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 // 09/05/02 Improved performance
45 // 01/17/03 Fixed to call error support when x=1024.0
46 // 03/31/05 Reformatted delimiters between data tables
49 //==============================================================
50 // double exp2(double)
52 // Overview of operation
53 //==============================================================
58 // Let x= (K + fh + fl + r), where
59 // K is an integer, fh= 0.b1 b2 b3 b4 b5,
60 // fl= 2^{-5}* 0.b6 b7 b8 b8 b10 (fh, fl >= 0),
62 // Th is a table that stores 2^fh (32 entries) rounded to
63 // double extended precision (only mantissa is stored)
64 // Tl is a table that stores 2^fl (32 entries) rounded to
65 // double extended precision (only mantissa is stored)
67 // 2^x is approximated as
68 // 2^K * Th [ f ] * Tl [ f ] * (1+c1*r+c2*r^2+c3*r^3+c4*r^4)
70 // Note: We use the following trick to speed up conversion from FP to integer:
72 // Let x = K + r, where K is an integer, and |r| <= 0.5
73 // Let N be the number of significand bits for the FP format used
74 // ( N=64 for double-extended, N=53 for double)
76 // Then let y = 1.5 * 2^(N-1) + x for RN mode
77 // K = y - 1.5 * 2^(N-1)
80 // If we want to obtain the integer part and the first m fractional bits of x,
81 // we can use the same trick, but with a constant of 1.5 * 2^(N-1-m):
84 // f = 0.b_1 b_2 ... b_m
87 // Then let y = 1.5 * 2^(N-1-m) + x for RN mode
88 // (K+f) = y - 1.5 * 2^(N-1-m)
93 //==============================================================
100 //==============================================================
137 GR_Parameter_RESULT = r39
138 GR_Parameter_TAG = r40
174 //==============================================================
180 LOCAL_OBJECT_START(poly_coeffs)
182 data8 0x3fac6b08d704a0c0, 0x3f83b2ab6fba4e77 // C_3 and C_4
183 data8 0xb17217f7d1cf79ab, 0x00003ffe // C_1
184 data8 0xf5fdeffc162c7541, 0x00003ffc // C_2
185 LOCAL_OBJECT_END(poly_coeffs)
188 LOCAL_OBJECT_START(T_table)
190 // 2^{0.00000 b6 b7 b8 b9 b10}
191 data8 0x8000000000000000, 0x8016302f17467628
192 data8 0x802c6436d0e04f50, 0x80429c17d77c18ed
193 data8 0x8058d7d2d5e5f6b0, 0x806f17687707a7af
194 data8 0x80855ad965e88b83, 0x809ba2264dada76a
195 data8 0x80b1ed4fd999ab6c, 0x80c83c56b50cf77f
196 data8 0x80de8f3b8b85a0af, 0x80f4e5ff089f763e
197 data8 0x810b40a1d81406d4, 0x81219f24a5baa59d
198 data8 0x813801881d886f7b, 0x814e67cceb90502c
199 data8 0x8164d1f3bc030773, 0x817b3ffd3b2f2e47
200 data8 0x8191b1ea15813bfd, 0x81a827baf7838b78
201 data8 0x81bea1708dde6055, 0x81d51f0b8557ec1c
202 data8 0x81eba08c8ad4536f, 0x820225f44b55b33b
203 data8 0x8218af4373fc25eb, 0x822f3c7ab205c89a
204 data8 0x8245cd9ab2cec048, 0x825c62a423d13f0c
205 data8 0x8272fb97b2a5894c, 0x828998760d01faf3
206 data8 0x82a0393fe0bb0ca8, 0x82b6ddf5dbc35906
208 // 2^{0.b1 b2 b3 b4 b5}
209 data8 0x8000000000000000, 0x82cd8698ac2ba1d7
210 data8 0x85aac367cc487b14, 0x88980e8092da8527
211 data8 0x8b95c1e3ea8bd6e6, 0x8ea4398b45cd53c0
212 data8 0x91c3d373ab11c336, 0x94f4efa8fef70961
213 data8 0x9837f0518db8a96f, 0x9b8d39b9d54e5538
214 data8 0x9ef5326091a111ad, 0xa27043030c496818
215 data8 0xa5fed6a9b15138ea, 0xa9a15ab4ea7c0ef8
216 data8 0xad583eea42a14ac6, 0xb123f581d2ac258f
217 data8 0xb504f333f9de6484, 0xb8fbaf4762fb9ee9
218 data8 0xbd08a39f580c36be, 0xc12c4cca66709456
219 data8 0xc5672a115506dadd, 0xc9b9bd866e2f27a2
220 data8 0xce248c151f8480e3, 0xd2a81d91f12ae45a
221 data8 0xd744fccad69d6af4, 0xdbfbb797daf23755
222 data8 0xe0ccdeec2a94e111, 0xe5b906e77c8348a8
223 data8 0xeac0c6e7dd24392e, 0xefe4b99bdcdaf5cb
224 data8 0xf5257d152486cc2c, 0xfa83b2db722a033a
225 LOCAL_OBJECT_END(T_table)
230 GLOBAL_LIBM_ENTRY(exp2)
234 alloc r32= ar.pfs, 1, 4, 4, 0
235 // will continue only for non-zero normal/denormal numbers
236 fclass.nm p12, p0= f8, 0x1b
237 // GR_TBL_START= pointer to C_1...C_4 followed by T_table
238 addl GR_TBL_START= @ltoff(poly_coeffs), gp
241 mov GR_OF_LIMIT= 0xffff + 10 // Exponent of overflow limit
242 movl GR_ROUNDVAL= 0x5a400000 // 1.5*2^(63-10) (SP)
246 // Form special constant 1.5*2^(63-10) to give integer part and first 10
247 // fractional bits of x
249 setf.s FR_ROUNDVAL= GR_ROUNDVAL // Form special constant
250 fcmp.lt.s1 p6, p8= f8, f0 // X<0 ?
254 ld8 GR_COEFF_START= [ GR_TBL_START ] // Load pointer to coeff table
256 (p12) br.cond.spnt SPECIAL_exp2 // Branch if nan, inf, zero
261 setf.exp FR_OF_LIMIT= GR_OF_LIMIT // Set overflow limit
262 movl GR_UF_LIMIT= 0xc4866000 // (-2^10-51) = -1075
267 ldfpd FR_COEFF3, FR_COEFF4= [ GR_COEFF_START ], 16 // load C_3, C_4
268 fma.s0 f8= f8, f1, f0 // normalize x
274 setf.s FR_UF_LIMIT= GR_UF_LIMIT // Set underflow limit
275 ldfe FR_COEFF1= [ GR_COEFF_START ], 16 // load C_1
276 mov GR_EXP_CORR= 0xffff-126
281 ldfe FR_COEFF2= [ GR_COEFF_START ], 16 // load C_2
282 fma.s1 FR_KF0= f8, f1, FR_ROUNDVAL // y= x + 1.5*2^(63-10)
289 fms.s1 FR_KF= FR_KF0, f1, FR_ROUNDVAL // (K+f)
295 getf.sig GR_KF0= FR_KF0 // (K+f)*2^10= round_to_int(y)
296 fcmp.ge.s1 p12, p7= f8, FR_OF_LIMIT // x >= overflow threshold ?
297 add GR_LOG_TBL= 256, GR_COEFF_START // Pointer to high T_table
302 and GR_F_low= GR_KF0, GR_MASK_low // f_low
303 and GR_F_high= GR_MASK, GR_KF0 // f_high*32
304 shr GR_K= GR_KF0, 10 // K
309 shladd GR_Flow_ADDR= GR_F_low, 3, GR_COEFF_START // address of 2^{f_low}
310 add GR_BIAS= GR_K, GR_EXP_CORR // K= bias-2*63
311 shr GR_Fh= GR_F_high, 5 // f_high
316 setf.exp FR_2_TO_K= GR_BIAS // 2^{K-126}
317 fnma.s1 FR_R= FR_KF, f1, f8 // r= x - (K+f)
318 shladd GR_Fh_ADDR= GR_Fh, 3, GR_LOG_TBL // address of 2^{f_high}
321 ldf8 FR_T_low= [ GR_Flow_ADDR ] // load T_low= 2^{f_low}
322 movl GR_EMIN= 0xc47f8000 // EMIN= -1022
327 ldf8 FR_T_high= [ GR_Fh_ADDR ] // load T_high= 2^{f_high}
328 (p7) fcmp.lt.s1 p12, p7= f8, FR_UF_LIMIT // x<underflow threshold ?
334 setf.s FR_EXPMIN= GR_EMIN // FR_EXPMIN= EMIN
335 fma.s1 FR_P34= FR_COEFF4, FR_R, FR_COEFF3 // P34= C_3+C_4*r
340 fma.s1 FR_R2= FR_R, FR_R, f0 // r*r
341 (p12) br.cond.spnt OUT_RANGE_exp2
347 fma.s1 FR_P12= FR_COEFF2, FR_R, FR_COEFF1 // P12= C_1+C_2*r
354 fma.s1 FR_T_low_K= FR_T_low, FR_2_TO_K, f0 // T= 2^{K-126}*T_low
361 fma.s1 FR_P14= FR_R2, FR_P34, FR_P12 // P14= P12+r2*P34
368 fma.s1 FR_T= FR_T_low_K, FR_T_high, f0 // T= T*T_high
375 fcmp.lt.s0 p6, p8= f8, FR_EXPMIN // underflow (x<EMIN) ?
382 fma.s1 FR_P= FR_P14, FR_R, f0 // P= P14*r
389 fma.d.s0 f8= FR_P, FR_T, FR_T // result= T+T*P
390 (p8) br.ret.sptk b0 // return
395 (p6) mov GR_Parameter_TAG= 162
397 (p6) br.cond.sptk __libm_error_region
405 fclass.m p6, p0= f8, 0x22 // x= -Infinity ?
412 fclass.m p7, p0= f8, 0x21 // x= +Infinity ?
419 fclass.m p8, p0= f8, 0x7 // x= +/-Zero ?
424 (p6) mov f8= f0 // exp2(-Infinity)= 0
432 (p7) br.ret.spnt b0 // exp2(+Infinity)= +Infinity
438 (p8) mov f8= f1 // exp2(+/-0)= 1
445 fma.d.s0 f8= f8, f1, f0 // Remaining cases: NaNs
456 (p8) mov GR_EXPMAX= 0x1fffe
463 (p8) mov GR_Parameter_TAG= 161
464 (p8) setf.exp FR_R= GR_EXPMAX
471 (p8) fma.d.s0 f8= FR_R, FR_R, f0 // Create overflow
476 (p6) mov GR_Parameter_TAG= 162
477 (p6) mov GR_EXPMAX= 1
484 (p6) setf.exp FR_R= GR_EXPMAX
491 (p6) fma.d.s0 f8= FR_R, FR_R, f0 // Create underflow
496 GLOBAL_LIBM_END(exp2)
499 LOCAL_LIBM_ENTRY(__libm_error_region)
503 add GR_Parameter_Y= -32, sp // Parameter 2 value
505 .save ar.pfs, GR_SAVE_PFS
506 mov GR_SAVE_PFS= ar.pfs // Save ar.pfs
511 add sp= -64, sp // Create new stack
513 mov GR_SAVE_GP= gp // Save gp
518 stfd [ GR_Parameter_Y ]= FR_Y, 16 // STORE Parameter 2 on stack
519 add GR_Parameter_X= 16, sp // Parameter 1 address
521 mov GR_SAVE_B0= b0 // Save b0
527 stfd [ GR_Parameter_X ]= FR_X // STORE Parameter 1 on stack
528 add GR_Parameter_RESULT= 0, GR_Parameter_Y // Parameter 3 address
532 stfd [ GR_Parameter_Y ]= FR_RESULT // STORE Parameter 3 on stack
533 add GR_Parameter_Y= -16, GR_Parameter_Y
534 br.call.sptk b0= __libm_error_support# // Call error handling function
539 add GR_Parameter_RESULT= 48, sp
546 ldfd f8= [ GR_Parameter_RESULT ] // Get return result off stack
548 add sp= 64, sp // Restore stack pointer
549 mov b0= GR_SAVE_B0 // Restore return address
554 mov gp= GR_SAVE_GP // Restore gp
555 mov ar.pfs= GR_SAVE_PFS // Restore ar.pfs
556 br.ret.sptk b0 // Return
561 LOCAL_LIBM_END(__libm_error_region)
563 .type __libm_error_support#, @function
564 .global __libm_error_support#