4 // Copyright (c) 2000 - 2005, 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
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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
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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.
41 //*********************************************************************
42 // 02/02/00 Original 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 // 08/21/00 Improvements to save 2 cycles on main path, and shorten x=0 case
47 // 12/07/00 Widen main path, shorten x=inf, nan paths
48 // 03/15/01 Fix monotonicity problem around x=0 for round to +inf
49 // 02/05/02 Corrected uninitialize predicate in POSSIBLE_UNDERFLOW path
50 // 05/20/02 Cleaned up namespace and sf0 syntax
51 // 07/26/02 Algorithm changed, accuracy improved
52 // 09/26/02 support of higher precision inputs added, underflow threshold
54 // 11/15/02 Improved performance on Itanium 2, added possible over/under paths
55 // 05/30/03 Set inexact flag on unmasked overflow/underflow
56 // 03/31/05 Reformatted delimiters between data tables
60 //*********************************************************************
63 // Overview of operation
64 //*********************************************************************
65 // Take the input x. w is "how many log2/128 in x?"
71 // x = n*log2 + (log2/64)*j + R
73 // So, exp(x) = 2^n * 2^(j/64)* exp(R)
78 // actually all the entries of 2^(j/64) table are stored in DP and
79 // with exponent bits set to 0 -> multiplication on 2^n can be
80 // performed by doing logical "or" operation with bits presenting 2^n
82 // exp(R) = 1 + (exp(R) - 1)
83 // P = exp(R) - 1 approximated by Taylor series of 3rd degree
84 // P = A3*R^3 + A2*R^2 + R, A3 = 1/6, A2 = 1/2
87 // The final result is reconstructed as follows
91 //*********************************************************************
95 // expf(+qnan) = +qnan
96 // expf(-qnan) = -qnan
97 // expf(+snan) = +qnan
98 // expf(-snan) = -qnan
103 // Overflow and Underflow
104 //*********************************************************************
105 // expf(x) = largest single normal when
106 // x = 88.72283 = 0x42b17217
108 // expf(x) = smallest single normal when
109 // x = -87.33654 = 0xc2aeac4f
111 // expf(x) = largest round-to-nearest single zero when
112 // x = -103.97208 = 0xc2cff1b5
116 //*********************************************************************
117 // Floating Point registers used:
119 // f6,f7, f9 -> f15, f32 -> f40
121 // General registers used:
124 // Predicate registers used:
128 //*********************************************************************
129 // integer registers used
150 GR_Parameter_RESULT = r37
151 GR_Parameter_TAG = r38
153 // floating point registers used
170 fMIN_SGL_OFLOW_ARG = f34
171 fMAX_SGL_ZERO_ARG = f35
172 fMAX_SGL_NORM_ARG = f36
173 fMIN_SGL_NORM_ARG = f37
184 LOCAL_OBJECT_START(_expf_table)
185 data4 0x42b17218 // Smallest sgl arg to overflow sgl result, +88.7228
186 data4 0xc2cff1b5 // Largest sgl for rnd-to-nearest 0 result, -103.9720
187 data4 0x42b17217 // Largest sgl arg to give normal sgl result, +88.7228
188 data4 0xc2aeac4f // Smallest sgl arg to give normal sgl result, -87.3365
190 // 2^(j/64) table, j goes from 0 to 63
191 data8 0x0000000000000000 // 2^(0/64)
192 data8 0x00002C9A3E778061 // 2^(1/64)
193 data8 0x000059B0D3158574 // 2^(2/64)
194 data8 0x0000874518759BC8 // 2^(3/64)
195 data8 0x0000B5586CF9890F // 2^(4/64)
196 data8 0x0000E3EC32D3D1A2 // 2^(5/64)
197 data8 0x00011301D0125B51 // 2^(6/64)
198 data8 0x0001429AAEA92DE0 // 2^(7/64)
199 data8 0x000172B83C7D517B // 2^(8/64)
200 data8 0x0001A35BEB6FCB75 // 2^(9/64)
201 data8 0x0001D4873168B9AA // 2^(10/64)
202 data8 0x0002063B88628CD6 // 2^(11/64)
203 data8 0x0002387A6E756238 // 2^(12/64)
204 data8 0x00026B4565E27CDD // 2^(13/64)
205 data8 0x00029E9DF51FDEE1 // 2^(14/64)
206 data8 0x0002D285A6E4030B // 2^(15/64)
207 data8 0x000306FE0A31B715 // 2^(16/64)
208 data8 0x00033C08B26416FF // 2^(17/64)
209 data8 0x000371A7373AA9CB // 2^(18/64)
210 data8 0x0003A7DB34E59FF7 // 2^(19/64)
211 data8 0x0003DEA64C123422 // 2^(20/64)
212 data8 0x0004160A21F72E2A // 2^(21/64)
213 data8 0x00044E086061892D // 2^(22/64)
214 data8 0x000486A2B5C13CD0 // 2^(23/64)
215 data8 0x0004BFDAD5362A27 // 2^(24/64)
216 data8 0x0004F9B2769D2CA7 // 2^(25/64)
217 data8 0x0005342B569D4F82 // 2^(26/64)
218 data8 0x00056F4736B527DA // 2^(27/64)
219 data8 0x0005AB07DD485429 // 2^(28/64)
220 data8 0x0005E76F15AD2148 // 2^(29/64)
221 data8 0x0006247EB03A5585 // 2^(30/64)
222 data8 0x0006623882552225 // 2^(31/64)
223 data8 0x0006A09E667F3BCD // 2^(32/64)
224 data8 0x0006DFB23C651A2F // 2^(33/64)
225 data8 0x00071F75E8EC5F74 // 2^(34/64)
226 data8 0x00075FEB564267C9 // 2^(35/64)
227 data8 0x0007A11473EB0187 // 2^(36/64)
228 data8 0x0007E2F336CF4E62 // 2^(37/64)
229 data8 0x00082589994CCE13 // 2^(38/64)
230 data8 0x000868D99B4492ED // 2^(39/64)
231 data8 0x0008ACE5422AA0DB // 2^(40/64)
232 data8 0x0008F1AE99157736 // 2^(41/64)
233 data8 0x00093737B0CDC5E5 // 2^(42/64)
234 data8 0x00097D829FDE4E50 // 2^(43/64)
235 data8 0x0009C49182A3F090 // 2^(44/64)
236 data8 0x000A0C667B5DE565 // 2^(45/64)
237 data8 0x000A5503B23E255D // 2^(46/64)
238 data8 0x000A9E6B5579FDBF // 2^(47/64)
239 data8 0x000AE89F995AD3AD // 2^(48/64)
240 data8 0x000B33A2B84F15FB // 2^(49/64)
241 data8 0x000B7F76F2FB5E47 // 2^(50/64)
242 data8 0x000BCC1E904BC1D2 // 2^(51/64)
243 data8 0x000C199BDD85529C // 2^(52/64)
244 data8 0x000C67F12E57D14B // 2^(53/64)
245 data8 0x000CB720DCEF9069 // 2^(54/64)
246 data8 0x000D072D4A07897C // 2^(55/64)
247 data8 0x000D5818DCFBA487 // 2^(56/64)
248 data8 0x000DA9E603DB3285 // 2^(57/64)
249 data8 0x000DFC97337B9B5F // 2^(58/64)
250 data8 0x000E502EE78B3FF6 // 2^(59/64)
251 data8 0x000EA4AFA2A490DA // 2^(60/64)
252 data8 0x000EFA1BEE615A27 // 2^(61/64)
253 data8 0x000F50765B6E4540 // 2^(62/64)
254 data8 0x000FA7C1819E90D8 // 2^(63/64)
255 LOCAL_OBJECT_END(_expf_table)
259 GLOBAL_IEEE754_ENTRY(expf)
262 addl rTblAddr = @ltoff(_expf_table),gp
263 movl r64DivLn2 = 0x40571547652B82FE // 64/ln(2)
266 addl rA3 = 0x3E2AA, r0 // high bits of 1.0/6.0 rounded to SP
267 movl rRightShifter = 0x43E8000000000000 // DP Right Shifter
272 // point to the beginning of the table
273 ld8 rTblAddr = [rTblAddr]
274 fclass.m p14, p0 = f8, 0x22 // test for -INF
275 shl rA3 = rA3, 12 // 0x3E2AA000, approx to 1.0/6.0 in SP
279 fnorm.s1 fNormX = f8 // normalized x
280 addl rExpHalf = 0xFFFE, r0 // exponent of 1/2
285 setf.d f64DivLn2 = r64DivLn2 // load 64/ln(2) to FP reg
286 fclass.m p15, p0 = f8, 0x1e1 // test for NaT,NaN,+Inf
290 // load Right Shifter to FP reg
291 setf.d fRightShifter = rRightShifter
292 movl rLn2Div64 = 0x3F862E42FEFA39EF // DP ln(2)/64 in GR
298 fcmp.eq.s1 p13, p0 = f0, f8 // test for x = 0.0
302 setf.s fA3 = rA3 // load A3 to FP reg
303 (p14) fma.s.s0 f8 = f0, f1, f0 // result if x = -inf
304 (p14) br.ret.spnt b0 // exit here if x = -inf
309 setf.exp fA2 = rExpHalf // load A2 to FP reg
310 fcmp.eq.s0 p6, p0 = f8, f0 // Dummy to flag denorm
314 setf.d fLn2Div64 = rLn2Div64 // load ln(2)/64 to FP reg
315 (p15) fma.s.s0 f8 = f8, f1, f0 // result if x = NaT,NaN,+Inf
316 (p15) br.ret.spnt b0 // exit here if x = NaT,NaN,+Inf
321 // overflow and underflow_zero threshold
322 ldfps fMIN_SGL_OFLOW_ARG, fMAX_SGL_ZERO_ARG = [rTblAddr], 8
323 (p13) fma.s.s0 f8 = f1, f1, f0 // result if x = 0.0
324 (p13) br.ret.spnt b0 // exit here if x =0.0
328 // max normal and underflow_denorm threshold
330 ldfps fMAX_SGL_NORM_ARG, fMIN_SGL_NORM_ARG = [rTblAddr], 8
338 // x*(64/ln(2)) + Right Shifter
339 fma.s1 fNint = fNormX, f64DivLn2, fRightShifter
344 // Divide arguments into the following categories:
345 // Certain Underflow p11 - -inf < x <= MAX_SGL_ZERO_ARG
346 // Possible Underflow p13 - MAX_SGL_ZERO_ARG < x < MIN_SGL_NORM_ARG
347 // Certain Safe - MIN_SGL_NORM_ARG <= x <= MAX_SGL_NORM_ARG
348 // Possible Overflow p14 - MAX_SGL_NORM_ARG < x < MIN_SGL_OFLOW_ARG
349 // Certain Overflow p15 - MIN_SGL_OFLOW_ARG <= x < +inf
351 // If the input is really a single arg, then there will never be
352 // "Possible Overflow" arguments.
357 // check for overflow
358 fcmp.ge.s1 p15, p0 = fNormX, fMIN_SGL_OFLOW_ARG
365 // check for underflow and tiny (+0) result
366 fcmp.le.s1 p11, p0 = fNormX, fMAX_SGL_ZERO_ARG
371 fms.s1 fN = fNint, f1, fRightShifter // n in FP register
372 // branch out if overflow
373 (p15) br.cond.spnt EXP_CERTAIN_OVERFLOW
378 getf.sig rNJ = fNint // bits of n, j
379 // check for underflow and deno result
380 fcmp.lt.s1 p13, p0 = fNormX, fMIN_SGL_NORM_ARG
381 // branch out if underflow and tiny (+0) result
382 (p11) br.cond.spnt EXP_CERTAIN_UNDERFLOW
388 // check for possible overflow
389 fcmp.gt.s1 p14, p0 = fNormX, fMAX_SGL_NORM_ARG
390 extr.u rJ = rNJ, 0, 6 // bits of j
393 addl rN = 0xFFFF - 63, rNJ // biased and shifted n
394 fnma.s1 fR = fLn2Div64, fN, fNormX // R = x - N*ln(2)/64
400 shladd rJ = rJ, 3, rTblAddr // address in the 2^(j/64) table
402 shr rN = rN, 6 // biased n
409 shl rN = rN, 52 // 2^n bits in DP format
414 or rN = rN, rJ // bits of 2^n * 2^(j/64) in DP format
421 setf.d fT = rN // 2^n * 2^(j/64)
422 fma.s1 fP = fA3, fR, fA2 // A3*R + A2
427 fma.s1 fRSqr = fR, fR, f0 // R^2
434 fma.s1 fP = fP, fRSqr, fR // P = (A3*R + A2)*R^2 + R
441 // branch out if possible underflow
442 (p13) br.cond.spnt EXP_POSSIBLE_UNDERFLOW
443 // branch out if possible overflow result
444 (p14) br.cond.spnt EXP_POSSIBLE_OVERFLOW
450 // final result in the absence of over- and underflow
451 fma.s.s0 f8 = fP, fT, fT
452 // exit here in the absence of over- and underflow
457 EXP_POSSIBLE_OVERFLOW:
459 // Here if fMAX_SGL_NORM_ARG < x < fMIN_SGL_OFLOW_ARG
460 // This cannot happen if input is a single, only if input higher precision.
461 // Overflow is a possibility, not a certainty.
463 // Recompute result using status field 2 with user's rounding mode,
464 // and wre set. If result is larger than largest single, then we have
468 mov rGt_ln = 0x1007f // Exponent for largest single + 1 ulp
469 fsetc.s2 0x7F,0x42 // Get user's round mode, set wre
475 setf.exp fGt_pln = rGt_ln // Create largest single + 1 ulp
476 fma.s.s2 fWre_urm_f8 = fP, fT, fT // Result with wre set
483 fsetc.s2 0x7F,0x40 // Turn off wre in sf2
490 fcmp.ge.s1 p6, p0 = fWre_urm_f8, fGt_pln // Test for overflow
498 (p6) br.cond.spnt EXP_CERTAIN_OVERFLOW // Branch if overflow
504 fma.s.s0 f8 = fP, fT, fT
505 br.ret.sptk b0 // Exit if really no overflow
510 EXP_CERTAIN_OVERFLOW:
512 addl r17ones_m1 = 0x1FFFE, r0
514 setf.exp fTmp = r17ones_m1
520 alloc r32=ar.pfs,0,3,4,0
521 fmerge.s FR_X = f8,f8
525 mov GR_Parameter_TAG = 16
526 fma.s.s0 FR_RESULT = fTmp, fTmp, fTmp // Set I,O and +INF result
527 br.cond.sptk __libm_error_region
531 EXP_POSSIBLE_UNDERFLOW:
533 // Here if fMAX_SGL_ZERO_ARG < x < fMIN_SGL_NORM_ARG
534 // Underflow is a possibility, not a certainty
536 // We define an underflow when the answer with
538 // is zero (tiny numbers become zero)
540 // Notice (from below) that if we have an unlimited exponent range,
541 // then there is an extra machine number E between the largest denormal and
542 // the smallest normal.
544 // So if with unbounded exponent we round to E or below, then we are
545 // tiny and underflow has occurred.
547 // But notice that you can be in a situation where we are tiny, namely
548 // rounded to E, but when the exponent is bounded we round to smallest
549 // normal. So the answer can be the smallest normal with underflow.
552 // -----+--------------------+--------------------+-----
554 // 1.1...10 2^-3fff 1.1...11 2^-3fff 1.0...00 2^-3ffe
555 // 0.1...11 2^-3ffe (biased, 1)
556 // largest dn smallest normal
560 fsetc.s2 0x7F,0x41 // Get user's round mode, set ftz
567 fma.s.s2 fFtz_urm_f8 = fP, fT, fT // Result with ftz set
574 fsetc.s2 0x7F,0x40 // Turn off ftz in sf2
581 fcmp.eq.s1 p6, p7 = fFtz_urm_f8, f0 // Test for underflow
586 fma.s.s0 f8 = fP, fT, fT // Compute result, set I, maybe U
593 (p6) br.cond.spnt EXP_UNDERFLOW_COMMON // Branch if really underflow
594 (p7) br.ret.sptk b0 // Exit if really no underflow
598 EXP_CERTAIN_UNDERFLOW:
599 // Here if x < fMAX_SGL_ZERO_ARG
600 // Result will be zero (or smallest denorm if round to +inf) with I, U set
604 setf.exp fTmp = rTmp // Form small normal
611 fmerge.se fTmp = fTmp, f64DivLn2 // Small with non-trial signif
618 fma.s.s0 f8 = fTmp, fTmp, f0 // Set I,U, tiny (+0.0) result
619 br.cond.sptk EXP_UNDERFLOW_COMMON
623 EXP_UNDERFLOW_COMMON:
624 // Determine if underflow result is zero or nonzero
626 alloc r32=ar.pfs,0,3,4,0
627 fcmp.eq.s1 p6, p0 = f8, f0
634 fmerge.s FR_X = fNormX,fNormX
635 (p6) br.cond.spnt EXP_UNDERFLOW_ZERO
639 EXP_UNDERFLOW_NONZERO:
640 // Here if x < fMIN_SGL_NORM_ARG and result nonzero;
643 mov GR_Parameter_TAG = 17
644 nop.f 0 // FR_RESULT already set
645 br.cond.sptk __libm_error_region
650 // Here if x < fMIN_SGL_NORM_ARG and result zero;
653 mov GR_Parameter_TAG = 17
654 nop.f 0 // FR_RESULT already set
655 br.cond.sptk __libm_error_region
659 GLOBAL_IEEE754_END(expf)
662 LOCAL_LIBM_ENTRY(__libm_error_region)
665 add GR_Parameter_Y=-32,sp // Parameter 2 value
667 .save ar.pfs,GR_SAVE_PFS
668 mov GR_SAVE_PFS=ar.pfs // Save ar.pfs
672 add sp=-64,sp // Create new stack
674 mov GR_SAVE_GP=gp // Save gp
677 stfs [GR_Parameter_Y] = FR_Y,16 // Store Parameter 2 on stack
678 add GR_Parameter_X = 16,sp // Parameter 1 address
680 mov GR_SAVE_B0=b0 // Save b0
684 stfs [GR_Parameter_X] = FR_X // Store Parameter 1 on stack
686 add GR_Parameter_RESULT = 0,GR_Parameter_Y // Parameter 3 address
689 stfs [GR_Parameter_Y] = FR_RESULT // Store Parameter 3 on stack
690 add GR_Parameter_Y = -16,GR_Parameter_Y
691 br.call.sptk b0=__libm_error_support# // Call error handling function
695 add GR_Parameter_RESULT = 48,sp
701 ldfs f8 = [GR_Parameter_RESULT] // Get return result off stack
703 add sp = 64,sp // Restore stack pointer
704 mov b0 = GR_SAVE_B0 // Restore return address
707 mov gp = GR_SAVE_GP // Restore gp
708 mov ar.pfs = GR_SAVE_PFS // Restore ar.pfs
709 br.ret.sptk b0 // Return
712 LOCAL_LIBM_END(__libm_error_region)
715 .type __libm_error_support#,@function
716 .global __libm_error_support#