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.file "expf.s"
// Copyright (c) 2000, 2001, Intel Corporation// All rights reserved.//// Contributed 2/2/2000 by John Harrison, Ted Kubaska, Bob Norin, Shane Story,// and Ping Tak Peter Tang of the Computational Software Lab, Intel Corporation.//// WARRANTY DISCLAIMER//// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INTEL OR ITS// CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,// EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,// PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
// PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY
// OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY OR TORT (INCLUDING// NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS// SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.//// Intel Corporation is the author of this code, and requests that all// problem reports or change requests be submitted to it directly at// http://developer.intel.com/opensource.
// History//==============================================================// 4/04/00 Unwind update// 4/04/00 Unwind support added// 8/15/00 Bundle added after call to __libm_error_support to properly// set [the previously overwritten] GR_Parameter_RESULT.// 8/21/00 Improvements to save 2 cycles on main path, and shorten x=0 case// 12/07/00 Widen main path, shorten x=inf, nan paths// 3/15/01 Fix monotonicity problem around x=0 for round to +inf//// Assembly macros//==============================================================// integer registers used
exp_GR_0x0f = r33 exp_GR_0xf0 = r34
EXP_AD_P_1 = r36 EXP_AD_P_2 = r37 EXP_AD_T1 = r38 EXP_AD_T2 = r39 exp_GR_Mint = r40
exp_GR_Mint_p_128 = r41 exp_GR_Ind1 = r42 EXP_AD_M1 = r43 exp_GR_Ind2 = r44 EXP_AD_M2 = r45
exp_GR_min_oflow = r46 exp_GR_max_zero = r47 exp_GR_max_norm = r48 exp_GR_max_uflow = r49 exp_GR_min_norm = r50
exp_GR_17ones = r51 exp_GR_gt_ln = r52 exp_GR_T2_size = r53 exp_GR_bias = r54 exp_GR_signexp = r55 exp_GR_17ones_m1 = r56 exp_GR_one = r57 exp_GR_exp = r58
GR_SAVE_B0 = r53GR_SAVE_PFS = r55GR_SAVE_GP = r54
GR_Parameter_X = r59GR_Parameter_Y = r60GR_Parameter_RESULT = r61GR_Parameter_TAG = r62
FR_X = f10FR_Y = f1FR_RESULT = f8
// floating point registers used
EXP_MIN_SGL_OFLOW_ARG = f11 EXP_MAX_SGL_ZERO_ARG = f12 EXP_MAX_SGL_NORM_ARG = f13 EXP_MAX_SGL_UFLOW_ARG = f14 EXP_MIN_SGL_NORM_ARG = f15
exp_coeff_P5 = f32 exp_coeff_P6 = f33 exp_coeff_P3 = f34 exp_coeff_P4 = f35
exp_coeff_P1 = f36 exp_coeff_P2 = f37 exp_Mx = f38 exp_Mfloat = f39 exp_R = f40
exp_P1 = f41 exp_P2 = f42 exp_P3 = f43 exp_Rsq = f44 exp_R4 = f45
exp_P4 = f46 exp_P5 = f47 exp_P6 = f48 exp_P7 = f49 exp_T1 = f50
exp_T2 = f51 exp_T = f52 exp_A = f53 exp_norm_f8 = f54 exp_wre_urm_f8 = f55
exp_ftz_urm_f8 = f56 exp_gt_pln = f57
.data
.align 16
exp_coeff_1_table:data8 0x3F56F35FDE4F8563 // p5data8 0x3F2A378BEFECCFDD // p6data8 0x3FE00000258C581D // p1data8 0x3FC555557AE7B3D4 // p2
exp_coeff_2_table:data8 0x3FA5551BB6592FAE // p3data8 0x3F8110E8EBFFD485 // p4
exp_T2_table:data8 0xa175cf9cd7d85844 , 0x00003f46 // exp(-128)data8 0xdb7279415a1f9eed , 0x00003f47 // exp(-127)data8 0x95213b242bd8ca5f , 0x00003f49 // exp(-126)data8 0xcab03c968c989f83 , 0x00003f4a // exp(-125)data8 0x89bdb674702961ad , 0x00003f4c // exp(-124)data8 0xbb35a2eec278be35 , 0x00003f4d // exp(-123)data8 0xfe71b17f373e7e7a , 0x00003f4e // exp(-122)data8 0xace9a6ec52a39b63 , 0x00003f50 // exp(-121)data8 0xeb03423fe393cf1c , 0x00003f51 // exp(-120)data8 0x9fb52c5bcaef1693 , 0x00003f53 // exp(-119)data8 0xd910b6377ed60bf1 , 0x00003f54 // exp(-118)data8 0x9382dad8a9fdbfe4 , 0x00003f56 // exp(-117)data8 0xc87d0a84dea869a3 , 0x00003f57 // exp(-116)data8 0x883efb4c6d1087b0 , 0x00003f59 // exp(-115)data8 0xb92d7373dce9a502 , 0x00003f5a // exp(-114)data8 0xfbaeb020577fb0cb , 0x00003f5b // exp(-113)
exp_T1_table:data8 0x8000000000000000 , 0x00003fff // exp(16 * 0)data8 0x87975e8540010249 , 0x00004016 // exp(16 * 1) data8 0x8fa1fe625b3163ec , 0x0000402d // exp(16 * 2) data8 0x9826b576512a59d7 , 0x00004044 // exp(16 * 3) data8 0xa12cc167acbe6902 , 0x0000405b // exp(16 * 4) data8 0xaabbcdcc279f59e4 , 0x00004072 // exp(16 * 5) data8 0xb4dbfaadc045d16f , 0x00004089 // exp(16 * 6) data8 0xbf95e372ccdbf146 , 0x000040a0 // exp(16 * 7) data8 0xcaf2a62eea10bbfb , 0x000040b7 // exp(16 * 8) data8 0xd6fbeb62fddbd340 , 0x000040ce // exp(16 * 9) data8 0xe3bbee32e4a440ea , 0x000040e5 // exp(16 * 10)data8 0xf13d8517c34199a8 , 0x000040fc // exp(16 * 11)data8 0xff8c2b166241eedd , 0x00004113 // exp(16 * 12)data8 0x875a04c0b38d6129 , 0x0000412b // exp(16 * 13)data8 0x8f610127db6774d7 , 0x00004142 // exp(16 * 14)data8 0x97e1dd87e5c20bb6 , 0x00004159 // exp(16 * 15)
// Argument Reduction// exp_Mx = (int)f8 ==> The value of f8 rounded to int is placed into the// significand of exp_Mx as a two's// complement number.
// Later we want to have exp_Mx in a general register. Do this with a getf.sig// and call the general register exp_GR_Mint
// exp_Mfloat = (float)(int)f8 ==> the two's complement number in// significand of exp_Mx is turned// into a floating point number.// R = 1 - exp_Mfloat ==> reduced argument
// Core Approximation// Calculate a series in R// R * p6 + p5// R * p4 + p3// R * p2 + p1// R^2// R^4// R^2(R * p6 + p5) + (R * p4 + p3)// R^2(R * p2 + p1)// R^4(R^2(R * p6 + p5) + (R * p4 + p3)) + (R^2(R * p2 + p1))// R + 1// exp(R) = (1 + R) + R^4(R^2(R * p6 + p5) + (R * p4 + p3)) + (R^2(R * p2 + p1))// exp(R) = 1 + R + R^2 * p1 + R^3 * p2 + R^4 * p3 + R^5 * p4 + R^6 * p5 + R^7 * p6
// Reconstruction// signficand of exp_Mx is two's complement,// -103 < x < 89// The smallest single denormal is 2^-149 = ssdn// For e^x = ssdn// x = log(ssdn) = -103.279// But with rounding result goes to ssdn until -103.972079// The largest single normal is 1.<23 1's> 2^126 ~ 2^127 = lsn// For e^x = lsn// x = log(lsn) = 88.7228//// expf overflows when x > 42b17218 = 88.7228// expf returns largest single denormal when x = c2aeac50// expf goes to zero when x < c2cff1b5
// Consider range of 8-bit two's complement, -128 ---> 127// Add 128; range becomes 0 ---> 255
// The number (=i) in 0 ---> 255 is used as offset into two tables.
// i = abcd efgh = abcd * 16 + efgh = i1 * 16 + i2
// i1 = (exp_GR_Mint + 128) & 0xf0 (show 0xf0 as -0x10 to avoid assembler error)// (The immediate in the AND is an 8-bit two's complement)// i1 = i1 + start of T1 table (EXP_AD_T1)// Note that the entries in T1 are double-extended numbers on 16-byte boundaries// and that i1 is already shifted left by 16 after the AND.
// i2 must be shifted left by 4 before adding to the start of the table.// i2 = ((exp_GR_Mint + 128) & 0x0f) << 4// i2 = i2 + start of T2 table (EXP_AD_T2)
// T = T1 * T2// A = T * (1 + R)// answer = T * (R^2 * p1 + R^3 * p2 + R^4 * p3 + R^5 * p4 + R^6 * p5 + R^7 * p6) +// T * (1 + R)// = T * exp(R)
.global expf#
.section .text.proc expf#
.align 32expf:
{ .mfi alloc r32 = ar.pfs,1,26,4,0 fcvt.fx.s1 exp_Mx = f8 mov exp_GR_17ones = 0x1FFFF}{ .mlx addl EXP_AD_P_1 = @ltoff(exp_coeff_1_table),gp movl exp_GR_min_oflow = 0x42b17218 };;
// Fnorm done to take any enabled faults{ .mfi ld8 EXP_AD_P_1 = [EXP_AD_P_1] fclass.m p6,p0 = f8, 0x07 //@zero nop.i 999}{ .mfi add exp_GR_max_norm = -1, exp_GR_min_oflow // 0x42b17217 fnorm exp_norm_f8 = f8 nop.i 999};;
{ .mfi setf.s EXP_MIN_SGL_OFLOW_ARG = exp_GR_min_oflow // 0x42b17218 fclass.m p7,p0 = f8, 0x22 // Test for x=-inf mov exp_GR_0xf0 = 0x0f0}{ .mlx setf.s EXP_MAX_SGL_NORM_ARG = exp_GR_max_norm movl exp_GR_max_zero = 0xc2cff1b5 };;
{ .mlx mov exp_GR_0x0f = 0x00f movl exp_GR_max_uflow = 0xc2aeac50 }{ .mfb nop.m 999(p6) fma.s f8 = f1,f1,f0(p6) br.ret.spnt b0 // quick exit for x=0};;
{ .mfi setf.s EXP_MAX_SGL_ZERO_ARG = exp_GR_max_zero fclass.m p8,p0 = f8, 0x21 // Test for x=+inf adds exp_GR_min_norm = 1, exp_GR_max_uflow // 0xc2aeac51}{ .mfb ldfpd exp_coeff_P5,exp_coeff_P6 = [EXP_AD_P_1],16(p7) fma.s f8 = f0,f0,f0(p7) br.ret.spnt b0 // quick exit for x=-inf};;
{ .mmf ldfpd exp_coeff_P1,exp_coeff_P2 = [EXP_AD_P_1],16 setf.s EXP_MAX_SGL_UFLOW_ARG = exp_GR_max_uflow fclass.m p9,p0 = f8, 0xc3 // Test for x=nan};;
{ .mmb ldfpd exp_coeff_P3,exp_coeff_P4 = [EXP_AD_P_1],16 setf.s EXP_MIN_SGL_NORM_ARG = exp_GR_min_norm(p8) br.ret.spnt b0 // quick exit for x=+inf};;
// EXP_AD_P_1 now points to exp_T2_table{ .mfi mov exp_GR_T2_size = 0x100 fcvt.xf exp_Mfloat = exp_Mx nop.i 999};;
{ .mfb getf.sig exp_GR_Mint = exp_Mx(p9) fmerge.s f8 = exp_norm_f8, exp_norm_f8(p9) br.ret.spnt b0 // quick exit for x=nan};;
{ .mmi getf.exp exp_GR_signexp = exp_norm_f8 mov EXP_AD_T2 = EXP_AD_P_1 add EXP_AD_T1 = exp_GR_T2_size,EXP_AD_P_1 ;;
}
{ .mmi adds exp_GR_Mint_p_128 = 0x80,exp_GR_Mint ;;
and exp_GR_Ind1 = exp_GR_Mint_p_128, exp_GR_0xf0 and exp_GR_Ind2 = exp_GR_Mint_p_128, exp_GR_0x0f ;;
}
// Divide arguments into the following categories:// Certain Underflow/zero p11 - -inf < x <= MAX_SGL_ZERO_ARG // Certain Underflow p12 - MAX_SGL_ZERO_ARG < x <= MAX_SGL_UFLOW_ARG // Possible Underflow p13 - MAX_SGL_UFLOW_ARG < x < MIN_SGL_NORM_ARG// Certain Safe - MIN_SGL_NORM_ARG <= x <= MAX_SGL_NORM_ARG// Possible Overflow p14 - MAX_SGL_NORM_ARG < x < MIN_SGL_OFLOW_ARG// Certain Overflow p15 - MIN_SGL_OFLOW_ARG <= x < +inf//// If the input is really a single arg, then there will never be "Possible// Underflow" or "Possible Overflow" arguments.//
{ .mfi add EXP_AD_M1 = exp_GR_Ind1,EXP_AD_T1 fcmp.ge.s1 p15,p14 = exp_norm_f8,EXP_MIN_SGL_OFLOW_ARG nop.i 999}{ .mfi shladd EXP_AD_M2 = exp_GR_Ind2,4,EXP_AD_T2 fms.s1 exp_R = f1,f8,exp_Mfloat nop.i 999 ;;
}
{ .mfi ldfe exp_T1 = [EXP_AD_M1] fcmp.le.s1 p11,p12 = exp_norm_f8,EXP_MAX_SGL_ZERO_ARG mov exp_GR_bias = 0xffff};;
{ .mfb ldfe exp_T2 = [EXP_AD_M2](p14) fcmp.gt.s1 p14,p0 = exp_norm_f8,EXP_MAX_SGL_NORM_ARG(p15) br.cond.spnt EXP_CERTAIN_OVERFLOW ;;
}
{ .mfb and exp_GR_exp = exp_GR_signexp, exp_GR_17ones(p12) fcmp.le.s1 p12,p0 = exp_norm_f8,EXP_MAX_SGL_UFLOW_ARG(p11) br.cond.spnt EXP_CERTAIN_UNDERFLOW_ZERO};;
{ .mfi sub exp_GR_exp = exp_GR_exp, exp_GR_bias(p13) fcmp.lt.s1 p13,p0 = exp_norm_f8,EXP_MIN_SGL_NORM_ARG nop.i 999};;
{ .mfi nop.m 999 fma.s1 exp_Rsq = exp_R,exp_R,f0 nop.i 999}{ .mfi nop.m 999 fma.s1 exp_P3 = exp_R,exp_coeff_P2,exp_coeff_P1 nop.i 999 };;
{ .mfi nop.m 999 fma.s1 exp_P1 = exp_R,exp_coeff_P6,exp_coeff_P5 nop.i 999 }{ .mfi nop.m 999 fma.s1 exp_P2 = exp_R,exp_coeff_P4,exp_coeff_P3 nop.i 999};;
{ .mfi nop.m 999 fma.s1 exp_P7 = f1,exp_R,f1 nop.i 999};;
{ .mfi nop.m 999 fma.s1 exp_P5 = exp_Rsq,exp_P3,f0 nop.i 999}{ .mfi nop.m 999 fma.s1 exp_R4 = exp_Rsq,exp_Rsq,f0 nop.i 999 };;
{ .mfi nop.m 999 fma.s1 exp_T = exp_T1,exp_T2,f0 nop.i 999 }{ .mfi nop.m 999 fma.s1 exp_P4 = exp_Rsq,exp_P1,exp_P2 nop.i 999 };;
// There is a potential monotonicity problem in round to +inf. If x is // very small and negative, exp_P7 (and exp_A) will round to 1.0, // whereas exp_P6 will be very small, but positive. This would result in the // final calculation rounding to 3f800001. To fix this, we test for |x|<=2^63.// If so, the result will be set to 1+x.
{ .mfi cmp.ge p6,p7 = -63, exp_GR_exp fma.s1 exp_A = exp_T,exp_P7,f0 nop.i 999}{ .mfi nop.m 999 fma.s1 exp_P6 = exp_R4,exp_P4,exp_P5 nop.i 999};;
{ .bbb(p12) br.cond.spnt EXP_CERTAIN_UNDERFLOW(p13) br.cond.spnt EXP_POSSIBLE_UNDERFLOW(p14) br.cond.spnt EXP_POSSIBLE_OVERFLOW};;
{ .mfi nop.m 999(p6) fma.s f8 = f1,f1,exp_norm_f8 nop.i 999}{ .mfb nop.m 999(p7) fma.s f8 = exp_T,exp_P6,exp_A br.ret.sptk b0};;
EXP_POSSIBLE_OVERFLOW:
// We got an answer. EXP_MAX_SGL_NORM_ARG < x < EXP_MIN_SGL_OFLOW_ARG// overflow is a possibility, not a certainty// Set wre in s2 and perform the last operation with s2
// We define an overflow when the answer with// WRE set// user-defined rounding mode// is lsn +1
// Is the exponent 1 more than the largest single?// If so, go to ERROR RETURN, else (no overflow) get the answer and// leave.
// Largest single is FE (biased single)// FE - 7F + FFFF = 1007E
// Create + largest_single_plus_ulp// Create - largest_single_plus_ulp
// Calculate answer with WRE set.
// Cases when answer is lsn+1 are as follows:
// midpoint// |// lsn | lsn+1// --+----------|----------+------------// |// +inf +inf -inf// RN RN// RZ// exp_gt_pln contains the floating point number lsn+1.// The setf.exp puts 0x1007f in the exponent and 0x800... in the significand.
// If the answer is >= lsn+1, we have overflowed.// Then p6 is TRUE. Set the overflow tag, save input in FR_X,// do the final calculation for IEEE result, and branch to error return.
{ .mfi mov exp_GR_gt_ln = 0x1007F fsetc.s2 0x7F,0x42 nop.i 999};;
{ .mfi setf.exp exp_gt_pln = exp_GR_gt_ln fma.s.s2 exp_wre_urm_f8 = exp_T, exp_P6, exp_A nop.i 999};;
{ .mfi nop.m 999 fsetc.s2 0x7F,0x40 nop.i 999};;
{ .mfi nop.m 999 fcmp.ge.unc.s1 p6, p0 = exp_wre_urm_f8, exp_gt_pln nop.i 999};;
{ .mfb nop.m 999 nop.f 999(p6) br.cond.spnt EXP_CERTAIN_OVERFLOW // Branch if really overflow};;
{ .mfb nop.m 999 fma.s f8 = exp_T, exp_P6, exp_A br.ret.sptk b0 // Exit if really no overflow};;
EXP_CERTAIN_OVERFLOW:{ .mmi sub exp_GR_17ones_m1 = exp_GR_17ones, r0, 1 ;;
setf.exp f9 = exp_GR_17ones_m1 nop.i 999 ;;
}
{ .mfi nop.m 999 fmerge.s FR_X = f8,f8 nop.i 999}{ .mfb mov GR_Parameter_TAG = 16 fma.s FR_RESULT = f9, f9, f0 // Set I,O and +INF result br.cond.sptk __libm_error_region ;;
}
EXP_POSSIBLE_UNDERFLOW:
// We got an answer. EXP_MAX_SGL_UFLOW_ARG < x < EXP_MIN_SGL_NORM_ARG// underflow is a possibility, not a certainty
// We define an underflow when the answer with// ftz set// is zero (tiny numbers become zero)
// Notice (from below) that if we have an unlimited exponent range,// then there is an extra machine number E between the largest denormal and// the smallest normal.
// So if with unbounded exponent we round to E or below, then we are// tiny and underflow has occurred.
// But notice that you can be in a situation where we are tiny, namely// rounded to E, but when the exponent is bounded we round to smallest// normal. So the answer can be the smallest normal with underflow.
// E// -----+--------------------+--------------------+-----// | | |// 1.1...10 2^-7f 1.1...11 2^-7f 1.0...00 2^-7e // 0.1...11 2^-7e (biased, 1)// largest dn smallest normal
// If the answer is = 0, we have underflowed.// Then p6 is TRUE. Set the underflow tag, save input in FR_X,// do the final calculation for IEEE result, and branch to error return.
{ .mfi nop.m 999 fsetc.s2 0x7F,0x41 nop.i 999};;
{ .mfi nop.m 999 fma.s.s2 exp_ftz_urm_f8 = exp_T, exp_P6, exp_A nop.i 999};;
{ .mfi nop.m 999 fsetc.s2 0x7F,0x40 nop.i 999};;
{ .mfi nop.m 999 fcmp.eq.unc.s1 p6, p0 = exp_ftz_urm_f8, f0 nop.i 999};;
{ .mfb nop.m 999 nop.f 999(p6) br.cond.spnt EXP_CERTAIN_UNDERFLOW // Branch if really underflow };;
{ .mfb nop.m 999 fma.s f8 = exp_T, exp_P6, exp_A br.ret.sptk b0 // Exit if really no underflow};;
EXP_CERTAIN_UNDERFLOW:{ .mfi nop.m 999 fmerge.s FR_X = f8,f8 nop.i 999}{ .mfb mov GR_Parameter_TAG = 17 fma.s FR_RESULT = exp_T, exp_P6, exp_A // Set I,U and tiny result br.cond.sptk __libm_error_region ;;
}
EXP_CERTAIN_UNDERFLOW_ZERO:{ .mmi mov exp_GR_one = 1 ;;
setf.exp f9 = exp_GR_one nop.i 999 ;;
}
{ .mfi nop.m 999 fmerge.s FR_X = f8,f8 nop.i 999}{ .mfb mov GR_Parameter_TAG = 17 fma.s FR_RESULT = f9, f9, f0 // Set I,U and tiny (+0.0) result br.cond.sptk __libm_error_region ;;
}
.endp expf
.proc __libm_error_region__libm_error_region:.prologue{ .mfi add GR_Parameter_Y=-32,sp // Parameter 2 value nop.f 999.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{ .mfi stfs [GR_Parameter_X] = FR_X // Store Parameter 1 on stack nop.f 0 add GR_Parameter_RESULT = 0,GR_Parameter_Y // Parameter 3 address}{ .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 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};;
.endp __libm_error_region
.type __libm_error_support#,@function
.global __libm_error_support#
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