4 // Copyright (c) 2000 - 2005, Intel Corporation
5 // All rights reserved.
7 // Contributed 2000 by the Intel Numerics Group, 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://www.intel.com/software/products/opensource/libraries/num.htm.
41 //==============================================================
42 // 08/25/00 Initial version
43 // 05/20/02 Cleaned up namespace and sf0 syntax
44 // 09/05/02 Improved performance and accuracy
45 // 01/17/03 Fixed to call error support when x=128.0
46 // 03/31/05 Reformatted delimiters between data tables
49 //==============================================================
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)
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
169 //==============================================================
175 LOCAL_OBJECT_START(poly_coeffs)
177 data8 0xb17217f7d1cf79ab, 0x00003ffe // C_1
178 data8 0xf5fdeffc162c7541, 0x00003ffc // C_2
179 LOCAL_OBJECT_END(poly_coeffs)
182 LOCAL_OBJECT_START(T_table)
184 // 2^{0.00000 b6 b7 b8 b9 b10}
185 data8 0x8000000000000000, 0x8016302f17467628
186 data8 0x802c6436d0e04f50, 0x80429c17d77c18ed
187 data8 0x8058d7d2d5e5f6b0, 0x806f17687707a7af
188 data8 0x80855ad965e88b83, 0x809ba2264dada76a
189 data8 0x80b1ed4fd999ab6c, 0x80c83c56b50cf77f
190 data8 0x80de8f3b8b85a0af, 0x80f4e5ff089f763e
191 data8 0x810b40a1d81406d4, 0x81219f24a5baa59d
192 data8 0x813801881d886f7b, 0x814e67cceb90502c
193 data8 0x8164d1f3bc030773, 0x817b3ffd3b2f2e47
194 data8 0x8191b1ea15813bfd, 0x81a827baf7838b78
195 data8 0x81bea1708dde6055, 0x81d51f0b8557ec1c
196 data8 0x81eba08c8ad4536f, 0x820225f44b55b33b
197 data8 0x8218af4373fc25eb, 0x822f3c7ab205c89a
198 data8 0x8245cd9ab2cec048, 0x825c62a423d13f0c
199 data8 0x8272fb97b2a5894c, 0x828998760d01faf3
200 data8 0x82a0393fe0bb0ca8, 0x82b6ddf5dbc35906
202 // 2^{0.b1 b2 b3 b4 b5}
203 data8 0x8000000000000000, 0x82cd8698ac2ba1d7
204 data8 0x85aac367cc487b14, 0x88980e8092da8527
205 data8 0x8b95c1e3ea8bd6e6, 0x8ea4398b45cd53c0
206 data8 0x91c3d373ab11c336, 0x94f4efa8fef70961
207 data8 0x9837f0518db8a96f, 0x9b8d39b9d54e5538
208 data8 0x9ef5326091a111ad, 0xa27043030c496818
209 data8 0xa5fed6a9b15138ea, 0xa9a15ab4ea7c0ef8
210 data8 0xad583eea42a14ac6, 0xb123f581d2ac258f
211 data8 0xb504f333f9de6484, 0xb8fbaf4762fb9ee9
212 data8 0xbd08a39f580c36be, 0xc12c4cca66709456
213 data8 0xc5672a115506dadd, 0xc9b9bd866e2f27a2
214 data8 0xce248c151f8480e3, 0xd2a81d91f12ae45a
215 data8 0xd744fccad69d6af4, 0xdbfbb797daf23755
216 data8 0xe0ccdeec2a94e111, 0xe5b906e77c8348a8
217 data8 0xeac0c6e7dd24392e, 0xefe4b99bdcdaf5cb
218 data8 0xf5257d152486cc2c, 0xfa83b2db722a033a
219 LOCAL_OBJECT_END(T_table)
224 GLOBAL_LIBM_ENTRY(exp2f)
228 alloc r32= ar.pfs, 1, 4, 4, 0
229 // will continue only for non-zero normal/denormal numbers
230 fclass.nm p12, p0= f8, 0x1b
231 // GR_TBL_START= pointer to C_1...C_2 followed by T_table
232 addl GR_TBL_START= @ltoff(poly_coeffs), gp
235 mov GR_OF_LIMIT= 0xffff + 7 // Exponent of overflow limit
236 movl GR_ROUNDVAL= 0x5a400000 // 1.5*2^(63-10) (SP)
240 // Form special constant 1.5*2^(63-10) to give integer part and first 10
241 // fractional bits of x
243 setf.s FR_ROUNDVAL= GR_ROUNDVAL // Form special constant
244 fcmp.lt.s1 p6, p8= f8, f0 // X<0 ?
248 ld8 GR_COEFF_START= [ GR_TBL_START ] // Load pointer to coeff table
250 (p12) br.cond.spnt SPECIAL_exp2 // Branch if nan, inf, zero
255 setf.exp FR_OF_LIMIT= GR_OF_LIMIT // Set overflow limit
256 movl GR_UF_LIMIT= 0xc3160000 // (-2^7-22) = -150
261 ldfe FR_COEFF1= [ GR_COEFF_START ], 16 // load C_1
262 fma.s0 f8= f8, f1, f0 // normalize x
268 ldfe FR_COEFF2= [ GR_COEFF_START ], 16 // load C_2
269 setf.s FR_UF_LIMIT= GR_UF_LIMIT // Set underflow limit
270 mov GR_EXP_CORR= 0xffff-126
276 fma.s1 FR_KF0= f8, f1, FR_ROUNDVAL // y= x + 1.5*2^(63-10)
283 fms.s1 FR_KF= FR_KF0, f1, FR_ROUNDVAL // (K+f)
289 getf.sig GR_KF0= FR_KF0 // (K+f)*2^10= round_to_int(y)
290 fcmp.ge.s1 p12, p7= f8, FR_OF_LIMIT // x >= overflow threshold ?
291 add GR_LOG_TBL= 256, GR_COEFF_START // Pointer to high T_table
296 and GR_F_low= GR_KF0, GR_MASK_low // f_low
297 and GR_F_high= GR_MASK, GR_KF0 // f_high*32
298 shr GR_K= GR_KF0, 10 // K
303 shladd GR_Flow_ADDR= GR_F_low, 3, GR_COEFF_START // address of 2^{f_low}
304 add GR_BIAS= GR_K, GR_EXP_CORR // K= bias-2*63
305 shr GR_Fh= GR_F_high, 5 // f_high
310 setf.exp FR_2_TO_K= GR_BIAS // 2^{K-126}
311 fnma.s1 FR_R= FR_KF, f1, f8 // r= x - (K+f)
312 shladd GR_Fh_ADDR= GR_Fh, 3, GR_LOG_TBL // address of 2^{f_high}
315 ldf8 FR_T_low= [ GR_Flow_ADDR ] // load T_low= 2^{f_low}
316 movl GR_EMIN= 0xc2fc0000 // EMIN= -126
321 ldf8 FR_T_high= [ GR_Fh_ADDR ] // load T_high= 2^{f_high}
322 (p7) fcmp.lt.s1 p12, p7= f8, FR_UF_LIMIT // x<underflow threshold ?
328 setf.s FR_EXPMIN= GR_EMIN // FR_EXPMIN= EMIN
329 fma.s1 FR_P12= FR_COEFF2, FR_R, FR_COEFF1 // P12= C_1+C_2*r
330 (p12) br.cond.spnt OUT_RANGE_exp2
336 fma.s1 FR_T_low_K= FR_T_low, FR_2_TO_K, f0 // T= 2^{K-126}*T_low
343 fma.s1 FR_P= FR_R, FR_P12, f0 // P= P12+r
350 fma.s1 FR_T= FR_T_low_K, FR_T_high, f0 // T= T*T_high
357 fcmp.lt.s0 p6, p8= f8, FR_EXPMIN // underflow (x<EMIN) ?
364 fma.s.s0 f8= FR_P, FR_T, FR_T // result= T+T*P
365 (p8) br.ret.sptk b0 // return
370 (p6) mov GR_Parameter_TAG= 164
372 (p6) br.cond.sptk __libm_error_region
380 fclass.m p6, p0= f8, 0x22 // x= -Infinity ?
387 fclass.m p7, p0= f8, 0x21 // x= +Infinity ?
394 fclass.m p8, p0= f8, 0x7 // x= +/-Zero ?
399 (p6) mov f8= f0 // exp2(-Infinity)= 0
407 (p7) br.ret.spnt b0 // exp2(+Infinity)= +Infinity
413 (p8) mov f8= f1 // exp2(+/-0)= 1
420 fma.s.s0 f8= f8, f1, f0 // Remaining cases: NaNs
431 (p8) mov GR_EXPMAX= 0x1fffe
438 (p8) mov GR_Parameter_TAG= 163
439 (p8) setf.exp FR_R= GR_EXPMAX
446 (p8) fma.s.s0 f8= FR_R, FR_R, f0 // Create overflow
451 (p6) mov GR_Parameter_TAG= 164
452 (p6) mov GR_EXPMAX= 1
459 (p6) setf.exp FR_R= GR_EXPMAX
466 (p6) fma.s.s0 f8= FR_R, FR_R, f0 // Create underflow
471 GLOBAL_LIBM_END(exp2f)
474 LOCAL_LIBM_ENTRY(__libm_error_region)
478 add GR_Parameter_Y= -32, sp // Parameter 2 value
480 .save ar.pfs, GR_SAVE_PFS
481 mov GR_SAVE_PFS= ar.pfs // Save ar.pfs
486 add sp= -64, sp // Create new stack
488 mov GR_SAVE_GP= gp // Save gp
493 stfs [ GR_Parameter_Y ]= FR_Y, 16 // STORE Parameter 2 on stack
494 add GR_Parameter_X= 16, sp // Parameter 1 address
496 mov GR_SAVE_B0= b0 // Save b0
502 stfs [ GR_Parameter_X ]= FR_X // STORE Parameter 1 on stack
503 add GR_Parameter_RESULT= 0, GR_Parameter_Y // Parameter 3 address
507 stfs [ GR_Parameter_Y ]= FR_RESULT // STORE Parameter 3 on stack
508 add GR_Parameter_Y= -16, GR_Parameter_Y
509 br.call.sptk b0= __libm_error_support# // Call error handling function
514 add GR_Parameter_RESULT= 48, sp
521 ldfs f8= [ GR_Parameter_RESULT ] // Get return result off stack
523 add sp= 64, sp // Restore stack pointer
524 mov b0= GR_SAVE_B0 // Restore return address
529 mov gp= GR_SAVE_GP // Restore gp
530 mov ar.pfs= GR_SAVE_PFS // Restore ar.pfs
531 br.ret.sptk b0 // Return
536 LOCAL_LIBM_END(__libm_error_region)
538 .type __libm_error_support#, @function
539 .global __libm_error_support#