4 // Copyright (c) 2000 - 2003, Intel Corporation
5 // All rights reserved.
8 // Redistribution and use in source and binary forms, with or without
9 // modification, are permitted provided that the following conditions are
12 // * Redistributions of source code must retain the above copyright
13 // notice, this list of conditions and the following disclaimer.
15 // * Redistributions in binary form must reproduce the above copyright
16 // notice, this list of conditions and the following disclaimer in the
17 // documentation and/or other materials provided with the distribution.
19 // * The name of Intel Corporation may not be used to endorse or promote
20 // products derived from this software without specific prior written
23 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
24 // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
25 // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
26 // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INTEL OR ITS
27 // CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
28 // EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
29 // PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
30 // PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY
31 // OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY OR TORT (INCLUDING
32 // NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
33 // SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
35 // Intel Corporation is the author of this code, and requests that all
36 // problem reports or change requests be submitted to it directly at
37 // http://www.intel.com/software/products/opensource/libraries/num.htm.
39 //*********************************************************************
42 // 02/02/00 Initial version
43 // 04/04/00 Unwind support added
44 // 08/15/00 Bundle added after call to __libm_error_support to properly
45 // set [the previously overwritten] GR_Parameter_RESULT.
46 // 05/21/01 Removed logl and log10l, putting them in a separate file
47 // 06/29/01 Improved speed of all paths
48 // 05/20/02 Cleaned up namespace and sf0 syntax
49 // 02/10/03 Reordered header: .section, .global, .proc, .align;
50 // used data8 for long double table values
52 //*********************************************************************
54 //*********************************************************************
56 // Function: log1pl(x) = ln(x+1), for double-extended precision x values
58 //*********************************************************************
62 // Floating-Point Registers: f8 (Input and Return Value)
65 // General Purpose Registers:
67 // r53-r56 (Used to pass arguments to error handling routine)
69 // Predicate Registers: p6-p13
71 //*********************************************************************
73 // IEEE Special Conditions:
75 // Denormal fault raised on denormal inputs
76 // Overflow exceptions cannot occur
77 // Underflow exceptions raised when appropriate for log1p
78 // Inexact raised when appropriate by algorithm
81 // log1pl(-inf) = QNaN
82 // log1pl(+/-0) = +/-0
84 // log1pl(SNaN) = QNaN
85 // log1pl(QNaN) = QNaN
86 // log1pl(EM_special Values) = QNaN
88 //*********************************************************************
92 // The method consists of three cases.
94 // If |X| < 2^(-80) use case log1p_small;
95 // else |X| < 2^(-7) use case log_near1;
96 // else use case log_regular;
100 // log1pl( X ) = logl( X+1 ) can be approximated by X
104 // log1pl( X ) = log( X+1 ) can be approximated by a simple polynomial
105 // in W = X. This polynomial resembles the truncated Taylor
106 // series W - W^/2 + W^3/3 - ...
110 // Here we use a table lookup method. The basic idea is that in
111 // order to compute logl(Arg) = log1pl (Arg-1) for an argument Arg in [1,2),
112 // we construct a value G such that G*Arg is close to 1 and that
113 // logl(1/G) is obtainable easily from a table of values calculated
116 // logl(Arg) = logl(1/G) + logl(G*Arg)
117 // = logl(1/G) + logl(1 + (G*Arg - 1))
119 // Because |G*Arg - 1| is small, the second term on the right hand
120 // side can be approximated by a short polynomial. We elaborate
121 // this method in four steps.
123 // Step 0: Initialization
125 // We need to calculate logl( X+1 ). Obtain N, S_hi such that
127 // X+1 = 2^N * ( S_hi + S_lo ) exactly
129 // where S_hi in [1,2) and S_lo is a correction to S_hi in the sense
130 // that |S_lo| <= ulp(S_hi).
132 // Step 1: Argument Reduction
134 // Based on S_hi, obtain G_1, G_2, G_3 from a table and calculate
136 // G := G_1 * G_2 * G_3
137 // r := (G * S_hi - 1) + G * S_lo
139 // These G_j's have the property that the product is exactly
140 // representable and that |r| < 2^(-12) as a result.
142 // Step 2: Approximation
145 // logl(1 + r) is approximated by a short polynomial poly(r).
147 // Step 3: Reconstruction
150 // Finally, log1pl( X ) = logl( X+1 ) is given by
152 // logl( X+1 ) = logl( 2^N * (S_hi + S_lo) )
153 // ~=~ N*logl(2) + logl(1/G) + logl(1 + r)
154 // ~=~ N*logl(2) + logl(1/G) + poly(r).
156 // **** Algorithm ****
160 // Although log1pl(X) is basically X, we would like to preserve the inexactness
161 // nature as well as consistent behavior under different rounding modes.
162 // We can do this by computing the result as
164 // log1pl(X) = X - X*X
169 // Here we compute a simple polynomial. To exploit parallelism, we split
170 // the polynomial into two portions.
176 // Y_hi := W + Wsq*(P_1 + W*(P_2 + W*(P_3 + W*P_4))
177 // Y_lo := W6*(P_5 + W*(P_6 + W*(P_7 + W*P_8)))
181 // We present the algorithm in four steps.
183 // Step 0. Initialization
184 // ----------------------
187 // N := unbaised exponent of Z
188 // S_hi := 2^(-N) * Z
189 // S_lo := 2^(-N) * { (max(X,1)-Z) + min(X,1) }
191 // Step 1. Argument Reduction
192 // --------------------------
196 // Z = 2^N * S_hi = 2^N * 1.d_1 d_2 d_3 ... d_63
198 // We obtain G_1, G_2, G_3 by the following steps.
201 // Define X_0 := 1.d_1 d_2 ... d_14. This is extracted
204 // Define A_1 := 1.d_1 d_2 d_3 d_4. This is X_0 truncated
207 // Define index_1 := [ d_1 d_2 d_3 d_4 ].
209 // Fetch Z_1 := (1/A_1) rounded UP in fixed point with
210 // fixed point lsb = 2^(-15).
211 // Z_1 looks like z_0.z_1 z_2 ... z_15
212 // Note that the fetching is done using index_1.
213 // A_1 is actually not needed in the implementation
214 // and is used here only to explain how is the value
217 // Fetch G_1 := (1/A_1) truncated to 21 sig. bits.
218 // floating pt. Again, fetching is done using index_1. A_1
219 // explains how G_1 is defined.
221 // Calculate X_1 := X_0 * Z_1 truncated to lsb = 2^(-14)
222 // = 1.0 0 0 0 d_5 ... d_14
223 // This is accomplished by integer multiplication.
224 // It is proved that X_1 indeed always begin
225 // with 1.0000 in fixed point.
228 // Define A_2 := 1.0 0 0 0 d_5 d_6 d_7 d_8. This is X_1
229 // truncated to lsb = 2^(-8). Similar to A_1,
230 // A_2 is not needed in actual implementation. It
231 // helps explain how some of the values are defined.
233 // Define index_2 := [ d_5 d_6 d_7 d_8 ].
235 // Fetch Z_2 := (1/A_2) rounded UP in fixed point with
236 // fixed point lsb = 2^(-15). Fetch done using index_2.
237 // Z_2 looks like z_0.z_1 z_2 ... z_15
239 // Fetch G_2 := (1/A_2) truncated to 21 sig. bits.
242 // Calculate X_2 := X_1 * Z_2 truncated to lsb = 2^(-14)
243 // = 1.0 0 0 0 0 0 0 0 d_9 d_10 ... d_14
244 // This is accomplished by integer multiplication.
245 // It is proved that X_2 indeed always begin
246 // with 1.00000000 in fixed point.
249 // Define A_3 := 1.0 0 0 0 0 0 0 0 d_9 d_10 d_11 d_12 d_13 1.
250 // This is 2^(-14) + X_2 truncated to lsb = 2^(-13).
252 // Define index_3 := [ d_9 d_10 d_11 d_12 d_13 ].
254 // Fetch G_3 := (1/A_3) truncated to 21 sig. bits.
255 // floating pt. Fetch is done using index_3.
257 // Compute G := G_1 * G_2 * G_3.
259 // This is done exactly since each of G_j only has 21 sig. bits.
263 // r := (G*S_hi - 1) + G*S_lo using 2 FMA operations.
265 // Thus r approximates G*(S_hi + S_lo) - 1 to within a couple of
269 // Step 2. Approximation
270 // ---------------------
272 // This step computes an approximation to logl( 1 + r ) where r is the
273 // reduced argument just obtained. It is proved that |r| <= 1.9*2^(-13);
274 // thus logl(1+r) can be approximated by a short polynomial:
276 // logl(1+r) ~=~ poly = r + Q1 r^2 + ... + Q4 r^5
279 // Step 3. Reconstruction
280 // ----------------------
282 // This step computes the desired result of logl(X+1):
284 // logl(X+1) = logl( 2^N * (S_hi + S_lo) )
285 // = N*logl(2) + logl( S_hi + S_lo) )
286 // = N*logl(2) + logl(1/G) +
287 // logl(1 + G * ( S_hi + S_lo ) - 1 )
289 // logl(2), logl(1/G_j) are stored as pairs of (single,double) numbers:
290 // log2_hi, log2_lo, log1byGj_hi, log1byGj_lo. The high parts are
291 // single-precision numbers and the low parts are double precision
292 // numbers. These have the property that
294 // N*log2_hi + SUM ( log1byGj_hi )
296 // is computable exactly in double-extended precision (64 sig. bits).
299 // Y_hi := N*log2_hi + SUM ( log1byGj_hi )
300 // Y_lo := poly_hi + [ poly_lo +
301 // ( SUM ( log1byGj_lo ) + N*log2_lo ) ]
307 // ************* DO NOT CHANGE THE ORDER OF THESE TABLES *************
309 // P_8, P_7, P_6, P_5, P_4, P_3, P_2, and P_1
311 LOCAL_OBJECT_START(Constants_P)
312 //data4 0xEFD62B15,0xE3936754,0x00003FFB,0x00000000
313 //data4 0xA5E56381,0x8003B271,0x0000BFFC,0x00000000
314 //data4 0x73282DB0,0x9249248C,0x00003FFC,0x00000000
315 //data4 0x47305052,0xAAAAAA9F,0x0000BFFC,0x00000000
316 //data4 0xCCD17FC9,0xCCCCCCCC,0x00003FFC,0x00000000
317 //data4 0x00067ED5,0x80000000,0x0000BFFD,0x00000000
318 //data4 0xAAAAAAAA,0xAAAAAAAA,0x00003FFD,0x00000000
319 //data4 0xFFFFFFFE,0xFFFFFFFF,0x0000BFFD,0x00000000
320 data8 0xE3936754EFD62B15,0x00003FFB
321 data8 0x8003B271A5E56381,0x0000BFFC
322 data8 0x9249248C73282DB0,0x00003FFC
323 data8 0xAAAAAA9F47305052,0x0000BFFC
324 data8 0xCCCCCCCCCCD17FC9,0x00003FFC
325 data8 0x8000000000067ED5,0x0000BFFD
326 data8 0xAAAAAAAAAAAAAAAA,0x00003FFD
327 data8 0xFFFFFFFFFFFFFFFE,0x0000BFFD
328 LOCAL_OBJECT_END(Constants_P)
330 // log2_hi, log2_lo, Q_4, Q_3, Q_2, and Q_1
332 LOCAL_OBJECT_START(Constants_Q)
333 //data4 0x00000000,0xB1721800,0x00003FFE,0x00000000
334 //data4 0x4361C4C6,0x82E30865,0x0000BFE2,0x00000000
335 //data4 0x328833CB,0xCCCCCAF2,0x00003FFC,0x00000000
336 //data4 0xA9D4BAFB,0x80000077,0x0000BFFD,0x00000000
337 //data4 0xAAABE3D2,0xAAAAAAAA,0x00003FFD,0x00000000
338 //data4 0xFFFFDAB7,0xFFFFFFFF,0x0000BFFD,0x00000000
339 data8 0xB172180000000000,0x00003FFE
340 data8 0x82E308654361C4C6,0x0000BFE2
341 data8 0xCCCCCAF2328833CB,0x00003FFC
342 data8 0x80000077A9D4BAFB,0x0000BFFD
343 data8 0xAAAAAAAAAAABE3D2,0x00003FFD
344 data8 0xFFFFFFFFFFFFDAB7,0x0000BFFD
345 LOCAL_OBJECT_END(Constants_Q)
347 // 1/ln10_hi, 1/ln10_lo
349 LOCAL_OBJECT_START(Constants_1_by_LN10)
350 //data4 0x37287195,0xDE5BD8A9,0x00003FFD,0x00000000
351 //data4 0xACCF70C8,0xD56EAABE,0x00003FBB,0x00000000
352 data8 0xDE5BD8A937287195,0x00003FFD
353 data8 0xD56EAABEACCF70C8,0x00003FBB
354 LOCAL_OBJECT_END(Constants_1_by_LN10)
359 LOCAL_OBJECT_START(Constants_Z_1)
376 LOCAL_OBJECT_END(Constants_Z_1)
378 // G1 and H1 - IEEE single and h1 - IEEE double
380 LOCAL_OBJECT_START(Constants_G_H_h1)
381 data4 0x3F800000,0x00000000
382 data8 0x0000000000000000
383 data4 0x3F70F0F0,0x3D785196
384 data8 0x3DA163A6617D741C
385 data4 0x3F638E38,0x3DF13843
386 data8 0x3E2C55E6CBD3D5BB
387 data4 0x3F579430,0x3E2FF9A0
388 data8 0xBE3EB0BFD86EA5E7
389 data4 0x3F4CCCC8,0x3E647FD6
390 data8 0x3E2E6A8C86B12760
391 data4 0x3F430C30,0x3E8B3AE7
392 data8 0x3E47574C5C0739BA
393 data4 0x3F3A2E88,0x3EA30C68
394 data8 0x3E20E30F13E8AF2F
395 data4 0x3F321640,0x3EB9CEC8
396 data8 0xBE42885BF2C630BD
397 data4 0x3F2AAAA8,0x3ECF9927
398 data8 0x3E497F3497E577C6
399 data4 0x3F23D708,0x3EE47FC5
400 data8 0x3E3E6A6EA6B0A5AB
401 data4 0x3F1D89D8,0x3EF8947D
402 data8 0xBDF43E3CD328D9BE
403 data4 0x3F17B420,0x3F05F3A1
404 data8 0x3E4094C30ADB090A
405 data4 0x3F124920,0x3F0F4303
406 data8 0xBE28FBB2FC1FE510
407 data4 0x3F0D3DC8,0x3F183EBF
408 data8 0x3E3A789510FDE3FA
409 data4 0x3F088888,0x3F20EC80
410 data8 0x3E508CE57CC8C98F
411 data4 0x3F042108,0x3F29516A
412 data8 0xBE534874A223106C
413 LOCAL_OBJECT_END(Constants_G_H_h1)
417 LOCAL_OBJECT_START(Constants_Z_2)
434 LOCAL_OBJECT_END(Constants_Z_2)
436 // G2 and H2 - IEEE single and h2 - IEEE double
438 LOCAL_OBJECT_START(Constants_G_H_h2)
439 data4 0x3F800000,0x00000000
440 data8 0x0000000000000000
441 data4 0x3F7F00F8,0x3B7F875D
442 data8 0x3DB5A11622C42273
443 data4 0x3F7E03F8,0x3BFF015B
444 data8 0x3DE620CF21F86ED3
445 data4 0x3F7D08E0,0x3C3EE393
446 data8 0xBDAFA07E484F34ED
447 data4 0x3F7C0FC0,0x3C7E0586
448 data8 0xBDFE07F03860BCF6
449 data4 0x3F7B1880,0x3C9E75D2
450 data8 0x3DEA370FA78093D6
451 data4 0x3F7A2328,0x3CBDC97A
452 data8 0x3DFF579172A753D0
453 data4 0x3F792FB0,0x3CDCFE47
454 data8 0x3DFEBE6CA7EF896B
455 data4 0x3F783E08,0x3CFC15D0
456 data8 0x3E0CF156409ECB43
457 data4 0x3F774E38,0x3D0D874D
458 data8 0xBE0B6F97FFEF71DF
459 data4 0x3F766038,0x3D1CF49B
460 data8 0xBE0804835D59EEE8
461 data4 0x3F757400,0x3D2C531D
462 data8 0x3E1F91E9A9192A74
463 data4 0x3F748988,0x3D3BA322
464 data8 0xBE139A06BF72A8CD
465 data4 0x3F73A0D0,0x3D4AE46F
466 data8 0x3E1D9202F8FBA6CF
467 data4 0x3F72B9D0,0x3D5A1756
468 data8 0xBE1DCCC4BA796223
469 data4 0x3F71D488,0x3D693B9D
470 data8 0xBE049391B6B7C239
471 LOCAL_OBJECT_END(Constants_G_H_h2)
473 // G3 and H3 - IEEE single and h3 - IEEE double
475 LOCAL_OBJECT_START(Constants_G_H_h3)
476 data4 0x3F7FFC00,0x38800100
477 data8 0x3D355595562224CD
478 data4 0x3F7FF400,0x39400480
479 data8 0x3D8200A206136FF6
480 data4 0x3F7FEC00,0x39A00640
481 data8 0x3DA4D68DE8DE9AF0
482 data4 0x3F7FE400,0x39E00C41
483 data8 0xBD8B4291B10238DC
484 data4 0x3F7FDC00,0x3A100A21
485 data8 0xBD89CCB83B1952CA
486 data4 0x3F7FD400,0x3A300F22
487 data8 0xBDB107071DC46826
488 data4 0x3F7FCC08,0x3A4FF51C
489 data8 0x3DB6FCB9F43307DB
490 data4 0x3F7FC408,0x3A6FFC1D
491 data8 0xBD9B7C4762DC7872
492 data4 0x3F7FBC10,0x3A87F20B
493 data8 0xBDC3725E3F89154A
494 data4 0x3F7FB410,0x3A97F68B
495 data8 0xBD93519D62B9D392
496 data4 0x3F7FAC18,0x3AA7EB86
497 data8 0x3DC184410F21BD9D
498 data4 0x3F7FA420,0x3AB7E101
499 data8 0xBDA64B952245E0A6
500 data4 0x3F7F9C20,0x3AC7E701
501 data8 0x3DB4B0ECAABB34B8
502 data4 0x3F7F9428,0x3AD7DD7B
503 data8 0x3D9923376DC40A7E
504 data4 0x3F7F8C30,0x3AE7D474
505 data8 0x3DC6E17B4F2083D3
506 data4 0x3F7F8438,0x3AF7CBED
507 data8 0x3DAE314B811D4394
508 data4 0x3F7F7C40,0x3B03E1F3
509 data8 0xBDD46F21B08F2DB1
510 data4 0x3F7F7448,0x3B0BDE2F
511 data8 0xBDDC30A46D34522B
512 data4 0x3F7F6C50,0x3B13DAAA
513 data8 0x3DCB0070B1F473DB
514 data4 0x3F7F6458,0x3B1BD766
515 data8 0xBDD65DDC6AD282FD
516 data4 0x3F7F5C68,0x3B23CC5C
517 data8 0xBDCDAB83F153761A
518 data4 0x3F7F5470,0x3B2BC997
519 data8 0xBDDADA40341D0F8F
520 data4 0x3F7F4C78,0x3B33C711
521 data8 0x3DCD1BD7EBC394E8
522 data4 0x3F7F4488,0x3B3BBCC6
523 data8 0xBDC3532B52E3E695
524 data4 0x3F7F3C90,0x3B43BAC0
525 data8 0xBDA3961EE846B3DE
526 data4 0x3F7F34A0,0x3B4BB0F4
527 data8 0xBDDADF06785778D4
528 data4 0x3F7F2CA8,0x3B53AF6D
529 data8 0x3DCC3ED1E55CE212
530 data4 0x3F7F24B8,0x3B5BA620
531 data8 0xBDBA31039E382C15
532 data4 0x3F7F1CC8,0x3B639D12
533 data8 0x3D635A0B5C5AF197
534 data4 0x3F7F14D8,0x3B6B9444
535 data8 0xBDDCCB1971D34EFC
536 data4 0x3F7F0CE0,0x3B7393BC
537 data8 0x3DC7450252CD7ADA
538 data4 0x3F7F04F0,0x3B7B8B6D
539 data8 0xBDB68F177D7F2A42
540 LOCAL_OBJECT_END(Constants_G_H_h3)
543 // Floating Point Registers
602 FR_Output_X_tmp = f76
608 FR_2_to_minus_N = f82
615 // General Purpose Registers
646 // Added for unwind support
654 GR_Parameter_RESULT = r55
655 GR_Parameter_TAG = r56
658 GLOBAL_IEEE754_ENTRY(log1pl)
660 alloc r32 = ar.pfs,0,21,4,0
661 fclass.m p6, p0 = FR_Input_X, 0x1E3 // Test for natval, nan, inf
665 addl GR_ad_z_1 = @ltoff(Constants_Z_1#),gp
666 fma.s1 FR_Z = FR_Input_X, f1, f1 // x+1
673 fmerge.ns FR_Neg_One = f1, f1 // Form -1.0
678 fnorm.s1 FR_X_Prime = FR_Input_X // Normalize x
684 ld8 GR_ad_z_1 = [GR_ad_z_1] // Get pointer to Constants_Z_1
686 mov GR_exp_2tom7 = 0x0fff8 // Exponent of 2^-7
691 getf.sig GR_signif = FR_Z // Get significand of x+1
692 fcmp.eq.s1 p9, p0 = FR_Input_X, f0 // Test for x=0
693 (p6) br.cond.spnt LOG1P_special // Branch for nan, inf, natval
698 add GR_ad_tbl_1 = 0x040, GR_ad_z_1 // Point to Constants_G_H_h1
699 fcmp.lt.s1 p13, p0 = FR_X_Prime, FR_Neg_One // Test for x<-1
700 add GR_ad_p = -0x100, GR_ad_z_1 // Point to Constants_P
703 add GR_ad_z_2 = 0x140, GR_ad_z_1 // Point to Constants_Z_2
705 add GR_ad_tbl_2 = 0x180, GR_ad_z_1 // Point to Constants_G_H_h2
710 add GR_ad_q = 0x080, GR_ad_p // Point to Constants_Q
711 fcmp.eq.s1 p8, p0 = FR_X_Prime, FR_Neg_One // Test for x=-1
712 extr.u GR_Index1 = GR_signif, 59, 4 // Get high 4 bits of signif
715 add GR_ad_tbl_3 = 0x280, GR_ad_z_1 // Point to Constants_G_H_h3
717 (p9) br.ret.spnt b0 // Exit if x=0, return input
722 shladd GR_ad_z_1 = GR_Index1, 2, GR_ad_z_1 // Point to Z_1
723 fclass.nm p10, p0 = FR_Input_X, 0x1FF // Test for unsupported
724 extr.u GR_X_0 = GR_signif, 49, 15 // Get high 15 bits of significand
727 ldfe FR_P8 = [GR_ad_p],16 // Load P_8 for near1 path
728 fsub.s1 FR_W = FR_X_Prime, f0 // W = x
729 add GR_ad_ln10 = 0x060, GR_ad_q // Point to Constants_1_by_LN10
734 ld4 GR_Z_1 = [GR_ad_z_1] // Load Z_1
735 fmax.s1 FR_AA = FR_X_Prime, f1 // For S_lo, form AA = max(X,1.0)
736 mov GR_exp_mask = 0x1FFFF // Create exponent mask
739 shladd GR_ad_tbl_1 = GR_Index1, 4, GR_ad_tbl_1 // Point to G_1
740 mov GR_Bias = 0x0FFFF // Create exponent bias
741 (p13) br.cond.spnt LOG1P_LT_Minus_1 // Branch if x<-1
746 ldfps FR_G, FR_H = [GR_ad_tbl_1],8 // Load G_1, H_1
747 fmerge.se FR_S_hi = f1,FR_Z // Form |x+1|
748 (p8) br.cond.spnt LOG1P_EQ_Minus_1 // Branch if x=-1
753 getf.exp GR_N = FR_Z // Get N = exponent of x+1
754 ldfd FR_h = [GR_ad_tbl_1] // Load h_1
755 (p10) br.cond.spnt LOG1P_unsupported // Branch for unsupported type
760 ldfe FR_log2_hi = [GR_ad_q],16 // Load log2_hi
761 fcmp.eq.s0 p8, p0 = FR_Input_X, f0 // Dummy op to flag denormals
762 pmpyshr2.u GR_X_1 = GR_X_0,GR_Z_1,15 // Get bits 30-15 of X_0 * Z_1
767 // For performance, don't use result of pmpyshr2.u for 4 cycles.
770 ldfe FR_log2_lo = [GR_ad_q],16 // Load log2_lo
771 sub GR_N = GR_N, GR_Bias
772 mov GR_exp_2tom80 = 0x0ffaf // Exponent of 2^-80
777 ldfe FR_Q4 = [GR_ad_q],16 // Load Q4
778 fms.s1 FR_S_lo = FR_AA, f1, FR_Z // Form S_lo = AA - Z
779 sub GR_minus_N = GR_Bias, GR_N // Form exponent of 2^(-N)
784 ldfe FR_Q3 = [GR_ad_q],16 // Load Q3
785 setf.sig FR_float_N = GR_N // Put integer N into rightmost significand
786 fmin.s1 FR_BB = FR_X_Prime, f1 // For S_lo, form BB = min(X,1.0)
791 getf.exp GR_M = FR_W // Get signexp of w = x
792 ldfe FR_Q2 = [GR_ad_q],16 // Load Q2
793 extr.u GR_Index2 = GR_X_1, 6, 4 // Extract bits 6-9 of X_1
798 ldfe FR_Q1 = [GR_ad_q] // Load Q1
799 shladd GR_ad_z_2 = GR_Index2, 2, GR_ad_z_2 // Point to Z_2
800 add GR_ad_p2 = 0x30,GR_ad_p // Point to P_4
805 ld4 GR_Z_2 = [GR_ad_z_2] // Load Z_2
806 shladd GR_ad_tbl_2 = GR_Index2, 4, GR_ad_tbl_2 // Point to G_2
807 and GR_M = GR_exp_mask, GR_M // Get exponent of w = x
812 ldfps FR_G2, FR_H2 = [GR_ad_tbl_2],8 // Load G_2, H_2
813 cmp.lt p8, p9 = GR_M, GR_exp_2tom7 // Test |x| < 2^-7
814 cmp.lt p7, p0 = GR_M, GR_exp_2tom80 // Test |x| < 2^-80
818 // Small path is separate code
819 // p7 is for the small path: |x| < 2^-80
820 // near1 and regular paths are merged.
821 // p8 is for the near1 path: |x| < 2^-7
822 // p9 is for regular path: |x| >= 2^-7
825 ldfd FR_h2 = [GR_ad_tbl_2] // Load h_2
830 (p9) setf.exp FR_2_to_minus_N = GR_minus_N // Form 2^(-N)
831 (p7) fnma.s0 f8 = FR_X_Prime, FR_X_Prime, FR_X_Prime // Result x - x*x
832 (p7) br.ret.spnt b0 // Branch if |x| < 2^-80
837 (p8) ldfe FR_P7 = [GR_ad_p],16 // Load P_7 for near1 path
838 (p8) ldfe FR_P4 = [GR_ad_p2],16 // Load P_4 for near1 path
839 (p9) pmpyshr2.u GR_X_2 = GR_X_1,GR_Z_2,15 // Get bits 30-15 of X_1 * Z_2
844 // For performance, don't use result of pmpyshr2.u for 4 cycles.
847 (p8) ldfe FR_P6 = [GR_ad_p],16 // Load P_6 for near1 path
848 (p8) ldfe FR_P3 = [GR_ad_p2],16 // Load P_3 for near1 path
849 (p9) fma.s1 FR_S_lo = FR_S_lo, f1, FR_BB // S_lo = S_lo + BB
854 (p8) ldfe FR_P5 = [GR_ad_p],16 // Load P_5 for near1 path
855 (p8) ldfe FR_P2 = [GR_ad_p2],16 // Load P_2 for near1 path
856 (p8) fmpy.s1 FR_wsq = FR_W, FR_W // wsq = w * w for near1 path
861 (p8) ldfe FR_P1 = [GR_ad_p2],16 ;; // Load P_1 for near1 path
863 (p9) extr.u GR_Index3 = GR_X_2, 1, 5 // Extract bits 1-5 of X_2
868 (p9) shladd GR_ad_tbl_3 = GR_Index3, 4, GR_ad_tbl_3 // Point to G_3
869 (p9) fcvt.xf FR_float_N = FR_float_N
875 (p9) ldfps FR_G3, FR_H3 = [GR_ad_tbl_3],8 // Load G_3, H_3
882 (p9) ldfd FR_h3 = [GR_ad_tbl_3] // Load h_3
883 (p9) fmpy.s1 FR_G = FR_G, FR_G2 // G = G_1 * G_2
888 (p9) fadd.s1 FR_H = FR_H, FR_H2 // H = H_1 + H_2
896 (p9) fadd.s1 FR_h = FR_h, FR_h2 // h = h_1 + h_2
902 (p8) fmpy.s1 FR_w4 = FR_wsq, FR_wsq // w4 = w^4 for near1 path
907 (p8) fma.s1 FR_p87 = FR_W, FR_P8, FR_P7 // p87 = w * P8 + P7
914 (p9) fma.s1 FR_S_lo = FR_S_lo, FR_2_to_minus_N, f0 // S_lo = S_lo * 2^(-N)
919 (p8) fma.s1 FR_p43 = FR_W, FR_P4, FR_P3 // p43 = w * P4 + P3
926 (p9) fmpy.s1 FR_G = FR_G, FR_G3 // G = (G_1 * G_2) * G_3
931 (p9) fadd.s1 FR_H = FR_H, FR_H3 // H = (H_1 + H_2) + H_3
938 (p9) fadd.s1 FR_h = FR_h, FR_h3 // h = (h_1 + h_2) + h_3
943 (p8) fmpy.s1 FR_w6 = FR_w4, FR_wsq // w6 = w^6 for near1 path
950 (p8) fma.s1 FR_p432 = FR_W, FR_p43, FR_P2 // p432 = w * p43 + P2
955 (p8) fma.s1 FR_p876 = FR_W, FR_p87, FR_P6 // p876 = w * p87 + P6
962 (p9) fms.s1 FR_r = FR_G, FR_S_hi, f1 // r = G * S_hi - 1
967 (p9) fma.s1 FR_Y_hi = FR_float_N, FR_log2_hi, FR_H // Y_hi = N * log2_hi + H
974 (p9) fma.s1 FR_h = FR_float_N, FR_log2_lo, FR_h // h = N * log2_lo + h
981 (p9) fma.s1 FR_r = FR_G, FR_S_lo, FR_r // r = G * S_lo + (G * S_hi - 1)
988 (p8) fma.s1 FR_p4321 = FR_W, FR_p432, FR_P1 // p4321 = w * p432 + P1
993 (p8) fma.s1 FR_p8765 = FR_W, FR_p876, FR_P5 // p8765 = w * p876 + P5
1000 (p9) fma.s1 FR_poly_lo = FR_r, FR_Q4, FR_Q3 // poly_lo = r * Q4 + Q3
1005 (p9) fmpy.s1 FR_rsq = FR_r, FR_r // rsq = r * r
1012 (p8) fma.s1 FR_Y_lo = FR_wsq, FR_p4321, f0 // Y_lo = wsq * p4321
1017 (p8) fma.s1 FR_Y_hi = FR_W, f1, f0 // Y_hi = w for near1 path
1024 (p9) fma.s1 FR_poly_lo = FR_poly_lo, FR_r, FR_Q2 // poly_lo = poly_lo * r + Q2
1029 (p9) fma.s1 FR_rcub = FR_rsq, FR_r, f0 // rcub = r^3
1036 (p8) fma.s1 FR_Y_lo = FR_w6, FR_p8765,FR_Y_lo // Y_lo = w6 * p8765 + w2 * p4321
1043 (p9) fma.s1 FR_poly_hi = FR_Q1, FR_rsq, FR_r // poly_hi = Q1 * rsq + r
1050 (p9) fma.s1 FR_poly_lo = FR_poly_lo, FR_rcub, FR_h // poly_lo = poly_lo*r^3 + h
1057 (p9) fadd.s1 FR_Y_lo = FR_poly_hi, FR_poly_lo // Y_lo = poly_hi + poly_lo
1062 // Remainder of code is common for near1 and regular paths
1065 fadd.s0 f8 = FR_Y_lo,FR_Y_hi // Result=Y_lo+Y_hi
1066 br.ret.sptk b0 // Common exit for 2^-80 < x < inf
1074 // If x=-1 raise divide by zero and return -inf
1077 mov GR_Parameter_TAG = 138
1078 fsub.s1 FR_Output_X_tmp = f0, f1
1085 frcpa.s0 FR_Output_X_tmp, p8 = FR_Output_X_tmp, f0
1086 br.cond.sptk __libm_error_region
1093 fclass.m.unc p8, p0 = FR_Input_X, 0x1E1 // Test for natval, nan, +inf
1099 // For SNaN raise invalid and return QNaN.
1100 // For QNaN raise invalid and return QNaN.
1101 // For +Inf return +Inf.
1105 (p8) fmpy.s0 f8 = FR_Input_X, f1
1106 (p8) br.ret.sptk b0 // Return for natval, nan, +inf
1111 // For -Inf raise invalid and return QNaN.
1114 mov GR_Parameter_TAG = 139
1115 fmpy.s0 FR_Output_X_tmp = FR_Input_X, f0
1116 br.cond.sptk __libm_error_region
1123 // Return generated NaN or other value.
1127 fmpy.s0 f8 = FR_Input_X, f0
1132 // Here if -inf < x < -1
1135 // Deal with x < -1 in a special way - raise
1136 // invalid and produce QNaN indefinite.
1139 mov GR_Parameter_TAG = 139
1140 frcpa.s0 FR_Output_X_tmp, p8 = f0, f0
1141 br.cond.sptk __libm_error_region
1146 GLOBAL_IEEE754_END(log1pl)
1147 libm_alias_ldouble_other (__log1p, log1p)
1149 LOCAL_LIBM_ENTRY(__libm_error_region)
1152 add GR_Parameter_Y=-32,sp // Parameter 2 value
1154 .save ar.pfs,GR_SAVE_PFS
1155 mov GR_SAVE_PFS=ar.pfs // Save ar.pfs
1159 add sp=-64,sp // Create new stack
1161 mov GR_SAVE_GP=gp // Save gp
1164 stfe [GR_Parameter_Y] = FR_Y,16 // Save Parameter 2 on stack
1165 add GR_Parameter_X = 16,sp // Parameter 1 address
1166 .save b0, GR_SAVE_B0
1167 mov GR_SAVE_B0=b0 // Save b0
1171 stfe [GR_Parameter_X] = FR_X // Store Parameter 1 on stack
1172 add GR_Parameter_RESULT = 0,GR_Parameter_Y
1173 nop.b 0 // Parameter 3 address
1176 stfe [GR_Parameter_Y] = FR_RESULT // Store Parameter 3 on stack
1177 add GR_Parameter_Y = -16,GR_Parameter_Y
1178 br.call.sptk b0=__libm_error_support# // Call error handling function
1183 add GR_Parameter_RESULT = 48,sp
1186 ldfe f8 = [GR_Parameter_RESULT] // Get return result off stack
1188 add sp = 64,sp // Restore stack pointer
1189 mov b0 = GR_SAVE_B0 // Restore return address
1192 mov gp = GR_SAVE_GP // Restore gp
1193 mov ar.pfs = GR_SAVE_PFS // Restore ar.pfs
1194 br.ret.sptk b0 // Return
1197 LOCAL_LIBM_END(__libm_error_region#)
1199 .type __libm_error_support#,@function
1200 .global __libm_error_support#
projects / thirdparty / glibc.git / blob