4 // Copyright (c) 2000 - 2005, Intel Corporation
5 // All rights reserved.
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9 // modification, are permitted provided that the following conditions are
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16 // notice, this list of conditions and the following disclaimer in the
17 // documentation and/or other materials provided with the distribution.
19 // * The name of Intel Corporation may not be used to endorse or promote
20 // products derived from this software without specific prior written
23 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
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25 // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
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33 // SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
35 // Intel Corporation is the author of this code, and requests that all
36 // problem reports or change requests be submitted to it directly at
37 // http://www.intel.com/software/products/opensource/libraries/num.htm.
40 //==============================================================
41 // 08/25/00 Initial version
42 // 05/20/02 Cleaned up namespace and sf0 syntax
43 // 09/05/02 Improved performance and accuracy
44 // 01/17/03 Fixed to call error support when x=128.0
45 // 03/31/05 Reformatted delimiters between data tables
48 //==============================================================
51 // Overview of operation
52 //==============================================================
57 // Let x= (K + fh + fl + r), where
58 // K is an integer, fh= 0.b1 b2 b3 b4 b5,
59 // fl= 2^{-5}* 0.b6 b7 b8 b8 b10 (fh, fl >= 0),
61 // Th is a table that stores 2^fh (32 entries) rounded to
62 // double extended precision (only mantissa is stored)
63 // Tl is a table that stores 2^fl (32 entries) rounded to
64 // double extended precision (only mantissa is stored)
66 // 2^x is approximated as
67 // 2^K * Th [ f ] * Tl [ f ] * (1+c1*r+c2*r^2)
69 // Note: We use the following trick to speed up conversion from FP to integer:
71 // Let x = K + r, where K is an integer, and |r| <= 0.5
72 // Let N be the number of significand bits for the FP format used
73 // ( N=64 for double-extended, N=53 for double)
75 // Then let y = 1.5 * 2^(N-1) + x for RN mode
76 // K = y - 1.5 * 2^(N-1)
79 // If we want to obtain the integer part and the first m fractional bits of x,
80 // we can use the same trick, but with a constant of 1.5 * 2^(N-1-m):
83 // f = 0.b_1 b_2 ... b_m
86 // Then let y = 1.5 * 2^(N-1-m) + x for RN mode
87 // (K+f) = y - 1.5 * 2^(N-1-m)
92 //==============================================================
99 //==============================================================
136 GR_Parameter_RESULT = r39
137 GR_Parameter_TAG = r40
168 //==============================================================
174 LOCAL_OBJECT_START(poly_coeffs)
176 data8 0xb17217f7d1cf79ab, 0x00003ffe // C_1
177 data8 0xf5fdeffc162c7541, 0x00003ffc // C_2
178 LOCAL_OBJECT_END(poly_coeffs)
181 LOCAL_OBJECT_START(T_table)
183 // 2^{0.00000 b6 b7 b8 b9 b10}
184 data8 0x8000000000000000, 0x8016302f17467628
185 data8 0x802c6436d0e04f50, 0x80429c17d77c18ed
186 data8 0x8058d7d2d5e5f6b0, 0x806f17687707a7af
187 data8 0x80855ad965e88b83, 0x809ba2264dada76a
188 data8 0x80b1ed4fd999ab6c, 0x80c83c56b50cf77f
189 data8 0x80de8f3b8b85a0af, 0x80f4e5ff089f763e
190 data8 0x810b40a1d81406d4, 0x81219f24a5baa59d
191 data8 0x813801881d886f7b, 0x814e67cceb90502c
192 data8 0x8164d1f3bc030773, 0x817b3ffd3b2f2e47
193 data8 0x8191b1ea15813bfd, 0x81a827baf7838b78
194 data8 0x81bea1708dde6055, 0x81d51f0b8557ec1c
195 data8 0x81eba08c8ad4536f, 0x820225f44b55b33b
196 data8 0x8218af4373fc25eb, 0x822f3c7ab205c89a
197 data8 0x8245cd9ab2cec048, 0x825c62a423d13f0c
198 data8 0x8272fb97b2a5894c, 0x828998760d01faf3
199 data8 0x82a0393fe0bb0ca8, 0x82b6ddf5dbc35906
201 // 2^{0.b1 b2 b3 b4 b5}
202 data8 0x8000000000000000, 0x82cd8698ac2ba1d7
203 data8 0x85aac367cc487b14, 0x88980e8092da8527
204 data8 0x8b95c1e3ea8bd6e6, 0x8ea4398b45cd53c0
205 data8 0x91c3d373ab11c336, 0x94f4efa8fef70961
206 data8 0x9837f0518db8a96f, 0x9b8d39b9d54e5538
207 data8 0x9ef5326091a111ad, 0xa27043030c496818
208 data8 0xa5fed6a9b15138ea, 0xa9a15ab4ea7c0ef8
209 data8 0xad583eea42a14ac6, 0xb123f581d2ac258f
210 data8 0xb504f333f9de6484, 0xb8fbaf4762fb9ee9
211 data8 0xbd08a39f580c36be, 0xc12c4cca66709456
212 data8 0xc5672a115506dadd, 0xc9b9bd866e2f27a2
213 data8 0xce248c151f8480e3, 0xd2a81d91f12ae45a
214 data8 0xd744fccad69d6af4, 0xdbfbb797daf23755
215 data8 0xe0ccdeec2a94e111, 0xe5b906e77c8348a8
216 data8 0xeac0c6e7dd24392e, 0xefe4b99bdcdaf5cb
217 data8 0xf5257d152486cc2c, 0xfa83b2db722a033a
218 LOCAL_OBJECT_END(T_table)
223 WEAK_LIBM_ENTRY(exp2f)
227 alloc r32= ar.pfs, 1, 4, 4, 0
228 // will continue only for non-zero normal/denormal numbers
229 fclass.nm p12, p0= f8, 0x1b
230 // GR_TBL_START= pointer to C_1...C_2 followed by T_table
231 addl GR_TBL_START= @ltoff(poly_coeffs), gp
234 mov GR_OF_LIMIT= 0xffff + 7 // Exponent of overflow limit
235 movl GR_ROUNDVAL= 0x5a400000 // 1.5*2^(63-10) (SP)
239 // Form special constant 1.5*2^(63-10) to give integer part and first 10
240 // fractional bits of x
242 setf.s FR_ROUNDVAL= GR_ROUNDVAL // Form special constant
243 fcmp.lt.s1 p6, p8= f8, f0 // X<0 ?
247 ld8 GR_COEFF_START= [ GR_TBL_START ] // Load pointer to coeff table
249 (p12) br.cond.spnt SPECIAL_exp2 // Branch if nan, inf, zero
254 setf.exp FR_OF_LIMIT= GR_OF_LIMIT // Set overflow limit
255 movl GR_UF_LIMIT= 0xc3160000 // (-2^7-22) = -150
260 ldfe FR_COEFF1= [ GR_COEFF_START ], 16 // load C_1
261 fma.s0 f8= f8, f1, f0 // normalize x
267 ldfe FR_COEFF2= [ GR_COEFF_START ], 16 // load C_2
268 setf.s FR_UF_LIMIT= GR_UF_LIMIT // Set underflow limit
269 mov GR_EXP_CORR= 0xffff-126
275 fma.s1 FR_KF0= f8, f1, FR_ROUNDVAL // y= x + 1.5*2^(63-10)
282 fms.s1 FR_KF= FR_KF0, f1, FR_ROUNDVAL // (K+f)
288 getf.sig GR_KF0= FR_KF0 // (K+f)*2^10= round_to_int(y)
289 fcmp.ge.s1 p12, p7= f8, FR_OF_LIMIT // x >= overflow threshold ?
290 add GR_LOG_TBL= 256, GR_COEFF_START // Pointer to high T_table
295 and GR_F_low= GR_KF0, GR_MASK_low // f_low
296 and GR_F_high= GR_MASK, GR_KF0 // f_high*32
297 shr GR_K= GR_KF0, 10 // K
302 shladd GR_Flow_ADDR= GR_F_low, 3, GR_COEFF_START // address of 2^{f_low}
303 add GR_BIAS= GR_K, GR_EXP_CORR // K= bias-2*63
304 shr GR_Fh= GR_F_high, 5 // f_high
309 setf.exp FR_2_TO_K= GR_BIAS // 2^{K-126}
310 fnma.s1 FR_R= FR_KF, f1, f8 // r= x - (K+f)
311 shladd GR_Fh_ADDR= GR_Fh, 3, GR_LOG_TBL // address of 2^{f_high}
314 ldf8 FR_T_low= [ GR_Flow_ADDR ] // load T_low= 2^{f_low}
315 movl GR_EMIN= 0xc2fc0000 // EMIN= -126
320 ldf8 FR_T_high= [ GR_Fh_ADDR ] // load T_high= 2^{f_high}
321 (p7) fcmp.lt.s1 p12, p7= f8, FR_UF_LIMIT // x<underflow threshold ?
327 setf.s FR_EXPMIN= GR_EMIN // FR_EXPMIN= EMIN
328 fma.s1 FR_P12= FR_COEFF2, FR_R, FR_COEFF1 // P12= C_1+C_2*r
329 (p12) br.cond.spnt OUT_RANGE_exp2
335 fma.s1 FR_T_low_K= FR_T_low, FR_2_TO_K, f0 // T= 2^{K-126}*T_low
342 fma.s1 FR_P= FR_R, FR_P12, f0 // P= P12+r
349 fma.s1 FR_T= FR_T_low_K, FR_T_high, f0 // T= T*T_high
356 fcmp.lt.s0 p6, p8= f8, FR_EXPMIN // underflow (x<EMIN) ?
363 fma.s.s0 f8= FR_P, FR_T, FR_T // result= T+T*P
364 (p8) br.ret.sptk b0 // return
369 (p6) mov GR_Parameter_TAG= 164
371 (p6) br.cond.sptk __libm_error_region
379 fclass.m p6, p0= f8, 0x22 // x= -Infinity ?
386 fclass.m p7, p0= f8, 0x21 // x= +Infinity ?
393 fclass.m p8, p0= f8, 0x7 // x= +/-Zero ?
398 (p6) mov f8= f0 // exp2(-Infinity)= 0
406 (p7) br.ret.spnt b0 // exp2(+Infinity)= +Infinity
412 (p8) mov f8= f1 // exp2(+/-0)= 1
419 fma.s.s0 f8= f8, f1, f0 // Remaining cases: NaNs
430 (p8) mov GR_EXPMAX= 0x1fffe
437 (p8) mov GR_Parameter_TAG= 163
438 (p8) setf.exp FR_R= GR_EXPMAX
445 (p8) fma.s.s0 f8= FR_R, FR_R, f0 // Create overflow
450 (p6) mov GR_Parameter_TAG= 164
451 (p6) mov GR_EXPMAX= 1
458 (p6) setf.exp FR_R= GR_EXPMAX
465 (p6) fma.s.s0 f8= FR_R, FR_R, f0 // Create underflow
471 libm_alias_float_other (__exp2, exp2)
473 .symver exp2f,exp2f@@GLIBC_2.27
475 .set __exp2f_compat,__exp2f
476 .symver __exp2f_compat,exp2f@GLIBC_2.2
480 LOCAL_LIBM_ENTRY(__libm_error_region)
484 add GR_Parameter_Y= -32, sp // Parameter 2 value
486 .save ar.pfs, GR_SAVE_PFS
487 mov GR_SAVE_PFS= ar.pfs // Save ar.pfs
492 add sp= -64, sp // Create new stack
494 mov GR_SAVE_GP= gp // Save gp
499 stfs [ GR_Parameter_Y ]= FR_Y, 16 // STORE Parameter 2 on stack
500 add GR_Parameter_X= 16, sp // Parameter 1 address
502 mov GR_SAVE_B0= b0 // Save b0
508 stfs [ GR_Parameter_X ]= FR_X // STORE Parameter 1 on stack
509 add GR_Parameter_RESULT= 0, GR_Parameter_Y // Parameter 3 address
513 stfs [ GR_Parameter_Y ]= FR_RESULT // STORE Parameter 3 on stack
514 add GR_Parameter_Y= -16, GR_Parameter_Y
515 br.call.sptk b0= __libm_error_support# // Call error handling function
520 add GR_Parameter_RESULT= 48, sp
527 ldfs f8= [ GR_Parameter_RESULT ] // Get return result off stack
529 add sp= 64, sp // Restore stack pointer
530 mov b0= GR_SAVE_B0 // Restore return address
535 mov gp= GR_SAVE_GP // Restore gp
536 mov ar.pfs= GR_SAVE_PFS // Restore ar.pfs
537 br.ret.sptk b0 // Return
542 LOCAL_LIBM_END(__libm_error_region)
544 .type __libm_error_support#, @function
545 .global __libm_error_support#