4 // Copyright (c) 2000 - 2005, Intel Corporation
5 // All rights reserved.
7 // Contributed 2000 by the Intel Numerics Group, Intel Corporation
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21 // products derived from this software without specific prior written
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41 //==============================================================
42 // 08/25/00 Initial version
43 // 05/20/02 Cleaned up namespace and sf0 syntax
44 // 09/06/02 Improved performance; no inexact flags on exact cases
45 // 01/29/03 Added missing } to bundle templates
46 // 12/16/04 Call error handling on underflow.
47 // 03/31/05 Reformatted delimiters between data tables
50 //==============================================================
51 // double exp10(double)
53 // Overview of operation
54 //==============================================================
59 // Let x= (K + fh + fl + r)/log2(10), where
60 // K is an integer, fh= 0.b1 b2 b3 b4 b5,
61 // fl= 2^{-5}* 0.b6 b7 b8 b8 b10 (fh, fl >= 0),
63 // Th is a table that stores 2^fh (32 entries) rounded to
64 // double extended precision (only mantissa is stored)
65 // Tl is a table that stores 2^fl (32 entries) rounded to
66 // double extended precision (only mantissa is stored)
68 // 10^x is approximated as
69 // 2^K * Th [ f ] * Tl [ f ] * (1+c1*e+c1*r+c2*r^2+c3*r^3+c4*r^4),
70 // where e= (x*log2(10)_hi-RN(x*log2(10)_hi))+log2(10)_lo*x
72 // Note there are only 22 non-zero values that produce an exact result:
73 // 1.0, 2.0, ... 22.0.
74 // We test for these cases and use s1 to avoid setting the inexact flag.
77 //==============================================================
84 //==============================================================
123 GR_Parameter_RESULT = r39
124 GR_Parameter_TAG = r40
171 //==============================================================
177 LOCAL_OBJECT_START(poly_coeffs)
179 data8 0xd49a784bcd1b8afe, 0x00003fcb // log2(10)*2^(10-63)
180 data8 0x9257edfe9b5fb698, 0x3fbf // log2(10)_low (bits 64...127)
181 data8 0x3fac6b08d704a0c0, 0x3f83b2ab6fba4e77 // C_3 and C_4
182 data8 0xb17217f7d1cf79ab, 0x00003ffe // C_1
183 data8 0xf5fdeffc162c7541, 0x00003ffc // C_2
184 LOCAL_OBJECT_END(poly_coeffs)
187 LOCAL_OBJECT_START(T_table)
189 // 2^{0.00000 b6 b7 b8 b9 b10}
190 data8 0x8000000000000000, 0x8016302f17467628
191 data8 0x802c6436d0e04f50, 0x80429c17d77c18ed
192 data8 0x8058d7d2d5e5f6b0, 0x806f17687707a7af
193 data8 0x80855ad965e88b83, 0x809ba2264dada76a
194 data8 0x80b1ed4fd999ab6c, 0x80c83c56b50cf77f
195 data8 0x80de8f3b8b85a0af, 0x80f4e5ff089f763e
196 data8 0x810b40a1d81406d4, 0x81219f24a5baa59d
197 data8 0x813801881d886f7b, 0x814e67cceb90502c
198 data8 0x8164d1f3bc030773, 0x817b3ffd3b2f2e47
199 data8 0x8191b1ea15813bfd, 0x81a827baf7838b78
200 data8 0x81bea1708dde6055, 0x81d51f0b8557ec1c
201 data8 0x81eba08c8ad4536f, 0x820225f44b55b33b
202 data8 0x8218af4373fc25eb, 0x822f3c7ab205c89a
203 data8 0x8245cd9ab2cec048, 0x825c62a423d13f0c
204 data8 0x8272fb97b2a5894c, 0x828998760d01faf3
205 data8 0x82a0393fe0bb0ca8, 0x82b6ddf5dbc35906
207 // 2^{0.b1 b2 b3 b4 b5}
208 data8 0x8000000000000000, 0x82cd8698ac2ba1d7
209 data8 0x85aac367cc487b14, 0x88980e8092da8527
210 data8 0x8b95c1e3ea8bd6e6, 0x8ea4398b45cd53c0
211 data8 0x91c3d373ab11c336, 0x94f4efa8fef70961
212 data8 0x9837f0518db8a96f, 0x9b8d39b9d54e5538
213 data8 0x9ef5326091a111ad, 0xa27043030c496818
214 data8 0xa5fed6a9b15138ea, 0xa9a15ab4ea7c0ef8
215 data8 0xad583eea42a14ac6, 0xb123f581d2ac258f
216 data8 0xb504f333f9de6484, 0xb8fbaf4762fb9ee9
217 data8 0xbd08a39f580c36be, 0xc12c4cca66709456
218 data8 0xc5672a115506dadd, 0xc9b9bd866e2f27a2
219 data8 0xce248c151f8480e3, 0xd2a81d91f12ae45a
220 data8 0xd744fccad69d6af4, 0xdbfbb797daf23755
221 data8 0xe0ccdeec2a94e111, 0xe5b906e77c8348a8
222 data8 0xeac0c6e7dd24392e, 0xefe4b99bdcdaf5cb
223 data8 0xf5257d152486cc2c, 0xfa83b2db722a033a
224 LOCAL_OBJECT_END(T_table)
229 GLOBAL_IEEE754_ENTRY(exp10)
233 alloc r32= ar.pfs, 1, 4, 4, 0
234 // will continue only for non-zero normal/denormal numbers
235 fclass.nm.unc p12, p7= f8, 0x1b
236 mov GR_BIAS53= 0xffff+63-10
239 // GR_TBL_START= pointer to log2(10), C_1...C_4 followed by T_table
240 addl GR_TBL_START= @ltoff(poly_coeffs), gp
241 movl GR_ROUNDVAL= 0x3fc00000 // 1.5 (SP)
246 ld8 GR_COEFF_START= [ GR_TBL_START ] // Load pointer to coeff table
247 fcmp.lt.s1 p6, p8= f8, f0 // X<0 ?
253 setf.exp FR_2P53= GR_BIAS53 // 2^{63-10}
254 movl GR_UF_LIMIT= 0xc07439b746e36b52 // (-2^10-51) / log2(10)
257 setf.s FR_ROUNDVAL= GR_ROUNDVAL
258 movl GR_OF_LIMIT= 0x40734413509f79fe // Overflow threshold
263 ldfe FR_LOG2_10= [ GR_COEFF_START ], 16 // load log2(10)*2^(10-63)
264 movl GR_SNORM_LIMIT= 0xc0733a7146f72a41 // Smallest normal threshold
269 (p12) br.cond.spnt SPECIAL_exp10 // Branch if nan, inf, zero
274 ldfe FR_L2_10_low= [ GR_COEFF_START ], 16 // load log2(10)_low
275 setf.d FR_OF_LIMIT= GR_OF_LIMIT // Set overflow limit
276 fma.s0 f8= f8, f1, f0 // normalize x
281 ldfpd FR_COEFF3, FR_COEFF4= [ GR_COEFF_START ], 16 // load C_3, C_4
282 (p8) fcvt.fx.s1 FR_int_x = f8 // Convert x to integer
286 setf.d FR_UF_LIMIT= GR_UF_LIMIT // Set underflow limit
287 fma.s1 FR_KF0= f8, FR_LOG2_10, FR_ROUNDVAL // y= (x*log2(10)*2^10 +
288 // 1.5*2^63) * 2^(-63)
289 mov GR_EXP_CORR= 0xffff-126
294 setf.d FR_SNORM_LIMIT= GR_SNORM_LIMIT // Set smallest normal limit
295 fma.s1 FR_L2_10_high= FR_LOG2_10, FR_2P53, f0 // FR_LOG2_10= log2(10)_hi
301 ldfe FR_COEFF1= [ GR_COEFF_START ], 16 // load C_1
302 fms.s1 FR_KF= FR_KF0, f1, FR_ROUNDVAL // (K+f)*2^(10-63)
308 ldfe FR_COEFF2= [ GR_COEFF_START ], 16 // load C_2
309 fma.s1 FR_LOG2_10= f8, FR_L2_10_high, f0 // y0= x*log2(10)_hi
315 getf.sig GR_KF0= FR_KF0 // (K+f)*2^10= round_to_int(y)
316 (p8) movl GR_exact_limit= 0x41b00000 // Largest x for exact result,
322 add GR_LOG_TBL= 256, GR_COEFF_START // Pointer to high T_table
323 fcmp.gt.s1 p12, p7= f8, FR_OF_LIMIT // x>overflow threshold ?
329 (p8) setf.s FR_exact_limit = GR_exact_limit // Largest x for exact result
330 (p8) fcvt.xf FR_int_x = FR_int_x // Integral part of x
331 shr GR_K= GR_KF0, 10 // K
334 and GR_F_high= GR_MASK, GR_KF0 // f_high*32
335 fnma.s1 FR_R= FR_KF, FR_2P53, FR_LOG2_10 // r= x*log2(10)-2^{63-10}*
336 // [ (K+f)*2^{10-63} ]
337 and GR_F_low= GR_KF0, GR_MASK_low // f_low
342 shladd GR_Flow_ADDR= GR_F_low, 3, GR_COEFF_START // address of 2^{f_low}
343 add GR_BIAS= GR_K, GR_EXP_CORR // K= bias-2*63
344 shr GR_Fh= GR_F_high, 5 // f_high
349 setf.exp FR_2_TO_K= GR_BIAS // 2^{K-126}
350 (p7) fcmp.lt.s1 p12, p7= f8, FR_UF_LIMIT // x<underflow threshold ?
351 shladd GR_Fh_ADDR= GR_Fh, 3, GR_LOG_TBL // address of 2^{f_high}
354 ldf8 FR_T_low= [ GR_Flow_ADDR ] // load T_low= 2^{f_low}
355 fms.s1 FR_DX_L210= f8, FR_L2_10_high, FR_LOG2_10 // x*log2(10)_hi-
362 ldf8 FR_T_high= [ GR_Fh_ADDR ] // load T_high= 2^{f_high}
363 fma.s1 FR_P34= FR_COEFF4, FR_R, FR_COEFF3 // P34= C_3+C_4*r
368 fma.s1 FR_R2= FR_R, FR_R, f0 // r*r
369 (p12) br.cond.spnt OUT_RANGE_exp10
375 // e= (x*log2(10)_hi-RN(x*log2(10)_hi))+log2(10)_lo*x
376 fma.s1 FR_E0= f8, FR_L2_10_low, FR_DX_L210
377 cmp.eq p7,p9= r0,r0 // Assume inexact result
381 fma.s1 FR_P12= FR_COEFF2, FR_R, FR_COEFF1 // P12= C_1+C_2*r
388 (p8) fcmp.eq.s1 p9,p7= FR_int_x, f8 // Test x positive integer
393 fma.s1 FR_T_low_K= FR_T_low, FR_2_TO_K, f0 // T= 2^{K-126}*T_low
400 fcmp.ge.s1 p11,p0= f8, FR_SNORM_LIMIT // Test x for normal range
407 fma.s1 FR_E= FR_E0, FR_COEFF1, f0 // E= C_1*e
412 fma.s1 FR_P14= FR_R2, FR_P34, FR_P12 // P14= P12+r2*P34
417 // If x a positive integer, will it produce an exact result?
418 // p7 result will be inexact
419 // p9 result will be exact
422 (p9) fcmp.le.s1 p9,p7= f8, FR_exact_limit // Test x gives exact result
427 fma.s1 FR_T= FR_T_low_K, FR_T_high, f0 // T= T*T_high
434 fma.s1 FR_P= FR_P14, FR_R, FR_E // P= P14*r+E
439 .pred.rel "mutex",p7,p9
442 (p7) fma.d.s0 f8= FR_P, FR_T, FR_T // result= T+T*P, inexact set
447 (p9) fma.d.s1 f8= FR_P, FR_T, FR_T // result= T+T*P, exact use s1
448 (p11) br.ret.sptk b0 // return, if result normal
452 // Here if result in denormal range (and not zero)
455 mov GR_Parameter_TAG= 265
456 br.cond.sptk __libm_error_region // Branch to error handling
463 fclass.m p6, p0= f8, 0x22 // x= -Infinity ?
470 fclass.m p7, p0= f8, 0x21 // x= +Infinity ?
477 fclass.m p8, p0= f8, 0x7 // x= +/-Zero ?
482 (p6) mov f8= f0 // exp10(-Infinity)= 0
490 (p7) br.ret.spnt b0 // exp10(+Infinity)= +Infinity
496 (p8) mov f8= f1 // exp10(+/-0)= 1
503 fma.d.s0 f8= f8, f1, f0 // Remaining cases: NaNs
514 .pred.rel "mutex",p6,p8
516 (p8) mov GR_EXPMAX= 0x1fffe
517 (p6) mov GR_EXPMAX= 1
523 setf.exp FR_R= GR_EXPMAX
524 (p8) mov GR_Parameter_TAG= 166
525 (p6) mov GR_Parameter_TAG= 265
531 fma.d.s0 f8= FR_R, FR_R, f0 // Create overflow/underflow
532 br.cond.sptk __libm_error_region // Branch to error handling
536 GLOBAL_IEEE754_END(exp10)
537 weak_alias (exp10, pow10)
540 LOCAL_LIBM_ENTRY(__libm_error_region)
544 add GR_Parameter_Y= -32, sp // Parameter 2 value
546 .save ar.pfs, GR_SAVE_PFS
547 mov GR_SAVE_PFS= ar.pfs // Save ar.pfs
552 add sp= -64, sp // Create new stack
554 mov GR_SAVE_GP= gp // Save gp
559 stfd [ GR_Parameter_Y ]= FR_Y, 16 // STORE Parameter 2 on stack
560 add GR_Parameter_X= 16, sp // Parameter 1 address
562 mov GR_SAVE_B0= b0 // Save b0
568 stfd [ GR_Parameter_X ]= FR_X // STORE Parameter 1 on stack
569 add GR_Parameter_RESULT= 0, GR_Parameter_Y // Parameter 3 address
573 stfd [ GR_Parameter_Y ]= FR_RESULT // STORE Parameter 3 on stack
574 add GR_Parameter_Y= -16, GR_Parameter_Y
575 br.call.sptk b0= __libm_error_support# // Call error handling function
580 add GR_Parameter_RESULT= 48, sp
587 ldfd f8= [ GR_Parameter_RESULT ] // Get return result off stack
589 add sp= 64, sp // Restore stack pointer
590 mov b0= GR_SAVE_B0 // Restore return address
595 mov gp= GR_SAVE_GP // Restore gp
596 mov ar.pfs= GR_SAVE_PFS // Restore ar.pfs
597 br.ret.sptk b0 // Return
602 LOCAL_LIBM_END(__libm_error_region)
604 .type __libm_error_support#, @function
605 .global __libm_error_support#