4 // Copyright (c) 2000 - 2003, Intel Corporation
5 // All rights reserved.
7 // Contributed 2000 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 // 02/02/00 Initial version
44 // 04/04/00 Unwind support added
45 // 08/15/00 Bundle added after call to __libm_error_support to properly
46 // set [the previously overwritten] GR_Parameter_RESULT.
47 // 05/21/01 Removed logl and log10l, putting them in a separate file
48 // 06/29/01 Improved speed of all paths
49 // 05/20/02 Cleaned up namespace and sf0 syntax
50 // 02/10/03 Reordered header: .section, .global, .proc, .align;
51 // used data8 for long double table values
53 //*********************************************************************
55 //*********************************************************************
57 // Function: log1pl(x) = ln(x+1), for double-extended precision x values
59 //*********************************************************************
63 // Floating-Point Registers: f8 (Input and Return Value)
66 // General Purpose Registers:
68 // r53-r56 (Used to pass arguments to error handling routine)
70 // Predicate Registers: p6-p13
72 //*********************************************************************
74 // IEEE Special Conditions:
76 // Denormal fault raised on denormal inputs
77 // Overflow exceptions cannot occur
78 // Underflow exceptions raised when appropriate for log1p
79 // Inexact raised when appropriate by algorithm
82 // log1pl(-inf) = QNaN
83 // log1pl(+/-0) = +/-0
85 // log1pl(SNaN) = QNaN
86 // log1pl(QNaN) = QNaN
87 // log1pl(EM_special Values) = QNaN
89 //*********************************************************************
93 // The method consists of three cases.
95 // If |X| < 2^(-80) use case log1p_small;
96 // else |X| < 2^(-7) use case log_near1;
97 // else use case log_regular;
101 // log1pl( X ) = logl( X+1 ) can be approximated by X
105 // log1pl( X ) = log( X+1 ) can be approximated by a simple polynomial
106 // in W = X. This polynomial resembles the truncated Taylor
107 // series W - W^/2 + W^3/3 - ...
111 // Here we use a table lookup method. The basic idea is that in
112 // order to compute logl(Arg) = log1pl (Arg-1) for an argument Arg in [1,2),
113 // we construct a value G such that G*Arg is close to 1 and that
114 // logl(1/G) is obtainable easily from a table of values calculated
117 // logl(Arg) = logl(1/G) + logl(G*Arg)
118 // = logl(1/G) + logl(1 + (G*Arg - 1))
120 // Because |G*Arg - 1| is small, the second term on the right hand
121 // side can be approximated by a short polynomial. We elaborate
122 // this method in four steps.
124 // Step 0: Initialization
126 // We need to calculate logl( X+1 ). Obtain N, S_hi such that
128 // X+1 = 2^N * ( S_hi + S_lo ) exactly
130 // where S_hi in [1,2) and S_lo is a correction to S_hi in the sense
131 // that |S_lo| <= ulp(S_hi).
133 // Step 1: Argument Reduction
135 // Based on S_hi, obtain G_1, G_2, G_3 from a table and calculate
137 // G := G_1 * G_2 * G_3
138 // r := (G * S_hi - 1) + G * S_lo
140 // These G_j's have the property that the product is exactly
141 // representable and that |r| < 2^(-12) as a result.
143 // Step 2: Approximation
146 // logl(1 + r) is approximated by a short polynomial poly(r).
148 // Step 3: Reconstruction
151 // Finally, log1pl( X ) = logl( X+1 ) is given by
153 // logl( X+1 ) = logl( 2^N * (S_hi + S_lo) )
154 // ~=~ N*logl(2) + logl(1/G) + logl(1 + r)
155 // ~=~ N*logl(2) + logl(1/G) + poly(r).
157 // **** Algorithm ****
161 // Although log1pl(X) is basically X, we would like to preserve the inexactness
162 // nature as well as consistent behavior under different rounding modes.
163 // We can do this by computing the result as
165 // log1pl(X) = X - X*X
170 // Here we compute a simple polynomial. To exploit parallelism, we split
171 // the polynomial into two portions.
177 // Y_hi := W + Wsq*(P_1 + W*(P_2 + W*(P_3 + W*P_4))
178 // Y_lo := W6*(P_5 + W*(P_6 + W*(P_7 + W*P_8)))
182 // We present the algorithm in four steps.
184 // Step 0. Initialization
185 // ----------------------
188 // N := unbaised exponent of Z
189 // S_hi := 2^(-N) * Z
190 // S_lo := 2^(-N) * { (max(X,1)-Z) + min(X,1) }
192 // Step 1. Argument Reduction
193 // --------------------------
197 // Z = 2^N * S_hi = 2^N * 1.d_1 d_2 d_3 ... d_63
199 // We obtain G_1, G_2, G_3 by the following steps.
202 // Define X_0 := 1.d_1 d_2 ... d_14. This is extracted
205 // Define A_1 := 1.d_1 d_2 d_3 d_4. This is X_0 truncated
208 // Define index_1 := [ d_1 d_2 d_3 d_4 ].
210 // Fetch Z_1 := (1/A_1) rounded UP in fixed point with
211 // fixed point lsb = 2^(-15).
212 // Z_1 looks like z_0.z_1 z_2 ... z_15
213 // Note that the fetching is done using index_1.
214 // A_1 is actually not needed in the implementation
215 // and is used here only to explain how is the value
218 // Fetch G_1 := (1/A_1) truncated to 21 sig. bits.
219 // floating pt. Again, fetching is done using index_1. A_1
220 // explains how G_1 is defined.
222 // Calculate X_1 := X_0 * Z_1 truncated to lsb = 2^(-14)
223 // = 1.0 0 0 0 d_5 ... d_14
224 // This is accomplised by integer multiplication.
225 // It is proved that X_1 indeed always begin
226 // with 1.0000 in fixed point.
229 // Define A_2 := 1.0 0 0 0 d_5 d_6 d_7 d_8. This is X_1
230 // truncated to lsb = 2^(-8). Similar to A_1,
231 // A_2 is not needed in actual implementation. It
232 // helps explain how some of the values are defined.
234 // Define index_2 := [ d_5 d_6 d_7 d_8 ].
236 // Fetch Z_2 := (1/A_2) rounded UP in fixed point with
237 // fixed point lsb = 2^(-15). Fetch done using index_2.
238 // Z_2 looks like z_0.z_1 z_2 ... z_15
240 // Fetch G_2 := (1/A_2) truncated to 21 sig. bits.
243 // Calculate X_2 := X_1 * Z_2 truncated to lsb = 2^(-14)
244 // = 1.0 0 0 0 0 0 0 0 d_9 d_10 ... d_14
245 // This is accomplised by integer multiplication.
246 // It is proved that X_2 indeed always begin
247 // with 1.00000000 in fixed point.
250 // Define A_3 := 1.0 0 0 0 0 0 0 0 d_9 d_10 d_11 d_12 d_13 1.
251 // This is 2^(-14) + X_2 truncated to lsb = 2^(-13).
253 // Define index_3 := [ d_9 d_10 d_11 d_12 d_13 ].
255 // Fetch G_3 := (1/A_3) truncated to 21 sig. bits.
256 // floating pt. Fetch is done using index_3.
258 // Compute G := G_1 * G_2 * G_3.
260 // This is done exactly since each of G_j only has 21 sig. bits.
264 // r := (G*S_hi - 1) + G*S_lo using 2 FMA operations.
266 // Thus r approximates G*(S_hi + S_lo) - 1 to within a couple of
270 // Step 2. Approximation
271 // ---------------------
273 // This step computes an approximation to logl( 1 + r ) where r is the
274 // reduced argument just obtained. It is proved that |r| <= 1.9*2^(-13);
275 // thus logl(1+r) can be approximated by a short polynomial:
277 // logl(1+r) ~=~ poly = r + Q1 r^2 + ... + Q4 r^5
280 // Step 3. Reconstruction
281 // ----------------------
283 // This step computes the desired result of logl(X+1):
285 // logl(X+1) = logl( 2^N * (S_hi + S_lo) )
286 // = N*logl(2) + logl( S_hi + S_lo) )
287 // = N*logl(2) + logl(1/G) +
288 // logl(1 + G * ( S_hi + S_lo ) - 1 )
290 // logl(2), logl(1/G_j) are stored as pairs of (single,double) numbers:
291 // log2_hi, log2_lo, log1byGj_hi, log1byGj_lo. The high parts are
292 // single-precision numbers and the low parts are double precision
293 // numbers. These have the property that
295 // N*log2_hi + SUM ( log1byGj_hi )
297 // is computable exactly in double-extended precision (64 sig. bits).
300 // Y_hi := N*log2_hi + SUM ( log1byGj_hi )
301 // Y_lo := poly_hi + [ poly_lo +
302 // ( SUM ( log1byGj_lo ) + N*log2_lo ) ]
308 // ************* DO NOT CHANGE THE ORDER OF THESE TABLES *************
310 // P_8, P_7, P_6, P_5, P_4, P_3, P_2, and P_1
312 LOCAL_OBJECT_START(Constants_P)
313 //data4 0xEFD62B15,0xE3936754,0x00003FFB,0x00000000
314 //data4 0xA5E56381,0x8003B271,0x0000BFFC,0x00000000
315 //data4 0x73282DB0,0x9249248C,0x00003FFC,0x00000000
316 //data4 0x47305052,0xAAAAAA9F,0x0000BFFC,0x00000000
317 //data4 0xCCD17FC9,0xCCCCCCCC,0x00003FFC,0x00000000
318 //data4 0x00067ED5,0x80000000,0x0000BFFD,0x00000000
319 //data4 0xAAAAAAAA,0xAAAAAAAA,0x00003FFD,0x00000000
320 //data4 0xFFFFFFFE,0xFFFFFFFF,0x0000BFFD,0x00000000
321 data8 0xE3936754EFD62B15,0x00003FFB
322 data8 0x8003B271A5E56381,0x0000BFFC
323 data8 0x9249248C73282DB0,0x00003FFC
324 data8 0xAAAAAA9F47305052,0x0000BFFC
325 data8 0xCCCCCCCCCCD17FC9,0x00003FFC
326 data8 0x8000000000067ED5,0x0000BFFD
327 data8 0xAAAAAAAAAAAAAAAA,0x00003FFD
328 data8 0xFFFFFFFFFFFFFFFE,0x0000BFFD
329 LOCAL_OBJECT_END(Constants_P)
331 // log2_hi, log2_lo, Q_4, Q_3, Q_2, and Q_1
333 LOCAL_OBJECT_START(Constants_Q)
334 //data4 0x00000000,0xB1721800,0x00003FFE,0x00000000
335 //data4 0x4361C4C6,0x82E30865,0x0000BFE2,0x00000000
336 //data4 0x328833CB,0xCCCCCAF2,0x00003FFC,0x00000000
337 //data4 0xA9D4BAFB,0x80000077,0x0000BFFD,0x00000000
338 //data4 0xAAABE3D2,0xAAAAAAAA,0x00003FFD,0x00000000
339 //data4 0xFFFFDAB7,0xFFFFFFFF,0x0000BFFD,0x00000000
340 data8 0xB172180000000000,0x00003FFE
341 data8 0x82E308654361C4C6,0x0000BFE2
342 data8 0xCCCCCAF2328833CB,0x00003FFC
343 data8 0x80000077A9D4BAFB,0x0000BFFD
344 data8 0xAAAAAAAAAAABE3D2,0x00003FFD
345 data8 0xFFFFFFFFFFFFDAB7,0x0000BFFD
346 LOCAL_OBJECT_END(Constants_Q)
348 // 1/ln10_hi, 1/ln10_lo
350 LOCAL_OBJECT_START(Constants_1_by_LN10)
351 //data4 0x37287195,0xDE5BD8A9,0x00003FFD,0x00000000
352 //data4 0xACCF70C8,0xD56EAABE,0x00003FBB,0x00000000
353 data8 0xDE5BD8A937287195,0x00003FFD
354 data8 0xD56EAABEACCF70C8,0x00003FBB
355 LOCAL_OBJECT_END(Constants_1_by_LN10)
360 LOCAL_OBJECT_START(Constants_Z_1)
377 LOCAL_OBJECT_END(Constants_Z_1)
379 // G1 and H1 - IEEE single and h1 - IEEE double
381 LOCAL_OBJECT_START(Constants_G_H_h1)
382 data4 0x3F800000,0x00000000
383 data8 0x0000000000000000
384 data4 0x3F70F0F0,0x3D785196
385 data8 0x3DA163A6617D741C
386 data4 0x3F638E38,0x3DF13843
387 data8 0x3E2C55E6CBD3D5BB
388 data4 0x3F579430,0x3E2FF9A0
389 data8 0xBE3EB0BFD86EA5E7
390 data4 0x3F4CCCC8,0x3E647FD6
391 data8 0x3E2E6A8C86B12760
392 data4 0x3F430C30,0x3E8B3AE7
393 data8 0x3E47574C5C0739BA
394 data4 0x3F3A2E88,0x3EA30C68
395 data8 0x3E20E30F13E8AF2F
396 data4 0x3F321640,0x3EB9CEC8
397 data8 0xBE42885BF2C630BD
398 data4 0x3F2AAAA8,0x3ECF9927
399 data8 0x3E497F3497E577C6
400 data4 0x3F23D708,0x3EE47FC5
401 data8 0x3E3E6A6EA6B0A5AB
402 data4 0x3F1D89D8,0x3EF8947D
403 data8 0xBDF43E3CD328D9BE
404 data4 0x3F17B420,0x3F05F3A1
405 data8 0x3E4094C30ADB090A
406 data4 0x3F124920,0x3F0F4303
407 data8 0xBE28FBB2FC1FE510
408 data4 0x3F0D3DC8,0x3F183EBF
409 data8 0x3E3A789510FDE3FA
410 data4 0x3F088888,0x3F20EC80
411 data8 0x3E508CE57CC8C98F
412 data4 0x3F042108,0x3F29516A
413 data8 0xBE534874A223106C
414 LOCAL_OBJECT_END(Constants_G_H_h1)
418 LOCAL_OBJECT_START(Constants_Z_2)
435 LOCAL_OBJECT_END(Constants_Z_2)
437 // G2 and H2 - IEEE single and h2 - IEEE double
439 LOCAL_OBJECT_START(Constants_G_H_h2)
440 data4 0x3F800000,0x00000000
441 data8 0x0000000000000000
442 data4 0x3F7F00F8,0x3B7F875D
443 data8 0x3DB5A11622C42273
444 data4 0x3F7E03F8,0x3BFF015B
445 data8 0x3DE620CF21F86ED3
446 data4 0x3F7D08E0,0x3C3EE393
447 data8 0xBDAFA07E484F34ED
448 data4 0x3F7C0FC0,0x3C7E0586
449 data8 0xBDFE07F03860BCF6
450 data4 0x3F7B1880,0x3C9E75D2
451 data8 0x3DEA370FA78093D6
452 data4 0x3F7A2328,0x3CBDC97A
453 data8 0x3DFF579172A753D0
454 data4 0x3F792FB0,0x3CDCFE47
455 data8 0x3DFEBE6CA7EF896B
456 data4 0x3F783E08,0x3CFC15D0
457 data8 0x3E0CF156409ECB43
458 data4 0x3F774E38,0x3D0D874D
459 data8 0xBE0B6F97FFEF71DF
460 data4 0x3F766038,0x3D1CF49B
461 data8 0xBE0804835D59EEE8
462 data4 0x3F757400,0x3D2C531D
463 data8 0x3E1F91E9A9192A74
464 data4 0x3F748988,0x3D3BA322
465 data8 0xBE139A06BF72A8CD
466 data4 0x3F73A0D0,0x3D4AE46F
467 data8 0x3E1D9202F8FBA6CF
468 data4 0x3F72B9D0,0x3D5A1756
469 data8 0xBE1DCCC4BA796223
470 data4 0x3F71D488,0x3D693B9D
471 data8 0xBE049391B6B7C239
472 LOCAL_OBJECT_END(Constants_G_H_h2)
474 // G3 and H3 - IEEE single and h3 - IEEE double
476 LOCAL_OBJECT_START(Constants_G_H_h3)
477 data4 0x3F7FFC00,0x38800100
478 data8 0x3D355595562224CD
479 data4 0x3F7FF400,0x39400480
480 data8 0x3D8200A206136FF6
481 data4 0x3F7FEC00,0x39A00640
482 data8 0x3DA4D68DE8DE9AF0
483 data4 0x3F7FE400,0x39E00C41
484 data8 0xBD8B4291B10238DC
485 data4 0x3F7FDC00,0x3A100A21
486 data8 0xBD89CCB83B1952CA
487 data4 0x3F7FD400,0x3A300F22
488 data8 0xBDB107071DC46826
489 data4 0x3F7FCC08,0x3A4FF51C
490 data8 0x3DB6FCB9F43307DB
491 data4 0x3F7FC408,0x3A6FFC1D
492 data8 0xBD9B7C4762DC7872
493 data4 0x3F7FBC10,0x3A87F20B
494 data8 0xBDC3725E3F89154A
495 data4 0x3F7FB410,0x3A97F68B
496 data8 0xBD93519D62B9D392
497 data4 0x3F7FAC18,0x3AA7EB86
498 data8 0x3DC184410F21BD9D
499 data4 0x3F7FA420,0x3AB7E101
500 data8 0xBDA64B952245E0A6
501 data4 0x3F7F9C20,0x3AC7E701
502 data8 0x3DB4B0ECAABB34B8
503 data4 0x3F7F9428,0x3AD7DD7B
504 data8 0x3D9923376DC40A7E
505 data4 0x3F7F8C30,0x3AE7D474
506 data8 0x3DC6E17B4F2083D3
507 data4 0x3F7F8438,0x3AF7CBED
508 data8 0x3DAE314B811D4394
509 data4 0x3F7F7C40,0x3B03E1F3
510 data8 0xBDD46F21B08F2DB1
511 data4 0x3F7F7448,0x3B0BDE2F
512 data8 0xBDDC30A46D34522B
513 data4 0x3F7F6C50,0x3B13DAAA
514 data8 0x3DCB0070B1F473DB
515 data4 0x3F7F6458,0x3B1BD766
516 data8 0xBDD65DDC6AD282FD
517 data4 0x3F7F5C68,0x3B23CC5C
518 data8 0xBDCDAB83F153761A
519 data4 0x3F7F5470,0x3B2BC997
520 data8 0xBDDADA40341D0F8F
521 data4 0x3F7F4C78,0x3B33C711
522 data8 0x3DCD1BD7EBC394E8
523 data4 0x3F7F4488,0x3B3BBCC6
524 data8 0xBDC3532B52E3E695
525 data4 0x3F7F3C90,0x3B43BAC0
526 data8 0xBDA3961EE846B3DE
527 data4 0x3F7F34A0,0x3B4BB0F4
528 data8 0xBDDADF06785778D4
529 data4 0x3F7F2CA8,0x3B53AF6D
530 data8 0x3DCC3ED1E55CE212
531 data4 0x3F7F24B8,0x3B5BA620
532 data8 0xBDBA31039E382C15
533 data4 0x3F7F1CC8,0x3B639D12
534 data8 0x3D635A0B5C5AF197
535 data4 0x3F7F14D8,0x3B6B9444
536 data8 0xBDDCCB1971D34EFC
537 data4 0x3F7F0CE0,0x3B7393BC
538 data8 0x3DC7450252CD7ADA
539 data4 0x3F7F04F0,0x3B7B8B6D
540 data8 0xBDB68F177D7F2A42
541 LOCAL_OBJECT_END(Constants_G_H_h3)
544 // Floating Point Registers
603 FR_Output_X_tmp = f76
609 FR_2_to_minus_N = f82
616 // General Purpose Registers
647 // Added for unwind support
655 GR_Parameter_RESULT = r55
656 GR_Parameter_TAG = r56
659 GLOBAL_IEEE754_ENTRY(log1pl)
661 alloc r32 = ar.pfs,0,21,4,0
662 fclass.m p6, p0 = FR_Input_X, 0x1E3 // Test for natval, nan, inf
666 addl GR_ad_z_1 = @ltoff(Constants_Z_1#),gp
667 fma.s1 FR_Z = FR_Input_X, f1, f1 // x+1
674 fmerge.ns FR_Neg_One = f1, f1 // Form -1.0
679 fnorm.s1 FR_X_Prime = FR_Input_X // Normalize x
685 ld8 GR_ad_z_1 = [GR_ad_z_1] // Get pointer to Constants_Z_1
687 mov GR_exp_2tom7 = 0x0fff8 // Exponent of 2^-7
692 getf.sig GR_signif = FR_Z // Get significand of x+1
693 fcmp.eq.s1 p9, p0 = FR_Input_X, f0 // Test for x=0
694 (p6) br.cond.spnt LOG1P_special // Branch for nan, inf, natval
699 add GR_ad_tbl_1 = 0x040, GR_ad_z_1 // Point to Constants_G_H_h1
700 fcmp.lt.s1 p13, p0 = FR_X_Prime, FR_Neg_One // Test for x<-1
701 add GR_ad_p = -0x100, GR_ad_z_1 // Point to Constants_P
704 add GR_ad_z_2 = 0x140, GR_ad_z_1 // Point to Constants_Z_2
706 add GR_ad_tbl_2 = 0x180, GR_ad_z_1 // Point to Constants_G_H_h2
711 add GR_ad_q = 0x080, GR_ad_p // Point to Constants_Q
712 fcmp.eq.s1 p8, p0 = FR_X_Prime, FR_Neg_One // Test for x=-1
713 extr.u GR_Index1 = GR_signif, 59, 4 // Get high 4 bits of signif
716 add GR_ad_tbl_3 = 0x280, GR_ad_z_1 // Point to Constants_G_H_h3
718 (p9) br.ret.spnt b0 // Exit if x=0, return input
723 shladd GR_ad_z_1 = GR_Index1, 2, GR_ad_z_1 // Point to Z_1
724 fclass.nm p10, p0 = FR_Input_X, 0x1FF // Test for unsupported
725 extr.u GR_X_0 = GR_signif, 49, 15 // Get high 15 bits of significand
728 ldfe FR_P8 = [GR_ad_p],16 // Load P_8 for near1 path
729 fsub.s1 FR_W = FR_X_Prime, f0 // W = x
730 add GR_ad_ln10 = 0x060, GR_ad_q // Point to Constants_1_by_LN10
735 ld4 GR_Z_1 = [GR_ad_z_1] // Load Z_1
736 fmax.s1 FR_AA = FR_X_Prime, f1 // For S_lo, form AA = max(X,1.0)
737 mov GR_exp_mask = 0x1FFFF // Create exponent mask
740 shladd GR_ad_tbl_1 = GR_Index1, 4, GR_ad_tbl_1 // Point to G_1
741 mov GR_Bias = 0x0FFFF // Create exponent bias
742 (p13) br.cond.spnt LOG1P_LT_Minus_1 // Branch if x<-1
747 ldfps FR_G, FR_H = [GR_ad_tbl_1],8 // Load G_1, H_1
748 fmerge.se FR_S_hi = f1,FR_Z // Form |x+1|
749 (p8) br.cond.spnt LOG1P_EQ_Minus_1 // Branch if x=-1
754 getf.exp GR_N = FR_Z // Get N = exponent of x+1
755 ldfd FR_h = [GR_ad_tbl_1] // Load h_1
756 (p10) br.cond.spnt LOG1P_unsupported // Branch for unsupported type
761 ldfe FR_log2_hi = [GR_ad_q],16 // Load log2_hi
762 fcmp.eq.s0 p8, p0 = FR_Input_X, f0 // Dummy op to flag denormals
763 pmpyshr2.u GR_X_1 = GR_X_0,GR_Z_1,15 // Get bits 30-15 of X_0 * Z_1
768 // For performance, don't use result of pmpyshr2.u for 4 cycles.
771 ldfe FR_log2_lo = [GR_ad_q],16 // Load log2_lo
772 sub GR_N = GR_N, GR_Bias
773 mov GR_exp_2tom80 = 0x0ffaf // Exponent of 2^-80
778 ldfe FR_Q4 = [GR_ad_q],16 // Load Q4
779 fms.s1 FR_S_lo = FR_AA, f1, FR_Z // Form S_lo = AA - Z
780 sub GR_minus_N = GR_Bias, GR_N // Form exponent of 2^(-N)
785 ldfe FR_Q3 = [GR_ad_q],16 // Load Q3
786 setf.sig FR_float_N = GR_N // Put integer N into rightmost significand
787 fmin.s1 FR_BB = FR_X_Prime, f1 // For S_lo, form BB = min(X,1.0)
792 getf.exp GR_M = FR_W // Get signexp of w = x
793 ldfe FR_Q2 = [GR_ad_q],16 // Load Q2
794 extr.u GR_Index2 = GR_X_1, 6, 4 // Extract bits 6-9 of X_1
799 ldfe FR_Q1 = [GR_ad_q] // Load Q1
800 shladd GR_ad_z_2 = GR_Index2, 2, GR_ad_z_2 // Point to Z_2
801 add GR_ad_p2 = 0x30,GR_ad_p // Point to P_4
806 ld4 GR_Z_2 = [GR_ad_z_2] // Load Z_2
807 shladd GR_ad_tbl_2 = GR_Index2, 4, GR_ad_tbl_2 // Point to G_2
808 and GR_M = GR_exp_mask, GR_M // Get exponent of w = x
813 ldfps FR_G2, FR_H2 = [GR_ad_tbl_2],8 // Load G_2, H_2
814 cmp.lt p8, p9 = GR_M, GR_exp_2tom7 // Test |x| < 2^-7
815 cmp.lt p7, p0 = GR_M, GR_exp_2tom80 // Test |x| < 2^-80
819 // Small path is separate code
820 // p7 is for the small path: |x| < 2^-80
821 // near1 and regular paths are merged.
822 // p8 is for the near1 path: |x| < 2^-7
823 // p9 is for regular path: |x| >= 2^-7
826 ldfd FR_h2 = [GR_ad_tbl_2] // Load h_2
831 (p9) setf.exp FR_2_to_minus_N = GR_minus_N // Form 2^(-N)
832 (p7) fnma.s0 f8 = FR_X_Prime, FR_X_Prime, FR_X_Prime // Result x - x*x
833 (p7) br.ret.spnt b0 // Branch if |x| < 2^-80
838 (p8) ldfe FR_P7 = [GR_ad_p],16 // Load P_7 for near1 path
839 (p8) ldfe FR_P4 = [GR_ad_p2],16 // Load P_4 for near1 path
840 (p9) pmpyshr2.u GR_X_2 = GR_X_1,GR_Z_2,15 // Get bits 30-15 of X_1 * Z_2
845 // For performance, don't use result of pmpyshr2.u for 4 cycles.
848 (p8) ldfe FR_P6 = [GR_ad_p],16 // Load P_6 for near1 path
849 (p8) ldfe FR_P3 = [GR_ad_p2],16 // Load P_3 for near1 path
850 (p9) fma.s1 FR_S_lo = FR_S_lo, f1, FR_BB // S_lo = S_lo + BB
855 (p8) ldfe FR_P5 = [GR_ad_p],16 // Load P_5 for near1 path
856 (p8) ldfe FR_P2 = [GR_ad_p2],16 // Load P_2 for near1 path
857 (p8) fmpy.s1 FR_wsq = FR_W, FR_W // wsq = w * w for near1 path
862 (p8) ldfe FR_P1 = [GR_ad_p2],16 ;; // Load P_1 for near1 path
864 (p9) extr.u GR_Index3 = GR_X_2, 1, 5 // Extract bits 1-5 of X_2
869 (p9) shladd GR_ad_tbl_3 = GR_Index3, 4, GR_ad_tbl_3 // Point to G_3
870 (p9) fcvt.xf FR_float_N = FR_float_N
876 (p9) ldfps FR_G3, FR_H3 = [GR_ad_tbl_3],8 // Load G_3, H_3
883 (p9) ldfd FR_h3 = [GR_ad_tbl_3] // Load h_3
884 (p9) fmpy.s1 FR_G = FR_G, FR_G2 // G = G_1 * G_2
889 (p9) fadd.s1 FR_H = FR_H, FR_H2 // H = H_1 + H_2
897 (p9) fadd.s1 FR_h = FR_h, FR_h2 // h = h_1 + h_2
903 (p8) fmpy.s1 FR_w4 = FR_wsq, FR_wsq // w4 = w^4 for near1 path
908 (p8) fma.s1 FR_p87 = FR_W, FR_P8, FR_P7 // p87 = w * P8 + P7
915 (p9) fma.s1 FR_S_lo = FR_S_lo, FR_2_to_minus_N, f0 // S_lo = S_lo * 2^(-N)
920 (p8) fma.s1 FR_p43 = FR_W, FR_P4, FR_P3 // p43 = w * P4 + P3
927 (p9) fmpy.s1 FR_G = FR_G, FR_G3 // G = (G_1 * G_2) * G_3
932 (p9) fadd.s1 FR_H = FR_H, FR_H3 // H = (H_1 + H_2) + H_3
939 (p9) fadd.s1 FR_h = FR_h, FR_h3 // h = (h_1 + h_2) + h_3
944 (p8) fmpy.s1 FR_w6 = FR_w4, FR_wsq // w6 = w^6 for near1 path
951 (p8) fma.s1 FR_p432 = FR_W, FR_p43, FR_P2 // p432 = w * p43 + P2
956 (p8) fma.s1 FR_p876 = FR_W, FR_p87, FR_P6 // p876 = w * p87 + P6
963 (p9) fms.s1 FR_r = FR_G, FR_S_hi, f1 // r = G * S_hi - 1
968 (p9) fma.s1 FR_Y_hi = FR_float_N, FR_log2_hi, FR_H // Y_hi = N * log2_hi + H
975 (p9) fma.s1 FR_h = FR_float_N, FR_log2_lo, FR_h // h = N * log2_lo + h
982 (p9) fma.s1 FR_r = FR_G, FR_S_lo, FR_r // r = G * S_lo + (G * S_hi - 1)
989 (p8) fma.s1 FR_p4321 = FR_W, FR_p432, FR_P1 // p4321 = w * p432 + P1
994 (p8) fma.s1 FR_p8765 = FR_W, FR_p876, FR_P5 // p8765 = w * p876 + P5
1001 (p9) fma.s1 FR_poly_lo = FR_r, FR_Q4, FR_Q3 // poly_lo = r * Q4 + Q3
1006 (p9) fmpy.s1 FR_rsq = FR_r, FR_r // rsq = r * r
1013 (p8) fma.s1 FR_Y_lo = FR_wsq, FR_p4321, f0 // Y_lo = wsq * p4321
1018 (p8) fma.s1 FR_Y_hi = FR_W, f1, f0 // Y_hi = w for near1 path
1025 (p9) fma.s1 FR_poly_lo = FR_poly_lo, FR_r, FR_Q2 // poly_lo = poly_lo * r + Q2
1030 (p9) fma.s1 FR_rcub = FR_rsq, FR_r, f0 // rcub = r^3
1037 (p8) fma.s1 FR_Y_lo = FR_w6, FR_p8765,FR_Y_lo // Y_lo = w6 * p8765 + w2 * p4321
1044 (p9) fma.s1 FR_poly_hi = FR_Q1, FR_rsq, FR_r // poly_hi = Q1 * rsq + r
1051 (p9) fma.s1 FR_poly_lo = FR_poly_lo, FR_rcub, FR_h // poly_lo = poly_lo*r^3 + h
1058 (p9) fadd.s1 FR_Y_lo = FR_poly_hi, FR_poly_lo // Y_lo = poly_hi + poly_lo
1063 // Remainder of code is common for near1 and regular paths
1066 fadd.s0 f8 = FR_Y_lo,FR_Y_hi // Result=Y_lo+Y_hi
1067 br.ret.sptk b0 // Common exit for 2^-80 < x < inf
1075 // If x=-1 raise divide by zero and return -inf
1078 mov GR_Parameter_TAG = 138
1079 fsub.s1 FR_Output_X_tmp = f0, f1
1086 frcpa.s0 FR_Output_X_tmp, p8 = FR_Output_X_tmp, f0
1087 br.cond.sptk __libm_error_region
1094 fclass.m.unc p8, p0 = FR_Input_X, 0x1E1 // Test for natval, nan, +inf
1100 // For SNaN raise invalid and return QNaN.
1101 // For QNaN raise invalid and return QNaN.
1102 // For +Inf return +Inf.
1106 (p8) fmpy.s0 f8 = FR_Input_X, f1
1107 (p8) br.ret.sptk b0 // Return for natval, nan, +inf
1112 // For -Inf raise invalid and return QNaN.
1115 mov GR_Parameter_TAG = 139
1116 fmpy.s0 FR_Output_X_tmp = FR_Input_X, f0
1117 br.cond.sptk __libm_error_region
1124 // Return generated NaN or other value.
1128 fmpy.s0 f8 = FR_Input_X, f0
1133 // Here if -inf < x < -1
1136 // Deal with x < -1 in a special way - raise
1137 // invalid and produce QNaN indefinite.
1140 mov GR_Parameter_TAG = 139
1141 frcpa.s0 FR_Output_X_tmp, p8 = f0, f0
1142 br.cond.sptk __libm_error_region
1147 GLOBAL_IEEE754_END(log1pl)
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