FF694A6058D9592,0x40136EEBB003A92B // R3,R2
738 data8 0x3FF83FE966AF5360,0x3C90323B6D1FE86D // R1,R0
739 // near -3.1435808883499798405
740 data8 0x405C11371268DA38,0x4039D4D2977D2C23 // R3,R2
741 data8 0x401F20A65F2FAC62,0x3CDE9605E3AE7A62 // R1,R0
742 // near -2.7476826467274126919
743 data8 0xC034185AC31314FF,0x4023267F3C28DFE3 // R3,R2
744 data8 0xBFFEA12DA904B194,0x3CA8FB8530BA7689 // R1,R0
745 // near -2.7476826467274126919
746 data8 0xC0AD25359E70C888,0x406F76DEAEA1B8C6 // R3,R2
747 data8 0xC034B99D966C5644,0xBCBDDC0336980B58 // R1,R0
748 LOCAL_OBJECT_END(lgammaf_data)
750 //*********************************************************************
753 GLOBAL_LIBM_ENTRY(__libm_lgammaf)
755 getf.exp GR_SignExp = f8
756 frcpa.s1 FR_InvX,p0 = f1,f8
757 mov GR_ExpOf2 = 0x10000
760 addl GR_ad_Data = @ltoff(lgammaf_data),gp
761 fcvt.fx.s1 FR_int_N = f8
762 mov GR_ExpMask = 0x1ffff
766 fclass.m p13,p0 = f8,0x1EF // is x NaTVal, NaN,
767 // +/-0, +/-INF or +/-deno?
768 mov GR_ExpBias = 0xffff
771 ld8 GR_ad_Data = [GR_ad_Data]
772 fma.s1 FR_Xp1 = f8,f1,f1
773 mov GR_StirlBound = 0x1000C
776 setf.exp FR_2 = GR_ExpOf2
777 fmerge.se FR_x = f1,f8
778 dep.z GR_Ind = GR_SignExp,3,4
781 cmp.eq p8,p0 = GR_SignExp,GR_ExpBias
782 fcvt.fx.trunc.s1 FR_int_Ntrunc = f8
783 and GR_Exp = GR_ExpMask,GR_SignExp
786 add GR_ad_C650 = 0xB20,GR_ad_Data
787 fcmp.lt.s1 p14,p15 = f8,f0
788 extr.u GR_Ind4T = GR_Sig,55,8
791 sub GR_PureExp = GR_Exp,GR_ExpBias
792 fnorm.s1 FR_NormX = f8
793 // jump if x is NaTVal, NaN, +/-0, +/-INF or +/-deno
794 (p13) br.cond.spnt lgammaf_spec
798 ldfpd FR_P1,FR_LocalMin = [GR_ad_C650],16
799 fms.s1 FR_xm2 = f8,f1,f1
800 add GR_ad_Co = 0x820,GR_ad_Data
803 ldfpd FR_P3,FR_P2 = [GR_ad_Data],16
804 cmp.ltu p9,p0 = GR_SignExp,GR_ExpBias
805 // jump if x is from the interval [1; 2)
806 (p8) br.cond.spnt lgammaf_1_2
809 setf.sig FR_int_Ln = GR_PureExp
810 fms.s1 FR_r = FR_InvX,f8,f1
811 shladd GR_ad_Co = GR_Ind,3,GR_ad_Co
814 ldfpd FR_LnSqrt2Pi,FR_3 = [GR_ad_C650],16
815 cmp.lt p13,p12 = GR_Exp,GR_StirlBound
816 // jump if x is from the interval (0; 1)
817 (p9) br.cond.spnt lgammaf_0_1
820 ldfpd FR_Ln2,FR_05 = [GR_ad_Data],16
821 fma.s1 FR_Xp2 = f1,f1,FR_Xp1 // (x+2)
822 shladd GR_ad_C650 = GR_Ind,2,GR_ad_C650
825 add GR_ad_Ce = 0x20,GR_ad_Co
827 add GR_ad_C43 = 0x30,GR_ad_Co
830 // load coefficients of polynomial approximation
831 // of ln(GAMMA(x)), 2 <= x < 2^13
832 (p13) ldfpd FR_A10,FR_A9 = [GR_ad_Co],16
833 fcvt.xf FR_N = FR_int_N
834 cmp.eq.unc p6,p7 = GR_ExpOf2,GR_SignExp
837 (p13) ldfpd FR_A8,FR_A7 = [GR_ad_Ce]
838 (p14) cmp.le.unc p9,p0 = GR_StirlBound,GR_Exp
839 // jump if x is less or equal to -2^13
840 (p9) br.cond.spnt lgammaf_negstirling
842 .pred.rel "mutex",p6,p7
844 (p13) ldfpd FR_A6,FR_A5 = [GR_ad_C650],16
845 (p6) fma.s1 FR_x = f0,f0,FR_NormX
846 shladd GR_ad_T = GR_Ind4T,3,GR_ad_Data
849 (p13) ldfpd FR_A4,FR_A3 = [GR_ad_C43]
850 (p7) fms.s1 FR_x = FR_x,f1,f1
851 (p14) mov GR_ReqBound = 0x20005
854 (p13) ldfpd FR_A2,FR_A1 = [GR_ad_Co],16
855 fms.s1 FR_xm2 = FR_xm2,f1,f1
856 (p14) extr.u GR_Arg = GR_Sig,60,4
859 mov GR_SignOfGamma = 1 // set sign of gamma(x) to 1
860 fcvt.xf FR_Ntrunc = FR_int_Ntrunc
864 ldfd FR_T = [GR_ad_T]
865 fma.s1 FR_r2 = FR_r,FR_r,f0
866 shl GR_ReqBound = GR_ReqBound,3
869 add GR_ad_Co = 0xCA0,GR_ad_Data
870 fnma.s1 FR_Req = FR_Xp1,FR_NormX,f0 // -x*(x+1)
871 (p14) shladd GR_Arg = GR_Exp,4,GR_Arg
874 (p13) ldfd FR_A0 = [GR_ad_C650]
875 fma.s1 FR_Xp3 = FR_2,f1,FR_Xp1 // (x+3)
876 (p14) cmp.le.unc p9,p0 = GR_Arg,GR_ReqBound
879 (p14) add GR_ad_Ce = 0x20,GR_ad_Co
880 fma.s1 FR_Xp4 = FR_2,FR_2,FR_NormX // (x+4)
881 (p15) add GR_ad_OvfBound = 0xBB8,GR_ad_Data
884 // load coefficients of polynomial approximation
885 // of ln(sin(Pi*xf)/(Pi*xf)), |xf| <= 0.5
886 (p14) ldfpd FR_S16,FR_S14 = [GR_ad_Co],16
887 (p14) fms.s1 FR_Xf = FR_NormX,f1,FR_N // xf = x - [x]
888 (p14) sub GR_SignOfGamma = r0,GR_SignOfGamma // set sign of
892 (p14) ldfpd FR_S12,FR_S10 = [GR_ad_Ce],16
893 fma.s1 FR_Xp5 = FR_2,FR_2,FR_Xp1 // (x+5)
894 // jump if x is from the interval (-9; 0)
895 (p9) br.cond.spnt lgammaf_negrecursion
898 (p14) ldfpd FR_S8,FR_S6 = [GR_ad_Co],16
899 fma.s1 FR_P32 = FR_P3,FR_r,FR_P2
903 (p14) ldfpd FR_S4,FR_S2 = [GR_ad_Ce],16
904 fma.s1 FR_x2 = FR_x,FR_x,f0
905 // jump if x is from the interval (-2^13; -9)
906 (p14) br.cond.spnt lgammaf_negpoly
909 ldfd FR_OverflowBound = [GR_ad_OvfBound]
910 (p12) fcvt.xf FR_N = FR_int_Ln
911 // set p9 if signgum is 32-bit int
912 // set p10 if signgum is 64-bit int
913 cmp.eq p10,p9 = 8,r34
917 (p12) fma.s1 FR_P10 = FR_P1,FR_r,f1
920 .pred.rel "mutex",p6,p7
921 .pred.rel "mutex",p9,p10
923 // store sign of gamma(x) as 32-bit int
924 (p9) st4 [r33] = GR_SignOfGamma
925 (p6) fma.s1 FR_xx = FR_x,FR_xm2,f0
929 // store sign of gamma(x) as 64-bit int
930 (p10) st8 [r33] = GR_SignOfGamma
931 (p7) fma.s1 FR_xx = f0,f0,FR_x
936 (p13) fma.s1 FR_A9 = FR_A10,FR_x,FR_A9
941 (p13) fma.s1 FR_A7 = FR_A8,FR_x,FR_A7
946 (p13) fma.s1 FR_A5 = FR_A6,FR_x,FR_A5
951 (p13) fma.s1 FR_A3 = FR_A4,FR_x,FR_A3
956 (p15) fcmp.eq.unc.s1 p8,p0 = FR_NormX,FR_2 // is input argument 2.0?
961 (p13) fma.s1 FR_A1 = FR_A2,FR_x,FR_A1
966 (p12) fma.s1 FR_T = FR_N,FR_Ln2,FR_T
971 (p12) fma.s1 FR_P32 = FR_P32,FR_r2,FR_P10
976 (p13) fma.s1 FR_x4 = FR_x2,FR_x2,f0
981 (p13) fma.s1 FR_x3 = FR_x2,FR_xx,f0
986 (p13) fma.s1 FR_A7 = FR_A9,FR_x2,FR_A7
991 (p8) fma.s.s0 f8 = f0,f0,f0
992 (p8) br.ret.spnt b0 // fast exit for 2.0
996 (p6) fma.s1 FR_A0 = FR_A0,FR_xm2,f0
1001 (p13) fma.s1 FR_A3 = FR_A5,FR_x2,FR_A3
1006 (p15) fcmp.le.unc.s1 p8,p0 = FR_OverflowBound,FR_NormX // overflow test
1011 (p12) fms.s1 FR_xm05 = FR_NormX,f1,FR_05
1016 (p12) fma.s1 FR_Ln = FR_P32,FR_r,FR_T
1021 (p12) fms.s1 FR_LnSqrt2Pi = FR_LnSqrt2Pi,f1,FR_NormX
1026 (p13) fma.s1 FR_A0 = FR_A1,FR_xx,FR_A0
1031 (p13) fma.s1 FR_A3 = FR_A7,FR_x4,FR_A3
1032 // jump if result overflows
1033 (p8) br.cond.spnt lgammaf_overflow
1035 .pred.rel "mutex",p12,p13
1038 (p12) fma.s.s0 f8 = FR_Ln,FR_xm05,FR_LnSqrt2Pi
1043 (p13) fma.s.s0 f8 = FR_A3,FR_x3,FR_A0
1046 // branch for calculating of ln(GAMMA(x)) for 0 < x < 1
1047 //---------------------------------------------------------------------
1051 getf.sig GR_Ind = FR_Xp1
1052 fma.s1 FR_r2 = FR_r,FR_r,f0
1053 mov GR_fff7 = 0xFFF7
1056 ldfpd FR_Ln2,FR_05 = [GR_ad_Data],16
1057 fma.s1 FR_P32 = FR_P3,FR_r,FR_P2
1058 // input argument cann't be equal to 1.0
1059 cmp.eq p0,p14 = r0,r0
1062 getf.exp GR_Exp = FR_w
1063 fcvt.xf FR_N = FR_int_Ln
1064 add GR_ad_Co = 0xCE0,GR_ad_Data
1067 shladd GR_ad_T = GR_Ind4T,3,GR_ad_Data
1068 fma.s1 FR_P10 = FR_P1,FR_r,f1
1069 add GR_ad_Ce = 0xD00,GR_ad_Data
1072 ldfd FR_T = [GR_ad_T]
1073 fma.s1 FR_w2 = FR_w,FR_w,f0
1074 extr.u GR_Ind = GR_Ind,61,2
1078 fma.s1 FR_Q32 = FR_P3,FR_w,FR_P2
1079 //// add GR_ad_C0 = 0xB30,GR_ad_Data
1080 add GR_ad_C0 = 0xB38,GR_ad_Data
1083 and GR_Exp = GR_Exp,GR_ExpMask
1085 shladd GR_IndX8 = GR_Ind,3,r0
1088 shladd GR_IndX2 = GR_Ind,1,r0
1089 fma.s1 FR_Q10 = FR_P1,FR_w,f1
1090 cmp.eq p6,p15 = 0,GR_Ind
1093 shladd GR_ad_Co = GR_IndX8,3,GR_ad_Co
1094 (p6) fma.s1 FR_x = f0,f0,FR_NormX
1095 shladd GR_ad_C0 = GR_IndX2,4,GR_ad_C0
1098 shladd GR_ad_Ce = GR_IndX8,3,GR_ad_Ce
1100 (p15) cmp.eq.unc p7,p8 = 1,GR_Ind
1102 .pred.rel "mutex",p7,p8
1104 ldfpd FR_A8,FR_A7 = [GR_ad_Co],16
1105 (p7) fms.s1 FR_x = FR_NormX,f1,FR_LocalMin
1106 cmp.ge p10,p11 = GR_Exp,GR_fff7
1109 ldfpd FR_A6,FR_A5 = [GR_ad_Ce],16
1110 (p8) fma.s1 FR_x = f1,f1,FR_NormX
1111 br.cond.sptk lgamma_0_2_core
1113 // branch for calculating of ln(GAMMA(x)) for 1 <= x < 2
1114 //---------------------------------------------------------------------
1118 add GR_ad_Co = 0xCF0,GR_ad_Data
1119 fcmp.eq.s1 p14,p0 = f1,FR_NormX // is input argument 1.0?
1120 extr.u GR_Ind = GR_Sig,61,2
1123 add GR_ad_Ce = 0xD10,GR_ad_Data
1125 //// add GR_ad_C0 = 0xB40,GR_ad_Data
1126 add GR_ad_C0 = 0xB48,GR_ad_Data
1129 shladd GR_IndX8 = GR_Ind,3,r0
1131 shladd GR_IndX2 = GR_Ind,1,r0
1134 cmp.eq p6,p15 = 0,GR_Ind // p6 <- x from [1;1.25)
1136 cmp.ne p9,p0 = r0,r0
1139 shladd GR_ad_Co = GR_IndX8,3,GR_ad_Co
1140 (p6) fms.s1 FR_x = FR_NormX,f1,f1 // reduced x for [1;1.25)
1141 shladd GR_ad_C0 = GR_IndX2,4,GR_ad_C0
1144 shladd GR_ad_Ce = GR_IndX8,3,GR_ad_Ce
1145 (p14) fma.s.s0 f8 = f0,f0,f0
1146 (p15) cmp.eq.unc p7,p8 = 1,GR_Ind // p7 <- x from [1.25;1.5)
1148 .pred.rel "mutex",p7,p8
1150 ldfpd FR_A8,FR_A7 = [GR_ad_Co],16
1151 (p7) fms.s1 FR_x = FR_xm2,f1,FR_LocalMin
1155 ldfpd FR_A6,FR_A5 = [GR_ad_Ce],16
1156 (p8) fma.s1 FR_x = f0,f0,FR_NormX
1157 (p9) cmp.eq.unc p10,p11 = r0,r0
1161 ldfpd FR_A4,FR_A3 = [GR_ad_Co],16
1162 ldfpd FR_A2,FR_A1 = [GR_ad_Ce],16
1163 mov GR_SignOfGamma = 1 // set sign of gamma(x) to 1
1166 // add GR_ad_C0 = 8,GR_ad_C0
1167 ldfd FR_A0 = [GR_ad_C0]
1169 // set p13 if signgum is 32-bit int
1170 // set p15 if signgum is 64-bit int
1171 cmp.eq p15,p13 = 8,r34
1173 .pred.rel "mutex",p13,p15
1175 // store sign of gamma(x)
1176 (p13) st4 [r33] = GR_SignOfGamma // as 32-bit int
1177 (p15) st8 [r33] = GR_SignOfGamma // as 64-bit int
1178 (p11) fma.s1 FR_Q32 = FR_Q32,FR_w2,FR_Q10
1182 (p10) fma.s1 FR_P32 = FR_P32,FR_r2,FR_P10
1183 (p14) br.ret.spnt b0 // fast exit for 1.0
1187 (p10) fma.s1 FR_T = FR_N,FR_Ln2,FR_T
1188 cmp.eq p6,p7 = 0,GR_Ind // p6 <- x from [1;1.25)
1192 fma.s1 FR_x2 = FR_x,FR_x,f0
1193 cmp.eq p8,p0 = r0,r0 // set p8 to 1 that means we on [1;2]
1197 (p11) fma.s1 FR_Ln = FR_Q32,FR_w,f0
1205 .pred.rel "mutex",p6,p7
1208 (p6) fma.s1 FR_xx = f0,f0,FR_x
1213 (p7) fma.s1 FR_xx = f0,f0,f1
1218 fma.s1 FR_A7 = FR_A8,FR_x,FR_A7
1223 fma.s1 FR_A5 = FR_A6,FR_x,FR_A5
1224 (p9) cmp.ne p8,p0 = r0,r0 // set p8 to 0 that means we on [0;1]
1228 fma.s1 FR_A3 = FR_A4,FR_x,FR_A3
1233 fma.s1 FR_A1 = FR_A2,FR_x,FR_A1
1238 fma.s1 FR_x4 = FR_x2,FR_x2,f0
1243 (p10) fma.s1 FR_Ln = FR_P32,FR_r,FR_T
1248 fma.s1 FR_A5 = FR_A7,FR_x2,FR_A5
1253 fma.s1 FR_A1 = FR_A3,FR_x2,FR_A1
1256 .pred.rel "mutex",p9,p8
1259 (p9) fms.d.s1 FR_A0 = FR_A0,FR_xx,FR_Ln
1264 (p8) fms.s1 FR_A0 = FR_A0,FR_xx,f0
1269 fma.d.s1 FR_A1 = FR_A5,FR_x4,FR_A1
1277 .pred.rel "mutex",p6,p7
1280 (p6) fma.s.s0 f8 = FR_A1,FR_x2,FR_A0
1285 (p7) fma.s.s0 f8 = FR_A1,FR_x,FR_A0
1288 // branch for calculating of ln(GAMMA(x)) for -9 < x < 1
1289 //---------------------------------------------------------------------
1291 lgammaf_negrecursion:
1293 getf.sig GR_N = FR_int_Ntrunc
1294 fms.s1 FR_1pXf = FR_Xp2,f1,FR_Ntrunc // 1 + (x+1) - [x]
1298 add GR_ad_Co = 0xCE0,GR_ad_Data
1299 fms.s1 FR_Xf = FR_Xp1,f1,FR_Ntrunc // (x+1) - [x]
1303 add GR_ad_Ce = 0xD00,GR_ad_Data
1304 fma.s1 FR_Xp6 = FR_2,FR_2,FR_Xp2 // (x+6)
1305 add GR_ad_C0 = 0xB30,GR_ad_Data
1308 sub GR_Neg2 = r0,GR_Neg2
1309 fma.s1 FR_Xp7 = FR_2,FR_3,FR_Xp1 // (x+7)
1310 sub GR_Neg4 = r0,GR_Neg4
1313 cmp.ne p8,p0 = r0,GR_N
1314 fcmp.eq.s1 p13,p0 = FR_NormX,FR_Ntrunc
1315 and GR_IntNum = 0xF,GR_N
1318 cmp.lt p6,p0 = GR_N,GR_Neg2
1319 fma.s1 FR_Xp8 = FR_2,FR_3,FR_Xp2 // (x+8)
1320 cmp.lt p7,p0 = GR_N,GR_Neg4
1323 getf.d GR_Arg = FR_NormX
1324 (p6) fma.s1 FR_Xp2 = FR_Xp2,FR_Xp3,f0
1325 (p8) tbit.z.unc p14,p15 = GR_IntNum,0
1328 sub GR_RootInd = 0xE,GR_IntNum
1329 (p7) fma.s1 FR_Xp4 = FR_Xp4,FR_Xp5,f0
1330 add GR_ad_Root = 0xDE0,GR_ad_Data
1333 shladd GR_ad_Root = GR_RootInd,3,GR_ad_Root
1334 fms.s1 FR_x = FR_Xp1,f1,FR_Ntrunc // (x+1) - [x]
1340 (p13) br.cond.spnt lgammaf_singularity
1342 .pred.rel "mutex",p14,p15
1344 cmp.gt p6,p0 = 0xA,GR_IntNum
1345 (p14) fma.s1 FR_Req = FR_Req,FR_Xf,f0
1346 cmp.gt p7,p0 = 0xD,GR_IntNum
1349 (p15) mov GR_SignOfGamma = 1 // set sign of gamma(x) to 1
1350 (p15) fnma.s1 FR_Req = FR_Req,FR_Xf,f0
1351 cmp.leu p0,p13 = 2,GR_RootInd
1355 (p6) fma.s1 FR_Xp6 = FR_Xp6,FR_Xp7,f0
1356 (p13) add GR_ad_RootCo = 0xE00,GR_ad_Data
1360 fcmp.eq.s1 p12,p11 = FR_1pXf,FR_2
1364 getf.sig GR_Sig = FR_1pXf
1365 fcmp.le.s1 p9,p0 = FR_05,FR_Xf
1369 (p13) shladd GR_RootInd = GR_RootInd,4,r0
1370 (p7) fma.s1 FR_Xp2 = FR_Xp2,FR_Xp4,f0
1371 (p8) cmp.gt.unc p10,p0 = 0x9,GR_IntNum
1373 .pred.rel "mutex",p11,p12
1376 (p10) fma.s1 FR_Req = FR_Req,FR_Xp8,f0
1377 (p11) extr.u GR_Ind = GR_Sig,61,2
1380 (p13) add GR_RootInd = GR_RootInd,GR_RootInd
1382 (p12) mov GR_Ind = 3
1385 shladd GR_IndX2 = GR_Ind,1,r0
1387 cmp.gt p14,p0 = 2,GR_Ind
1390 shladd GR_IndX8 = GR_Ind,3,r0
1392 cmp.eq p6,p0 = 1,GR_Ind
1394 .pred.rel "mutex",p6,p9
1396 shladd GR_ad_Co = GR_IndX8,3,GR_ad_Co
1397 (p6) fms.s1 FR_x = FR_Xf,f1,FR_LocalMin
1398 cmp.gt p10,p0 = 0xB,GR_IntNum
1401 shladd GR_ad_Ce = GR_IndX8,3,GR_ad_Ce
1402 (p9) fma.s1 FR_x = f0,f0,FR_1pXf
1403 shladd GR_ad_C0 = GR_IndX2,4,GR_ad_C0
1406 // load coefficients of polynomial approximation
1407 // of ln(GAMMA(x)), 1 <= x < 2
1408 ldfpd FR_A8,FR_A7 = [GR_ad_Co],16
1409 (p10) fma.s1 FR_Xp2 = FR_Xp2,FR_Xp6,f0
1410 add GR_ad_C0 = 8,GR_ad_C0
1413 ldfpd FR_A6,FR_A5 = [GR_ad_Ce],16
1415 (p14) add GR_ad_Root = 0x10,GR_ad_Root
1418 ldfpd FR_A4,FR_A3 = [GR_ad_Co],16
1420 add GR_ad_RootCe = 0xE10,GR_ad_Data
1423 ldfpd FR_A2,FR_A1 = [GR_ad_Ce],16
1425 (p14) add GR_RootInd = 0x40,GR_RootInd
1428 ldfd FR_A0 = [GR_ad_C0]
1429 (p13) add GR_ad_RootCo = GR_ad_RootCo,GR_RootInd
1430 (p13) add GR_ad_RootCe = GR_ad_RootCe,GR_RootInd
1433 (p13) ld8 GR_Root = [GR_ad_Root]
1434 (p13) ldfd FR_Root = [GR_ad_Root]
1435 mov GR_ExpBias = 0xffff
1439 fma.s1 FR_x2 = FR_x,FR_x,f0
1443 (p8) cmp.gt.unc p10,p0 = 0xF,GR_IntNum
1444 movl GR_Dx = 0x000000014F8B588E
1447 // load coefficients of polynomial approximation
1448 // of ln(GAMMA(x)), x is close to one of negative roots
1449 (p13) ldfpd FR_R3,FR_R2 = [GR_ad_RootCo]
1450 // argumenth for logarithm
1451 (p10) fma.s1 FR_Req = FR_Req,FR_Xp2,f0
1452 mov GR_ExpMask = 0x1ffff
1455 (p13) ldfpd FR_R1,FR_R0 = [GR_ad_RootCe]
1457 // set p9 if signgum is 32-bit int
1458 // set p8 if signgum is 64-bit int
1459 cmp.eq p8,p9 = 8,r34
1461 .pred.rel "mutex",p9,p8
1463 (p9) st4 [r33] = GR_SignOfGamma // as 32-bit int
1464 fma.s1 FR_A7 = FR_A8,FR_x,FR_A7
1465 (p13) sub GR_Root = GR_Arg,GR_Root
1468 (p8) st8 [r33] = GR_SignOfGamma // as 64-bit int
1469 fma.s1 FR_A5 = FR_A6,FR_x,FR_A5
1474 fms.s1 FR_w = FR_Req,f1,f1
1475 (p13) add GR_Root = GR_Root,GR_Dx
1480 (p13) add GR_2xDx = GR_Dx,GR_Dx
1484 fma.s1 FR_A3 = FR_A4,FR_x,FR_A3
1489 fma.s1 FR_A1 = FR_A2,FR_x,FR_A1
1490 (p13) cmp.leu.unc p10,p0 = GR_Root,GR_2xDx
1494 frcpa.s1 FR_InvX,p0 = f1,FR_Req
1499 (p10) fms.s1 FR_rx = FR_NormX,f1,FR_Root
1503 getf.exp GR_SignExp = FR_Req
1504 fma.s1 FR_x4 = FR_x2,FR_x2,f0
1508 getf.sig GR_Sig = FR_Req
1509 fma.s1 FR_A5 = FR_A7,FR_x2,FR_A5
1513 sub GR_PureExp = GR_SignExp,GR_ExpBias
1514 fma.s1 FR_w2 = FR_w,FR_w,f0
1519 fma.s1 FR_Q32 = FR_P3,FR_w,FR_P2
1523 setf.sig FR_int_Ln = GR_PureExp
1524 fma.s1 FR_A1 = FR_A3,FR_x2,FR_A1
1525 extr.u GR_Ind4T = GR_Sig,55,8
1529 fma.s1 FR_Q10 = FR_P1,FR_w,f1
1533 shladd GR_ad_T = GR_Ind4T,3,GR_ad_Data
1534 fms.s1 FR_r = FR_InvX,FR_Req,f1
1539 (p10) fms.s1 FR_rx2 = FR_rx,FR_rx,f0
1543 ldfd FR_T = [GR_ad_T]
1544 (p10) fma.s1 FR_R2 = FR_R3,FR_rx,FR_R2
1549 (p10) fma.s1 FR_R0 = FR_R1,FR_rx,FR_R0
1553 getf.exp GR_Exp = FR_w
1554 fma.s1 FR_A1 = FR_A5,FR_x4,FR_A1
1555 mov GR_ExpMask = 0x1ffff
1559 fma.s1 FR_Q32 = FR_Q32, FR_w2,FR_Q10
1564 fma.s1 FR_r2 = FR_r,FR_r,f0
1565 mov GR_fff7 = 0xFFF7
1569 fma.s1 FR_P32 = FR_P3,FR_r,FR_P2
1574 fma.s1 FR_P10 = FR_P1,FR_r,f1
1575 and GR_Exp = GR_ExpMask,GR_Exp
1579 (p10) fma.s.s0 f8 = FR_R2,FR_rx2,FR_R0
1580 (p10) br.ret.spnt b0 // exit for arguments close to negative roots
1584 fcvt.xf FR_N = FR_int_Ln
1588 cmp.ge p14,p15 = GR_Exp,GR_fff7
1594 fma.s1 FR_A0 = FR_A1,FR_x,FR_A0
1599 (p15) fma.s1 FR_Ln = FR_Q32,FR_w,f0
1604 (p14) fma.s1 FR_P32 = FR_P32,FR_r2,FR_P10
1605 cmp.eq p6,p7 = 0,GR_Ind
1609 (p14) fma.s1 FR_T = FR_N,FR_Ln2,FR_T
1614 (p14) fma.s1 FR_Ln = FR_P32,FR_r,FR_T
1617 .pred.rel "mutex",p6,p7
1620 (p6) fms.s.s0 f8 = FR_A0,FR_x,FR_Ln
1625 (p7) fms.s.s0 f8 = FR_A0,f1,FR_Ln
1629 // branch for calculating of ln(GAMMA(x)) for x < -2^13
1630 //---------------------------------------------------------------------
1632 lgammaf_negstirling:
1634 shladd GR_ad_T = GR_Ind4T,3,GR_ad_Data
1635 fms.s1 FR_Xf = FR_NormX,f1,FR_N // xf = x - [x]
1636 mov GR_SingBound = 0x10016
1639 add GR_ad_Co = 0xCA0,GR_ad_Data
1640 fma.s1 FR_P32 = FR_P3,FR_r,FR_P2
1644 ldfd FR_T = [GR_ad_T]
1645 fcvt.xf FR_int_Ln = FR_int_Ln
1646 cmp.le p6,p0 = GR_SingBound,GR_Exp
1649 add GR_ad_Ce = 0x20,GR_ad_Co
1650 fma.s1 FR_r2 = FR_r,FR_r,f0
1651 (p6) br.cond.spnt lgammaf_singularity
1654 // load coefficients of polynomial approximation
1655 // of ln(sin(Pi*xf)/(Pi*xf)), |xf| <= 0.5
1656 ldfpd FR_S16,FR_S14 = [GR_ad_Co],16
1657 fma.s1 FR_P10 = FR_P1,FR_r,f1
1661 ldfpd FR_S12,FR_S10 = [GR_ad_Ce],16
1662 fms.s1 FR_xm05 = FR_NormX,f1,FR_05
1666 ldfpd FR_S8,FR_S6 = [GR_ad_Co],16
1667 ldfpd FR_S4,FR_S2 = [GR_ad_Ce],16
1671 getf.sig GR_N = FR_int_Ntrunc // signgam calculation
1672 fma.s1 FR_Xf2 = FR_Xf,FR_Xf,f0
1677 frcpa.s1 FR_InvXf,p0 = f1,FR_Xf
1681 getf.d GR_Arg = FR_Xf
1682 fcmp.eq.s1 p6,p0 = FR_NormX,FR_N
1683 mov GR_ExpBias = 0x3FF
1687 fma.s1 FR_T = FR_int_Ln,FR_Ln2,FR_T
1688 extr.u GR_Exp = GR_Arg,52,11
1692 fma.s1 FR_P32 = FR_P32,FR_r2,FR_P10
1696 sub GR_PureExp = GR_Exp,GR_ExpBias
1697 fma.s1 FR_S14 = FR_S16,FR_Xf2,FR_S14
1698 extr.u GR_Ind4T = GR_Arg,44,8
1701 mov GR_SignOfGamma = 1 // set signgam to -1
1702 fma.s1 FR_S10 = FR_S12,FR_Xf2,FR_S10
1703 (p6) br.cond.spnt lgammaf_singularity
1706 setf.sig FR_int_Ln = GR_PureExp
1707 fms.s1 FR_rf = FR_InvXf,FR_Xf,f1
1708 // set p14 if GR_N is even
1709 tbit.z p14,p0 = GR_N,0
1712 shladd GR_ad_T = GR_Ind4T,3,GR_ad_Data
1713 fma.s1 FR_Xf4 = FR_Xf2,FR_Xf2,f0
1717 (p14) sub GR_SignOfGamma = r0,GR_SignOfGamma // set signgam to -1
1718 fma.s1 FR_S6 = FR_S8,FR_Xf2,FR_S6
1722 // set p9 if signgum is 32-bit int
1723 // set p10 if signgum is 64-bit int
1724 cmp.eq p10,p9 = 8,r34
1725 fma.s1 FR_S2 = FR_S4,FR_Xf2,FR_S2
1729 ldfd FR_Tf = [GR_ad_T]
1730 fma.s1 FR_Ln = FR_P32,FR_r,FR_T
1735 fma.s1 FR_LnSqrt2Pi = FR_LnSqrt2Pi,f1,FR_NormX
1738 .pred.rel "mutex",p9,p10
1740 (p9) st4 [r33] = GR_SignOfGamma // as 32-bit int
1741 fma.s1 FR_rf2 = FR_rf,FR_rf,f0
1745 (p10) st8 [r33] = GR_SignOfGamma // as 64-bit int
1746 fma.s1 FR_S10 = FR_S14,FR_Xf4,FR_S10
1751 fma.s1 FR_P32f = FR_P3,FR_rf,FR_P2
1756 fma.s1 FR_Xf8 = FR_Xf4,FR_Xf4,f0
1761 fma.s1 FR_P10f = FR_P1,FR_rf,f1
1766 fma.s1 FR_S2 = FR_S6,FR_Xf4,FR_S2
1771 fms.s1 FR_Ln = FR_Ln,FR_xm05,FR_LnSqrt2Pi
1776 fcvt.xf FR_Nf = FR_int_Ln
1781 fma.s1 FR_S2 = FR_S10,FR_Xf8,FR_S2
1786 fma.s1 FR_Tf = FR_Nf,FR_Ln2,FR_Tf
1791 fma.s1 FR_P32f = FR_P32f,FR_rf2,FR_P10f // ??????
1796 fnma.s1 FR_Ln = FR_S2,FR_Xf2,FR_Ln
1801 fma.s1 FR_Lnf = FR_P32f,FR_rf,FR_Tf
1806 fms.s.s0 f8 = FR_Ln,f1,FR_Lnf
1809 // branch for calculating of ln(GAMMA(x)) for -2^13 < x < -9
1810 //---------------------------------------------------------------------
1814 getf.d GR_Arg = FR_Xf
1815 frcpa.s1 FR_InvXf,p0 = f1,FR_Xf
1816 mov GR_ExpBias = 0x3FF
1820 fma.s1 FR_Xf2 = FR_Xf,FR_Xf,f0
1824 getf.sig GR_N = FR_int_Ntrunc
1825 fcvt.xf FR_N = FR_int_Ln
1826 mov GR_SignOfGamma = 1
1830 fma.s1 FR_A9 = FR_A10,FR_x,FR_A9
1835 fma.s1 FR_P10 = FR_P1,FR_r,f1
1836 extr.u GR_Exp = GR_Arg,52,11
1840 fma.s1 FR_x4 = FR_x2,FR_x2,f0
1844 sub GR_PureExp = GR_Exp,GR_ExpBias
1845 fma.s1 FR_A7 = FR_A8,FR_x,FR_A7
1846 tbit.z p14,p0 = GR_N,0
1850 fma.s1 FR_A5 = FR_A6,FR_x,FR_A5
1854 setf.sig FR_int_Ln = GR_PureExp
1855 fma.s1 FR_A3 = FR_A4,FR_x,FR_A3
1860 fma.s1 FR_A1 = FR_A2,FR_x,FR_A1
1861 (p14) sub GR_SignOfGamma = r0,GR_SignOfGamma
1865 fms.s1 FR_rf = FR_InvXf,FR_Xf,f1
1870 fma.s1 FR_Xf4 = FR_Xf2,FR_Xf2,f0
1875 fma.s1 FR_S14 = FR_S16,FR_Xf2,FR_S14
1880 fma.s1 FR_S10 = FR_S12,FR_Xf2,FR_S10
1885 fma.s1 FR_T = FR_N,FR_Ln2,FR_T
1890 fma.s1 FR_P32 = FR_P32,FR_r2,FR_P10
1895 fma.s1 FR_S6 = FR_S8,FR_Xf2,FR_S6
1896 extr.u GR_Ind4T = GR_Arg,44,8
1900 fma.s1 FR_S2 = FR_S4,FR_Xf2,FR_S2
1905 fma.s1 FR_A7 = FR_A9,FR_x2,FR_A7
1909 shladd GR_ad_T = GR_Ind4T,3,GR_ad_Data
1910 fma.s1 FR_A3 = FR_A5,FR_x2,FR_A3
1915 fma.s1 FR_Xf8 = FR_Xf4,FR_Xf4,f0
1920 fma.s1 FR_rf2 = FR_rf,FR_rf,f0
1925 fma.s1 FR_P32f = FR_P3,FR_rf,FR_P2
1930 fma.s1 FR_P10f = FR_P1,FR_rf,f1
1934 ldfd FR_Tf = [GR_ad_T]
1935 fma.s1 FR_Ln = FR_P32,FR_r,FR_T
1940 fma.s1 FR_A0 = FR_A1,FR_x,FR_A0
1945 fma.s1 FR_S10 = FR_S14,FR_Xf4,FR_S10
1950 fma.s1 FR_S2 = FR_S6,FR_Xf4,FR_S2
1955 fcvt.xf FR_Nf = FR_int_Ln
1960 fma.s1 FR_A3 = FR_A7,FR_x4,FR_A3
1965 fcmp.eq.s1 p13,p0 = FR_NormX,FR_Ntrunc
1970 fnma.s1 FR_x3 = FR_x2,FR_x,f0 // -x^3
1975 fma.s1 FR_P32f = FR_P32f,FR_rf2,FR_P10f
1979 // set p9 if signgum is 32-bit int
1980 // set p10 if signgum is 64-bit int
1981 cmp.eq p10,p9 = 8,r34
1982 fma.s1 FR_S2 = FR_S10,FR_Xf8,FR_S2
1983 (p13) br.cond.spnt lgammaf_singularity
1985 .pred.rel "mutex",p9,p10
1987 (p9) st4 [r33] = GR_SignOfGamma // as 32-bit int
1988 (p10) st8 [r33] = GR_SignOfGamma // as 64-bit int
1989 fms.s1 FR_A0 = FR_A3,FR_x3,FR_A0 // -A3*x^3-A0
1993 fma.s1 FR_Tf = FR_Nf,FR_Ln2,FR_Tf
1998 fma.s1 FR_Ln = FR_S2,FR_Xf2,FR_Ln // S2*Xf^2+Ln
2003 fma.s1 FR_Lnf = FR_P32f,FR_rf,FR_Tf
2008 fms.s1 FR_Ln = FR_A0,f1,FR_Ln
2013 fms.s.s0 f8 = FR_Ln,f1,FR_Lnf
2016 // branch for handling +/-0, NaT, QNaN, +/-INF and denormalised numbers
2017 //---------------------------------------------------------------------
2021 getf.exp GR_SignExp = FR_NormX
2022 fclass.m p6,p0 = f8,0x21 // is arg +INF?
2023 mov GR_SignOfGamma = 1 // set signgam to 1
2026 getf.sig GR_Sig = FR_NormX
2027 fclass.m p7,p0 = f8,0xB // is x deno?
2028 // set p11 if signgum is 32-bit int
2029 // set p12 if signgum is 64-bit int
2030 cmp.eq p12,p11 = 8,r34
2032 .pred.rel "mutex",p11,p12
2034 // store sign of gamma(x) as 32-bit int
2035 (p11) st4 [r33] = GR_SignOfGamma
2036 fclass.m p8,p0 = f8,0x1C0 // is arg NaT or NaN?
2037 dep.z GR_Ind = GR_SignExp,3,4
2040 // store sign of gamma(x) as 64-bit int
2041 (p12) st8 [r33] = GR_SignOfGamma
2042 and GR_Exp = GR_ExpMask,GR_SignExp
2043 (p6) br.ret.spnt b0 // exit for +INF
2046 sub GR_PureExp = GR_Exp,GR_ExpBias
2047 fclass.m p9,p0 = f8,0x22 // is arg -INF?
2048 extr.u GR_Ind4T = GR_Sig,55,8
2052 (p7) fma.s0 FR_tmp = f1,f1,f8
2053 (p7) br.cond.sptk lgammaf_core
2057 (p8) fms.s.s0 f8 = f8,f1,f8
2058 (p8) br.ret.spnt b0 // exit for NaT and NaN
2062 (p9) fmerge.s f8 = f1,f8
2063 (p9) br.ret.spnt b0 // exit -INF
2065 // branch for handling negative integers and +/-0
2066 //---------------------------------------------------------------------
2068 lgammaf_singularity:
2070 mov GR_SignOfGamma = 1 // set signgam to 1
2071 fclass.m p6,p0 = f8,0x6 // is x -0?
2072 mov GR_TAG = 109 // negative
2075 mov GR_ad_SignGam = r33
2076 fma.s1 FR_X = f0,f0,f8
2081 frcpa.s0 f8,p0 = f1,f0
2082 // set p9 if signgum is 32-bit int
2083 // set p10 if signgum is 64-bit int
2084 cmp.eq p10,p9 = 8,r34
2088 (p6) sub GR_SignOfGamma = r0,GR_SignOfGamma
2089 br.cond.sptk lgammaf_libm_err
2091 // overflow (x > OVERFLOV_BOUNDARY)
2092 //---------------------------------------------------------------------
2102 fmerge.s FR_X = f8,f8
2103 mov GR_TAG = 108 // overflow
2106 mov GR_ad_SignGam = r33
2108 // set p9 if signgum is 32-bit int
2109 // set p10 if signgum is 64-bit int
2110 cmp.eq p10,p9 = 8,r34
2114 fma.s.s0 f8 = f9,f9,f0 // Set I,O and +INF result
2117 // gate to __libm_error_support#
2118 //---------------------------------------------------------------------
2122 alloc r32 = ar.pfs,1,4,4,0
2123 mov GR_Parameter_TAG = GR_TAG
2126 .pred.rel "mutex",p9,p10
2128 // store sign of gamma(x) as 32-bit int
2129 (p9) st4 [GR_ad_SignGam] = GR_SignOfGamma
2130 // store sign of gamma(x) as 64-bit int
2131 (p10) st8 [GR_ad_SignGam] = GR_SignOfGamma
2134 GLOBAL_LIBM_END(__libm_lgammaf)
2136 LOCAL_LIBM_ENTRY(__libm_error_region)
2139 add GR_Parameter_Y=-32,sp // Parameter 2 value
2141 .save ar.pfs,GR_SAVE_PFS
2142 mov GR_SAVE_PFS=ar.pfs // Save ar.pfs
2146 add sp=-64,sp // Create new stack
2148 mov GR_SAVE_GP=gp // Save gp
2151 stfs [GR_Parameter_Y] = FR_Y,16 // STORE Parameter 2 on stack
2152 add GR_Parameter_X = 16,sp // Parameter 1 address
2153 .save b0, GR_SAVE_B0
2154 mov GR_SAVE_B0=b0 // Save b0
2158 stfs [GR_Parameter_X] = FR_X // STORE Parameter 1
2160 add GR_Parameter_RESULT = 0,GR_Parameter_Y // Parameter 3 address
2164 stfs [GR_Parameter_Y] = FR_RESULT // STORE Parameter 3
2166 add GR_Parameter_Y = -16,GR_Parameter_Y
2167 br.call.sptk b0=__libm_error_support# // Call error handling
2173 add GR_Parameter_RESULT = 48,sp
2176 ldfs f8 = [GR_Parameter_RESULT] // Get return result off stack
2178 add sp = 64,sp // Restore stack pointer
2179 mov b0 = GR_SAVE_B0 // Restore return address
2182 mov gp = GR_SAVE_GP // Restore gp
2183 mov ar.pfs = GR_SAVE_PFS // Restore ar.pfs
2184 br.ret.sptk b0 // Return
2187 LOCAL_LIBM_END(__libm_error_region)
2188 .type __libm_error_support#,@function
2189 .global __libm_error_support#