.file "exp.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 //============================================================== // 2/02/00 Initial version // 3/07/00 exp(inf) = inf but now does NOT call error support // exp(-inf) = 0 but now does NOT call error support // 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. // 11/30/00 Reworked to shorten main path, widen main path to include all // args in normal range, and add quick exit for 0, nan, inf. // 12/05/00 Loaded constants earlier with setf to save 2 cycles. // 02/05/02 Corrected uninitialize predicate in POSSIBLE_UNDERFLOW path // 05/20/02 Cleaned up namespace and sf0 syntax // 09/07/02 Force inexact flag // 11/15/02 Split underflow path into zero/nonzero; eliminated fma in main path // 05/30/03 Set inexact flag on unmasked overflow/underflow // 03/31/05 Reformatted delimiters between data tables // API //============================================================== // double exp(double) // Overview of operation //============================================================== // Take the input x. w is "how many log2/128 in x?" // w = x * 128/log2 // n = int(w) // x = n log2/128 + r + delta // n = 128M + index_1 + 2^4 index_2 // x = M log2 + (log2/128) index_1 + (log2/8) index_2 + r + delta // exp(x) = 2^M 2^(index_1/128) 2^(index_2/8) exp(r) exp(delta) // Construct 2^M // Get 2^(index_1/128) from table_1; // Get 2^(index_2/8) from table_2; // Calculate exp(r) by 5th order polynomial // r = x - n (log2/128)_high // delta = - n (log2/128)_low // Calculate exp(delta) as 1 + delta // Special values //============================================================== // exp(+0) = 1.0 // exp(-0) = 1.0 // exp(+qnan) = +qnan // exp(-qnan) = -qnan // exp(+snan) = +qnan // exp(-snan) = -qnan // exp(-inf) = +0 // exp(+inf) = +inf // Overflow and Underflow //======================= // exp(x) = largest double normal when // x = 709.7827 = 0x40862e42fefa39ef // exp(x) = smallest double normal when // x = -708.396 = 0xc086232bdd7abcd2 // exp(x) = largest round-to-nearest single zero when // x = -745.1332 = 0xc0874910d52d3052 // Registers used //============================================================== // Floating Point registers used: // f8, input, output // f6 -> f15, f32 -> f49 // General registers used: // r14 -> r40 // Predicate registers used: // p6 -> p15 // Assembly macros //============================================================== rRshf = r14 rAD_TB1 = r15 rAD_T1 = r15 rAD_TB2 = r16 rAD_T2 = r16 rAD_P = r17 rN = r18 rIndex_1 = r19 rIndex_2_16 = r20 rM = r21 rBiased_M = r21 rIndex_1_16 = r21 rSig_inv_ln2 = r22 rExp_bias = r23 rExp_mask = r24 rTmp = r25 rRshf_2to56 = r26 rGt_ln = r27 rExp_2tom56 = r28 GR_SAVE_B0 = r33 GR_SAVE_PFS = r34 GR_SAVE_GP = r35 GR_SAVE_SP = r36 GR_Parameter_X = r37 GR_Parameter_Y = r38 GR_Parameter_RESULT = r39 GR_Parameter_TAG = r40 FR_X = f10 FR_Y = f1 FR_RESULT = f8 fRSHF_2TO56 = f6 fINV_LN2_2TO63 = f7 fW_2TO56_RSH = f9 f2TOM56 = f11 fP5 = f12 fP54 = f12 fP5432 = f12 fP4 = f13 fP3 = f14 fP32 = f14 fP2 = f15 fP = f15 fLn2_by_128_hi = f33 fLn2_by_128_lo = f34 fRSHF = f35 fNfloat = f36 fNormX = f37 fR = f38 fF = f39 fRsq = f40 f2M = f41 fS1 = f42 fT1 = f42 fS2 = f43 fT2 = f43 fS = f43 fWre_urm_f8 = f44 fFtz_urm_f8 = f44 fMIN_DBL_OFLOW_ARG = f45 fMAX_DBL_ZERO_ARG = f46 fMAX_DBL_NORM_ARG = f47 fMIN_DBL_NORM_ARG = f48 fGt_pln = f49 fTmp = f49 // Data tables //============================================================== RODATA .align 16 // ************* DO NOT CHANGE ORDER OF THESE TABLES ******************** // double-extended 1/ln(2) // 3fff b8aa 3b29 5c17 f0bb be87fed0691d3e88 // 3fff b8aa 3b29 5c17 f0bc // For speed the significand will be loaded directly with a movl and setf.sig // and the exponent will be bias+63 instead of bias+0. Thus subsequent // computations need to scale appropriately. // The constant 128/ln(2) is needed for the computation of w. This is also // obtained by scaling the computations. // // Two shifting constants are loaded directly with movl and setf.d. // 1. fRSHF_2TO56 = 1.1000..00 * 2^(63-7) // This constant is added to x*1/ln2 to shift the integer part of // x*128/ln2 into the rightmost bits of the significand. // The result of this fma is fW_2TO56_RSH. // 2. fRSHF = 1.1000..00 * 2^(63) // This constant is subtracted from fW_2TO56_RSH * 2^(-56) to give // the integer part of w, n, as a floating-point number. // The result of this fms is fNfloat. LOCAL_OBJECT_START(exp_table_1) data8 0x40862e42fefa39f0 // smallest dbl overflow arg, +709.7827 data8 0xc0874910d52d3052 // largest arg for rnd-to-nearest 0 result, -745.133 data8 0x40862e42fefa39ef // largest dbl arg to give normal dbl result, +709.7827 data8 0xc086232bdd7abcd2 // smallest dbl arg to give normal dbl result, -708.396 data8 0xb17217f7d1cf79ab , 0x00003ff7 // ln2/128 hi data8 0xc9e3b39803f2f6af , 0x00003fb7 // ln2/128 lo // // Table 1 is 2^(index_1/128) where // index_1 goes from 0 to 15 // data8 0x8000000000000000 , 0x00003FFF data8 0x80B1ED4FD999AB6C , 0x00003FFF data8 0x8164D1F3BC030773 , 0x00003FFF data8 0x8218AF4373FC25EC , 0x00003FFF data8 0x82CD8698AC2BA1D7 , 0x00003FFF data8 0x8383594EEFB6EE37 , 0x00003FFF data8 0x843A28C3ACDE4046 , 0x00003FFF data8 0x84F1F656379C1A29 , 0x00003FFF data8 0x85AAC367CC487B15 , 0x00003FFF data8 0x8664915B923FBA04 , 0x00003FFF data8 0x871F61969E8D1010 , 0x00003FFF data8 0x87DB357FF698D792 , 0x00003FFF data8 0x88980E8092DA8527 , 0x00003FFF data8 0x8955EE03618E5FDD , 0x00003FFF data8 0x8A14D575496EFD9A , 0x00003FFF data8 0x8AD4C6452C728924 , 0x00003FFF LOCAL_OBJECT_END(exp_table_1) // Table 2 is 2^(index_1/8) where // index_2 goes from 0 to 7 LOCAL_OBJECT_START(exp_table_2) data8 0x8000000000000000 , 0x00003FFF data8 0x8B95C1E3EA8BD6E7 , 0x00003FFF data8 0x9837F0518DB8A96F , 0x00003FFF data8 0xA5FED6A9B15138EA , 0x00003FFF data8 0xB504F333F9DE6484 , 0x00003FFF data8 0xC5672A115506DADD , 0x00003FFF data8 0xD744FCCAD69D6AF4 , 0x00003FFF data8 0xEAC0C6E7DD24392F , 0x00003FFF LOCAL_OBJECT_END(exp_table_2) LOCAL_OBJECT_START(exp_p_table) data8 0x3f8111116da21757 //P5 data8 0x3fa55555d787761c //P4 data8 0x3fc5555555555414 //P3 data8 0x3fdffffffffffd6a //P2 LOCAL_OBJECT_END(exp_p_table) .section .text GLOBAL_IEEE754_ENTRY(exp) { .mlx nop.m 0 movl rSig_inv_ln2 = 0xb8aa3b295c17f0bc // significand of 1/ln2 } { .mlx addl rAD_TB1 = @ltoff(exp_table_1), gp movl rRshf_2to56 = 0x4768000000000000 // 1.10000 2^(63+56) } ;; { .mfi ld8 rAD_TB1 = [rAD_TB1] fclass.m p8,p0 = f8,0x07 // Test for x=0 mov rExp_mask = 0x1ffff } { .mfi mov rExp_bias = 0xffff fnorm.s1 fNormX = f8 mov rExp_2tom56 = 0xffff-56 } ;; // 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 { .mfi setf.sig fINV_LN2_2TO63 = rSig_inv_ln2 // form 1/ln2 * 2^63 fclass.m p9,p0 = f8,0x22 // Test for x=-inf nop.i 0 } { .mlx setf.d fRSHF_2TO56 = rRshf_2to56 // Form const 1.100 * 2^(63+56) movl rRshf = 0x43e8000000000000 // 1.10000 2^63 for right shift } ;; { .mfi ldfpd fMIN_DBL_OFLOW_ARG, fMAX_DBL_ZERO_ARG = [rAD_TB1],16 fclass.m p10,p0 = f8,0x1e1 // Test for x=+inf, nan, NaT nop.i 0 } { .mfb setf.exp f2TOM56 = rExp_2tom56 // form 2^-56 for scaling Nfloat (p9) fma.d.s0 f8 = f0,f0,f0 // quick exit for x=-inf (p9) br.ret.spnt b0 } ;; { .mfi ldfpd fMAX_DBL_NORM_ARG, fMIN_DBL_NORM_ARG = [rAD_TB1],16 nop.f 0 nop.i 0 } { .mfb setf.d fRSHF = rRshf // Form right shift const 1.100 * 2^63 (p8) fma.d.s0 f8 = f1,f1,f0 // quick exit for x=0 (p8) br.ret.spnt b0 } ;; { .mfb ldfe fLn2_by_128_hi = [rAD_TB1],16 (p10) fma.d.s0 f8 = f8,f8,f0 // Result if x=+inf, nan, NaT (p10) br.ret.spnt b0 // quick exit for x=+inf, nan, NaT } ;; { .mfi ldfe fLn2_by_128_lo = [rAD_TB1],16 fcmp.eq.s0 p6,p0 = f8, f0 // Dummy to set D nop.i 0 } ;; // After that last load, rAD_TB1 points to the beginning of table 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 nop.m 0 fma.s1 fW_2TO56_RSH = fNormX, fINV_LN2_2TO63, fRSHF_2TO56 nop.i 0 } ;; // Divide arguments into the following categories: // Certain Underflow p11 - -inf < x <= MAX_DBL_ZERO_ARG // Possible Underflow p13 - MAX_DBL_ZERO_ARG < x < MIN_DBL_NORM_ARG // Certain Safe - MIN_DBL_NORM_ARG <= x <= MAX_DBL_NORM_ARG // Possible Overflow p14 - MAX_DBL_NORM_ARG < x < MIN_DBL_OFLOW_ARG // Certain Overflow p15 - MIN_DBL_OFLOW_ARG <= x < +inf // // If the input is really a double arg, then there will never be // "Possible Overflow" arguments. // { .mfi add rAD_TB2 = 0x100, rAD_TB1 fcmp.ge.s1 p15,p0 = fNormX,fMIN_DBL_OFLOW_ARG nop.i 0 } ;; { .mfi add rAD_P = 0x80, rAD_TB2 fcmp.le.s1 p11,p0 = fNormX,fMAX_DBL_ZERO_ARG nop.i 0 } ;; { .mfb ldfpd fP5, fP4 = [rAD_P] ,16 fcmp.gt.s1 p14,p0 = fNormX,fMAX_DBL_NORM_ARG (p15) br.cond.spnt EXP_CERTAIN_OVERFLOW } ;; // Nfloat = round_int(W) // The signficand of fW_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 rN. // Since fW_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 fNfloat. // Thus, fNfloat contains the floating point version of N { .mfb ldfpd fP3, fP2 = [rAD_P] fms.s1 fNfloat = fW_2TO56_RSH, f2TOM56, fRSHF (p11) br.cond.spnt EXP_CERTAIN_UNDERFLOW } ;; { .mfi getf.sig rN = fW_2TO56_RSH nop.f 0 nop.i 0 } ;; // rIndex_1 has index_1 // rIndex_2_16 has index_2 * 16 // rBiased_M has M // rIndex_1_16 has index_1 * 16 // rM has true M // r = x - Nfloat * ln2_by_128_hi // f = 1 - Nfloat * ln2_by_128_lo { .mfi and rIndex_1 = 0x0f, rN fnma.s1 fR = fNfloat, fLn2_by_128_hi, fNormX shr rM = rN, 0x7 } { .mfi and rIndex_2_16 = 0x70, rN fnma.s1 fF = fNfloat, fLn2_by_128_lo, f1 nop.i 0 } ;; // rAD_T1 has address of T1 // rAD_T2 has address if T2 { .mmi add rBiased_M = rExp_bias, rM add rAD_T2 = rAD_TB2, rIndex_2_16 shladd rAD_T1 = rIndex_1, 4, rAD_TB1 } ;; // Create Scale = 2^M { .mmi setf.exp f2M = rBiased_M ldfe fT2 = [rAD_T2] nop.i 0 } ;; // Load T1 and T2 { .mfi ldfe fT1 = [rAD_T1] fmpy.s0 fTmp = fLn2_by_128_lo, fLn2_by_128_lo // Force inexact nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fRsq = fR, fR, f0 nop.i 0 } { .mfi nop.m 0 fma.s1 fP54 = fR, fP5, fP4 nop.i 0 } ;; { .mfi nop.m 0 fcmp.lt.s1 p13,p0 = fNormX,fMIN_DBL_NORM_ARG nop.i 0 } { .mfi nop.m 0 fma.s1 fP32 = fR, fP3, fP2 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fP5432 = fRsq, fP54, fP32 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fS1 = f2M,fT1,f0 nop.i 0 } { .mfi nop.m 0 fma.s1 fS2 = fF,fT2,f0 nop.i 0 } ;; { .mfi nop.m 0 fma.s1 fP = fRsq, fP5432, fR nop.i 0 } { .mfi nop.m 0 fma.s1 fS = fS1,fS2,f0 nop.i 0 } ;; { .mbb nop.m 0 (p13) br.cond.spnt EXP_POSSIBLE_UNDERFLOW (p14) br.cond.spnt EXP_POSSIBLE_OVERFLOW } ;; { .mfb nop.m 0 fma.d.s0 f8 = fS, fP, fS br.ret.sptk b0 // Normal path exit } ;; EXP_POSSIBLE_OVERFLOW: // Here if fMAX_DBL_NORM_ARG < x < fMIN_DBL_OFLOW_ARG // This cannot happen if input is a double, only if input higher precision. // Overflow is a possibility, not a certainty. // Recompute result using status field 2 with user's rounding mode, // and wre set. If result is larger than largest double, then we have // overflow { .mfi mov rGt_ln = 0x103ff // Exponent for largest dbl + 1 ulp fsetc.s2 0x7F,0x42 // Get user's round mode, set wre nop.i 0 } ;; { .mfi setf.exp fGt_pln = rGt_ln // Create largest double + 1 ulp fma.d.s2 fWre_urm_f8 = fS, fP, fS // Result with wre set nop.i 0 } ;; { .mfi nop.m 0 fsetc.s2 0x7F,0x40 // Turn off wre in sf2 nop.i 0 } ;; { .mfi nop.m 0 fcmp.ge.s1 p6, p0 = fWre_urm_f8, fGt_pln // Test for overflow nop.i 0 } ;; { .mfb nop.m 0 nop.f 0 (p6) br.cond.spnt EXP_CERTAIN_OVERFLOW // Branch if overflow } ;; { .mfb nop.m 0 fma.d.s0 f8 = fS, fP, fS br.ret.sptk b0 // Exit if really no overflow } ;; EXP_CERTAIN_OVERFLOW: { .mmi sub rTmp = rExp_mask, r0, 1 ;; setf.exp fTmp = rTmp nop.i 0 } ;; { .mfi alloc r32=ar.pfs,1,4,4,0 fmerge.s FR_X = f8,f8 nop.i 0 } { .mfb mov GR_Parameter_TAG = 14 fma.d.s0 FR_RESULT = fTmp, fTmp, fTmp // Set I,O and +INF result br.cond.sptk __libm_error_region } ;; EXP_POSSIBLE_UNDERFLOW: // Here if fMAX_DBL_ZERO_ARG < x < fMIN_DBL_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^-3fff 1.1...11 2^-3fff 1.0...00 2^-3ffe // 0.1...11 2^-3ffe (biased, 1) // largest dn smallest normal { .mfi nop.m 0 fsetc.s2 0x7F,0x41 // Get user's round mode, set ftz nop.i 0 } ;; { .mfi nop.m 0 fma.d.s2 fFtz_urm_f8 = fS, fP, fS // Result with ftz set nop.i 0 } ;; { .mfi nop.m 0 fsetc.s2 0x7F,0x40 // Turn off ftz in sf2 nop.i 0 } ;; { .mfi nop.m 0 fcmp.eq.s1 p6, p7 = fFtz_urm_f8, f0 // Test for underflow nop.i 0 } { .mfi nop.m 0 fma.d.s0 f8 = fS, fP, fS // Compute result, set I, maybe U nop.i 0 } ;; { .mbb nop.m 0 (p6) br.cond.spnt EXP_UNDERFLOW_COMMON // Branch if really underflow (p7) br.ret.sptk b0 // Exit if really no underflow } ;; EXP_CERTAIN_UNDERFLOW: // Here if x < fMAX_DBL_ZERO_ARG // Result will be zero (or smallest denorm if round to +inf) with I, U set { .mmi mov rTmp = 1 ;; setf.exp fTmp = rTmp // Form small normal nop.i 0 } ;; { .mfi nop.m 0 fmerge.se fTmp = fTmp, fLn2_by_128_lo // Small with signif lsb 1 nop.i 0 } ;; { .mfb nop.m 0 fma.d.s0 f8 = fTmp, fTmp, f0 // Set I,U, tiny (+0.0) result br.cond.sptk EXP_UNDERFLOW_COMMON } ;; EXP_UNDERFLOW_COMMON: // Determine if underflow result is zero or nonzero { .mfi alloc r32=ar.pfs,1,4,4,0 fcmp.eq.s1 p6, p0 = f8, f0 nop.i 0 } ;; { .mfb nop.m 0 fmerge.s FR_X = fNormX,fNormX (p6) br.cond.spnt EXP_UNDERFLOW_ZERO } ;; EXP_UNDERFLOW_NONZERO: // Here if x < fMIN_DBL_NORM_ARG and result nonzero; // I, U are set { .mfb mov GR_Parameter_TAG = 15 nop.f 0 // FR_RESULT already set br.cond.sptk __libm_error_region } ;; EXP_UNDERFLOW_ZERO: // Here if x < fMIN_DBL_NORM_ARG and result zero; // I, U are set { .mfb mov GR_Parameter_TAG = 15 nop.f 0 // FR_RESULT already set br.cond.sptk __libm_error_region } ;; GLOBAL_IEEE754_END(exp) 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#