3 // Copyright (C) 2000, 2001, Intel Corporation
4 // All rights reserved.
6 // Contributed 2/2/2000 by John Harrison, Ted Kubaska, Bob Norin, Shane Story,
7 // and Ping Tak Peter Tang of the Computational Software Lab, 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://developer.intel.com/opensource.
41 //==============================================================
42 // 2/02/00 Initial version
43 // 3/07/00 exp(inf) = inf but now does NOT call error support
44 // exp(-inf) = 0 but now does NOT call error support
45 // 4/04/00 Unwind support added
46 // 8/15/00 Bundle added after call to __libm_error_support to properly
47 // set [the previously overwritten] GR_Parameter_RESULT.
48 // 11/30/00 Reworked to shorten main path, widen main path to include all
49 // args in normal range, and add quick exit for 0, nan, inf.
50 // 12/05/00 Loaded constants earlier with setf to save 2 cycles.
53 //==============================================================
56 // Overview of operation
57 //==============================================================
58 // Take the input x. w is "how many log2/128 in x?"
61 // x = n log2/128 + r + delta
63 // n = 128M + index_1 + 2^4 index_2
64 // x = M log2 + (log2/128) index_1 + (log2/8) index_2 + r + delta
66 // exp(x) = 2^M 2^(index_1/128) 2^(index_2/8) exp(r) exp(delta)
68 // Get 2^(index_1/128) from table_1;
69 // Get 2^(index_2/8) from table_2;
70 // Calculate exp(r) by series
71 // r = x - n (log2/128)_high
72 // delta = - n (log2/128)_low
73 // Calculate exp(delta) as 1 + delta
77 //==============================================================
89 // Overfow and Underfow
90 //=======================
91 // exp(-x) = smallest double normal when
92 // x = -708.396 = c086232bdd7abcd2
94 // exp(x) = largest double normal when
95 // x = 709.7827 = 40862e42fefa39ef
100 //==============================================================
101 // Floating Point registers used:
103 // f9 -> f15, f32 -> f60
105 // General registers used:
108 // Predicate registers used:
111 #include "libm_support.h"
114 //==============================================================
123 exp_GR_index_2_16 = r39
125 exp_GR_biased_M = r40
126 exp_GR_index_1_16 = r41
129 exp_GR_sig_inv_ln2 = r44
135 exp_GR_rshf_2to56 = r49
138 exp_GR_exp_2tom56 = r51
140 exp_GR_17ones_m1 = r52
149 GR_Parameter_RESULT = r59
150 GR_Parameter_TAG = r60
158 EXP_INV_LN2_2TO63 = f7
166 exp_ln2_by_128_hi = f33
167 exp_ln2_by_128_lo = f34
182 EXP_MIN_DBL_OFLOW_ARG = f45
183 EXP_MAX_DBL_ZERO_ARG = f46
184 EXP_MAX_DBL_NORM_ARG = f47
185 EXP_MAX_DBL_UFLOW_ARG = f48
186 EXP_MIN_DBL_NORM_ARG = f49
205 //==============================================================
215 // ************* DO NOT CHANGE ORDER OF THESE TABLES ********************
217 // double-extended 1/ln(2)
218 // 3fff b8aa 3b29 5c17 f0bb be87fed0691d3e88
219 // 3fff b8aa 3b29 5c17 f0bc
220 // For speed the significand will be loaded directly with a movl and setf.sig
221 // and the exponent will be bias+63 instead of bias+0. Thus subsequent
222 // computations need to scale appropriately.
223 // The constant 128/ln(2) is needed for the computation of w. This is also
224 // obtained by scaling the computations.
226 // Two shifting constants are loaded directly with movl and setf.d.
227 // 1. EXP_RSHF_2TO56 = 1.1000..00 * 2^(63-7)
228 // This constant is added to x*1/ln2 to shift the integer part of
229 // x*128/ln2 into the rightmost bits of the significand.
230 // The result of this fma is EXP_W_2TO56_RSH.
231 // 2. EXP_RSHF = 1.1000..00 * 2^(63)
232 // This constant is subtracted from EXP_W_2TO56_RSH * 2^(-56) to give
233 // the integer part of w, n, as a floating-point number.
234 // The result of this fms is EXP_Nfloat.
238 ASM_TYPE_DIRECTIVE(exp_table_1,@object)
239 data8 0x40862e42fefa39f0 // smallest dbl overflow arg
240 data8 0xc0874c0000000000 // approx largest arg for zero result
241 data8 0x40862e42fefa39ef // largest dbl arg to give normal dbl result
242 data8 0xc086232bdd7abcd3 // largest dbl underflow arg
243 data8 0xc086232bdd7abcd2 // smallest dbl arg to give normal dbl result
245 data8 0xb17217f7d1cf79ab , 0x00003ff7 // ln2/128 hi
246 data8 0xc9e3b39803f2f6af , 0x00003fb7 // ln2/128 lo
248 // Table 1 is 2^(index_1/128) where
249 // index_1 goes from 0 to 15
251 data8 0x8000000000000000 , 0x00003FFF
252 data8 0x80B1ED4FD999AB6C , 0x00003FFF
253 data8 0x8164D1F3BC030773 , 0x00003FFF
254 data8 0x8218AF4373FC25EC , 0x00003FFF
255 data8 0x82CD8698AC2BA1D7 , 0x00003FFF
256 data8 0x8383594EEFB6EE37 , 0x00003FFF
257 data8 0x843A28C3ACDE4046 , 0x00003FFF
258 data8 0x84F1F656379C1A29 , 0x00003FFF
259 data8 0x85AAC367CC487B15 , 0x00003FFF
260 data8 0x8664915B923FBA04 , 0x00003FFF
261 data8 0x871F61969E8D1010 , 0x00003FFF
262 data8 0x87DB357FF698D792 , 0x00003FFF
263 data8 0x88980E8092DA8527 , 0x00003FFF
264 data8 0x8955EE03618E5FDD , 0x00003FFF
265 data8 0x8A14D575496EFD9A , 0x00003FFF
266 data8 0x8AD4C6452C728924 , 0x00003FFF
267 ASM_SIZE_DIRECTIVE(exp_table_1)
269 // Table 2 is 2^(index_1/8) where
270 // index_2 goes from 0 to 7
272 ASM_TYPE_DIRECTIVE(exp_table_2,@object)
273 data8 0x8000000000000000 , 0x00003FFF
274 data8 0x8B95C1E3EA8BD6E7 , 0x00003FFF
275 data8 0x9837F0518DB8A96F , 0x00003FFF
276 data8 0xA5FED6A9B15138EA , 0x00003FFF
277 data8 0xB504F333F9DE6484 , 0x00003FFF
278 data8 0xC5672A115506DADD , 0x00003FFF
279 data8 0xD744FCCAD69D6AF4 , 0x00003FFF
280 data8 0xEAC0C6E7DD24392F , 0x00003FFF
281 ASM_SIZE_DIRECTIVE (exp_table_2)
285 ASM_TYPE_DIRECTIVE(exp_p_table,@object)
286 data8 0x3f8111116da21757 //P_4
287 data8 0x3fa55555d787761c //P_3
288 data8 0x3fc5555555555414 //P_2
289 data8 0x3fdffffffffffd6a //P_1
290 ASM_SIZE_DIRECTIVE(exp_p_table)
301 .global __ieee754_exp#
306 alloc r32=ar.pfs,1,24,4,0
307 movl exp_GR_sig_inv_ln2 = 0xb8aa3b295c17f0bc // significand of 1/ln2
310 addl EXP_AD_TB1 = @ltoff(exp_table_1), gp
311 movl exp_GR_rshf_2to56 = 0x4768000000000000 ;; // 1.10000 2^(63+56)
315 // We do this fnorm right at the beginning to take any enabled
316 // faults and to normalize any input unnormals so that SWA is not taken.
318 ld8 EXP_AD_TB1 = [EXP_AD_TB1]
319 fclass.m p8,p0 = f8,0x07 // Test for x=0
320 mov exp_GR_17ones = 0x1FFFF
323 mov exp_TB1_size = 0x100
324 fnorm EXP_NORM_f8 = f8
325 mov exp_GR_exp_2tom56 = 0xffff-56
329 // Form two constants we need
330 // 1/ln2 * 2^63 to compute w = x * 1/ln2 * 128
331 // 1.1000..000 * 2^(63+63-7) to right shift int(w) into the significand
334 setf.sig EXP_INV_LN2_2TO63 = exp_GR_sig_inv_ln2 // form 1/ln2 * 2^63
335 setf.d EXP_RSHF_2TO56 = exp_GR_rshf_2to56 // Form const 1.100 * 2^(63+56)
336 fclass.m p9,p0 = f8,0x22 // Test for x=-inf
341 setf.exp EXP_2TOM56 = exp_GR_exp_2tom56 // form 2^-56 for scaling Nfloat
342 movl exp_GR_rshf = 0x43e8000000000000 // 1.10000 2^63 for right shift
345 mov exp_TB2_size = 0x80
346 (p8) fma.d f8 = f1,f1,f0 // quick exit for x=0
352 ldfpd EXP_MIN_DBL_OFLOW_ARG, EXP_MAX_DBL_ZERO_ARG = [EXP_AD_TB1],16
353 fclass.m p10,p0 = f8,0x21 // Test for x=+inf
358 (p9) fma.d f8 = f0,f0,f0 // quick exit for x=-inf
364 ldfpd EXP_MAX_DBL_NORM_ARG, EXP_MAX_DBL_UFLOW_ARG = [EXP_AD_TB1],16
365 setf.d EXP_RSHF = exp_GR_rshf // Form right shift const 1.100 * 2^63
366 fclass.m p11,p0 = f8,0xc3 // Test for x=nan
371 ldfd EXP_MIN_DBL_NORM_ARG = [EXP_AD_TB1],16
373 (p10) br.ret.spnt b0 // quick exit for x=+inf
378 ldfe exp_ln2_by_128_hi = [EXP_AD_TB1],16
386 ldfe exp_ln2_by_128_lo = [EXP_AD_TB1],16
387 (p11) fmerge.s f8 = EXP_NORM_f8, EXP_NORM_f8
388 (p11) br.ret.spnt b0 // quick exit for x=nan
392 // After that last load, EXP_AD_TB1 points to the beginning of table 1
394 // W = X * Inv_log2_by_128
395 // By adding 1.10...0*2^63 we shift and get round_int(W) in significand.
396 // We actually add 1.10...0*2^56 to X * Inv_log2 to do the same thing.
400 fma.s1 EXP_W_2TO56_RSH = EXP_NORM_f8, EXP_INV_LN2_2TO63, EXP_RSHF_2TO56
406 // Divide arguments into the following categories:
407 // Certain Underflow/zero p11 - -inf < x <= MAX_DBL_ZERO_ARG
408 // Certain Underflow p12 - MAX_DBL_ZERO_ARG < x <= MAX_DBL_UFLOW_ARG
409 // Possible Underflow p13 - MAX_DBL_UFLOW_ARG < x < MIN_DBL_NORM_ARG
410 // Certain Safe - MIN_DBL_NORM_ARG <= x <= MAX_DBL_NORM_ARG
411 // Possible Overflow p14 - MAX_DBL_NORM_ARG < x < MIN_DBL_OFLOW_ARG
412 // Certain Overflow p15 - MIN_DBL_OFLOW_ARG <= x < +inf
414 // If the input is really a double arg, then there will never be "Possible
415 // Underflow" or "Possible Overflow" arguments.
419 add EXP_AD_TB2 = exp_TB1_size, EXP_AD_TB1
420 fcmp.ge.s1 p15,p14 = EXP_NORM_f8,EXP_MIN_DBL_OFLOW_ARG
426 add EXP_AD_P = exp_TB2_size, EXP_AD_TB2
427 fcmp.le.s1 p11,p12 = EXP_NORM_f8,EXP_MAX_DBL_ZERO_ARG
433 ldfpd exp_P4, exp_P3 = [EXP_AD_P] ,16
434 (p14) fcmp.gt.unc.s1 p14,p0 = EXP_NORM_f8,EXP_MAX_DBL_NORM_ARG
435 (p15) br.cond.spnt L(EXP_CERTAIN_OVERFLOW)
440 // Nfloat = round_int(W)
441 // The signficand of EXP_W_2TO56_RSH contains the rounded integer part of W,
442 // as a twos complement number in the lower bits (that is, it may be negative).
443 // That twos complement number (called N) is put into exp_GR_N.
445 // Since EXP_W_2TO56_RSH is scaled by 2^56, it must be multiplied by 2^-56
446 // before the shift constant 1.10000 * 2^63 is subtracted to yield EXP_Nfloat.
447 // Thus, EXP_Nfloat contains the floating point version of N
452 (p12) fcmp.le.unc p12,p0 = EXP_NORM_f8,EXP_MAX_DBL_UFLOW_ARG
456 ldfpd exp_P2, exp_P1 = [EXP_AD_P]
457 fms.s1 EXP_Nfloat = EXP_W_2TO56_RSH, EXP_2TOM56, EXP_RSHF
458 (p11) br.cond.spnt L(EXP_CERTAIN_UNDERFLOW_ZERO)
463 getf.sig exp_GR_N = EXP_W_2TO56_RSH
464 (p13) fcmp.lt.unc p13,p0 = EXP_NORM_f8,EXP_MIN_DBL_NORM_ARG
470 // exp_GR_index_1 has index_1
471 // exp_GR_index_2_16 has index_2 * 16
472 // exp_GR_biased_M has M
473 // exp_GR_index_1_16 has index_1 * 16
477 and exp_GR_index_1 = 0x0f, exp_GR_N
478 fnma.s1 exp_r = EXP_Nfloat, exp_ln2_by_128_hi, EXP_NORM_f8
479 shr r2 = exp_GR_N, 0x7
482 and exp_GR_index_2_16 = 0x70, exp_GR_N
483 fnma.s1 exp_f = EXP_Nfloat, exp_ln2_by_128_lo, f1
489 // EXP_AD_T1 has address of T1
490 // EXP_AD_T2 has address if T2
493 addl exp_GR_biased_M = 0xffff, r2
494 add EXP_AD_T2 = EXP_AD_TB2, exp_GR_index_2_16
495 shladd EXP_AD_T1 = exp_GR_index_1, 4, EXP_AD_TB1
500 // Create Scale = 2^M
501 // r = x - Nfloat * ln2_by_128_hi
502 // f = 1 - Nfloat * ln2_by_128_lo
505 setf.exp EXP_2M = exp_GR_biased_M
506 ldfe exp_T2 = [EXP_AD_T2]
513 ldfe exp_T1 = [EXP_AD_T1]
522 fma.s1 exp_rsq = exp_r, exp_r, f0
527 fma.s1 exp_rP4pP3 = exp_r, exp_P4, exp_P3
536 fma.s1 exp_rcube = exp_r, exp_rsq, f0
541 fma.s1 exp_P_lo = exp_r, exp_rP4pP3, exp_P2
549 fma.s1 exp_P_hi = exp_rsq, exp_P1, exp_r
554 fma.s1 exp_S2 = exp_f,exp_T2,f0
561 fma.s1 exp_S1 = EXP_2M,exp_T1,f0
569 fma.s1 exp_P = exp_rcube, exp_P_lo, exp_P_hi
576 fma.s1 exp_S = exp_S1,exp_S2,f0
582 (p12) br.cond.spnt L(EXP_CERTAIN_UNDERFLOW)
583 (p13) br.cond.spnt L(EXP_POSSIBLE_UNDERFLOW)
584 (p14) br.cond.spnt L(EXP_POSSIBLE_OVERFLOW)
591 fma.d f8 = exp_S, exp_P, exp_S
592 br.ret.sptk b0 ;; // Normal path exit
596 L(EXP_POSSIBLE_OVERFLOW):
598 // We got an answer. EXP_MAX_DBL_NORM_ARG < x < EXP_MIN_DBL_OFLOW_ARG
599 // overflow is a possibility, not a certainty
609 fma.d.s2 exp_wre_urm_f8 = exp_S, exp_P, exp_S
613 // We define an overflow when the answer with
615 // user-defined rounding mode
618 // Is the exponent 1 more than the largest double?
619 // If so, go to ERROR RETURN, else get the answer and
622 // Largest double is 7FE (biased double)
623 // 7FE - 3FF + FFFF = 103FE
624 // Create + largest_double_plus_ulp
625 // Create - largest_double_plus_ulp
626 // Calculate answer with WRE set.
628 // Cases when answer is ldn+1 are as follows:
630 // --+----------|----------+------------
639 mov exp_GR_gt_ln = 0x103ff ;;
643 setf.exp exp_gt_pln = exp_GR_gt_ln
650 fcmp.ge.unc.s1 p6, p0 = exp_wre_urm_f8, exp_gt_pln
657 (p6) br.cond.spnt L(EXP_CERTAIN_OVERFLOW) ;; // Branch if really overflow
662 fma.d f8 = exp_S, exp_P, exp_S
663 br.ret.sptk b0 ;; // Exit if really no overflow
666 L(EXP_CERTAIN_OVERFLOW):
668 sub exp_GR_17ones_m1 = exp_GR_17ones, r0, 1 ;;
669 setf.exp f9 = exp_GR_17ones_m1
675 fmerge.s FR_X = f8,f8
679 mov GR_Parameter_TAG = 14
680 fma.d FR_RESULT = f9, f9, f0 // Set I,O and +INF result
681 br.cond.sptk __libm_error_region ;;
684 L(EXP_POSSIBLE_UNDERFLOW):
686 // We got an answer. EXP_MAX_DBL_UFLOW_ARG < x < EXP_MIN_DBL_NORM_ARG
687 // underflow is a possibility, not a certainty
689 // We define an underflow when the answer with
691 // is zero (tiny numbers become zero)
693 // Notice (from below) that if we have an unlimited exponent range,
694 // then there is an extra machine number E between the largest denormal and
695 // the smallest normal.
697 // So if with unbounded exponent we round to E or below, then we are
698 // tiny and underflow has occurred.
700 // But notice that you can be in a situation where we are tiny, namely
701 // rounded to E, but when the exponent is bounded we round to smallest
702 // normal. So the answer can be the smallest normal with underflow.
705 // -----+--------------------+--------------------+-----
707 // 1.1...10 2^-3fff 1.1...11 2^-3fff 1.0...00 2^-3ffe
708 // 0.1...11 2^-3ffe (biased, 1)
709 // largest dn smallest normal
718 fma.d.s2 exp_ftz_urm_f8 = exp_S, exp_P, exp_S
728 fcmp.eq.unc.s1 p6, p0 = exp_ftz_urm_f8, f0
734 (p6) br.cond.spnt L(EXP_CERTAIN_UNDERFLOW) ;; // Branch if really underflow
738 fma.d f8 = exp_S, exp_P, exp_S
739 br.ret.sptk b0 ;; // Exit if really no underflow
742 L(EXP_CERTAIN_UNDERFLOW):
745 fmerge.s FR_X = f8,f8
749 mov GR_Parameter_TAG = 15
750 fma.d FR_RESULT = exp_S, exp_P, exp_S // Set I,U and tiny result
751 br.cond.sptk __libm_error_region ;;
754 L(EXP_CERTAIN_UNDERFLOW_ZERO):
756 mov exp_GR_one = 1 ;;
757 setf.exp f9 = exp_GR_one
763 fmerge.s FR_X = f8,f8
767 mov GR_Parameter_TAG = 15
768 fma.d FR_RESULT = f9, f9, f0 // Set I,U and tiny (+0.0) result
769 br.cond.sptk __libm_error_region ;;
773 ASM_SIZE_DIRECTIVE(exp)
776 .proc __libm_error_region
780 add GR_Parameter_Y=-32,sp // Parameter 2 value
782 .save ar.pfs,GR_SAVE_PFS
783 mov GR_SAVE_PFS=ar.pfs // Save ar.pfs
787 add sp=-64,sp // Create new stack
789 mov GR_SAVE_GP=gp // Save gp
792 stfd [GR_Parameter_Y] = FR_Y,16 // STORE Parameter 2 on stack
793 add GR_Parameter_X = 16,sp // Parameter 1 address
795 mov GR_SAVE_B0=b0 // Save b0
799 stfd [GR_Parameter_X] = FR_X // STORE Parameter 1 on stack
800 add GR_Parameter_RESULT = 0,GR_Parameter_Y // Parameter 3 address
804 stfd [GR_Parameter_Y] = FR_RESULT // STORE Parameter 3 on stack
805 add GR_Parameter_Y = -16,GR_Parameter_Y
806 br.call.sptk b0=__libm_error_support# // Call error handling function
811 add GR_Parameter_RESULT = 48,sp
814 ldfd f8 = [GR_Parameter_RESULT] // Get return result off stack
816 add sp = 64,sp // Restore stack pointer
817 mov b0 = GR_SAVE_B0 // Restore return address
820 mov gp = GR_SAVE_GP // Restore gp
821 mov ar.pfs = GR_SAVE_PFS // Restore ar.pfs
822 br.ret.sptk b0 // Return
825 .endp __libm_error_region
826 ASM_SIZE_DIRECTIVE(__libm_error_region)
827 .type __libm_error_support#,@function
828 .global __libm_error_support#