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
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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 // 05/21/01 Extracted logl and log10l from log1pl.s file, and optimized
44 // 06/20/01 Fixed error tag for x=-inf.
45 // 05/20/02 Cleaned up namespace and sf0 syntax
46 // 02/10/03 Reordered header: .section, .global, .proc, .align;
47 // used data8 for long double table values
49 //*********************************************************************
51 //*********************************************************************
53 // Function: Combined logl(x) and log10l(x) where
54 // logl(x) = ln(x), for double-extended precision x values
55 // log10l(x) = log (x), 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-p14
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 // (Error Handling Routine called for underflow)
79 // Inexact raised when appropriate by algorithm
86 // logl(EM_special Values) = QNaN
88 // log10l(-inf) = QNaN
89 // log10l(+/-0) = -inf
90 // log10l(SNaN) = QNaN
91 // log10l(QNaN) = QNaN
92 // log10l(EM_special Values) = QNaN
94 //*********************************************************************
98 // The method consists of two cases.
100 // If |X-1| < 2^(-7) use case log_near1;
101 // else use case log_regular;
105 // logl( 1 + X ) can be approximated by a simple polynomial
106 // in W = X-1. 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) for an argument Arg in [1,2), we
113 // 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 ). Obtain N, S_hi such that
128 // X = 2^N * S_hi exactly
130 // where S_hi in [1,2)
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)
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, logl( X ) is given by
152 // logl( X ) = logl( 2^N * S_hi )
153 // ~=~ N*logl(2) + logl(1/G) + logl(1 + r)
154 // ~=~ N*logl(2) + logl(1/G) + poly(r).
156 // **** Algorithm ****
160 // Here we compute a simple polynomial. To exploit parallelism, we split
161 // the polynomial into two portions.
167 // Y_hi := W + Wsq*(P_1 + W*(P_2 + W*(P_3 + W*P_4))
168 // Y_lo := W6*(P_5 + W*(P_6 + W*(P_7 + W*P_8)))
172 // We present the algorithm in four steps.
174 // Step 0. Initialization
175 // ----------------------
178 // N := unbaised exponent of Z
179 // S_hi := 2^(-N) * Z
181 // Step 1. Argument Reduction
182 // --------------------------
186 // Z = 2^N * S_hi = 2^N * 1.d_1 d_2 d_3 ... d_63
188 // We obtain G_1, G_2, G_3 by the following steps.
191 // Define X_0 := 1.d_1 d_2 ... d_14. This is extracted
194 // Define A_1 := 1.d_1 d_2 d_3 d_4. This is X_0 truncated
197 // Define index_1 := [ d_1 d_2 d_3 d_4 ].
199 // Fetch Z_1 := (1/A_1) rounded UP in fixed point with
200 // fixed point lsb = 2^(-15).
201 // Z_1 looks like z_0.z_1 z_2 ... z_15
202 // Note that the fetching is done using index_1.
203 // A_1 is actually not needed in the implementation
204 // and is used here only to explain how is the value
207 // Fetch G_1 := (1/A_1) truncated to 21 sig. bits.
208 // floating pt. Again, fetching is done using index_1. A_1
209 // explains how G_1 is defined.
211 // Calculate X_1 := X_0 * Z_1 truncated to lsb = 2^(-14)
212 // = 1.0 0 0 0 d_5 ... d_14
213 // This is accomplished by integer multiplication.
214 // It is proved that X_1 indeed always begin
215 // with 1.0000 in fixed point.
218 // Define A_2 := 1.0 0 0 0 d_5 d_6 d_7 d_8. This is X_1
219 // truncated to lsb = 2^(-8). Similar to A_1,
220 // A_2 is not needed in actual implementation. It
221 // helps explain how some of the values are defined.
223 // Define index_2 := [ d_5 d_6 d_7 d_8 ].
225 // Fetch Z_2 := (1/A_2) rounded UP in fixed point with
226 // fixed point lsb = 2^(-15). Fetch done using index_2.
227 // Z_2 looks like z_0.z_1 z_2 ... z_15
229 // Fetch G_2 := (1/A_2) truncated to 21 sig. bits.
232 // Calculate X_2 := X_1 * Z_2 truncated to lsb = 2^(-14)
233 // = 1.0 0 0 0 0 0 0 0 d_9 d_10 ... d_14
234 // This is accomplished by integer multiplication.
235 // It is proved that X_2 indeed always begin
236 // with 1.00000000 in fixed point.
239 // Define A_3 := 1.0 0 0 0 0 0 0 0 d_9 d_10 d_11 d_12 d_13 1.
240 // This is 2^(-14) + X_2 truncated to lsb = 2^(-13).
242 // Define index_3 := [ d_9 d_10 d_11 d_12 d_13 ].
244 // Fetch G_3 := (1/A_3) truncated to 21 sig. bits.
245 // floating pt. Fetch is done using index_3.
247 // Compute G := G_1 * G_2 * G_3.
249 // This is done exactly since each of G_j only has 21 sig. bits.
256 // Step 2. Approximation
257 // ---------------------
259 // This step computes an approximation to logl( 1 + r ) where r is the
260 // reduced argument just obtained. It is proved that |r| <= 1.9*2^(-13);
261 // thus logl(1+r) can be approximated by a short polynomial:
263 // logl(1+r) ~=~ poly = r + Q1 r^2 + ... + Q4 r^5
266 // Step 3. Reconstruction
267 // ----------------------
269 // This step computes the desired result of logl(X):
271 // logl(X) = logl( 2^N * S_hi )
272 // = N*logl(2) + logl( S_hi )
273 // = N*logl(2) + logl(1/G) +
274 // logl(1 + G*S_hi - 1 )
276 // logl(2), logl(1/G_j) are stored as pairs of (single,double) numbers:
277 // log2_hi, log2_lo, log1byGj_hi, log1byGj_lo. The high parts are
278 // single-precision numbers and the low parts are double precision
279 // numbers. These have the property that
281 // N*log2_hi + SUM ( log1byGj_hi )
283 // is computable exactly in double-extended precision (64 sig. bits).
286 // Y_hi := N*log2_hi + SUM ( log1byGj_hi )
287 // Y_lo := poly_hi + [ poly_lo +
288 // ( SUM ( log1byGj_lo ) + N*log2_lo ) ]
294 // ************* DO NOT CHANGE THE ORDER OF THESE TABLES *************
296 // P_8, P_7, P_6, P_5, P_4, P_3, P_2, and P_1
298 LOCAL_OBJECT_START(Constants_P)
299 data8 0xE3936754EFD62B15,0x00003FFB
300 data8 0x8003B271A5E56381,0x0000BFFC
301 data8 0x9249248C73282DB0,0x00003FFC
302 data8 0xAAAAAA9F47305052,0x0000BFFC
303 data8 0xCCCCCCCCCCD17FC9,0x00003FFC
304 data8 0x8000000000067ED5,0x0000BFFD
305 data8 0xAAAAAAAAAAAAAAAA,0x00003FFD
306 data8 0xFFFFFFFFFFFFFFFE,0x0000BFFD
307 LOCAL_OBJECT_END(Constants_P)
309 // log2_hi, log2_lo, Q_4, Q_3, Q_2, and Q_1
311 LOCAL_OBJECT_START(Constants_Q)
312 data8 0xB172180000000000,0x00003FFE
313 data8 0x82E308654361C4C6,0x0000BFE2
314 data8 0xCCCCCAF2328833CB,0x00003FFC
315 data8 0x80000077A9D4BAFB,0x0000BFFD
316 data8 0xAAAAAAAAAAABE3D2,0x00003FFD
317 data8 0xFFFFFFFFFFFFDAB7,0x0000BFFD
318 LOCAL_OBJECT_END(Constants_Q)
320 // 1/ln10_hi, 1/ln10_lo
322 LOCAL_OBJECT_START(Constants_1_by_LN10)
323 data8 0xDE5BD8A937287195,0x00003FFD
324 data8 0xD56EAABEACCF70C8,0x00003FBB
325 LOCAL_OBJECT_END(Constants_1_by_LN10)
330 LOCAL_OBJECT_START(Constants_Z_1)
347 LOCAL_OBJECT_END(Constants_Z_1)
349 // G1 and H1 - IEEE single and h1 - IEEE double
351 LOCAL_OBJECT_START(Constants_G_H_h1)
352 data4 0x3F800000,0x00000000
353 data8 0x0000000000000000
354 data4 0x3F70F0F0,0x3D785196
355 data8 0x3DA163A6617D741C
356 data4 0x3F638E38,0x3DF13843
357 data8 0x3E2C55E6CBD3D5BB
358 data4 0x3F579430,0x3E2FF9A0
359 data8 0xBE3EB0BFD86EA5E7
360 data4 0x3F4CCCC8,0x3E647FD6
361 data8 0x3E2E6A8C86B12760
362 data4 0x3F430C30,0x3E8B3AE7
363 data8 0x3E47574C5C0739BA
364 data4 0x3F3A2E88,0x3EA30C68
365 data8 0x3E20E30F13E8AF2F
366 data4 0x3F321640,0x3EB9CEC8
367 data8 0xBE42885BF2C630BD
368 data4 0x3F2AAAA8,0x3ECF9927
369 data8 0x3E497F3497E577C6
370 data4 0x3F23D708,0x3EE47FC5
371 data8 0x3E3E6A6EA6B0A5AB
372 data4 0x3F1D89D8,0x3EF8947D
373 data8 0xBDF43E3CD328D9BE
374 data4 0x3F17B420,0x3F05F3A1
375 data8 0x3E4094C30ADB090A
376 data4 0x3F124920,0x3F0F4303
377 data8 0xBE28FBB2FC1FE510
378 data4 0x3F0D3DC8,0x3F183EBF
379 data8 0x3E3A789510FDE3FA
380 data4 0x3F088888,0x3F20EC80
381 data8 0x3E508CE57CC8C98F
382 data4 0x3F042108,0x3F29516A
383 data8 0xBE534874A223106C
384 LOCAL_OBJECT_END(Constants_G_H_h1)
388 LOCAL_OBJECT_START(Constants_Z_2)
405 LOCAL_OBJECT_END(Constants_Z_2)
407 // G2 and H2 - IEEE single and h2 - IEEE double
409 LOCAL_OBJECT_START(Constants_G_H_h2)
410 data4 0x3F800000,0x00000000
411 data8 0x0000000000000000
412 data4 0x3F7F00F8,0x3B7F875D
413 data8 0x3DB5A11622C42273
414 data4 0x3F7E03F8,0x3BFF015B
415 data8 0x3DE620CF21F86ED3
416 data4 0x3F7D08E0,0x3C3EE393
417 data8 0xBDAFA07E484F34ED
418 data4 0x3F7C0FC0,0x3C7E0586
419 data8 0xBDFE07F03860BCF6
420 data4 0x3F7B1880,0x3C9E75D2
421 data8 0x3DEA370FA78093D6
422 data4 0x3F7A2328,0x3CBDC97A
423 data8 0x3DFF579172A753D0
424 data4 0x3F792FB0,0x3CDCFE47
425 data8 0x3DFEBE6CA7EF896B
426 data4 0x3F783E08,0x3CFC15D0
427 data8 0x3E0CF156409ECB43
428 data4 0x3F774E38,0x3D0D874D
429 data8 0xBE0B6F97FFEF71DF
430 data4 0x3F766038,0x3D1CF49B
431 data8 0xBE0804835D59EEE8
432 data4 0x3F757400,0x3D2C531D
433 data8 0x3E1F91E9A9192A74
434 data4 0x3F748988,0x3D3BA322
435 data8 0xBE139A06BF72A8CD
436 data4 0x3F73A0D0,0x3D4AE46F
437 data8 0x3E1D9202F8FBA6CF
438 data4 0x3F72B9D0,0x3D5A1756
439 data8 0xBE1DCCC4BA796223
440 data4 0x3F71D488,0x3D693B9D
441 data8 0xBE049391B6B7C239
442 LOCAL_OBJECT_END(Constants_G_H_h2)
444 // G3 and H3 - IEEE single and h3 - IEEE double
446 LOCAL_OBJECT_START(Constants_G_H_h3)
447 data4 0x3F7FFC00,0x38800100
448 data8 0x3D355595562224CD
449 data4 0x3F7FF400,0x39400480
450 data8 0x3D8200A206136FF6
451 data4 0x3F7FEC00,0x39A00640
452 data8 0x3DA4D68DE8DE9AF0
453 data4 0x3F7FE400,0x39E00C41
454 data8 0xBD8B4291B10238DC
455 data4 0x3F7FDC00,0x3A100A21
456 data8 0xBD89CCB83B1952CA
457 data4 0x3F7FD400,0x3A300F22
458 data8 0xBDB107071DC46826
459 data4 0x3F7FCC08,0x3A4FF51C
460 data8 0x3DB6FCB9F43307DB
461 data4 0x3F7FC408,0x3A6FFC1D
462 data8 0xBD9B7C4762DC7872
463 data4 0x3F7FBC10,0x3A87F20B
464 data8 0xBDC3725E3F89154A
465 data4 0x3F7FB410,0x3A97F68B
466 data8 0xBD93519D62B9D392
467 data4 0x3F7FAC18,0x3AA7EB86
468 data8 0x3DC184410F21BD9D
469 data4 0x3F7FA420,0x3AB7E101
470 data8 0xBDA64B952245E0A6
471 data4 0x3F7F9C20,0x3AC7E701
472 data8 0x3DB4B0ECAABB34B8
473 data4 0x3F7F9428,0x3AD7DD7B
474 data8 0x3D9923376DC40A7E
475 data4 0x3F7F8C30,0x3AE7D474
476 data8 0x3DC6E17B4F2083D3
477 data4 0x3F7F8438,0x3AF7CBED
478 data8 0x3DAE314B811D4394
479 data4 0x3F7F7C40,0x3B03E1F3
480 data8 0xBDD46F21B08F2DB1
481 data4 0x3F7F7448,0x3B0BDE2F
482 data8 0xBDDC30A46D34522B
483 data4 0x3F7F6C50,0x3B13DAAA
484 data8 0x3DCB0070B1F473DB
485 data4 0x3F7F6458,0x3B1BD766
486 data8 0xBDD65DDC6AD282FD
487 data4 0x3F7F5C68,0x3B23CC5C
488 data8 0xBDCDAB83F153761A
489 data4 0x3F7F5470,0x3B2BC997
490 data8 0xBDDADA40341D0F8F
491 data4 0x3F7F4C78,0x3B33C711
492 data8 0x3DCD1BD7EBC394E8
493 data4 0x3F7F4488,0x3B3BBCC6
494 data8 0xBDC3532B52E3E695
495 data4 0x3F7F3C90,0x3B43BAC0
496 data8 0xBDA3961EE846B3DE
497 data4 0x3F7F34A0,0x3B4BB0F4
498 data8 0xBDDADF06785778D4
499 data4 0x3F7F2CA8,0x3B53AF6D
500 data8 0x3DCC3ED1E55CE212
501 data4 0x3F7F24B8,0x3B5BA620
502 data8 0xBDBA31039E382C15
503 data4 0x3F7F1CC8,0x3B639D12
504 data8 0x3D635A0B5C5AF197
505 data4 0x3F7F14D8,0x3B6B9444
506 data8 0xBDDCCB1971D34EFC
507 data4 0x3F7F0CE0,0x3B7393BC
508 data8 0x3DC7450252CD7ADA
509 data4 0x3F7F04F0,0x3B7B8B6D
510 data8 0xBDB68F177D7F2A42
511 LOCAL_OBJECT_END(Constants_G_H_h3)
514 // Floating Point Registers
573 FR_Output_X_tmp = f76
580 // General Purpose Registers
608 // Added for unwind support
616 GR_Parameter_RESULT = r55
617 GR_Parameter_TAG = r56
621 GLOBAL_IEEE754_ENTRY(logl)
623 alloc r32 = ar.pfs,0,21,4,0
624 fclass.m p6, p0 = FR_Input_X, 0x1E3 // Test for natval, nan, inf
625 cmp.eq p7, p14 = r0, r0 // Set p7 if logl
628 addl GR_ad_z_1 = @ltoff(Constants_Z_1#),gp
629 fnorm.s1 FR_X_Prime = FR_Input_X // Normalize x
630 br.cond.sptk LOGL_BEGIN
634 GLOBAL_IEEE754_END(logl)
635 libm_alias_ldouble_other (__log, log)
638 GLOBAL_IEEE754_ENTRY(log10l)
640 alloc r32 = ar.pfs,0,21,4,0
641 fclass.m p6, p0 = FR_Input_X, 0x1E3 // Test for natval, nan, inf
642 cmp.ne p7, p14 = r0, r0 // Set p14 if log10l
645 addl GR_ad_z_1 = @ltoff(Constants_Z_1#),gp
646 fnorm.s1 FR_X_Prime = FR_Input_X // Normalize x
652 // Common code for logl and log10
655 ld8 GR_ad_z_1 = [GR_ad_z_1] // Get pointer to Constants_Z_1
656 fclass.m p10, p0 = FR_Input_X, 0x0b // Test for denormal
657 mov GR_exp_2tom7 = 0x0fff8 // Exponent of 2^-7
662 getf.sig GR_signif = FR_Input_X // Get significand of x
663 fcmp.eq.s1 p9, p0 = FR_Input_X, f1 // Test for x=1.0
664 (p6) br.cond.spnt LOGL_64_special // Branch for nan, inf, natval
669 add GR_ad_tbl_1 = 0x040, GR_ad_z_1 // Point to Constants_G_H_h1
670 fcmp.lt.s1 p13, p0 = FR_Input_X, f0 // Test for x<0
671 add GR_ad_p = -0x100, GR_ad_z_1 // Point to Constants_P
674 add GR_ad_z_2 = 0x140, GR_ad_z_1 // Point to Constants_Z_2
675 add GR_ad_tbl_2 = 0x180, GR_ad_z_1 // Point to Constants_G_H_h2
676 (p10) br.cond.spnt LOGL_64_denormal // Branch for denormal
682 add GR_ad_q = 0x080, GR_ad_p // Point to Constants_Q
683 fcmp.eq.s1 p8, p0 = FR_Input_X, f0 // Test for x=0
684 extr.u GR_Index1 = GR_signif, 59, 4 // Get high 4 bits of signif
687 add GR_ad_tbl_3 = 0x280, GR_ad_z_1 // Point to Constants_G_H_h3
688 (p9) fma.s0 f8 = FR_Input_X, f0, f0 // If x=1, return +0.0
689 (p9) br.ret.spnt b0 // Exit if x=1
694 shladd GR_ad_z_1 = GR_Index1, 2, GR_ad_z_1 // Point to Z_1
695 fclass.nm p10, p0 = FR_Input_X, 0x1FF // Test for unsupported
696 extr.u GR_X_0 = GR_signif, 49, 15 // Get high 15 bits of significand
699 ldfe FR_P8 = [GR_ad_p],16 // Load P_8 for near1 path
700 fsub.s1 FR_W = FR_X_Prime, f1 // W = x - 1
701 add GR_ad_ln10 = 0x060, GR_ad_q // Point to Constants_1_by_LN10
706 ld4 GR_Z_1 = [GR_ad_z_1] // Load Z_1
708 mov GR_exp_mask = 0x1FFFF // Create exponent mask
711 shladd GR_ad_tbl_1 = GR_Index1, 4, GR_ad_tbl_1 // Point to G_1
712 mov GR_Bias = 0x0FFFF // Create exponent bias
713 (p13) br.cond.spnt LOGL_64_negative // Branch if x<0
718 ldfps FR_G, FR_H = [GR_ad_tbl_1],8 // Load G_1, H_1
719 fmerge.se FR_S_hi = f1,FR_X_Prime // Form |x|
720 (p8) br.cond.spnt LOGL_64_zero // Branch if x=0
725 getf.exp GR_N = FR_X_Prime // Get N = exponent of x
726 ldfd FR_h = [GR_ad_tbl_1] // Load h_1
727 (p10) br.cond.spnt LOGL_64_unsupported // Branch for unsupported type
732 ldfe FR_log2_hi = [GR_ad_q],16 // Load log2_hi
733 fcmp.eq.s0 p8, p0 = FR_Input_X, f0 // Dummy op to flag denormals
734 pmpyshr2.u GR_X_1 = GR_X_0,GR_Z_1,15 // Get bits 30-15 of X_0 * Z_1
739 // For performance, don't use result of pmpyshr2.u for 4 cycles.
742 ldfe FR_log2_lo = [GR_ad_q],16 // Load log2_lo
743 (p14) ldfe FR_1LN10_hi = [GR_ad_ln10],16 // If log10l, load 1/ln10_hi
744 sub GR_N = GR_N, GR_Bias
749 ldfe FR_Q4 = [GR_ad_q],16 // Load Q4
750 (p14) ldfe FR_1LN10_lo = [GR_ad_ln10] // If log10l, load 1/ln10_lo
756 ldfe FR_Q3 = [GR_ad_q],16 // Load Q3
757 setf.sig FR_float_N = GR_N // Put integer N into rightmost significand
763 getf.exp GR_M = FR_W // Get signexp of w = x - 1
764 ldfe FR_Q2 = [GR_ad_q],16 // Load Q2
765 extr.u GR_Index2 = GR_X_1, 6, 4 // Extract bits 6-9 of X_1
770 ldfe FR_Q1 = [GR_ad_q] // Load Q1
771 shladd GR_ad_z_2 = GR_Index2, 2, GR_ad_z_2 // Point to Z_2
772 add GR_ad_p2 = 0x30,GR_ad_p // Point to P_4
777 ld4 GR_Z_2 = [GR_ad_z_2] // Load Z_2
778 shladd GR_ad_tbl_2 = GR_Index2, 4, GR_ad_tbl_2 // Point to G_2
779 and GR_M = GR_exp_mask, GR_M // Get exponent of w = x - 1
784 ldfps FR_G2, FR_H2 = [GR_ad_tbl_2],8 // Load G_2, H_2
785 cmp.lt p8, p9 = GR_M, GR_exp_2tom7 // Test |x-1| < 2^-7
791 // p8 is for the near1 path: |x-1| < 2^-7
792 // p9 is for regular path: |x-1| >= 2^-7
795 ldfd FR_h2 = [GR_ad_tbl_2] // Load h_2
802 (p8) ldfe FR_P7 = [GR_ad_p],16 // Load P_7 for near1 path
803 (p8) ldfe FR_P4 = [GR_ad_p2],16 // Load P_4 for near1 path
804 (p9) pmpyshr2.u GR_X_2 = GR_X_1,GR_Z_2,15 // Get bits 30-15 of X_1 * Z_2
809 // For performance, don't use result of pmpyshr2.u for 4 cycles.
812 (p8) ldfe FR_P6 = [GR_ad_p],16 // Load P_6 for near1 path
813 (p8) ldfe FR_P3 = [GR_ad_p2],16 // Load P_3 for near1 path
819 (p8) ldfe FR_P5 = [GR_ad_p],16 // Load P_5 for near1 path
820 (p8) ldfe FR_P2 = [GR_ad_p2],16 // Load P_2 for near1 path
821 (p8) fmpy.s1 FR_wsq = FR_W, FR_W // wsq = w * w for near1 path
826 (p8) ldfe FR_P1 = [GR_ad_p2],16 ;; // Load P_1 for near1 path
828 (p9) extr.u GR_Index3 = GR_X_2, 1, 5 // Extract bits 1-5 of X_2
833 (p9) shladd GR_ad_tbl_3 = GR_Index3, 4, GR_ad_tbl_3 // Point to G_3
834 (p9) fcvt.xf FR_float_N = FR_float_N
840 (p9) ldfps FR_G3, FR_H3 = [GR_ad_tbl_3],8 // Load G_3, H_3
847 (p9) ldfd FR_h3 = [GR_ad_tbl_3] // Load h_3
848 (p9) fmpy.s1 FR_G = FR_G, FR_G2 // G = G_1 * G_2
853 (p9) fadd.s1 FR_H = FR_H, FR_H2 // H = H_1 + H_2
861 (p9) fadd.s1 FR_h = FR_h, FR_h2 // h = h_1 + h_2
867 (p8) fmpy.s1 FR_w4 = FR_wsq, FR_wsq // w4 = w^4 for near1 path
872 (p8) fma.s1 FR_p87 = FR_W, FR_P8, FR_P7 // p87 = w * P8 + P7
879 (p8) fma.s1 FR_p43 = FR_W, FR_P4, FR_P3 // p43 = w * P4 + P3
886 (p9) fmpy.s1 FR_G = FR_G, FR_G3 // G = (G_1 * G_2) * G_3
891 (p9) fadd.s1 FR_H = FR_H, FR_H3 // H = (H_1 + H_2) + H_3
898 (p9) fadd.s1 FR_h = FR_h, FR_h3 // h = (h_1 + h_2) + h_3
903 (p8) fmpy.s1 FR_w6 = FR_w4, FR_wsq // w6 = w^6 for near1 path
910 (p8) fma.s1 FR_p432 = FR_W, FR_p43, FR_P2 // p432 = w * p43 + P2
915 (p8) fma.s1 FR_p876 = FR_W, FR_p87, FR_P6 // p876 = w * p87 + P6
922 (p9) fms.s1 FR_r = FR_G, FR_S_hi, f1 // r = G * S_hi - 1
927 (p9) fma.s1 FR_Y_hi = FR_float_N, FR_log2_hi, FR_H // Y_hi = N * log2_hi + H
934 (p9) fma.s1 FR_h = FR_float_N, FR_log2_lo, FR_h // h = N * log2_lo + h
941 (p8) fma.s1 FR_p4321 = FR_W, FR_p432, FR_P1 // p4321 = w * p432 + P1
946 (p8) fma.s1 FR_p8765 = FR_W, FR_p876, FR_P5 // p8765 = w * p876 + P5
953 (p9) fma.s1 FR_poly_lo = FR_r, FR_Q4, FR_Q3 // poly_lo = r * Q4 + Q3
958 (p9) fmpy.s1 FR_rsq = FR_r, FR_r // rsq = r * r
965 (p8) fma.s1 FR_Y_lo = FR_wsq, FR_p4321, f0 // Y_lo = wsq * p4321
970 (p8) fma.s1 FR_Y_hi = FR_W, f1, f0 // Y_hi = w for near1 path
977 (p9) fma.s1 FR_poly_lo = FR_poly_lo, FR_r, FR_Q2 // poly_lo = poly_lo * r + Q2
982 (p9) fma.s1 FR_rcub = FR_rsq, FR_r, f0 // rcub = r^3
989 (p8) fma.s1 FR_Y_lo = FR_w6, FR_p8765,FR_Y_lo // Y_lo = w6 * p8765 + w2 * p4321
996 (p9) fma.s1 FR_poly_hi = FR_Q1, FR_rsq, FR_r // poly_hi = Q1 * rsq + r
1003 (p9) fma.s1 FR_poly_lo = FR_poly_lo, FR_rcub, FR_h // poly_lo = poly_lo*r^3 + h
1010 (p9) fadd.s1 FR_Y_lo = FR_poly_hi, FR_poly_lo // Y_lo = poly_hi + poly_lo
1015 // Remainder of code is common for near1 and regular paths
1018 (p7) fadd.s0 f8 = FR_Y_lo,FR_Y_hi // If logl, result=Y_lo+Y_hi
1023 (p14) fmpy.s1 FR_Output_X_tmp = FR_Y_lo,FR_1LN10_hi
1030 (p14) fma.s1 FR_Output_X_tmp = FR_Y_hi,FR_1LN10_lo,FR_Output_X_tmp
1037 (p14) fma.s0 f8 = FR_Y_hi,FR_1LN10_hi,FR_Output_X_tmp
1038 br.ret.sptk b0 // Common exit for 0 < x < inf
1046 // If x=+-0 raise divide by zero and return -inf
1049 (p7) mov GR_Parameter_TAG = 0
1050 fsub.s1 FR_Output_X_tmp = f0, f1
1056 (p14) mov GR_Parameter_TAG = 6
1057 frcpa.s0 FR_Output_X_tmp, p8 = FR_Output_X_tmp, f0
1058 br.cond.sptk __libm_error_region
1065 fclass.m.unc p8, p0 = FR_Input_X, 0x1E1 // Test for natval, nan, +inf
1071 // For SNaN raise invalid and return QNaN.
1072 // For QNaN raise invalid and return QNaN.
1073 // For +Inf return +Inf.
1077 (p8) fmpy.s0 f8 = FR_Input_X, f1
1078 (p8) br.ret.sptk b0 // Return for natval, nan, +inf
1083 // For -Inf raise invalid and return QNaN.
1086 (p7) mov GR_Parameter_TAG = 1
1093 (p14) mov GR_Parameter_TAG = 7
1094 fmpy.s0 FR_Output_X_tmp = FR_Input_X, f0
1095 br.cond.sptk __libm_error_region
1099 // Here if x denormal or unnormal
1102 getf.sig GR_signif = FR_X_Prime // Get significand of normalized input
1109 getf.exp GR_N = FR_X_Prime // Get exponent of normalized input
1111 br.cond.sptk LOGL_64_COMMON // Branch back to common code
1115 LOGL_64_unsupported:
1117 // Return generated NaN or other value.
1121 fmpy.s0 f8 = FR_Input_X, f0
1126 // Here if -inf < x < 0
1129 // Deal with x < 0 in a special way - raise
1130 // invalid and produce QNaN indefinite.
1133 (p7) mov GR_Parameter_TAG = 1
1134 frcpa.s0 FR_Output_X_tmp, p8 = f0, f0
1140 (p14) mov GR_Parameter_TAG = 7
1142 br.cond.sptk __libm_error_region
1147 GLOBAL_IEEE754_END(log10l)
1148 libm_alias_ldouble_other (__log10, log10)
1150 LOCAL_LIBM_ENTRY(__libm_error_region)
1153 add GR_Parameter_Y=-32,sp // Parameter 2 value
1155 .save ar.pfs,GR_SAVE_PFS
1156 mov GR_SAVE_PFS=ar.pfs // Save ar.pfs
1160 add sp=-64,sp // Create new stack
1162 mov GR_SAVE_GP=gp // Save gp
1165 stfe [GR_Parameter_Y] = FR_Y,16 // Save Parameter 2 on stack
1166 add GR_Parameter_X = 16,sp // Parameter 1 address
1167 .save b0, GR_SAVE_B0
1168 mov GR_SAVE_B0=b0 // Save b0
1172 stfe [GR_Parameter_X] = FR_X // Store Parameter 1 on stack
1173 add GR_Parameter_RESULT = 0,GR_Parameter_Y
1174 nop.b 0 // Parameter 3 address
1177 stfe [GR_Parameter_Y] = FR_RESULT // Store Parameter 3 on stack
1178 add GR_Parameter_Y = -16,GR_Parameter_Y
1179 br.call.sptk b0=__libm_error_support# // Call error handling function
1184 add GR_Parameter_RESULT = 48,sp
1187 ldfe f8 = [GR_Parameter_RESULT] // Get return result off stack
1189 add sp = 64,sp // Restore stack pointer
1190 mov b0 = GR_SAVE_B0 // Restore return address
1193 mov gp = GR_SAVE_GP // Restore gp
1194 mov ar.pfs = GR_SAVE_PFS // Restore ar.pfs
1195 br.ret.sptk b0 // Return
1198 LOCAL_LIBM_END(__libm_error_region#)
1200 .type __libm_error_support#,@function
1201 .global __libm_error_support#