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981 lines
29 KiB
ArmAsm
981 lines
29 KiB
ArmAsm
.file "erfcf.s"
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// Copyright (c) 2002 - 2005, Intel Corporation
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// All rights reserved.
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//
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// Contributed 2002 by the Intel Numerics Group, Intel Corporation
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//
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// Redistribution and use in source and binary forms, with or without
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// modification, are permitted provided that the following conditions are
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// met:
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//
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// * Redistributions of source code must retain the above copyright
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// notice, this list of conditions and the following disclaimer.
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//
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// * Redistributions in binary form must reproduce the above copyright
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// notice, this list of conditions and the following disclaimer in the
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// documentation and/or other materials provided with the distribution.
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//
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// * The name of Intel Corporation may not be used to endorse or promote
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// products derived from this software without specific prior written
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// permission.
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// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
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// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
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// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
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// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INTEL OR ITS
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// CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
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// EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
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// PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
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// PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY
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// OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY OR TORT (INCLUDING
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// NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
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// SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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//
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// Intel Corporation is the author of this code, and requests that all
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// problem reports or change requests be submitted to it directly at
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// http://www.intel.com/software/products/opensource/libraries/num.htm.
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//
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// History
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//==============================================================
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// 01/17/02 Initial version
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// 05/20/02 Cleaned up namespace and sf0 syntax
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// 02/06/03 Reordered header: .section, .global, .proc, .align
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// 03/31/05 Reformatted delimiters between data tables
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//
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// API
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//==============================================================
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// float erfcf(float)
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//
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// Overview of operation
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//==============================================================
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// 1. 0 <= x <= 10.06
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//
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// erfcf(x) = P15(x) * exp( -x^2 )
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//
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// Comment:
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//
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// Let x(0)=0, x(i) = 2^(i), i=1,...3, x(4)= 10.06
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//
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// Let x(i)<= x < x(i+1).
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// We can find i as exponent of argument x (let i = 0 for 0<= x < 2 )
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//
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// Let P15(x) - polynomial approximation of degree 15 for function
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// erfcf(x) * exp( x^2) and x(i) <= x <= x(i+1), i = 0,1,2,3
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// Polynomial coefficients we have in the table erfc_p_table.
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//
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// So we can find result for erfcf(x) as above.
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// Algorithm description for exp function see below.
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//
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// 2. -4.4 <= x < 0
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//
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// erfcf(x) = 2.0 - erfcf(-x)
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//
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// 3. x > 10.06
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//
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// erfcf(x) ~=~ 0.0
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//
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// 4. x < -4.4
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//
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// erfcf(x) ~=~ 2.0
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// Special values
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//==============================================================
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// erfcf(+0) = 1.0
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// erfcf(-0) = 1.0
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// erfcf(+qnan) = +qnan
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// erfcf(-qnan) = -qnan
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// erfcf(+snan) = +qnan
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// erfcf(-snan) = -qnan
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// erfcf(-inf) = 2.0
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// erfcf(+inf) = +0
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//==============================================================
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// Take double exp(double) from libm_64.
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//
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// Overview of operation
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//==============================================================
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// Take the input x. w is "how many log2/128 in x?"
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// w = x * 128/log2
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// n = int(w)
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// x = n log2/128 + r + delta
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// n = 128M + index_1 + 2^4 index_2
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// x = M log2 + (log2/128) index_1 + (log2/8) index_2 + r + delta
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// exp(x) = 2^M 2^(index_1/128) 2^(index_2/8) exp(r) exp(delta)
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// Construct 2^M
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// Get 2^(index_1/128) from table_1;
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// Get 2^(index_2/8) from table_2;
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// Calculate exp(r) by series
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// r = x - n (log2/128)_high
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// delta = - n (log2/128)_low
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// Calculate exp(delta) as 1 + delta
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//
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// Comment for erfcf:
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//
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// Let exp(r) = 1 + x + 0.5*x^2 + (1/6)*x^3
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// Let delta = 0.
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//==============================================================
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//
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// Registers used
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//==============================================================
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// Floating Point registers used:
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// f8, input
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// f6,f7,f9 -> f11, f32 -> f92
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// General registers used:
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// r14 -> r22,r32 -> r50
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// Predicate registers used:
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// p6 -> p15
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// Assembly macros
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//==============================================================
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EXP_AD_TB1 = r14
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exp_GR_sig_inv_ln2 = r15
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exp_TB1_size = r16
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exp_GR_rshf_2to56 = r17
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exp_GR_exp_2tom56 = r18
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exp_GR_rshf = r33
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EXP_AD_TB2 = r34
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EXP_AD_P = r35
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exp_GR_N = r36
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exp_GR_index_1 = r37
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exp_GR_index_2_16 = r38
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exp_GR_biased_M = r39
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EXP_AD_T1 = r40
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EXP_AD_T2 = r41
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exp_TB2_size = r42
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// GR for erfcf(x)
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//==============================================================
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GR_IndxPlusBias = r19
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GR_ExpMask = r20
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GR_BIAS = r21
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GR_ShftPi_bias = r22
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GR_P_POINT_1 = r43
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GR_P_POINT_2 = r44
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GR_P_POINT_3 = r45
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GR_P_POINT_4 = r46
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GR_ShftPi = r47
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GR_EpsNorm = r48
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GR_05 = r49
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GR_1_by_6 = r50
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// GR for __libm_support call
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//==============================================================
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GR_SAVE_B0 = r43
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GR_SAVE_PFS = r44
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GR_SAVE_GP = r45
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GR_SAVE_SP = r46
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GR_Parameter_X = r47
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GR_Parameter_Y = r48
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GR_Parameter_RESULT = r49
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GR_Parameter_TAG = r50
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// FR for exp(-x^2)
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//==============================================================
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FR_X = f10
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FR_Y = f1
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FR_RESULT = f8
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EXP_2TOM56 = f6
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EXP_INV_LN2_2TO63 = f7
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EXP_W_2TO56_RSH = f9
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exp_ln2_by_128_hi = f11
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EXP_RSHF_2TO56 = f32
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exp_ln2_by_128_lo = f33
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EXP_RSHF = f34
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EXP_Nfloat = f35
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exp_r = f36
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exp_rsq = f37
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EXP_2M = f38
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exp_S1 = f39
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exp_T1 = f40
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exp_P = f41
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exp_S = f42
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EXP_NORM_f8 = f43
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exp_S2 = f44
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exp_T2 = f45
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// FR for erfcf(x)
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//==============================================================
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FR_AbsArg = f46
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FR_Tmp = f47
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FR_Tmp1 = f48
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FR_Tmpf = f49
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FR_NormX = f50
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FR_A15 = f51
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FR_A14 = f52
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FR_A13 = f53
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FR_A12 = f54
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FR_A11 = f55
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FR_A10 = f56
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FR_A9 = f57
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FR_A8 = f58
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FR_A7 = f59
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FR_A6 = f60
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FR_A5 = f61
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FR_A4 = f62
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FR_A3 = f63
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FR_A2 = f64
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FR_A1 = f65
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FR_A0 = f66
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FR_P15_0_1 = f67
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FR_P15_1_1 = f68
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FR_P15_1_2 = f69
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FR_P15_2_1 = f70
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FR_P15_2_2 = f71
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FR_P15_3_1 = f72
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FR_P15_3_2 = f73
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FR_P15_4_1 = f74
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FR_P15_4_2 = f75
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FR_P15_7_1 = f76
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FR_P15_7_2 = f77
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FR_P15_8_1 = f78
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FR_P15_9_1 = f79
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FR_P15_9_2 = f80
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FR_P15_13_1 = f81
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FR_P15_14_1 = f82
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FR_P15_14_2 = f83
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FR_2 = f84
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FR_05 = f85
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FR_1_by_6 = f86
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FR_Pol = f87
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FR_Exp = f88
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FR_POS_ARG_ASYMP = f89
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FR_NEG_ARG_ASYMP = f90
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FR_UnfBound = f91
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FR_EpsNorm = f92
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// Data tables
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//==============================================================
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RODATA
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.align 16
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// ************* DO NOT CHANGE ORDER OF THESE TABLES ********************
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// double-extended 1/ln(2)
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// 3fff b8aa 3b29 5c17 f0bb be87fed0691d3e88
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// 3fff b8aa 3b29 5c17 f0bc
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// For speed the significand will be loaded directly with a movl and setf.sig
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// and the exponent will be bias+63 instead of bias+0. Thus subsequent
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// computations need to scale appropriately.
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// The constant 128/ln(2) is needed for the computation of w. This is also
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// obtained by scaling the computations.
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//
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// Two shifting constants are loaded directly with movl and setf.d.
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// 1. EXP_RSHF_2TO56 = 1.1000..00 * 2^(63-7)
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// This constant is added to x*1/ln2 to shift the integer part of
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// x*128/ln2 into the rightmost bits of the significand.
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// The result of this fma is EXP_W_2TO56_RSH.
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// 2. EXP_RSHF = 1.1000..00 * 2^(63)
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// This constant is subtracted from EXP_W_2TO56_RSH * 2^(-56) to give
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// the integer part of w, n, as a floating-point number.
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// The result of this fms is EXP_Nfloat.
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LOCAL_OBJECT_START(exp_table_1)
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data4 0x4120f5c3, 0x408ccccd //POS_ARG_ASYMP = 10.06, NEG_ARG_ASYMP = 4.4
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data4 0x41131Cdf, 0x00800000 //UnfBound ~=~ 9.1, EpsNorm ~=~ 1.1754944e-38
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//
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data8 0xb17217f7d1cf79ab , 0x00003ff7 // ln2/128 hi
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data8 0xc9e3b39803f2f6af , 0x00003fb7 // ln2/128 lo
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//
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// Table 1 is 2^(index_1/128) where
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// index_1 goes from 0 to 15
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//
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data8 0x8000000000000000 , 0x00003FFF
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data8 0x80B1ED4FD999AB6C , 0x00003FFF
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data8 0x8164D1F3BC030773 , 0x00003FFF
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data8 0x8218AF4373FC25EC , 0x00003FFF
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data8 0x82CD8698AC2BA1D7 , 0x00003FFF
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data8 0x8383594EEFB6EE37 , 0x00003FFF
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data8 0x843A28C3ACDE4046 , 0x00003FFF
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data8 0x84F1F656379C1A29 , 0x00003FFF
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data8 0x85AAC367CC487B15 , 0x00003FFF
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data8 0x8664915B923FBA04 , 0x00003FFF
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data8 0x871F61969E8D1010 , 0x00003FFF
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data8 0x87DB357FF698D792 , 0x00003FFF
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data8 0x88980E8092DA8527 , 0x00003FFF
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data8 0x8955EE03618E5FDD , 0x00003FFF
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data8 0x8A14D575496EFD9A , 0x00003FFF
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data8 0x8AD4C6452C728924 , 0x00003FFF
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LOCAL_OBJECT_END(exp_table_1)
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// Table 2 is 2^(index_1/8) where
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// index_2 goes from 0 to 7
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LOCAL_OBJECT_START(exp_table_2)
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data8 0x8000000000000000 , 0x00003FFF
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data8 0x8B95C1E3EA8BD6E7 , 0x00003FFF
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data8 0x9837F0518DB8A96F , 0x00003FFF
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data8 0xA5FED6A9B15138EA , 0x00003FFF
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data8 0xB504F333F9DE6484 , 0x00003FFF
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data8 0xC5672A115506DADD , 0x00003FFF
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data8 0xD744FCCAD69D6AF4 , 0x00003FFF
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data8 0xEAC0C6E7DD24392F , 0x00003FFF
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LOCAL_OBJECT_END(exp_table_2)
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LOCAL_OBJECT_START(erfc_p_table)
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// Pol_0
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data8 0xBEA3260C63CB0446 //A15 = -5.70673541831883454676e-07
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data8 0x3EE63D6178077654 //A14 = +1.06047480138940182343e-05
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data8 0xBF18646BC5FC70A7 //A13 = -9.30491237309283694347e-05
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data8 0x3F40F92F909117FE //A12 = +5.17986512144075019133e-04
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data8 0xBF611344289DE1E6 //A11 = -2.08438217390159994419e-03
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data8 0x3F7AF9FE6AD16DC0 //A10 = +6.58606893292862351928e-03
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data8 0xBF91D219E196CBA7 //A9 = -1.74030345858217321001e-02
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data8 0x3FA4AFDDA355854C //A8 = +4.04042493708041968315e-02
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data8 0xBFB5D465BB7025AE //A7 = -8.52721769916999425445e-02
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data8 0x3FC54C15A95B717D //A6 = +1.66384418195672549029e-01
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data8 0xBFD340A75B4B1AB5 //A5 = -3.00821150926292166899e-01
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data8 0x3FDFFFC0BFCD247F //A4 = +4.99984919839853542841e-01
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data8 0xBFE81270C361852B //A3 = -7.52251035312075583309e-01
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data8 0x3FEFFFFFC67295FC //A2 = +9.99999892800303301771e-01
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data8 0xBFF20DD74F8CD2BF //A1 = -1.12837916445020868099e+00
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data8 0x3FEFFFFFFFFE7C1D //A0 = +9.99999999988975570714e-01
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// Pol_1
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data8 0xBDE8EC4BDD953B56 //A15 = -1.81338928934942767144e-10
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data8 0x3E43607F269E2A1C //A14 = +9.02309090272196442358e-09
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data8 0xBE8C4D9E69C10E02 //A13 = -2.10875261143659275328e-07
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data8 0x3EC9CF2F84566725 //A12 = +3.07671055805877356583e-06
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data8 0xBF007980B1B46A4D //A11 = -3.14228438702169818945e-05
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data8 0x3F2F4C3AD6DEF24A //A10 = +2.38783056770846320260e-04
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data8 0xBF56F5129F8D30FA //A9 = -1.40120333363130546426e-03
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data8 0x3F7AA6C7ABFC38EE //A8 = +6.50671002200751820429e-03
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data8 0xBF98E7522CB84BEF //A7 = -2.43199195666185511109e-02
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data8 0x3FB2F68EB1C3D073 //A6 = +7.40746673580490638637e-02
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data8 0xBFC7C16055AC6385 //A5 = -1.85588876564704611769e-01
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data8 0x3FD8A707AEF5A440 //A4 = +3.85194702967570635211e-01
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data8 0xBFE547BFE39AE2EA //A3 = -6.65008492032112467310e-01
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data8 0x3FEE7C91BDF13578 //A2 = +9.52706213932898128515e-01
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data8 0xBFF1CB5B61F8C589 //A1 = -1.11214769621105541214e+00
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data8 0x3FEFEA56BC81FD37 //A0 = +9.97355812243688815239e-01
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// Pol_2
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data8 0xBD302724A12F46E0 //A15 = -5.73866382814058809406e-14
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data8 0x3D98889B75D3102E //A14 = +5.57829983681360947356e-12
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data8 0xBDF16EA15074A1E9 //A13 = -2.53671153922423457844e-10
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data8 0x3E3EC6E688CFEE5F //A12 = +7.16581828336436419561e-09
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data8 0xBE82E5ED44C52609 //A11 = -1.40802202239825487803e-07
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data8 0x3EC120BE5CE42353 //A10 = +2.04180535157522081699e-06
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data8 0xBEF7B8B0311A1911 //A9 = -2.26225266204633600888e-05
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data8 0x3F29A281F43FC238 //A8 = +1.95577968156184077632e-04
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data8 0xBF55E19858B3B7A4 //A7 = -1.33552434527526534043e-03
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data8 0x3F7DAC8C3D12E5FD //A6 = +7.24463253680473816303e-03
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data8 0xBF9FF9C04613FB47 //A5 = -3.12261622211693854028e-02
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data8 0x3FBB3D5DBF9D9366 //A4 = +1.06405123978743883370e-01
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data8 0xBFD224DE9F62C258 //A3 = -2.83500342989133623476e-01
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data8 0x3FE28A95CB8C6D3E //A2 = +5.79417131000276437708e-01
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data8 0xBFEC21205D358672 //A1 = -8.79043752717008257224e-01
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data8 0x3FEDAE44D5EDFE5B //A0 = +9.27523057776805771830e-01
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// Pol_3
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data8 0xBCA3BCA734AC82F1 //A15 = -1.36952437983096410260e-16
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data8 0x3D16740DC3990612 //A14 = +1.99425676175410093285e-14
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data8 0xBD77F4353812C46A //A13 = -1.36162367755616790260e-12
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data8 0x3DCFD0BE13C73DB4 //A12 = +5.78718761040355136007e-11
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data8 0xBE1D728DF71189B4 //A11 = -1.71406885583934105120e-09
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data8 0x3E64252C8CB710B5 //A10 = +3.75233795940731111303e-08
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data8 0xBEA514B93180F33D //A9 = -6.28261292774310809962e-07
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data8 0x3EE1381118CC7151 //A8 = +8.21066421390821904504e-06
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data8 0xBF1634404FB0FA72 //A7 = -8.47019436358372148764e-05
|
|
data8 0x3F46B2CBBCF0EB32 //A6 = +6.92700845213200923490e-04
|
|
data8 0xBF725C2B445E6D81 //A5 = -4.48243046949004063741e-03
|
|
data8 0x3F974E7CFA4D89D9 //A4 = +2.27603462002522228717e-02
|
|
data8 0xBFB6D7BAC2E342D1 //A3 = -8.92292714882032736443e-02
|
|
data8 0x3FD0D156AD9CE2A6 //A2 = +2.62777013343603696631e-01
|
|
data8 0xBFE1C228572AADB0 //A1 = -5.54950876471982857725e-01
|
|
data8 0x3FE8A739F48B9A3B //A0 = +7.70413377406675619766e-01
|
|
LOCAL_OBJECT_END(erfc_p_table)
|
|
|
|
|
|
.section .text
|
|
GLOBAL_LIBM_ENTRY(erfcf)
|
|
|
|
// Form index i for table erfc_p_table as exponent of x
|
|
// We use i + bias in real calculations
|
|
{ .mlx
|
|
getf.exp GR_IndxPlusBias = f8 // (sign + exp + bias) of x
|
|
movl exp_GR_sig_inv_ln2 = 0xb8aa3b295c17f0bc //signif.of 1/ln2
|
|
}
|
|
{ .mlx
|
|
addl EXP_AD_TB1 = @ltoff(exp_table_1), gp
|
|
movl exp_GR_rshf_2to56 = 0x4768000000000000 // 1.100 2^(63+56)
|
|
}
|
|
;;
|
|
|
|
// Form argument EXP_NORM_f8 for exp(-x^2)
|
|
{ .mfi
|
|
ld8 EXP_AD_TB1 = [EXP_AD_TB1]
|
|
fcmp.ge.s1 p6,p7 = f8, f0 // p6: x >= 0 ,p7: x<0
|
|
mov GR_BIAS = 0x0FFFF
|
|
}
|
|
{ .mfi
|
|
mov exp_GR_exp_2tom56 = 0xffff-56
|
|
fnma.s1 EXP_NORM_f8 = f8, f8, f0 // -x^2
|
|
mov GR_ExpMask = 0x1ffff
|
|
}
|
|
;;
|
|
|
|
// Form two constants we need
|
|
// 1/ln2 * 2^63 to compute w = x * 1/ln2 * 128
|
|
// 1.1000..000 * 2^(63+63-7) to right shift int(w) into the significand
|
|
|
|
// p9: x = 0,+inf,-inf,nan,unnorm.
|
|
// p10: x!= 0,+inf,-inf,nan,unnorm.
|
|
{ .mfi
|
|
setf.sig EXP_INV_LN2_2TO63 = exp_GR_sig_inv_ln2 // Form 1/ln2*2^63
|
|
fclass.m p9,p10 = f8,0xef
|
|
shl GR_ShftPi_bias = GR_BIAS, 7
|
|
}
|
|
{ .mfi
|
|
setf.d EXP_RSHF_2TO56 = exp_GR_rshf_2to56 //Const 1.10*2^(63+56)
|
|
nop.f 0
|
|
and GR_IndxPlusBias = GR_IndxPlusBias, GR_ExpMask // i + bias
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
alloc r32 = ar.pfs, 0, 15, 4, 0
|
|
(p6) fma.s1 FR_AbsArg = f1, f0, f8 // |x| if x >= 0
|
|
cmp.lt p15,p0 = GR_IndxPlusBias, GR_BIAS//p15: i < 0 (for |x|<1)
|
|
}
|
|
{ .mlx
|
|
setf.exp EXP_2TOM56 = exp_GR_exp_2tom56 //2^-56 for scaling Nfloat
|
|
movl exp_GR_rshf = 0x43e8000000000000 //1.10 2^63,right shift.
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
ldfps FR_POS_ARG_ASYMP, FR_NEG_ARG_ASYMP = [EXP_AD_TB1],8
|
|
nop.f 0
|
|
(p15) mov GR_IndxPlusBias = GR_BIAS //Let i = 0 if i < 0
|
|
}
|
|
{ .mlx
|
|
mov GR_P_POINT_3 = 0x1A0
|
|
movl GR_05 = 0x3fe0000000000000
|
|
}
|
|
;;
|
|
|
|
// Form shift GR_ShftPi from the beginning of erfc_p_table
|
|
// to the polynomial with number i
|
|
{ .mfi
|
|
ldfps FR_UnfBound, FR_EpsNorm = [EXP_AD_TB1],8
|
|
nop.f 0
|
|
shl GR_ShftPi = GR_IndxPlusBias, 7
|
|
}
|
|
{ .mfi
|
|
setf.d EXP_RSHF = exp_GR_rshf // Form right shift 1.100 * 2^63
|
|
(p7) fms.s1 FR_AbsArg = f1, f0, f8 // |x| if x < 0
|
|
mov exp_TB1_size = 0x100
|
|
}
|
|
;;
|
|
|
|
// Form pointer GR_P_POINT_3 to the beginning of erfc_p_table
|
|
{ .mfi
|
|
setf.d FR_05 = GR_05
|
|
nop.f 0
|
|
sub GR_ShftPi = GR_ShftPi,GR_ShftPi_bias
|
|
}
|
|
{ .mfb
|
|
add GR_P_POINT_3 = GR_P_POINT_3, EXP_AD_TB1
|
|
nop.f 0
|
|
(p9) br.cond.spnt SPECIAL // For x = 0,+inf,-inf,nan,unnorm
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
add GR_P_POINT_1 = GR_P_POINT_3, GR_ShftPi
|
|
nop.f 0
|
|
add GR_P_POINT_2 = GR_P_POINT_3, GR_ShftPi
|
|
}
|
|
{ .mfi
|
|
ldfe exp_ln2_by_128_hi = [EXP_AD_TB1],16
|
|
fma.s1 FR_NormX = f8,f1,f0
|
|
add GR_P_POINT_3 = GR_P_POINT_3, GR_ShftPi
|
|
}
|
|
;;
|
|
|
|
// Load coefficients for polynomial P15(x)
|
|
{ .mfi
|
|
ldfpd FR_A15, FR_A14 = [GR_P_POINT_1], 16
|
|
nop.f 0
|
|
add GR_P_POINT_3 = 0x30, GR_P_POINT_3
|
|
}
|
|
{ .mfi
|
|
ldfe exp_ln2_by_128_lo = [EXP_AD_TB1], 16
|
|
nop.f 0
|
|
add GR_P_POINT_2 = 0x20, GR_P_POINT_2
|
|
}
|
|
;;
|
|
|
|
// Now EXP_AD_TB1 points to the beginning of table 1
|
|
{ .mlx
|
|
ldfpd FR_A13, FR_A12 = [GR_P_POINT_1]
|
|
movl GR_1_by_6 = 0x3FC5555555555555
|
|
}
|
|
{ .mfi
|
|
add GR_P_POINT_4 = 0x30, GR_P_POINT_2
|
|
nop.f 0
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
ldfpd FR_A11, FR_A10 = [GR_P_POINT_2]
|
|
fma.s1 FR_2 = f1, f1, f1
|
|
mov exp_TB2_size = 0x80
|
|
}
|
|
{ .mfi
|
|
ldfpd FR_A9, FR_A8 = [GR_P_POINT_3],16
|
|
nop.f 0
|
|
add GR_P_POINT_1 = 0x60 ,GR_P_POINT_1
|
|
}
|
|
;;
|
|
|
|
// W = X * Inv_log2_by_128
|
|
// By adding 1.10...0*2^63 we shift and get round_int(W) in significand.
|
|
// We actually add 1.10...0*2^56 to X * Inv_log2 to do the same thing.
|
|
{ .mfi
|
|
ldfpd FR_A7, FR_A6 = [GR_P_POINT_3]
|
|
fma.s1 EXP_W_2TO56_RSH = EXP_NORM_f8,EXP_INV_LN2_2TO63,EXP_RSHF_2TO56
|
|
add EXP_AD_TB2 = exp_TB1_size, EXP_AD_TB1
|
|
|
|
}
|
|
{ .mfi
|
|
ldfpd FR_A5, FR_A4 = [GR_P_POINT_4], 16
|
|
nop.f 0
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
ldfpd FR_A3, FR_A2 = [GR_P_POINT_4]
|
|
fmerge.s FR_X = f8,f8
|
|
nop.i 0
|
|
}
|
|
{ .mfi
|
|
ldfpd FR_A1, FR_A0 = [GR_P_POINT_1]
|
|
nop.f 0
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
//p14: x < - NEG_ARG_ASYMP = -4.4 -> erfcf(x) ~=~ 2.0
|
|
{ .mfi
|
|
setf.d FR_1_by_6 = GR_1_by_6
|
|
(p7) fcmp.gt.unc.s1 p14,p0 = FR_AbsArg, FR_NEG_ARG_ASYMP //p7: x < 0
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
//p15: x > POS_ARG_ASYMP = 10.06 -> erfcf(x) ~=~ 0.0
|
|
{ .mfi
|
|
nop.m 0
|
|
(p6) fcmp.gt.unc.s1 p15,p0 = FR_AbsArg, FR_POS_ARG_ASYMP //p6: x > 0
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
nop.m 0
|
|
fcmp.le.s1 p8,p0 = FR_NormX, FR_UnfBound // p8: x <= UnfBound
|
|
nop.i 0
|
|
}
|
|
{ .mfb
|
|
nop.m 0
|
|
(p14) fnma.s.s0 FR_RESULT = FR_EpsNorm, FR_EpsNorm, FR_2//y = 2 if x <-4.4
|
|
(p14) br.ret.spnt b0
|
|
}
|
|
;;
|
|
|
|
// Nfloat = round_int(W)
|
|
// The signficand of EXP_W_2TO56_RSH contains the rounded integer part of W,
|
|
// as a twos complement number in the lower bits (that is, it may be negative).
|
|
// That twos complement number (called N) is put into exp_GR_N.
|
|
|
|
// Since EXP_W_2TO56_RSH is scaled by 2^56, it must be multiplied by 2^-56
|
|
// before the shift constant 1.10000 * 2^63 is subtracted to yield EXP_Nfloat.
|
|
// Thus, EXP_Nfloat contains the floating point version of N
|
|
|
|
{ .mfi
|
|
nop.m 0
|
|
fms.s1 EXP_Nfloat = EXP_W_2TO56_RSH, EXP_2TOM56, EXP_RSHF
|
|
nop.i 0
|
|
}
|
|
{ .mfb
|
|
(p15) mov GR_Parameter_TAG = 209
|
|
(p15) fma.s.s0 FR_RESULT = FR_EpsNorm,FR_EpsNorm,f0 //Result.for x>10.06
|
|
(p15) br.cond.spnt __libm_error_region
|
|
}
|
|
;;
|
|
|
|
// Now we can calculate polynomial P15(x)
|
|
{ .mfi
|
|
nop.m 0
|
|
fma.s1 FR_P15_1_1 = FR_AbsArg, FR_AbsArg, f0 // x ^2
|
|
nop.i 0
|
|
}
|
|
{ .mfi
|
|
nop.m 0
|
|
fma.s1 FR_P15_0_1 = FR_A15, FR_AbsArg, FR_A14
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
nop.m 0
|
|
fma.s1 FR_P15_1_2 = FR_A13, FR_AbsArg, FR_A12
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
getf.sig exp_GR_N = EXP_W_2TO56_RSH
|
|
fma.s1 FR_P15_2_1 = FR_A9, FR_AbsArg, FR_A8
|
|
nop.i 0
|
|
}
|
|
{ .mfi
|
|
nop.m 0
|
|
fma.s1 FR_P15_2_2 = FR_A11, FR_AbsArg, FR_A10
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
nop.m 0
|
|
fma.s1 FR_P15_3_1 = FR_A5, FR_AbsArg, FR_A4
|
|
nop.i 0
|
|
}
|
|
{ .mfi
|
|
nop.m 0
|
|
fma.s1 FR_P15_3_2 = FR_A7, FR_AbsArg, FR_A6
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
// exp_GR_index_1 has index_1
|
|
// exp_GR_index_2_16 has index_2 * 16
|
|
// exp_GR_biased_M has M
|
|
// exp_GR_index_1_16 has index_1 * 16
|
|
|
|
// r2 has true M
|
|
{ .mfi
|
|
and exp_GR_index_1 = 0x0f, exp_GR_N
|
|
fma.s1 FR_P15_4_1 = FR_A1, FR_AbsArg, FR_A0
|
|
shr r2 = exp_GR_N, 0x7
|
|
|
|
}
|
|
{ .mfi
|
|
and exp_GR_index_2_16 = 0x70, exp_GR_N
|
|
fma.s1 FR_P15_4_2 = FR_A3, FR_AbsArg, FR_A2
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
// EXP_AD_T1 has address of T1
|
|
// EXP_AD_T2 has address if T2
|
|
|
|
{ .mfi
|
|
add EXP_AD_T2 = EXP_AD_TB2, exp_GR_index_2_16
|
|
nop.f 0
|
|
shladd EXP_AD_T1 = exp_GR_index_1, 4, EXP_AD_TB1
|
|
}
|
|
{ .mfi
|
|
addl exp_GR_biased_M = 0xffff, r2
|
|
fnma.s1 exp_r = EXP_Nfloat, exp_ln2_by_128_hi, EXP_NORM_f8
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
// Create Scale = 2^M
|
|
// r = x - Nfloat * ln2_by_128_hi
|
|
|
|
{ .mfi
|
|
setf.exp EXP_2M = exp_GR_biased_M
|
|
fma.s1 FR_P15_7_1 = FR_P15_0_1, FR_P15_1_1, FR_P15_1_2
|
|
nop.i 0
|
|
}
|
|
{ .mfi
|
|
ldfe exp_T2 = [EXP_AD_T2]
|
|
nop.f 0
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
// Load T1 and T2
|
|
|
|
{ .mfi
|
|
ldfe exp_T1 = [EXP_AD_T1]
|
|
fma.s1 FR_P15_7_2 = FR_P15_1_1, FR_P15_1_1, f0 // x^4
|
|
nop.i 0
|
|
}
|
|
{ .mfi
|
|
nop.m 0
|
|
fma.s1 FR_P15_8_1 = FR_P15_1_1, FR_P15_2_2, FR_P15_2_1
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
nop.m 0
|
|
fma.s1 FR_P15_9_1 = FR_P15_1_1, FR_P15_4_2, FR_P15_4_1
|
|
nop.i 0
|
|
}
|
|
{ .mfi
|
|
nop.m 0
|
|
fma.s1 FR_P15_9_2 = FR_P15_1_1, FR_P15_3_2, FR_P15_3_1
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
nop.m 0
|
|
fma.s1 exp_P = FR_1_by_6, exp_r, FR_05
|
|
nop.i 0
|
|
}
|
|
{ .mfi
|
|
nop.m 0
|
|
fma.s1 exp_rsq = exp_r, exp_r, f0
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
nop.m 0
|
|
fma.s1 FR_P15_13_1 = FR_P15_7_2, FR_P15_7_1, FR_P15_8_1
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
nop.m 0
|
|
fma.s1 FR_P15_14_1 = FR_P15_7_2, FR_P15_9_2, FR_P15_9_1
|
|
nop.i 0
|
|
}
|
|
{ .mfi
|
|
nop.m 0
|
|
fma.s1 FR_P15_14_2 = FR_P15_7_2, FR_P15_7_2, f0 // x^8
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
nop.m 0
|
|
fma.s1 exp_P = exp_P, exp_rsq, exp_r
|
|
nop.i 0
|
|
}
|
|
{ .mfi
|
|
nop.m 0
|
|
fma.s1 exp_S1 = EXP_2M, exp_T2, f0
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
nop.m 0
|
|
fma.s1 FR_Pol = FR_P15_14_2, FR_P15_13_1, FR_P15_14_1 // P15(x)
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
nop.m 0
|
|
fma.s1 exp_S = exp_S1, exp_T1, f0
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
nop.m 0
|
|
fma.s1 FR_Exp = exp_S, exp_P, exp_S // exp(-x^2)
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
nop.m 0
|
|
fma.s.s0 FR_Tmpf = f8, f1, f0 // Flag d
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
//p6: result for 0 < x < = POS_ARG_ASYMP
|
|
//p7: result for - NEG_ARG_ASYMP <= x < 0
|
|
//p8: exit for - NEG_ARG_ASYMP <= x <= UnfBound, x!=0
|
|
.pred.rel "mutex",p6,p7
|
|
{ .mfi
|
|
nop.m 0
|
|
(p6) fma.s.s0 f8 = FR_Exp, FR_Pol, f0
|
|
nop.i 0
|
|
}
|
|
{ .mfb
|
|
mov GR_Parameter_TAG = 209
|
|
(p7) fnma.s.s0 f8 = FR_Exp, FR_Pol, FR_2
|
|
(p8) br.ret.sptk b0
|
|
}
|
|
;;
|
|
|
|
//p10: branch for UnfBound < x < = POS_ARG_ASYMP
|
|
{ .mfb
|
|
nop.m 0
|
|
nop.f 0
|
|
(p10) br.cond.spnt __libm_error_region
|
|
}
|
|
;;
|
|
|
|
//Only via (p9) br.cond.spnt SPECIAL for x = 0,+inf,-inf,nan,unnorm
|
|
SPECIAL:
|
|
|
|
{ .mfi
|
|
nop.m 0
|
|
fclass.m.unc p10,p0 = f8,0x07 // p10: x = 0
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
nop.m 0
|
|
fclass.m.unc p11,p0 = f8,0x21 // p11: x = +inf
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
nop.m 0
|
|
fclass.m.unc p12,p0 = f8,0x22 // p12 x = -inf
|
|
nop.i 0
|
|
}
|
|
{ .mfb
|
|
nop.m 0
|
|
(p10) fma.s.s0 f8 = f1, f1, f0
|
|
(p10) br.ret.sptk b0 // Quick exit for x = 0
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
nop.m 0
|
|
fclass.m.unc p13,p0 = f8,0xc3 // p13: x = nan
|
|
nop.i 0
|
|
}
|
|
{ .mfb
|
|
nop.m 0
|
|
(p11) fma.s.s0 f8 = f0, f1, f0
|
|
(p11) br.ret.spnt b0 // Quick exit for x = +inf
|
|
}
|
|
;;
|
|
{ .mfi
|
|
nop.m 0
|
|
fclass.m.unc p14,p0 = f8,0x0b // P14: x = unnormalized
|
|
nop.i 0
|
|
}
|
|
{ .mfb
|
|
nop.m 0
|
|
(p12) fma.s.s0 f8 = f1, f1, f1
|
|
(p12) br.ret.spnt b0 // Quick exit for x = -inf
|
|
}
|
|
;;
|
|
|
|
{ .mfb
|
|
nop.m 0
|
|
(p13) fma.s.s0 f8 = f8, f1, f0
|
|
(p13) br.ret.sptk b0 // Quick exit for x = nan
|
|
}
|
|
;;
|
|
|
|
{ .mfb
|
|
nop.m 0
|
|
(p14) fnma.s.s0 f8 = f8, f1, f1
|
|
(p14) br.ret.sptk b0 // Quick exit for x = unnormalized
|
|
}
|
|
;;
|
|
|
|
GLOBAL_LIBM_END(erfcf)
|
|
|
|
|
|
// Call via (p10) br.cond.spnt __libm_error_region
|
|
// for UnfBound < x < = POS_ARG_ASYMP
|
|
// and
|
|
//
|
|
// call via (p15) br.cond.spnt __libm_error_region
|
|
// for x > POS_ARG_ASYMP
|
|
|
|
LOCAL_LIBM_ENTRY(__libm_error_region)
|
|
.prologue
|
|
{ .mfi
|
|
add GR_Parameter_Y=-32,sp // Parameter 2 value
|
|
nop.f 0
|
|
.save ar.pfs,GR_SAVE_PFS
|
|
mov GR_SAVE_PFS=ar.pfs // Save ar.pfs
|
|
}
|
|
{ .mfi
|
|
.fframe 64
|
|
add sp=-64,sp // Create new stack
|
|
nop.f 0
|
|
mov GR_SAVE_GP=gp // Save gp
|
|
};;
|
|
{ .mmi
|
|
stfs [GR_Parameter_Y] = FR_Y,16 // STORE Parameter 2 on stack
|
|
add GR_Parameter_X = 16,sp // Parameter 1 address
|
|
.save b0, GR_SAVE_B0
|
|
mov GR_SAVE_B0=b0 // Save b0
|
|
};;
|
|
.body
|
|
{ .mib
|
|
stfs [GR_Parameter_X] = FR_X // STORE Parameter 1 on stack
|
|
add GR_Parameter_RESULT = 0,GR_Parameter_Y // Parameter 3 address
|
|
nop.b 0
|
|
}
|
|
{ .mib
|
|
stfs [GR_Parameter_Y] = FR_RESULT // STORE Parameter 3 on stack
|
|
add GR_Parameter_Y = -16,GR_Parameter_Y
|
|
br.call.sptk b0=__libm_error_support# // Call error handling function
|
|
};;
|
|
{ .mmi
|
|
nop.m 0
|
|
nop.m 0
|
|
add GR_Parameter_RESULT = 48,sp
|
|
};;
|
|
{ .mmi
|
|
ldfs f8 = [GR_Parameter_RESULT] // Get return result off stack
|
|
.restore sp
|
|
add sp = 64,sp // Restore stack pointer
|
|
mov b0 = GR_SAVE_B0 // Restore return address
|
|
};;
|
|
{ .mib
|
|
mov gp = GR_SAVE_GP // Restore gp
|
|
mov ar.pfs = GR_SAVE_PFS // Restore ar.pfs
|
|
br.ret.sptk b0 // Return
|
|
};;
|
|
|
|
LOCAL_LIBM_END(__libm_error_region)
|
|
.type __libm_error_support#,@function
|
|
.global __libm_error_support#
|