.file "exp2.s"// Copyright (c) 2000 - 2005, Intel Corporation// All rights reserved.//// Contributed 2000 by the Intel Numerics Group, Intel Corporation//// Redistribution and use in source and binary forms, with or without// modification, are permitted provided that the following conditions are// met://// * Redistributions of source code must retain the above copyright// notice, this list of conditions and the following disclaimer.//// * Redistributions in binary form must reproduce the above copyright// notice, this list of conditions and the following disclaimer in the// documentation and/or other materials provided with the distribution.//// * The name of Intel Corporation may not be used to endorse or promote// products derived from this software without specific prior written// permission.// 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://www.intel.com/software/products/opensource/libraries/num.htm.//// History//==============================================================// 08/25/00  Initial version// 05/20/02  Cleaned up namespace and sf0 syntax// 09/05/02  Improved performance// 01/17/03  Fixed to call error support when x=1024.0// 03/31/05  Reformatted delimiters between data tables//// API//==============================================================// double exp2(double)//// Overview of operation//==============================================================// Background//// Implementation//// Let x= (K + fh + fl + r), where// K is an integer, fh= 0.b1 b2 b3 b4 b5,// fl= 2^{-5}* 0.b6 b7 b8 b8 b10 (fh, fl >= 0),// and |r|<2^{-11}// Th is a table that stores 2^fh (32 entries) rounded to// double extended precision (only mantissa is stored)// Tl is a table that stores 2^fl (32 entries) rounded to// double extended precision (only mantissa is stored)//// 2^x is approximated as// 2^K * Th [ f ] * Tl [ f ] * (1+c1*r+c2*r^2+c3*r^3+c4*r^4)// Note: We use the following trick to speed up conversion from FP to integer://// Let  x = K + r, where K is an integer, and  |r| <= 0.5// Let N be the number of significand bits for the FP format used//   ( N=64 for double-extended, N=53 for double)//// Then let y = 1.5 * 2^(N-1)  +  x    for RN mode//          K = y -  1.5 * 2^(N-1)//          r  = x - K//// If we want to obtain the integer part and the first m fractional bits of x,// we can use the same trick, but with a constant of  1.5 * 2^(N-1-m)://// Let x = K + f + r// f = 0.b_1 b_2 ... b_m// |r| <= 2^(-m-1)//// Then let y = 1.5 * 2^(N-1-m)  +  x    for RN mode//          (K+f) = y -  1.5 * 2^(N-1-m)//          r  = x - K// Special values//==============================================================// exp2(0)= 1// exp2(+inf)= inf// exp2(-inf)= 0//// Registers used//==============================================================// r2-r3, r14-r40// f6-f15, f32-f45// p6-p8, p12//GR_TBL_START        = r2GR_LOG_TBL          = r3GR_OF_LIMIT         = r14GR_UF_LIMIT         = r15GR_EXP_CORR         = r16GR_F_low            = r17GR_F_high           = r18GR_K                = r19GR_Flow_ADDR        = r20GR_BIAS             = r21GR_Fh               = r22GR_Fh_ADDR          = r23GR_EXPMAX           = r24GR_EMIN             = r25GR_ROUNDVAL         = r26GR_MASK             = r27GR_KF0              = r28GR_MASK_low         = r29GR_COEFF_START      = r30GR_SAVE_B0          = r33GR_SAVE_PFS         = r34GR_SAVE_GP          = r35GR_SAVE_SP          = r36GR_Parameter_X      = r37GR_Parameter_Y      = r38GR_Parameter_RESULT = r39GR_Parameter_TAG    = r40FR_X                = f10FR_Y                = f1FR_RESULT           = f8FR_COEFF1           = f6FR_COEFF2           = f7FR_R                = f9FR_KF0              = f12FR_COEFF3           = f13FR_COEFF4           = f14FR_UF_LIMIT         = f15FR_OF_LIMIT         = f32FR_EXPMIN           = f33FR_ROUNDVAL         = f34FR_KF               = f35FR_2_TO_K           = f36FR_T_low            = f37FR_T_high           = f38FR_P34              = f39FR_R2               = f40FR_P12              = f41FR_T_low_K          = f42FR_P14              = f43FR_T                = f44FR_P                = f45// Data tables//==============================================================RODATA.align 16LOCAL_OBJECT_START(poly_coeffs)data8 0x3fac6b08d704a0c0, 0x3f83b2ab6fba4e77 // C_3 and C_4data8 0xb17217f7d1cf79ab, 0x00003ffe // C_1data8 0xf5fdeffc162c7541, 0x00003ffc // C_2LOCAL_OBJECT_END(poly_coeffs)LOCAL_OBJECT_START(T_table)// 2^{0.00000 b6 b7 b8 b9 b10}data8 0x8000000000000000, 0x8016302f17467628data8 0x802c6436d0e04f50, 0x80429c17d77c18eddata8 0x8058d7d2d5e5f6b0, 0x806f17687707a7afdata8 0x80855ad965e88b83, 0x809ba2264dada76adata8 0x80b1ed4fd999ab6c, 0x80c83c56b50cf77fdata8 0x80de8f3b8b85a0af, 0x80f4e5ff089f763edata8 0x810b40a1d81406d4, 0x81219f24a5baa59ddata8 0x813801881d886f7b, 0x814e67cceb90502cdata8 0x8164d1f3bc030773, 0x817b3ffd3b2f2e47data8 0x8191b1ea15813bfd, 0x81a827baf7838b78data8 0x81bea1708dde6055, 0x81d51f0b8557ec1cdata8 0x81eba08c8ad4536f, 0x820225f44b55b33bdata8 0x8218af4373fc25eb, 0x822f3c7ab205c89adata8 0x8245cd9ab2cec048, 0x825c62a423d13f0cdata8 0x8272fb97b2a5894c, 0x828998760d01faf3data8 0x82a0393fe0bb0ca8, 0x82b6ddf5dbc35906//// 2^{0.b1 b2 b3 b4 b5}data8 0x8000000000000000, 0x82cd8698ac2ba1d7data8 0x85aac367cc487b14, 0x88980e8092da8527data8 0x8b95c1e3ea8bd6e6, 0x8ea4398b45cd53c0data8 0x91c3d373ab11c336, 0x94f4efa8fef70961data8 0x9837f0518db8a96f, 0x9b8d39b9d54e5538data8 0x9ef5326091a111ad, 0xa27043030c496818data8 0xa5fed6a9b15138ea, 0xa9a15ab4ea7c0ef8data8 0xad583eea42a14ac6, 0xb123f581d2ac258fdata8 0xb504f333f9de6484, 0xb8fbaf4762fb9ee9data8 0xbd08a39f580c36be, 0xc12c4cca66709456data8 0xc5672a115506dadd, 0xc9b9bd866e2f27a2data8 0xce248c151f8480e3, 0xd2a81d91f12ae45adata8 0xd744fccad69d6af4, 0xdbfbb797daf23755data8 0xe0ccdeec2a94e111, 0xe5b906e77c8348a8data8 0xeac0c6e7dd24392e, 0xefe4b99bdcdaf5cbdata8 0xf5257d152486cc2c, 0xfa83b2db722a033aLOCAL_OBJECT_END(T_table).section .textGLOBAL_LIBM_ENTRY(exp2){.mfi       alloc r32= ar.pfs, 1, 4, 4, 0       // will continue only for non-zero normal/denormal numbers       fclass.nm p12, p0= f8, 0x1b       // GR_TBL_START= pointer to C_1...C_4 followed by T_table       addl GR_TBL_START= @ltoff(poly_coeffs), gp}{.mlx       mov GR_OF_LIMIT= 0xffff + 10              // Exponent of overflow limit       movl GR_ROUNDVAL= 0x5a400000              // 1.5*2^(63-10) (SP)};;// Form special constant 1.5*2^(63-10) to give integer part and first 10// fractional bits of x{.mfi       setf.s FR_ROUNDVAL= GR_ROUNDVAL           // Form special constant       fcmp.lt.s1 p6, p8= f8, f0                 // X<0 ?       nop.i 0}{.mfb       ld8 GR_COEFF_START= [ GR_TBL_START ]      // Load pointer to coeff table       nop.f 0 (p12) br.cond.spnt SPECIAL_exp2                 // Branch if nan, inf, zero};;{.mlx       setf.exp FR_OF_LIMIT= GR_OF_LIMIT         // Set overflow limit       movl GR_UF_LIMIT= 0xc4866000              // (-2^10-51) = -1075};;{.mfi       ldfpd FR_COEFF3, FR_COEFF4= [ GR_COEFF_START ], 16 // load C_3, C_4       fma.s0 f8= f8, f1, f0                     // normalize x       nop.i 0};;{.mmi       setf.s FR_UF_LIMIT= GR_UF_LIMIT           // Set underflow limit       ldfe FR_COEFF1= [ GR_COEFF_START ], 16    // load C_1       mov GR_EXP_CORR= 0xffff-126};;{.mfi       ldfe FR_COEFF2= [ GR_COEFF_START ], 16    // load C_2       fma.s1 FR_KF0= f8, f1, FR_ROUNDVAL        // y= x + 1.5*2^(63-10)       nop.i 0};;{.mfi       mov GR_MASK= 1023       fms.s1 FR_KF= FR_KF0, f1, FR_ROUNDVAL     // (K+f)       mov GR_MASK_low= 31};;{.mfi       getf.sig GR_KF0= FR_KF0                   // (K+f)*2^10= round_to_int(y)       fcmp.ge.s1 p12, p7= f8, FR_OF_LIMIT       // x >= overflow threshold ?       add GR_LOG_TBL= 256, GR_COEFF_START       // Pointer to high T_table};;{.mmi       and GR_F_low= GR_KF0, GR_MASK_low         // f_low       and GR_F_high= GR_MASK, GR_KF0            // f_high*32       shr GR_K= GR_KF0, 10                      // K};;{.mmi       shladd GR_Flow_ADDR= GR_F_low, 3, GR_COEFF_START // address of 2^{f_low}       add GR_BIAS= GR_K, GR_EXP_CORR            // K= bias-2*63       shr GR_Fh= GR_F_high, 5                   // f_high};;{.mfi       setf.exp FR_2_TO_K= GR_BIAS               // 2^{K-126}       fnma.s1 FR_R= FR_KF, f1, f8               // r= x - (K+f)       shladd GR_Fh_ADDR= GR_Fh, 3, GR_LOG_TBL   // address of 2^{f_high}}{.mlx       ldf8 FR_T_low= [ GR_Flow_ADDR ]           // load T_low= 2^{f_low}       movl GR_EMIN= 0xc47f8000                  // EMIN= -1022};;{.mfi       ldf8 FR_T_high= [ GR_Fh_ADDR ]            // load T_high= 2^{f_high} (p7)  fcmp.lt.s1 p12, p7= f8, FR_UF_LIMIT       // x<underflow threshold ?       nop.i 0};;{.mfi       setf.s FR_EXPMIN= GR_EMIN                 // FR_EXPMIN= EMIN       fma.s1 FR_P34= FR_COEFF4, FR_R, FR_COEFF3 // P34= C_3+C_4*r       nop.i 0}{.mfb       nop.m 0       fma.s1 FR_R2= FR_R, FR_R, f0              // r*r (p12) br.cond.spnt OUT_RANGE_exp2};;{.mfi       nop.m 0       fma.s1 FR_P12= FR_COEFF2, FR_R, FR_COEFF1 // P12= C_1+C_2*r       nop.i 0};;{.mfi       nop.m 0       fma.s1 FR_T_low_K= FR_T_low, FR_2_TO_K, f0 // T= 2^{K-126}*T_low       nop.i 0};;{.mfi       nop.m 0       fma.s1 FR_P14= FR_R2, FR_P34, FR_P12       // P14= P12+r2*P34       nop.i 0};;{.mfi       nop.m 0       fma.s1 FR_T= FR_T_low_K, FR_T_high, f0     // T= T*T_high       nop.i 0};;{.mfi       nop.m 0       fcmp.lt.s0 p6, p8= f8, FR_EXPMIN           // underflow (x<EMIN) ?       nop.i 0};;{.mfi       nop.m 0       fma.s1 FR_P= FR_P14, FR_R, f0              // P= P14*r       nop.i 0};;{.mfb       nop.m 0       fma.d.s0 f8= FR_P, FR_T, FR_T              // result= T+T*P (p8)  br.ret.sptk b0                             // return};;{.mfb (p6)  mov GR_Parameter_TAG= 162       nop.f 0 (p6)  br.cond.sptk __libm_error_region};;SPECIAL_exp2:{.mfi       nop.m 0       fclass.m p6, p0= f8, 0x22                  // x= -Infinity ?       nop.i 0};;{.mfi       nop.m 0       fclass.m p7, p0= f8, 0x21                  // x= +Infinity ?       nop.i 0};;{.mfi       nop.m 0       fclass.m p8, p0= f8, 0x7                   // x= +/-Zero ?       nop.i 0}{.mfb       nop.m 0 (p6)  mov f8= f0                                 // exp2(-Infinity)= 0 (p6)  br.ret.spnt b0};;{.mfb       nop.m 0       nop.f 0 (p7)  br.ret.spnt b0                             // exp2(+Infinity)= +Infinity};;{.mfb       nop.m 0 (p8)  mov f8= f1                                 // exp2(+/-0)= 1 (p8)  br.ret.spnt b0};;{.mfb       nop.m 0       fma.d.s0 f8= f8, f1, f0                    // Remaining cases: NaNs       br.ret.sptk b0};;OUT_RANGE_exp2:// overflow: p8= 1{.mii (p8)  mov GR_EXPMAX= 0x1fffe       nop.i 0       nop.i 0};;{.mmb (p8)  mov GR_Parameter_TAG= 161 (p8)  setf.exp FR_R= GR_EXPMAX       nop.b 999};;{.mfi       nop.m 999 (p8)  fma.d.s0 f8= FR_R, FR_R, f0                // Create overflow       nop.i 999}// underflow: p6= 1{.mii (p6)  mov GR_Parameter_TAG= 162 (p6)  mov GR_EXPMAX= 1       nop.i 0};;{.mmb       nop.m 0 (p6)  setf.exp FR_R= GR_EXPMAX       nop.b 999};;{.mfb       nop.m 999 (p6)  fma.d.s0 f8= FR_R, FR_R, f0                // Create underflow       nop.b 0};;GLOBAL_LIBM_END(exp2)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       stfd [ 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       stfd [ 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       stfd [ 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       add GR_Parameter_RESULT= 48, sp       nop.m 0       nop.i 0};;{.mmi       ldfd 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#