4 // Copyright (c) 2000 - 2005, Intel Corporation
5 // All rights reserved.
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37 // http://www.intel.com/software/products/opensource/libraries/num.htm.
40 //*********************************************************************
41 // 02/02/00 Initial Version
42 // 04/04/00 Unwind support added
43 // 08/15/00 Bundle added after call to __libm_error_support to properly
44 // set [the previously overwritten] GR_Parameter_RESULT.
45 // 07/07/01 Improved speed of all paths
46 // 05/20/02 Cleaned up namespace and sf0 syntax
47 // 11/20/02 Improved speed, algorithm based on expf
48 // 03/31/05 Reformatted delimiters between data tables
52 //*********************************************************************
53 // float expm1f(float)
55 // Overview of operation
56 //*********************************************************************
57 // 1. Inputs of Nan, Inf, Zero, NatVal handled with special paths
60 // Result = x, computed by x + x*x to handle appropriate flags and rounding
62 // 3. 2^-40 <= |x| < 2^-2
63 // Result determined by 8th order Taylor series polynomial
64 // expm1f(x) = x + A2*x^2 + ... + A8*x^8
67 // Here we know result is essentially -1 + eps, where eps only affects
68 // rounded result. Set I.
71 // Result overflows. Set I, O, and call error support
73 // 6. 2^-2 <= x < 88.7228 or -24.0 <= x < -2^-2
74 // This is the main path. The algorithm is described below:
76 // Take the input x. w is "how many log2/128 in x?"
82 // x = n*log2 + (log2/64)*j + R
84 // So, exp(x) = 2^n * 2^(j/64)* exp(R)
89 // actually all the entries of 2^(j/64) table are stored in DP and
90 // with exponent bits set to 0 -> multiplication on 2^n can be
91 // performed by doing logical "or" operation with bits presenting 2^n
93 // exp(R) = 1 + (exp(R) - 1)
94 // P = exp(R) - 1 approximated by Taylor series of 3rd degree
95 // P = A3*R^3 + A2*R^2 + R, A3 = 1/6, A2 = 1/2
98 // The final result is reconstructed as follows
99 // expm1f(x) = T*P + (T - 1.0)
102 //*********************************************************************
106 // expm1f(+qnan) = +qnan
107 // expm1f(-qnan) = -qnan
108 // expm1f(+snan) = +qnan
109 // expm1f(-snan) = -qnan
111 // expm1f(-inf) = -1.0
112 // expm1f(+inf) = +inf
114 // Overflow and Underflow
115 //*********************************************************************
116 // expm1f(x) = largest single normal when
117 // x = 88.7228 = 0x42b17217
119 // Underflow is handled as described in case 2 above.
123 //*********************************************************************
124 // Floating Point registers used:
126 // f6,f7, f9 -> f15, f32 -> f45
128 // General registers used:
131 // Predicate registers used:
135 //*********************************************************************
136 // integer registers used
160 GR_Parameter_RESULT = r37
161 GR_Parameter_TAG = r38
163 // floating point registers used
181 fMIN_SGL_OFLOW_ARG = f35
182 fMAX_SGL_NORM_ARG = f36
183 fMAX_SGL_MINUS_1_ARG = f37
204 LOCAL_OBJECT_START(_expf_table)
205 data8 0x3efa01a01a01a01a // A8 = 1/8!
206 data8 0x3f2a01a01a01a01a // A7 = 1/7!
207 data8 0x3f56c16c16c16c17 // A6 = 1/6!
208 data8 0x3f81111111111111 // A5 = 1/5!
209 data8 0x3fa5555555555555 // A4 = 1/4!
210 data8 0x3fc5555555555555 // A3 = 1/3!
212 data4 0x42b17218 // Smallest sgl arg to overflow sgl result
213 data4 0x42b17217 // Largest sgl arg to give sgl result
215 // 2^(j/64) table, j goes from 0 to 63
216 data8 0x0000000000000000 // 2^(0/64)
217 data8 0x00002C9A3E778061 // 2^(1/64)
218 data8 0x000059B0D3158574 // 2^(2/64)
219 data8 0x0000874518759BC8 // 2^(3/64)
220 data8 0x0000B5586CF9890F // 2^(4/64)
221 data8 0x0000E3EC32D3D1A2 // 2^(5/64)
222 data8 0x00011301D0125B51 // 2^(6/64)
223 data8 0x0001429AAEA92DE0 // 2^(7/64)
224 data8 0x000172B83C7D517B // 2^(8/64)
225 data8 0x0001A35BEB6FCB75 // 2^(9/64)
226 data8 0x0001D4873168B9AA // 2^(10/64)
227 data8 0x0002063B88628CD6 // 2^(11/64)
228 data8 0x0002387A6E756238 // 2^(12/64)
229 data8 0x00026B4565E27CDD // 2^(13/64)
230 data8 0x00029E9DF51FDEE1 // 2^(14/64)
231 data8 0x0002D285A6E4030B // 2^(15/64)
232 data8 0x000306FE0A31B715 // 2^(16/64)
233 data8 0x00033C08B26416FF // 2^(17/64)
234 data8 0x000371A7373AA9CB // 2^(18/64)
235 data8 0x0003A7DB34E59FF7 // 2^(19/64)
236 data8 0x0003DEA64C123422 // 2^(20/64)
237 data8 0x0004160A21F72E2A // 2^(21/64)
238 data8 0x00044E086061892D // 2^(22/64)
239 data8 0x000486A2B5C13CD0 // 2^(23/64)
240 data8 0x0004BFDAD5362A27 // 2^(24/64)
241 data8 0x0004F9B2769D2CA7 // 2^(25/64)
242 data8 0x0005342B569D4F82 // 2^(26/64)
243 data8 0x00056F4736B527DA // 2^(27/64)
244 data8 0x0005AB07DD485429 // 2^(28/64)
245 data8 0x0005E76F15AD2148 // 2^(29/64)
246 data8 0x0006247EB03A5585 // 2^(30/64)
247 data8 0x0006623882552225 // 2^(31/64)
248 data8 0x0006A09E667F3BCD // 2^(32/64)
249 data8 0x0006DFB23C651A2F // 2^(33/64)
250 data8 0x00071F75E8EC5F74 // 2^(34/64)
251 data8 0x00075FEB564267C9 // 2^(35/64)
252 data8 0x0007A11473EB0187 // 2^(36/64)
253 data8 0x0007E2F336CF4E62 // 2^(37/64)
254 data8 0x00082589994CCE13 // 2^(38/64)
255 data8 0x000868D99B4492ED // 2^(39/64)
256 data8 0x0008ACE5422AA0DB // 2^(40/64)
257 data8 0x0008F1AE99157736 // 2^(41/64)
258 data8 0x00093737B0CDC5E5 // 2^(42/64)
259 data8 0x00097D829FDE4E50 // 2^(43/64)
260 data8 0x0009C49182A3F090 // 2^(44/64)
261 data8 0x000A0C667B5DE565 // 2^(45/64)
262 data8 0x000A5503B23E255D // 2^(46/64)
263 data8 0x000A9E6B5579FDBF // 2^(47/64)
264 data8 0x000AE89F995AD3AD // 2^(48/64)
265 data8 0x000B33A2B84F15FB // 2^(49/64)
266 data8 0x000B7F76F2FB5E47 // 2^(50/64)
267 data8 0x000BCC1E904BC1D2 // 2^(51/64)
268 data8 0x000C199BDD85529C // 2^(52/64)
269 data8 0x000C67F12E57D14B // 2^(53/64)
270 data8 0x000CB720DCEF9069 // 2^(54/64)
271 data8 0x000D072D4A07897C // 2^(55/64)
272 data8 0x000D5818DCFBA487 // 2^(56/64)
273 data8 0x000DA9E603DB3285 // 2^(57/64)
274 data8 0x000DFC97337B9B5F // 2^(58/64)
275 data8 0x000E502EE78B3FF6 // 2^(59/64)
276 data8 0x000EA4AFA2A490DA // 2^(60/64)
277 data8 0x000EFA1BEE615A27 // 2^(61/64)
278 data8 0x000F50765B6E4540 // 2^(62/64)
279 data8 0x000FA7C1819E90D8 // 2^(63/64)
280 LOCAL_OBJECT_END(_expf_table)
284 GLOBAL_IEEE754_ENTRY(expm1f)
287 getf.exp rSignexp_x = f8 // Must recompute if x unorm
288 movl r64DivLn2 = 0x40571547652B82FE // 64/ln(2)
291 addl rTblAddr = @ltoff(_expf_table),gp
292 movl rRightShifter = 0x43E8000000000000 // DP Right Shifter
297 // point to the beginning of the table
298 ld8 rTblAddr = [rTblAddr]
299 fclass.m p14, p0 = f8 , 0x22 // test for -INF
300 mov rExp_mask = 0x1ffff // Exponent mask
304 fnorm.s1 fNormX = f8 // normalized x
310 setf.d f64DivLn2 = r64DivLn2 // load 64/ln(2) to FP reg
311 fclass.m p9, p0 = f8 , 0x0b // test for x unorm
312 mov rExp_bias = 0xffff // Exponent bias
315 // load Right Shifter to FP reg
316 setf.d fRightShifter = rRightShifter
317 movl rLn2Div64 = 0x3F862E42FEFA39EF // DP ln(2)/64 in GR
322 ldfpd fA8, fA7 = [rTblAddr], 16
323 fcmp.eq.s1 p13, p0 = f0, f8 // test for x = 0.0
324 mov rExp_half = 0xfffe
327 setf.d fLn2Div64 = rLn2Div64 // load ln(2)/64 to FP reg
329 (p9) br.cond.spnt EXPM1_UNORM // Branch if x unorm
335 ldfpd fA6, fA5 = [rTblAddr], 16
336 (p14) fms.s.s0 f8 = f0, f0, f1 // result if x = -inf
337 (p14) br.ret.spnt b0 // exit here if x = -inf
342 ldfpd fA4, fA3 = [rTblAddr], 16
343 fclass.m p15, p0 = f8 , 0x1e1 // test for NaT,NaN,+Inf
344 (p13) br.ret.spnt b0 // exit here if x =0.0, result is x
349 // overflow thresholds
350 ldfps fMIN_SGL_OFLOW_ARG, fMAX_SGL_NORM_ARG = [rTblAddr], 8
351 fma.s1 fXsq = fNormX, fNormX, f0 // x^2 for small path
352 and rExp_x = rExp_mask, rSignexp_x // Biased exponent of x
356 movl rM1_lim = 0xc1c00000 // Minus -1 limit (-24.0), SP
361 setf.exp fA2 = rExp_half
362 // x*(64/ln(2)) + Right Shifter
363 fma.s1 fNint = fNormX, f64DivLn2, fRightShifter
364 sub rExp_x = rExp_x, rExp_bias // True exponent of x
368 (p15) fma.s.s0 f8 = f8, f1, f0 // result if x = NaT,NaN,+Inf
369 (p15) br.ret.spnt b0 // exit here if x = NaT,NaN,+Inf
374 setf.s fMAX_SGL_MINUS_1_ARG = rM1_lim // -1 threshold, -24.0
376 cmp.gt p7, p8 = -2, rExp_x // Test |x| < 2^(-2)
381 (p7) cmp.gt.unc p6, p7 = -40, rExp_x // Test |x| < 2^(-40)
382 fma.s1 fA87 = fA8, fNormX, fA7 // Small path, A8*x+A7
387 fma.s1 fA65 = fA6, fNormX, fA5 // Small path, A6*x+A5
394 (p6) fma.s.s0 f8 = f8, f8, f8 // If x < 2^-40, result=x+x*x
395 (p6) br.ret.spnt b0 // Exit if x < 2^-40
401 // check for overflow
402 fcmp.gt.s1 p15, p14 = fNormX, fMIN_SGL_OFLOW_ARG
407 fms.s1 fN = fNint, f1, fRightShifter // n in FP register
414 (p7) fma.s1 fA43 = fA4, fNormX, fA3 // Small path, A4*x+A3
420 getf.sig rNJ = fNint // bits of n, j
421 (p7) fma.s1 fA8765 = fA87, fXsq, fA65 // Small path, A87*xsq+A65
426 (p7) fma.s1 fX3 = fXsq, fNormX, f0 // Small path, x^3
427 // branch out if overflow
428 (p15) br.cond.spnt EXPM1_CERTAIN_OVERFLOW
433 addl rN = 0xffff-63, rNJ // biased and shifted n
434 fnma.s1 fR = fLn2Div64, fN, fNormX // R = x - N*ln(2)/64
435 extr.u rJ = rNJ , 0 , 6 // bits of j
440 shladd rJ = rJ, 3, rTblAddr // address in the 2^(j/64) table
441 // check for certain -1
442 fcmp.le.s1 p13, p0 = fNormX, fMAX_SGL_MINUS_1_ARG
443 shr rN = rN, 6 // biased n
447 (p7) fma.s1 fA432 = fA43, fNormX, fA2 // Small path, A43*x+A2
455 shl rN = rN , 52 // 2^n bits in DP format
460 or rN = rN, rJ // bits of 2^n * 2^(j/64) in DP format
461 (p13) mov rTmp = 1 // Make small value for -1 path
467 setf.d fT = rN // 2^n
468 // check for possible overflow (only happens if input higher precision)
469 (p14) fcmp.gt.s1 p14, p0 = fNormX, fMAX_SGL_NORM_ARG
474 (p7) fma.s1 fA8765432 = fA8765, fX3, fA432 // A8765*x^3+A432
480 (p13) setf.exp fTmp = rTmp // Make small value for -1 path
481 fma.s1 fP = fA3, fR, fA2 // A3*R + A2
486 fma.s1 fRSqr = fR, fR, f0 // R^2
487 (p13) br.cond.spnt EXPM1_CERTAIN_MINUS_ONE // Branch if x < -24.0
493 (p7) fma.s.s0 f8 = fA8765432, fXsq, fNormX // Small path,
494 // result=xsq*A8765432+x
495 (p7) br.ret.spnt b0 // Exit if 2^-40 <= |x| < 2^-2
501 fma.s1 fP = fP, fRSqr, fR // P = (A3*R + A2)*Rsqr + R
508 fms.s1 fTm1 = fT, f1, f1 // T - 1.0
509 (p14) br.cond.spnt EXPM1_POSSIBLE_OVERFLOW
515 fma.s.s0 f8 = fP, fT, fTm1
516 br.ret.sptk b0 // Result for main path
517 // minus_one_limit < x < -2^-2
518 // and +2^-2 <= x < overflow_limit
525 getf.exp rSignexp_x = fNormX // Must recompute if x unorm
526 fcmp.eq.s0 p6, p0 = f8, f0 // Set D flag
527 br.cond.sptk EXPM1_COMMON
531 // here if result will be -1 and inexact, x <= -24.0
532 EXPM1_CERTAIN_MINUS_ONE:
535 fms.s.s0 f8 = fTmp, fTmp, f1 // Result -1, and Inexact set
540 EXPM1_POSSIBLE_OVERFLOW:
542 // Here if fMAX_SGL_NORM_ARG < x < fMIN_SGL_OFLOW_ARG
543 // This cannot happen if input is a single, only if input higher precision.
544 // Overflow is a possibility, not a certainty.
546 // Recompute result using status field 2 with user's rounding mode,
547 // and wre set. If result is larger than largest single, then we have
551 mov rGt_ln = 0x1007f // Exponent for largest sgl + 1 ulp
552 fsetc.s2 0x7F,0x42 // Get user's round mode, set wre
558 setf.exp fGt_pln = rGt_ln // Create largest single + 1 ulp
559 fma.s.s2 fWre_urm_f8 = fP, fT, fTm1 // Result with wre set
566 fsetc.s2 0x7F,0x40 // Turn off wre in sf2
573 fcmp.ge.s1 p6, p0 = fWre_urm_f8, fGt_pln // Test for overflow
581 (p6) br.cond.spnt EXPM1_CERTAIN_OVERFLOW // Branch if overflow
587 fma.s.s0 f8 = fP, fT, fTm1
588 br.ret.sptk b0 // Exit if really no overflow
593 EXPM1_CERTAIN_OVERFLOW:
595 addl rTmp = 0x1FFFE, r0;;
602 alloc r32 = ar.pfs, 0, 3, 4, 0 // get some registers
603 fmerge.s FR_X = fNormX,fNormX
607 mov GR_Parameter_TAG = 43
608 fma.s.s0 FR_RESULT = fTmp, fTmp, f0 // Set I,O and +INF result
609 br.cond.sptk __libm_error_region
613 GLOBAL_IEEE754_END(expm1f)
614 libm_alias_float_other (__expm1, expm1)
617 LOCAL_LIBM_ENTRY(__libm_error_region)
620 add GR_Parameter_Y=-32,sp // Parameter 2 value
622 .save ar.pfs,GR_SAVE_PFS
623 mov GR_SAVE_PFS=ar.pfs // Save ar.pfs
627 add sp=-64,sp // Create new stack
629 mov GR_SAVE_GP=gp // Save gp
632 stfs [GR_Parameter_Y] = FR_Y,16 // Store Parameter 2 on stack
633 add GR_Parameter_X = 16,sp // Parameter 1 address
635 mov GR_SAVE_B0=b0 // Save b0
639 stfs [GR_Parameter_X] = FR_X // Store Parameter 1 on stack
641 add GR_Parameter_RESULT = 0,GR_Parameter_Y // Parameter 3 address
644 stfs [GR_Parameter_Y] = FR_RESULT // Store Parameter 3 on stack
645 add GR_Parameter_Y = -16,GR_Parameter_Y
646 br.call.sptk b0=__libm_error_support# // Call error handling function
650 add GR_Parameter_RESULT = 48,sp
656 ldfs f8 = [GR_Parameter_RESULT] // Get return result off stack
658 add sp = 64,sp // Restore stack pointer
659 mov b0 = GR_SAVE_B0 // Restore return address
662 mov gp = GR_SAVE_GP // Restore gp
663 mov ar.pfs = GR_SAVE_PFS // Restore ar.pfs
664 br.ret.sptk b0 // Return
667 LOCAL_LIBM_END(__libm_error_region)
670 .type __libm_error_support#,@function
671 .global __libm_error_support#