📄 e_logl.s
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.file "logl.s" // Copyright (c) 2000 - 2003, 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: // 05/21/01 Extracted logl and log10l from log1pl.s file, and optimized // all paths.// 06/20/01 Fixed error tag for x=-inf.// 05/20/02 Cleaned up namespace and sf0 syntax// 02/10/03 Reordered header: .section, .global, .proc, .align;// used data8 for long double table values////*********************************************************************////*********************************************************************//// Function: Combined logl(x) and log10l(x) where// logl(x) = ln(x), for double-extended precision x values// log10l(x) = log (x), for double-extended precision x values// 10////*********************************************************************//// Resources Used://// Floating-Point Registers: f8 (Input and Return Value)// f34-f76//// General Purpose Registers:// r32-r56// r53-r56 (Used to pass arguments to error handling routine)//// Predicate Registers: p6-p14////*********************************************************************//// IEEE Special Conditions://// Denormal fault raised on denormal inputs// Overflow exceptions cannot occur // Underflow exceptions raised when appropriate for log1p // (Error Handling Routine called for underflow)// Inexact raised when appropriate by algorithm//// logl(inf) = inf// logl(-inf) = QNaN // logl(+/-0) = -inf // logl(SNaN) = QNaN// logl(QNaN) = QNaN// logl(EM_special Values) = QNaN// log10l(inf) = inf// log10l(-inf) = QNaN // log10l(+/-0) = -inf // log10l(SNaN) = QNaN// log10l(QNaN) = QNaN// log10l(EM_special Values) = QNaN////*********************************************************************//// Overview//// The method consists of two cases.//// If |X-1| < 2^(-7) use case log_near1;// else use case log_regular;//// Case log_near1://// logl( 1 + X ) can be approximated by a simple polynomial// in W = X-1. This polynomial resembles the truncated Taylor// series W - W^/2 + W^3/3 - ...// // Case log_regular://// Here we use a table lookup method. The basic idea is that in// order to compute logl(Arg) for an argument Arg in [1,2), we // construct a value G such that G*Arg is close to 1 and that// logl(1/G) is obtainable easily from a table of values calculated// beforehand. Thus//// logl(Arg) = logl(1/G) + logl(G*Arg)// = logl(1/G) + logl(1 + (G*Arg - 1))//// Because |G*Arg - 1| is small, the second term on the right hand// side can be approximated by a short polynomial. We elaborate// this method in four steps.//// Step 0: Initialization//// We need to calculate logl( X ). Obtain N, S_hi such that//// X = 2^N * S_hi exactly//// where S_hi in [1,2) //// Step 1: Argument Reduction//// Based on S_hi, obtain G_1, G_2, G_3 from a table and calculate//// G := G_1 * G_2 * G_3// r := (G * S_hi - 1)//// These G_j's have the property that the product is exactly // representable and that |r| < 2^(-12) as a result.//// Step 2: Approximation////// logl(1 + r) is approximated by a short polynomial poly(r).//// Step 3: Reconstruction////// Finally, logl( X ) is given by//// logl( X ) = logl( 2^N * S_hi )// ~=~ N*logl(2) + logl(1/G) + logl(1 + r)// ~=~ N*logl(2) + logl(1/G) + poly(r).//// **** Algorithm ****//// Case log_near1://// Here we compute a simple polynomial. To exploit parallelism, we split// the polynomial into two portions.// // W := X - 1// Wsq := W * W// W4 := Wsq*Wsq// W6 := W4*Wsq// Y_hi := W + Wsq*(P_1 + W*(P_2 + W*(P_3 + W*P_4))// Y_lo := W6*(P_5 + W*(P_6 + W*(P_7 + W*P_8)))//// Case log_regular://// We present the algorithm in four steps.//// Step 0. Initialization// ----------------------//// Z := X // N := unbaised exponent of Z// S_hi := 2^(-N) * Z//// Step 1. Argument Reduction// --------------------------//// Let//// Z = 2^N * S_hi = 2^N * 1.d_1 d_2 d_3 ... d_63//// We obtain G_1, G_2, G_3 by the following steps.////// Define X_0 := 1.d_1 d_2 ... d_14. This is extracted// from S_hi.//// Define A_1 := 1.d_1 d_2 d_3 d_4. This is X_0 truncated// to lsb = 2^(-4).//// Define index_1 := [ d_1 d_2 d_3 d_4 ].//// Fetch Z_1 := (1/A_1) rounded UP in fixed point with// fixed point lsb = 2^(-15).// Z_1 looks like z_0.z_1 z_2 ... z_15// Note that the fetching is done using index_1.// A_1 is actually not needed in the implementation// and is used here only to explain how is the value// Z_1 defined.//// Fetch G_1 := (1/A_1) truncated to 21 sig. bits.// floating pt. Again, fetching is done using index_1. A_1// explains how G_1 is defined.//// Calculate X_1 := X_0 * Z_1 truncated to lsb = 2^(-14)// = 1.0 0 0 0 d_5 ... d_14// This is accomplised by integer multiplication.// It is proved that X_1 indeed always begin// with 1.0000 in fixed point.////// Define A_2 := 1.0 0 0 0 d_5 d_6 d_7 d_8. This is X_1 // truncated to lsb = 2^(-8). Similar to A_1,// A_2 is not needed in actual implementation. It// helps explain how some of the values are defined.//// Define index_2 := [ d_5 d_6 d_7 d_8 ].//// Fetch Z_2 := (1/A_2) rounded UP in fixed point with// fixed point lsb = 2^(-15). Fetch done using index_2.// Z_2 looks like z_0.z_1 z_2 ... z_15//// Fetch G_2 := (1/A_2) truncated to 21 sig. bits.// floating pt.//// Calculate X_2 := X_1 * Z_2 truncated to lsb = 2^(-14)// = 1.0 0 0 0 0 0 0 0 d_9 d_10 ... d_14// This is accomplised by integer multiplication.// It is proved that X_2 indeed always begin// with 1.00000000 in fixed point.////// Define A_3 := 1.0 0 0 0 0 0 0 0 d_9 d_10 d_11 d_12 d_13 1.// This is 2^(-14) + X_2 truncated to lsb = 2^(-13).//// Define index_3 := [ d_9 d_10 d_11 d_12 d_13 ].//// Fetch G_3 := (1/A_3) truncated to 21 sig. bits.// floating pt. Fetch is done using index_3.//// Compute G := G_1 * G_2 * G_3. //// This is done exactly since each of G_j only has 21 sig. bits.//// Compute //// r := (G*S_hi - 1) ////// Step 2. Approximation// ---------------------//// This step computes an approximation to logl( 1 + r ) where r is the// reduced argument just obtained. It is proved that |r| <= 1.9*2^(-13);// thus logl(1+r) can be approximated by a short polynomial://// logl(1+r) ~=~ poly = r + Q1 r^2 + ... + Q4 r^5////// Step 3. Reconstruction// ----------------------//// This step computes the desired result of logl(X)://// logl(X) = logl( 2^N * S_hi )// = N*logl(2) + logl( S_hi )// = N*logl(2) + logl(1/G) +// logl(1 + G*S_hi - 1 )//// logl(2), logl(1/G_j) are stored as pairs of (single,double) numbers:// log2_hi, log2_lo, log1byGj_hi, log1byGj_lo. The high parts are// single-precision numbers and the low parts are double precision// numbers. These have the property that//// N*log2_hi + SUM ( log1byGj_hi )//// is computable exactly in double-extended precision (64 sig. bits).// Finally//// Y_hi := N*log2_hi + SUM ( log1byGj_hi )// Y_lo := poly_hi + [ poly_lo + // ( SUM ( log1byGj_lo ) + N*log2_lo ) ]//RODATA.align 64// ************* DO NOT CHANGE THE ORDER OF THESE TABLES *************// P_8, P_7, P_6, P_5, P_4, P_3, P_2, and P_1 LOCAL_OBJECT_START(Constants_P)data8 0xE3936754EFD62B15,0x00003FFBdata8 0x8003B271A5E56381,0x0000BFFCdata8 0x9249248C73282DB0,0x00003FFCdata8 0xAAAAAA9F47305052,0x0000BFFCdata8 0xCCCCCCCCCCD17FC9,0x00003FFCdata8 0x8000000000067ED5,0x0000BFFDdata8 0xAAAAAAAAAAAAAAAA,0x00003FFDdata8 0xFFFFFFFFFFFFFFFE,0x0000BFFDLOCAL_OBJECT_END(Constants_P)// log2_hi, log2_lo, Q_4, Q_3, Q_2, and Q_1 LOCAL_OBJECT_START(Constants_Q)data8 0xB172180000000000,0x00003FFEdata8 0x82E308654361C4C6,0x0000BFE2data8 0xCCCCCAF2328833CB,0x00003FFCdata8 0x80000077A9D4BAFB,0x0000BFFDdata8 0xAAAAAAAAAAABE3D2,0x00003FFDdata8 0xFFFFFFFFFFFFDAB7,0x0000BFFDLOCAL_OBJECT_END(Constants_Q)// 1/ln10_hi, 1/ln10_loLOCAL_OBJECT_START(Constants_1_by_LN10)data8 0xDE5BD8A937287195,0x00003FFDdata8 0xD56EAABEACCF70C8,0x00003FBBLOCAL_OBJECT_END(Constants_1_by_LN10)// Z1 - 16 bit fixed LOCAL_OBJECT_START(Constants_Z_1)data4 0x00008000data4 0x00007879data4 0x000071C8data4 0x00006BCBdata4 0x00006667data4 0x00006187data4 0x00005D18data4 0x0000590Cdata4 0x00005556data4 0x000051ECdata4 0x00004EC5data4 0x00004BDBdata4 0x00004925data4 0x0000469Fdata4 0x00004445data4 0x00004211LOCAL_OBJECT_END(Constants_Z_1)// G1 and H1 - IEEE single and h1 - IEEE doubleLOCAL_OBJECT_START(Constants_G_H_h1)data4 0x3F800000,0x00000000data8 0x0000000000000000data4 0x3F70F0F0,0x3D785196data8 0x3DA163A6617D741Cdata4 0x3F638E38,0x3DF13843data8 0x3E2C55E6CBD3D5BBdata4 0x3F579430,0x3E2FF9A0data8 0xBE3EB0BFD86EA5E7data4 0x3F4CCCC8,0x3E647FD6data8 0x3E2E6A8C86B12760data4 0x3F430C30,0x3E8B3AE7data8 0x3E47574C5C0739BAdata4 0x3F3A2E88,0x3EA30C68data8 0x3E20E30F13E8AF2Fdata4 0x3F321640,0x3EB9CEC8data8 0xBE42885BF2C630BDdata4 0x3F2AAAA8,0x3ECF9927data8 0x3E497F3497E577C6data4 0x3F23D708,0x3EE47FC5data8 0x3E3E6A6EA6B0A5ABdata4 0x3F1D89D8,0x3EF8947Ddata8 0xBDF43E3CD328D9BEdata4 0x3F17B420,0x3F05F3A1data8 0x3E4094C30ADB090Adata4 0x3F124920,0x3F0F4303data8 0xBE28FBB2FC1FE510data4 0x3F0D3DC8,0x3F183EBFdata8 0x3E3A789510FDE3FAdata4 0x3F088888,0x3F20EC80data8 0x3E508CE57CC8C98Fdata4 0x3F042108,0x3F29516Adata8 0xBE534874A223106CLOCAL_OBJECT_END(Constants_G_H_h1)// Z2 - 16 bit fixedLOCAL_OBJECT_START(Constants_Z_2)data4 0x00008000data4 0x00007F81data4 0x00007F02data4 0x00007E85data4 0x00007E08data4 0x00007D8Ddata4 0x00007D12data4 0x00007C98data4 0x00007C20data4 0x00007BA8data4 0x00007B31data4 0x00007ABB⌨️ 快捷键说明
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