.file "atanhl.s" // Copyright (c) 2001 - 2003, Intel Corporation// All rights reserved.//// Contributed 2001 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: // 09/10/01  Initial version// 12/11/01  Corrected .restore syntax// 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: atanhl(x) computes the principle value of the inverse // hyperbolic tangent of x.////*********************************************************************//// Resources Used:////    Floating-Point Registers: f8 (Input and Return Value)//                              f33-f73////    General Purpose Registers://      r32-r52//      r49-r52 (Used to pass arguments to error handling routine)////    Predicate Registers:      p6-p15////*********************************************************************//// IEEE Special Conditions:////    atanhl(inf) = QNaN//    atanhl(-inf) = QNaN //    atanhl(+/-0) = +/-0 //    atanhl(1) =  +inf //    atanhl(-1) =  -inf //    atanhl(|x|>1) = QNaN//    atanhl(SNaN) = QNaN//    atanhl(QNaN) = QNaN////*********************************************************************//// Overview//// The method consists of two cases.//// If      |x| < 1/32  use case atanhl_near_zero;// else                 use case atanhl_regular;//// Case atanhl_near_zero:////   atanhl(x) can be approximated by the Taylor series expansion//   up to order 17.//// Case atanhl_regular:////   Here we use formula atanhl(x) = sign(x)*log1pl(2*|x|/(1-|x|))/2 and//   calculation is subdivided into two stages. The first stage is //   calculating of X = 2*|x|/(1-|x|). The second one is calculating of //   sign(x)*log1pl(X)/2. To obtain required accuracy we use precise division//   algorythm output of which is a pair of two extended precision values those//   approximate result of division with accuracy higher than working//   precision. This pair is passed to modified log1pl function.//////   1. calculating of X = 2*|x|/(1-|x|)//   ( based on Peter Markstein's "IA-64 and Elementary Functions" book )//   ********************************************************************////     a = 2*|x|//     b = 1 - |x|//     b_lo = |x| - (1 - b)////     y = frcpa(b)         initial approximation of 1/b//     q = a*y              initial approximation of a/b//     //     e = 1 - b*y//     e2 = e + e^2//     e1 = e^2//     y1 = y + y*e2 = y + y*(e+e^2)////     e3 = e + e1^2//     y2 = y + y1*e3 = y + y*(e+e^2+..+e^6)////     r = a - b*q//     e = 1 - b*y2//     X = q + r*y2         high part of a/b////     y3 = y2 + y2*e4//     r1 = a - b*X//     r1 = r1 - b_lo*X//     X_lo = r1*y3         low part of a/b//  //   2. special log1p algorithm overview//   ***********************************////    Here we use a table lookup method. The basic idea is that in//    order to compute logl(Arg) = log1pl (Arg-1) 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 several steps.////    Step 0: Initialization//    ------//    We need to calculate logl(X + X_lo + 1). Obtain N, S_hi such that////      X + X_lo + 1 = 2^N * ( S_hi + S_lo )   exactly////    where S_hi in [1,2) and S_lo is a correction to S_hi in the sense//    that |S_lo| <= ulp(S_hi).////    For the special version of log1p we add X_lo to S_lo (S_lo = S_lo + X_lo)//    !-----------------------------------------------------------------------!////    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) + G * S_lo////    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, log1pl(X + X_lo) = logl(X + X_lo + 1) is given by////    logl(X + X_lo + 1) =  logl(2^N * (S_hi + S_lo))//                      ~=~ N*logl(2) + logl(1/G) + logl(1 + r)//                      ~=~ N*logl(2) + logl(1/G) + poly(r).////    For detailed description see log1p1 function, regular path.////*********************************************************************RODATA.align 64// ************* DO NOT CHANGE THE ORDER OF THESE TABLES *************LOCAL_OBJECT_START(Constants_TaylorSeries)data8  0xF0F0F0F0F0F0F0F1,0x00003FFA // C17data8  0x8888888888888889,0x00003FFB // C15data8  0x9D89D89D89D89D8A,0x00003FFB // C13data8  0xBA2E8BA2E8BA2E8C,0x00003FFB // C11data8  0xE38E38E38E38E38E,0x00003FFB // C9data8  0x9249249249249249,0x00003FFC // C7data8  0xCCCCCCCCCCCCCCCD,0x00003FFC // C5data8  0xAAAAAAAAAAAAAAAA,0x00003FFD // C3data4  0x3f000000                    // 1/2data4  0x00000000                    // pad data4  0x00000000data4  0x00000000LOCAL_OBJECT_END(Constants_TaylorSeries)LOCAL_OBJECT_START(Constants_Q)data4  0x00000000,0xB1721800,0x00003FFE,0x00000000 // log2_hidata4  0x4361C4C6,0x82E30865,0x0000BFE2,0x00000000 // log2_lodata4  0x328833CB,0xCCCCCAF2,0x00003FFC,0x00000000 // Q4data4  0xA9D4BAFB,0x80000077,0x0000BFFD,0x00000000 // Q3data4  0xAAABE3D2,0xAAAAAAAA,0x00003FFD,0x00000000 // Q2data4  0xFFFFDAB7,0xFFFFFFFF,0x0000BFFD,0x00000000 // Q1LOCAL_OBJECT_END(Constants_Q)// Z1 - 16 bit fixedLOCAL_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  0x00007ABBdata4  0x00007A45data4  0x000079D1data4  0x0000795Ddata4  0x000078EBLOCAL_OBJECT_END(Constants_Z_2)// G2 and H2 - IEEE single and h2 - IEEE doubleLOCAL_OBJECT_START(Constants_G_H_h2)data4  0x3F800000,0x00000000data8  0x0000000000000000data4  0x3F7F00F8,0x3B7F875Ddata8  0x3DB5A11622C42273data4  0x3F7E03F8,0x3BFF015Bdata8  0x3DE620CF21F86ED3data4  0x3F7D08E0,0x3C3EE393data8  0xBDAFA07E484F34EDdata4  0x3F7C0FC0,0x3C7E0586data8  0xBDFE07F03860BCF6data4  0x3F7B1880,0x3C9E75D2data8  0x3DEA370FA78093D6data4  0x3F7A2328,0x3CBDC97Adata8  0x3DFF579172A753D0data4  0x3F792FB0,0x3CDCFE47data8  0x3DFEBE6CA7EF896Bdata4  0x3F783E08,0x3CFC15D0data8  0x3E0CF156409ECB43data4  0x3F774E38,0x3D0D874Ddata8  0xBE0B6F97FFEF71DFdata4  0x3F766038,0x3D1CF49Bdata8  0xBE0804835D59EEE8data4  0x3F757400,0x3D2C531Ddata8  0x3E1F91E9A9192A74data4  0x3F748988,0x3D3BA322data8  0xBE139A06BF72A8CDdata4  0x3F73A0D0,0x3D4AE46Fdata8  0x3E1D9202F8FBA6CFdata4  0x3F72B9D0,0x3D5A1756data8  0xBE1DCCC4BA796223data4  0x3F71D488,0x3D693B9Ddata8  0xBE049391B6B7C239LOCAL_OBJECT_END(Constants_G_H_h2)// G3 and H3 - IEEE single and h3 - IEEE double LOCAL_OBJECT_START(Constants_G_H_h3)data4  0x3F7FFC00,0x38800100data8  0x3D355595562224CDdata4  0x3F7FF400,0x39400480data8  0x3D8200A206136FF6data4  0x3F7FEC00,0x39A00640data8  0x3DA4D68DE8DE9AF0data4  0x3F7FE400,0x39E00C41data8  0xBD8B4291B10238DCdata4  0x3F7FDC00,0x3A100A21data8  0xBD89CCB83B1952CAdata4  0x3F7FD400,0x3A300F22data8  0xBDB107071DC46826data4  0x3F7FCC08,0x3A4FF51Cdata8  0x3DB6FCB9F43307DBdata4  0x3F7FC408,0x3A6FFC1Ddata8  0xBD9B7C4762DC7872data4  0x3F7FBC10,0x3A87F20Bdata8  0xBDC3725E3F89154Adata4  0x3F7FB410,0x3A97F68Bdata8  0xBD93519D62B9D392data4  0x3F7FAC18,0x3AA7EB86data8  0x3DC184410F21BD9Ddata4  0x3F7FA420,0x3AB7E101data8  0xBDA64B952245E0A6data4  0x3F7F9C20,0x3AC7E701data8  0x3DB4B0ECAABB34B8data4  0x3F7F9428,0x3AD7DD7Bdata8  0x3D9923376DC40A7Edata4  0x3F7F8C30,0x3AE7D474data8  0x3DC6E17B4F2083D3data4  0x3F7F8438,0x3AF7CBEDdata8  0x3DAE314B811D4394data4  0x3F7F7C40,0x3B03E1F3data8  0xBDD46F21B08F2DB1data4  0x3F7F7448,0x3B0BDE2Fdata8  0xBDDC30A46D34522Bdata4  0x3F7F6C50,0x3B13DAAAdata8  0x3DCB0070B1F473DBdata4  0x3F7F6458,0x3B1BD766data8  0xBDD65DDC6AD282FDdata4  0x3F7F5C68,0x3B23CC5Cdata8  0xBDCDAB83F153761Adata4  0x3F7F5470,0x3B2BC997data8  0xBDDADA40341D0F8Fdata4  0x3F7F4C78,0x3B33C711data8  0x3DCD1BD7EBC394E8data4  0x3F7F4488,0x3B3BBCC6data8  0xBDC3532B52E3E695data4  0x3F7F3C90,0x3B43BAC0data8  0xBDA3961EE846B3DEdata4  0x3F7F34A0,0x3B4BB0F4data8  0xBDDADF06785778D4data4  0x3F7F2CA8,0x3B53AF6Ddata8  0x3DCC3ED1E55CE212