3245 lines
80 KiB
ArmAsm
3245 lines
80 KiB
ArmAsm
.file "tancotl.s"
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// Copyright (c) 2000 - 2003, Intel Corporation
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// All rights reserved.
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//
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// Contributed 2000 by the Intel Numerics Group, Intel Corporation
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//
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// Redistribution and use in source and binary forms, with or without
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// modification, are permitted provided that the following conditions are
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// met:
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//
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// * Redistributions of source code must retain the above copyright
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// notice, this list of conditions and the following disclaimer.
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//
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// * Redistributions in binary form must reproduce the above copyright
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// notice, this list of conditions and the following disclaimer in the
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// documentation and/or other materials provided with the distribution.
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//
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// * The name of Intel Corporation may not be used to endorse or promote
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// products derived from this software without specific prior written
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// permission.
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// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
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// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
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// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
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// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INTEL OR ITS
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// CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
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// EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
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// PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
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// PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY
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// OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY OR TORT (INCLUDING
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// NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
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// SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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//
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// Intel Corporation is the author of this code, and requests that all
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// problem reports or change requests be submitted to it directly at
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// http://www.intel.com/software/products/opensource/libraries/num.htm.
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//
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//*********************************************************************
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//
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// History:
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//
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// 02/02/00 (hand-optimized)
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// 04/04/00 Unwind support added
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// 12/28/00 Fixed false invalid flags
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// 02/06/02 Improved speed
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// 05/07/02 Changed interface to __libm_pi_by_2_reduce
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// 05/30/02 Added cotl
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// 02/10/03 Reordered header: .section, .global, .proc, .align;
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// used data8 for long double table values
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// 05/15/03 Reformatted data tables
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//
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//*********************************************************************
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//
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// Functions: tanl(x) = tangent(x), for double-extended precision x values
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// cotl(x) = cotangent(x), for double-extended precision x values
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//
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//*********************************************************************
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//
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// Resources Used:
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//
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// Floating-Point Registers: f8 (Input and Return Value)
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// f9-f15
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// f32-f121
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//
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// General Purpose Registers:
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// r14-r26,r32-r57
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//
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// Predicate Registers: p6-p15
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//
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//*********************************************************************
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//
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// IEEE Special Conditions for tanl:
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//
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// Denormal fault raised on denormal inputs
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// Overflow exceptions do not occur
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// Underflow exceptions raised when appropriate for tan
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// (No specialized error handling for this routine)
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// Inexact raised when appropriate by algorithm
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//
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// tanl(SNaN) = QNaN
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// tanl(QNaN) = QNaN
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// tanl(inf) = QNaN
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// tanl(+/-0) = +/-0
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//
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//*********************************************************************
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//
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// IEEE Special Conditions for cotl:
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//
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// Denormal fault raised on denormal inputs
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// Overflow exceptions occur at zero and near zero
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// Underflow exceptions do not occur
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// Inexact raised when appropriate by algorithm
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//
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// cotl(SNaN) = QNaN
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// cotl(QNaN) = QNaN
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// cotl(inf) = QNaN
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// cotl(+/-0) = +/-Inf and error handling is called
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//
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//*********************************************************************
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//
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// Below are mathematical and algorithmic descriptions for tanl.
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// For cotl we use next identity cot(x) = -tan(x + Pi/2).
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// So, to compute cot(x) we just need to increment N (N = N + 1)
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// and invert sign of the computed result.
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//
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//*********************************************************************
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//
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// Mathematical Description
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//
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// We consider the computation of FPTANL of Arg. Now, given
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//
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// Arg = N pi/2 + alpha, |alpha| <= pi/4,
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//
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// basic mathematical relationship shows that
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//
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// tan( Arg ) = tan( alpha ) if N is even;
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// = -cot( alpha ) otherwise.
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//
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// The value of alpha is obtained by argument reduction and
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// represented by two working precision numbers r and c where
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//
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// alpha = r + c accurately.
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//
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// The reduction method is described in a previous write up.
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// The argument reduction scheme identifies 4 cases. For Cases 2
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// and 4, because |alpha| is small, tan(r+c) and -cot(r+c) can be
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// computed very easily by 2 or 3 terms of the Taylor series
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// expansion as follows:
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//
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// Case 2:
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// -------
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//
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// tan(r + c) = r + c + r^3/3 ...accurately
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// -cot(r + c) = -1/(r+c) + r/3 ...accurately
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//
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// Case 4:
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// -------
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//
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// tan(r + c) = r + c + r^3/3 + 2r^5/15 ...accurately
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// -cot(r + c) = -1/(r+c) + r/3 + r^3/45 ...accurately
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//
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//
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// The only cases left are Cases 1 and 3 of the argument reduction
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// procedure. These two cases will be merged since after the
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// argument is reduced in either cases, we have the reduced argument
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// represented as r + c and that the magnitude |r + c| is not small
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// enough to allow the usage of a very short approximation.
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//
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// The greatest challenge of this task is that the second terms of
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// the Taylor series for tan(r) and -cot(r)
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//
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// r + r^3/3 + 2 r^5/15 + ...
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//
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// and
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//
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// -1/r + r/3 + r^3/45 + ...
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//
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// are not very small when |r| is close to pi/4 and the rounding
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// errors will be a concern if simple polynomial accumulation is
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// used. When |r| < 2^(-2), however, the second terms will be small
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// enough (5 bits or so of right shift) that a normal Horner
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// recurrence suffices. Hence there are two cases that we consider
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// in the accurate computation of tan(r) and cot(r), |r| <= pi/4.
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//
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// Case small_r: |r| < 2^(-2)
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// --------------------------
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//
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// Since Arg = N pi/4 + r + c accurately, we have
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//
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// tan(Arg) = tan(r+c) for N even,
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// = -cot(r+c) otherwise.
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//
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// Here for this case, both tan(r) and -cot(r) can be approximated
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// by simple polynomials:
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//
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// tan(r) = r + P1_1 r^3 + P1_2 r^5 + ... + P1_9 r^19
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// -cot(r) = -1/r + Q1_1 r + Q1_2 r^3 + ... + Q1_7 r^13
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//
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// accurately. Since |r| is relatively small, tan(r+c) and
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// -cot(r+c) can be accurately approximated by replacing r with
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// r+c only in the first two terms of the corresponding polynomials.
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//
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// Note that P1_1 (and Q1_1 for that matter) approximates 1/3 to
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// almost 64 sig. bits, thus
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//
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// P1_1 (r+c)^3 = P1_1 r^3 + c * r^2 accurately.
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//
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// Hence,
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//
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// tan(r+c) = r + P1_1 r^3 + P1_2 r^5 + ... + P1_9 r^19
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// + c*(1 + r^2)
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//
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// -cot(r+c) = -1/(r+c) + Q1_1 r + Q1_2 r^3 + ... + Q1_7 r^13
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// + Q1_1*c
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//
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//
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// Case normal_r: 2^(-2) <= |r| <= pi/4
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// ------------------------------------
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//
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// This case is more likely than the previous one if one considers
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// r to be uniformly distributed in [-pi/4 pi/4].
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//
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// The required calculation is either
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//
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// tan(r + c) = tan(r) + correction, or
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// -cot(r + c) = -cot(r) + correction.
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//
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// Specifically,
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//
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// tan(r + c) = tan(r) + c tan'(r) + O(c^2)
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// = tan(r) + c sec^2(r) + O(c^2)
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// = tan(r) + c SEC_sq ...accurately
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// as long as SEC_sq approximates sec^2(r)
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// to, say, 5 bits or so.
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//
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// Similarly,
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//
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// -cot(r + c) = -cot(r) - c cot'(r) + O(c^2)
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// = -cot(r) + c csc^2(r) + O(c^2)
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// = -cot(r) + c CSC_sq ...accurately
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// as long as CSC_sq approximates csc^2(r)
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// to, say, 5 bits or so.
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//
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// We therefore concentrate on accurately calculating tan(r) and
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// cot(r) for a working-precision number r, |r| <= pi/4 to within
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// 0.1% or so.
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//
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// We will employ a table-driven approach. Let
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//
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// r = sgn_r * 2^k * 1.b_1 b_2 ... b_5 ... b_63
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// = sgn_r * ( B + x )
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//
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// where
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//
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// B = 2^k * 1.b_1 b_2 ... b_5 1
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// x = |r| - B
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//
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// Now,
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// tan(B) + tan(x)
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// tan( B + x ) = ------------------------
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// 1 - tan(B)*tan(x)
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//
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// / \
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// | tan(B) + tan(x) |
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// = tan(B) + | ------------------------ - tan(B) |
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// | 1 - tan(B)*tan(x) |
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// \ /
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//
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// sec^2(B) * tan(x)
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// = tan(B) + ------------------------
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// 1 - tan(B)*tan(x)
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//
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// (1/[sin(B)*cos(B)]) * tan(x)
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// = tan(B) + --------------------------------
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// cot(B) - tan(x)
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//
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//
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// Clearly, the values of tan(B), cot(B) and 1/(sin(B)*cos(B)) are
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// calculated beforehand and stored in a table. Since
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//
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// |x| <= 2^k * 2^(-6) <= 2^(-7) (because k = -1, -2)
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//
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// a very short polynomial will be sufficient to approximate tan(x)
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// accurately. The details involved in computing the last expression
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// will be given in the next section on algorithm description.
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//
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//
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// Now, we turn to the case where cot( B + x ) is needed.
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//
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//
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// 1 - tan(B)*tan(x)
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// cot( B + x ) = ------------------------
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// tan(B) + tan(x)
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//
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// / \
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// | 1 - tan(B)*tan(x) |
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// = cot(B) + | ----------------------- - cot(B) |
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// | tan(B) + tan(x) |
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// \ /
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//
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// [tan(B) + cot(B)] * tan(x)
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// = cot(B) - ----------------------------
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// tan(B) + tan(x)
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//
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// (1/[sin(B)*cos(B)]) * tan(x)
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// = cot(B) - --------------------------------
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// tan(B) + tan(x)
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//
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//
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// Note that the values of tan(B), cot(B) and 1/(sin(B)*cos(B)) that
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// are needed are the same set of values needed in the previous
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// case.
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//
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// Finally, we can put all the ingredients together as follows:
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//
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// Arg = N * pi/2 + r + c ...accurately
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//
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// tan(Arg) = tan(r) + correction if N is even;
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// = -cot(r) + correction otherwise.
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//
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// For Cases 2 and 4,
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//
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// Case 2:
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// tan(Arg) = tan(r + c) = r + c + r^3/3 N even
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// = -cot(r + c) = -1/(r+c) + r/3 N odd
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// Case 4:
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// tan(Arg) = tan(r + c) = r + c + r^3/3 + 2r^5/15 N even
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// = -cot(r + c) = -1/(r+c) + r/3 + r^3/45 N odd
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//
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//
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// For Cases 1 and 3,
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//
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// Case small_r: |r| < 2^(-2)
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//
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// tan(Arg) = r + P1_1 r^3 + P1_2 r^5 + ... + P1_9 r^19
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// + c*(1 + r^2) N even
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//
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// = -1/(r+c) + Q1_1 r + Q1_2 r^3 + ... + Q1_7 r^13
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// + Q1_1*c N odd
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//
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// Case normal_r: 2^(-2) <= |r| <= pi/4
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//
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// tan(Arg) = tan(r) + c * sec^2(r) N even
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// = -cot(r) + c * csc^2(r) otherwise
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//
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// For N even,
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//
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// tan(Arg) = tan(r) + c*sec^2(r)
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// = tan( sgn_r * (B+x) ) + c * sec^2(|r|)
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// = sgn_r * ( tan(B+x) + sgn_r*c*sec^2(|r|) )
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// = sgn_r * ( tan(B+x) + sgn_r*c*sec^2(B) )
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//
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// since B approximates |r| to 2^(-6) in relative accuracy.
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//
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// / (1/[sin(B)*cos(B)]) * tan(x)
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// tan(Arg) = sgn_r * | tan(B) + --------------------------------
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// \ cot(B) - tan(x)
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// \
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// + CORR |
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// /
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// where
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//
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// CORR = sgn_r*c*tan(B)*SC_inv(B); SC_inv(B) = 1/(sin(B)*cos(B)).
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//
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// For N odd,
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//
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// tan(Arg) = -cot(r) + c*csc^2(r)
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// = -cot( sgn_r * (B+x) ) + c * csc^2(|r|)
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// = sgn_r * ( -cot(B+x) + sgn_r*c*csc^2(|r|) )
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// = sgn_r * ( -cot(B+x) + sgn_r*c*csc^2(B) )
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//
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// since B approximates |r| to 2^(-6) in relative accuracy.
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//
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// / (1/[sin(B)*cos(B)]) * tan(x)
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// tan(Arg) = sgn_r * | -cot(B) + --------------------------------
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// \ tan(B) + tan(x)
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// \
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// + CORR |
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// /
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// where
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//
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// CORR = sgn_r*c*cot(B)*SC_inv(B); SC_inv(B) = 1/(sin(B)*cos(B)).
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//
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//
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// The actual algorithm prescribes how all the mathematical formulas
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// are calculated.
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//
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//
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// 2. Algorithmic Description
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// ==========================
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//
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// 2.1 Computation for Cases 2 and 4.
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// ----------------------------------
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//
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// For Case 2, we use two-term polynomials.
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//
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// For N even,
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//
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// rsq := r * r
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// Poly := c + r * rsq * P1_1
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// Result := r + Poly ...in user-defined rounding
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//
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// For N odd,
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// S_hi := -frcpa(r) ...8 bits
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// S_hi := S_hi + S_hi*(1 + S_hi*r) ...16 bits
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// S_hi := S_hi + S_hi*(1 + S_hi*r) ...32 bits
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// S_hi := S_hi + S_hi*(1 + S_hi*r) ...64 bits
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// S_lo := S_hi*( (1 + S_hi*r) + S_hi*c )
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// ...S_hi + S_lo is -1/(r+c) to extra precision
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// S_lo := S_lo + Q1_1*r
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//
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// Result := S_hi + S_lo ...in user-defined rounding
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//
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// For Case 4, we use three-term polynomials
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//
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// For N even,
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//
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// rsq := r * r
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// Poly := c + r * rsq * (P1_1 + rsq * P1_2)
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// Result := r + Poly ...in user-defined rounding
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//
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// For N odd,
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// S_hi := -frcpa(r) ...8 bits
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// S_hi := S_hi + S_hi*(1 + S_hi*r) ...16 bits
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// S_hi := S_hi + S_hi*(1 + S_hi*r) ...32 bits
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// S_hi := S_hi + S_hi*(1 + S_hi*r) ...64 bits
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// S_lo := S_hi*( (1 + S_hi*r) + S_hi*c )
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// ...S_hi + S_lo is -1/(r+c) to extra precision
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// rsq := r * r
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// P := Q1_1 + rsq*Q1_2
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// S_lo := S_lo + r*P
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//
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// Result := S_hi + S_lo ...in user-defined rounding
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//
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//
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// Note that the coefficients P1_1, P1_2, Q1_1, and Q1_2 are
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// the same as those used in the small_r case of Cases 1 and 3
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// below.
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//
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//
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// 2.2 Computation for Cases 1 and 3.
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// ----------------------------------
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// This is further divided into the case of small_r,
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// where |r| < 2^(-2), and the case of normal_r, where |r| lies between
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// 2^(-2) and pi/4.
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//
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// Algorithm for the case of small_r
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// ---------------------------------
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//
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// For N even,
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// rsq := r * r
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// Poly1 := rsq*(P1_1 + rsq*(P1_2 + rsq*P1_3))
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// r_to_the_8 := rsq * rsq
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// r_to_the_8 := r_to_the_8 * r_to_the_8
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// Poly2 := P1_4 + rsq*(P1_5 + rsq*(P1_6 + ... rsq*P1_9))
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// CORR := c * ( 1 + rsq )
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// Poly := Poly1 + r_to_the_8*Poly2
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// Poly := r*Poly + CORR
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// Result := r + Poly ...in user-defined rounding
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// ...note that Poly1 and r_to_the_8 can be computed in parallel
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// ...with Poly2 (Poly1 is intentionally set to be much
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// ...shorter than Poly2 so that r_to_the_8 and CORR can be hidden)
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//
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// For N odd,
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// S_hi := -frcpa(r) ...8 bits
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// S_hi := S_hi + S_hi*(1 + S_hi*r) ...16 bits
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// S_hi := S_hi + S_hi*(1 + S_hi*r) ...32 bits
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// S_hi := S_hi + S_hi*(1 + S_hi*r) ...64 bits
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// S_lo := S_hi*( (1 + S_hi*r) + S_hi*c )
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// ...S_hi + S_lo is -1/(r+c) to extra precision
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// S_lo := S_lo + Q1_1*c
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//
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|
// ...S_hi and S_lo are computed in parallel with
|
|
// ...the following
|
|
// rsq := r*r
|
|
// P := Q1_1 + rsq*(Q1_2 + rsq*(Q1_3 + ... + rsq*Q1_7))
|
|
//
|
|
// Poly := r*P + S_lo
|
|
// Result := S_hi + Poly ...in user-defined rounding
|
|
//
|
|
//
|
|
// Algorithm for the case of normal_r
|
|
// ----------------------------------
|
|
//
|
|
// Here, we first consider the computation of tan( r + c ). As
|
|
// presented in the previous section,
|
|
//
|
|
// tan( r + c ) = tan(r) + c * sec^2(r)
|
|
// = sgn_r * [ tan(B+x) + CORR ]
|
|
// CORR = sgn_r * c * tan(B) * 1/[sin(B)*cos(B)]
|
|
//
|
|
// because sec^2(r) = sec^(|r|), and B approximate |r| to 6.5 bits.
|
|
//
|
|
// tan( r + c ) =
|
|
// / (1/[sin(B)*cos(B)]) * tan(x)
|
|
// sgn_r * | tan(B) + -------------------------------- +
|
|
// \ cot(B) - tan(x)
|
|
// \
|
|
// CORR |
|
|
|
|
// /
|
|
//
|
|
// The values of tan(B), cot(B) and 1/(sin(B)*cos(B)) are
|
|
// calculated beforehand and stored in a table. Specifically,
|
|
// the table values are
|
|
//
|
|
// tan(B) as T_hi + T_lo;
|
|
// cot(B) as C_hi + C_lo;
|
|
// 1/[sin(B)*cos(B)] as SC_inv
|
|
//
|
|
// T_hi, C_hi are in double-precision memory format;
|
|
// T_lo, C_lo are in single-precision memory format;
|
|
// SC_inv is in extended-precision memory format.
|
|
//
|
|
// The value of tan(x) will be approximated by a short polynomial of
|
|
// the form
|
|
//
|
|
// tan(x) as x + x * P, where
|
|
// P = x^2 * (P2_1 + x^2 * (P2_2 + x^2 * P2_3))
|
|
//
|
|
// Because |x| <= 2^(-7), cot(B) - x approximates cot(B) - tan(x)
|
|
// to a relative accuracy better than 2^(-20). Thus, a good
|
|
// initial guess of 1/( cot(B) - tan(x) ) to initiate the iterative
|
|
// division is:
|
|
//
|
|
// 1/(cot(B) - tan(x)) is approximately
|
|
// 1/(cot(B) - x) is
|
|
// tan(B)/(1 - x*tan(B)) is approximately
|
|
// T_hi / ( 1 - T_hi * x ) is approximately
|
|
//
|
|
// T_hi * [ 1 + (Thi * x) + (T_hi * x)^2 ]
|
|
//
|
|
// The calculation of tan(r+c) therefore proceed as follows:
|
|
//
|
|
// Tx := T_hi * x
|
|
// xsq := x * x
|
|
//
|
|
// V_hi := T_hi*(1 + Tx*(1 + Tx))
|
|
// P := xsq * (P1_1 + xsq*(P1_2 + xsq*P1_3))
|
|
// ...V_hi serves as an initial guess of 1/(cot(B) - tan(x))
|
|
// ...good to about 20 bits of accuracy
|
|
//
|
|
// tanx := x + x*P
|
|
// D := C_hi - tanx
|
|
// ...D is a double precision denominator: cot(B) - tan(x)
|
|
//
|
|
// V_hi := V_hi + V_hi*(1 - V_hi*D)
|
|
// ....V_hi approximates 1/(cot(B)-tan(x)) to 40 bits
|
|
//
|
|
// V_lo := V_hi * ( [ (1 - V_hi*C_hi) + V_hi*tanx ]
|
|
// - V_hi*C_lo ) ...observe all order
|
|
// ...V_hi + V_lo approximates 1/(cot(B) - tan(x))
|
|
// ...to extra accuracy
|
|
//
|
|
// ... SC_inv(B) * (x + x*P)
|
|
// ... tan(B) + ------------------------- + CORR
|
|
// ... cot(B) - (x + x*P)
|
|
// ...
|
|
// ... = tan(B) + SC_inv(B)*(x + x*P)*(V_hi + V_lo) + CORR
|
|
// ...
|
|
//
|
|
// Sx := SC_inv * x
|
|
// CORR := sgn_r * c * SC_inv * T_hi
|
|
//
|
|
// ...put the ingredients together to compute
|
|
// ... SC_inv(B) * (x + x*P)
|
|
// ... tan(B) + ------------------------- + CORR
|
|
// ... cot(B) - (x + x*P)
|
|
// ...
|
|
// ... = tan(B) + SC_inv(B)*(x + x*P)*(V_hi + V_lo) + CORR
|
|
// ...
|
|
// ... = T_hi + T_lo + CORR +
|
|
// ... Sx * V_hi + Sx * V_lo + Sx * P *(V_hi + V_lo)
|
|
//
|
|
// CORR := CORR + T_lo
|
|
// tail := V_lo + P*(V_hi + V_lo)
|
|
// tail := Sx * tail + CORR
|
|
// tail := Sx * V_hi + tail
|
|
// T_hi := sgn_r * T_hi
|
|
//
|
|
// ...T_hi + sgn_r*tail now approximate
|
|
// ...sgn_r*(tan(B+x) + CORR) accurately
|
|
//
|
|
// Result := T_hi + sgn_r*tail ...in user-defined
|
|
// ...rounding control
|
|
// ...It is crucial that independent paths be fully
|
|
// ...exploited for performance's sake.
|
|
//
|
|
//
|
|
// Next, we consider the computation of -cot( r + c ). As
|
|
// presented in the previous section,
|
|
//
|
|
// -cot( r + c ) = -cot(r) + c * csc^2(r)
|
|
// = sgn_r * [ -cot(B+x) + CORR ]
|
|
// CORR = sgn_r * c * cot(B) * 1/[sin(B)*cos(B)]
|
|
//
|
|
// because csc^2(r) = csc^(|r|), and B approximate |r| to 6.5 bits.
|
|
//
|
|
// -cot( r + c ) =
|
|
// / (1/[sin(B)*cos(B)]) * tan(x)
|
|
// sgn_r * | -cot(B) + -------------------------------- +
|
|
// \ tan(B) + tan(x)
|
|
// \
|
|
// CORR |
|
|
|
|
// /
|
|
//
|
|
// The values of tan(B), cot(B) and 1/(sin(B)*cos(B)) are
|
|
// calculated beforehand and stored in a table. Specifically,
|
|
// the table values are
|
|
//
|
|
// tan(B) as T_hi + T_lo;
|
|
// cot(B) as C_hi + C_lo;
|
|
// 1/[sin(B)*cos(B)] as SC_inv
|
|
//
|
|
// T_hi, C_hi are in double-precision memory format;
|
|
// T_lo, C_lo are in single-precision memory format;
|
|
// SC_inv is in extended-precision memory format.
|
|
//
|
|
// The value of tan(x) will be approximated by a short polynomial of
|
|
// the form
|
|
//
|
|
// tan(x) as x + x * P, where
|
|
// P = x^2 * (P2_1 + x^2 * (P2_2 + x^2 * P2_3))
|
|
//
|
|
// Because |x| <= 2^(-7), tan(B) + x approximates tan(B) + tan(x)
|
|
// to a relative accuracy better than 2^(-18). Thus, a good
|
|
// initial guess of 1/( tan(B) + tan(x) ) to initiate the iterative
|
|
// division is:
|
|
//
|
|
// 1/(tan(B) + tan(x)) is approximately
|
|
// 1/(tan(B) + x) is
|
|
// cot(B)/(1 + x*cot(B)) is approximately
|
|
// C_hi / ( 1 + C_hi * x ) is approximately
|
|
//
|
|
// C_hi * [ 1 - (C_hi * x) + (C_hi * x)^2 ]
|
|
//
|
|
// The calculation of -cot(r+c) therefore proceed as follows:
|
|
//
|
|
// Cx := C_hi * x
|
|
// xsq := x * x
|
|
//
|
|
// V_hi := C_hi*(1 - Cx*(1 - Cx))
|
|
// P := xsq * (P1_1 + xsq*(P1_2 + xsq*P1_3))
|
|
// ...V_hi serves as an initial guess of 1/(tan(B) + tan(x))
|
|
// ...good to about 18 bits of accuracy
|
|
//
|
|
// tanx := x + x*P
|
|
// D := T_hi + tanx
|
|
// ...D is a double precision denominator: tan(B) + tan(x)
|
|
//
|
|
// V_hi := V_hi + V_hi*(1 - V_hi*D)
|
|
// ....V_hi approximates 1/(tan(B)+tan(x)) to 40 bits
|
|
//
|
|
// V_lo := V_hi * ( [ (1 - V_hi*T_hi) - V_hi*tanx ]
|
|
// - V_hi*T_lo ) ...observe all order
|
|
// ...V_hi + V_lo approximates 1/(tan(B) + tan(x))
|
|
// ...to extra accuracy
|
|
//
|
|
// ... SC_inv(B) * (x + x*P)
|
|
// ... -cot(B) + ------------------------- + CORR
|
|
// ... tan(B) + (x + x*P)
|
|
// ...
|
|
// ... =-cot(B) + SC_inv(B)*(x + x*P)*(V_hi + V_lo) + CORR
|
|
// ...
|
|
//
|
|
// Sx := SC_inv * x
|
|
// CORR := sgn_r * c * SC_inv * C_hi
|
|
//
|
|
// ...put the ingredients together to compute
|
|
// ... SC_inv(B) * (x + x*P)
|
|
// ... -cot(B) + ------------------------- + CORR
|
|
// ... tan(B) + (x + x*P)
|
|
// ...
|
|
// ... =-cot(B) + SC_inv(B)*(x + x*P)*(V_hi + V_lo) + CORR
|
|
// ...
|
|
// ... =-C_hi - C_lo + CORR +
|
|
// ... Sx * V_hi + Sx * V_lo + Sx * P *(V_hi + V_lo)
|
|
//
|
|
// CORR := CORR - C_lo
|
|
// tail := V_lo + P*(V_hi + V_lo)
|
|
// tail := Sx * tail + CORR
|
|
// tail := Sx * V_hi + tail
|
|
// C_hi := -sgn_r * C_hi
|
|
//
|
|
// ...C_hi + sgn_r*tail now approximates
|
|
// ...sgn_r*(-cot(B+x) + CORR) accurately
|
|
//
|
|
// Result := C_hi + sgn_r*tail in user-defined rounding control
|
|
// ...It is crucial that independent paths be fully
|
|
// ...exploited for performance's sake.
|
|
//
|
|
// 3. Implementation Notes
|
|
// =======================
|
|
//
|
|
// Table entries T_hi, T_lo; C_hi, C_lo; SC_inv
|
|
//
|
|
// Recall that 2^(-2) <= |r| <= pi/4;
|
|
//
|
|
// r = sgn_r * 2^k * 1.b_1 b_2 ... b_63
|
|
//
|
|
// and
|
|
//
|
|
// B = 2^k * 1.b_1 b_2 b_3 b_4 b_5 1
|
|
//
|
|
// Thus, for k = -2, possible values of B are
|
|
//
|
|
// B = 2^(-2) * ( 1 + index/32 + 1/64 ),
|
|
// index ranges from 0 to 31
|
|
//
|
|
// For k = -1, however, since |r| <= pi/4 = 0.78...
|
|
// possible values of B are
|
|
//
|
|
// B = 2^(-1) * ( 1 + index/32 + 1/64 )
|
|
// index ranges from 0 to 19.
|
|
//
|
|
//
|
|
|
|
RODATA
|
|
.align 16
|
|
|
|
LOCAL_OBJECT_START(TANL_BASE_CONSTANTS)
|
|
|
|
tanl_table_1:
|
|
data8 0xA2F9836E4E44152A, 0x00003FFE // two_by_pi
|
|
data8 0xC84D32B0CE81B9F1, 0x00004016 // P_0
|
|
data8 0xC90FDAA22168C235, 0x00003FFF // P_1
|
|
data8 0xECE675D1FC8F8CBB, 0x0000BFBD // P_2
|
|
data8 0xB7ED8FBBACC19C60, 0x0000BF7C // P_3
|
|
LOCAL_OBJECT_END(TANL_BASE_CONSTANTS)
|
|
|
|
LOCAL_OBJECT_START(tanl_table_2)
|
|
data8 0xC90FDAA22168C234, 0x00003FFE // PI_BY_4
|
|
data8 0xA397E5046EC6B45A, 0x00003FE7 // Inv_P_0
|
|
data8 0x8D848E89DBD171A1, 0x0000BFBF // d_1
|
|
data8 0xD5394C3618A66F8E, 0x0000BF7C // d_2
|
|
data4 0x3E800000 // two**-2
|
|
data4 0xBE800000 // -two**-2
|
|
data4 0x00000000 // pad
|
|
data4 0x00000000 // pad
|
|
LOCAL_OBJECT_END(tanl_table_2)
|
|
|
|
LOCAL_OBJECT_START(tanl_table_p1)
|
|
data8 0xAAAAAAAAAAAAAABD, 0x00003FFD // P1_1
|
|
data8 0x8888888888882E6A, 0x00003FFC // P1_2
|
|
data8 0xDD0DD0DD0F0177B6, 0x00003FFA // P1_3
|
|
data8 0xB327A440646B8C6D, 0x00003FF9 // P1_4
|
|
data8 0x91371B251D5F7D20, 0x00003FF8 // P1_5
|
|
data8 0xEB69A5F161C67914, 0x00003FF6 // P1_6
|
|
data8 0xBEDD37BE019318D2, 0x00003FF5 // P1_7
|
|
data8 0x9979B1463C794015, 0x00003FF4 // P1_8
|
|
data8 0x8EBD21A38C6EB58A, 0x00003FF3 // P1_9
|
|
LOCAL_OBJECT_END(tanl_table_p1)
|
|
|
|
LOCAL_OBJECT_START(tanl_table_q1)
|
|
data8 0xAAAAAAAAAAAAAAB4, 0x00003FFD // Q1_1
|
|
data8 0xB60B60B60B5FC93E, 0x00003FF9 // Q1_2
|
|
data8 0x8AB355E00C9BBFBF, 0x00003FF6 // Q1_3
|
|
data8 0xDDEBBC89CBEE3D4C, 0x00003FF2 // Q1_4
|
|
data8 0xB3548A685F80BBB6, 0x00003FEF // Q1_5
|
|
data8 0x913625604CED5BF1, 0x00003FEC // Q1_6
|
|
data8 0xF189D95A8EE92A83, 0x00003FE8 // Q1_7
|
|
LOCAL_OBJECT_END(tanl_table_q1)
|
|
|
|
LOCAL_OBJECT_START(tanl_table_p2)
|
|
data8 0xAAAAAAAAAAAB362F, 0x00003FFD // P2_1
|
|
data8 0x88888886E97A6097, 0x00003FFC // P2_2
|
|
data8 0xDD108EE025E716A1, 0x00003FFA // P2_3
|
|
LOCAL_OBJECT_END(tanl_table_p2)
|
|
|
|
LOCAL_OBJECT_START(tanl_table_tm2)
|
|
//
|
|
// Entries T_hi double-precision memory format
|
|
// Index = 0,1,...,31 B = 2^(-2)*(1+Index/32+1/64)
|
|
// Entries T_lo single-precision memory format
|
|
// Index = 0,1,...,31 B = 2^(-2)*(1+Index/32+1/64)
|
|
//
|
|
data8 0x3FD09BC362400794
|
|
data4 0x23A05C32, 0x00000000
|
|
data8 0x3FD124A9DFFBC074
|
|
data4 0x240078B2, 0x00000000
|
|
data8 0x3FD1AE235BD4920F
|
|
data4 0x23826B8E, 0x00000000
|
|
data8 0x3FD2383515E2701D
|
|
data4 0x22D31154, 0x00000000
|
|
data8 0x3FD2C2E463739C2D
|
|
data4 0x2265C9E2, 0x00000000
|
|
data8 0x3FD34E36AFEEA48B
|
|
data4 0x245C05EB, 0x00000000
|
|
data8 0x3FD3DA317DBB35D1
|
|
data4 0x24749F2D, 0x00000000
|
|
data8 0x3FD466DA67321619
|
|
data4 0x2462CECE, 0x00000000
|
|
data8 0x3FD4F4371F94A4D5
|
|
data4 0x246D0DF1, 0x00000000
|
|
data8 0x3FD5824D740C3E6D
|
|
data4 0x240A85B5, 0x00000000
|
|
data8 0x3FD611234CB1E73D
|
|
data4 0x23F96E33, 0x00000000
|
|
data8 0x3FD6A0BEAD9EA64B
|
|
data4 0x247C5393, 0x00000000
|
|
data8 0x3FD73125B804FD01
|
|
data4 0x241F3B29, 0x00000000
|
|
data8 0x3FD7C25EAB53EE83
|
|
data4 0x2479989B, 0x00000000
|
|
data8 0x3FD8546FE6640EED
|
|
data4 0x23B343BC, 0x00000000
|
|
data8 0x3FD8E75FE8AF1892
|
|
data4 0x241454D1, 0x00000000
|
|
data8 0x3FD97B3553928BDA
|
|
data4 0x238613D9, 0x00000000
|
|
data8 0x3FDA0FF6EB9DE4DE
|
|
data4 0x22859FA7, 0x00000000
|
|
data8 0x3FDAA5AB99ECF92D
|
|
data4 0x237A6D06, 0x00000000
|
|
data8 0x3FDB3C5A6D8F1796
|
|
data4 0x23952F6C, 0x00000000
|
|
data8 0x3FDBD40A9CFB8BE4
|
|
data4 0x2280FC95, 0x00000000
|
|
data8 0x3FDC6CC387943100
|
|
data4 0x245D2EC0, 0x00000000
|
|
data8 0x3FDD068CB736C500
|
|
data4 0x23C4AD7D, 0x00000000
|
|
data8 0x3FDDA16DE1DDBC31
|
|
data4 0x23D076E6, 0x00000000
|
|
data8 0x3FDE3D6EEB515A93
|
|
data4 0x244809A6, 0x00000000
|
|
data8 0x3FDEDA97E6E9E5F1
|
|
data4 0x220856C8, 0x00000000
|
|
data8 0x3FDF78F11963CE69
|
|
data4 0x244BE993, 0x00000000
|
|
data8 0x3FE00C417D635BCE
|
|
data4 0x23D21799, 0x00000000
|
|
data8 0x3FE05CAB1C302CD3
|
|
data4 0x248A1B1D, 0x00000000
|
|
data8 0x3FE0ADB9DB6A1FA0
|
|
data4 0x23D53E33, 0x00000000
|
|
data8 0x3FE0FF724A20BA81
|
|
data4 0x24DB9ED5, 0x00000000
|
|
data8 0x3FE151D9153FA6F5
|
|
data4 0x24E9E451, 0x00000000
|
|
LOCAL_OBJECT_END(tanl_table_tm2)
|
|
|
|
LOCAL_OBJECT_START(tanl_table_tm1)
|
|
//
|
|
// Entries T_hi double-precision memory format
|
|
// Index = 0,1,...,19 B = 2^(-1)*(1+Index/32+1/64)
|
|
// Entries T_lo single-precision memory format
|
|
// Index = 0,1,...,19 B = 2^(-1)*(1+Index/32+1/64)
|
|
//
|
|
data8 0x3FE1CEC4BA1BE39E
|
|
data4 0x24B60F9E, 0x00000000
|
|
data8 0x3FE277E45ABD9B2D
|
|
data4 0x248C2474, 0x00000000
|
|
data8 0x3FE324180272B110
|
|
data4 0x247B8311, 0x00000000
|
|
data8 0x3FE3D38B890E2DF0
|
|
data4 0x24C55751, 0x00000000
|
|
data8 0x3FE4866D46236871
|
|
data4 0x24E5BC34, 0x00000000
|
|
data8 0x3FE53CEE45E044B0
|
|
data4 0x24001BA4, 0x00000000
|
|
data8 0x3FE5F74282EC06E4
|
|
data4 0x24B973DC, 0x00000000
|
|
data8 0x3FE6B5A125DF43F9
|
|
data4 0x24895440, 0x00000000
|
|
data8 0x3FE77844CAFD348C
|
|
data4 0x240021CA, 0x00000000
|
|
data8 0x3FE83F6BCEED6B92
|
|
data4 0x24C45372, 0x00000000
|
|
data8 0x3FE90B58A34F3665
|
|
data4 0x240DAD33, 0x00000000
|
|
data8 0x3FE9DC522C1E56B4
|
|
data4 0x24F846CE, 0x00000000
|
|
data8 0x3FEAB2A427041578
|
|
data4 0x2323FB6E, 0x00000000
|
|
data8 0x3FEB8E9F9DD8C373
|
|
data4 0x24B3090B, 0x00000000
|
|
data8 0x3FEC709B65C9AA7B
|
|
data4 0x2449F611, 0x00000000
|
|
data8 0x3FED58F4ACCF8435
|
|
data4 0x23616A7E, 0x00000000
|
|
data8 0x3FEE480F97635082
|
|
data4 0x24C2FEAE, 0x00000000
|
|
data8 0x3FEF3E57F0ACC544
|
|
data4 0x242CE964, 0x00000000
|
|
data8 0x3FF01E20F7E06E4B
|
|
data4 0x2480D3EE, 0x00000000
|
|
data8 0x3FF0A1258A798A69
|
|
data4 0x24DB8967, 0x00000000
|
|
LOCAL_OBJECT_END(tanl_table_tm1)
|
|
|
|
LOCAL_OBJECT_START(tanl_table_cm2)
|
|
//
|
|
// Entries C_hi double-precision memory format
|
|
// Index = 0,1,...,31 B = 2^(-2)*(1+Index/32+1/64)
|
|
// Entries C_lo single-precision memory format
|
|
// Index = 0,1,...,31 B = 2^(-2)*(1+Index/32+1/64)
|
|
//
|
|
data8 0x400ED3E2E63EFBD0
|
|
data4 0x259D94D4, 0x00000000
|
|
data8 0x400DDDB4C515DAB5
|
|
data4 0x245F0537, 0x00000000
|
|
data8 0x400CF57ABE19A79F
|
|
data4 0x25D4EA9F, 0x00000000
|
|
data8 0x400C1A06D15298ED
|
|
data4 0x24AE40A0, 0x00000000
|
|
data8 0x400B4A4C164B2708
|
|
data4 0x25A5AAB6, 0x00000000
|
|
data8 0x400A855A5285B068
|
|
data4 0x25524F18, 0x00000000
|
|
data8 0x4009CA5A3FFA549F
|
|
data4 0x24C999C0, 0x00000000
|
|
data8 0x4009188A646AF623
|
|
data4 0x254FD801, 0x00000000
|
|
data8 0x40086F3C6084D0E7
|
|
data4 0x2560F5FD, 0x00000000
|
|
data8 0x4007CDD2A29A76EE
|
|
data4 0x255B9D19, 0x00000000
|
|
data8 0x400733BE6C8ECA95
|
|
data4 0x25CB021B, 0x00000000
|
|
data8 0x4006A07E1F8DDC52
|
|
data4 0x24AB4722, 0x00000000
|
|
data8 0x4006139BC298AD58
|
|
data4 0x252764E2, 0x00000000
|
|
data8 0x40058CABBAD7164B
|
|
data4 0x24DAF5DB, 0x00000000
|
|
data8 0x40050B4BAE31A5D3
|
|
data4 0x25EA20F4, 0x00000000
|
|
data8 0x40048F2189F85A8A
|
|
data4 0x2583A3E8, 0x00000000
|
|
data8 0x400417DAA862380D
|
|
data4 0x25DCC4CC, 0x00000000
|
|
data8 0x4003A52B1088FCFE
|
|
data4 0x2430A492, 0x00000000
|
|
data8 0x400336CCCD3527D5
|
|
data4 0x255F77CF, 0x00000000
|
|
data8 0x4002CC7F5760766D
|
|
data4 0x25DA0BDA, 0x00000000
|
|
data8 0x4002660711CE02E3
|
|
data4 0x256FF4A2, 0x00000000
|
|
data8 0x4002032CD37BBE04
|
|
data4 0x25208AED, 0x00000000
|
|
data8 0x4001A3BD7F050775
|
|
data4 0x24B72DD6, 0x00000000
|
|
data8 0x40014789A554848A
|
|
data4 0x24AB4DAA, 0x00000000
|
|
data8 0x4000EE65323E81B7
|
|
data4 0x2584C440, 0x00000000
|
|
data8 0x4000982721CF1293
|
|
data4 0x25C9428D, 0x00000000
|
|
data8 0x400044A93D415EEB
|
|
data4 0x25DC8482, 0x00000000
|
|
data8 0x3FFFE78FBD72C577
|
|
data4 0x257F5070, 0x00000000
|
|
data8 0x3FFF4AC375EFD28E
|
|
data4 0x23EBBF7A, 0x00000000
|
|
data8 0x3FFEB2AF60B52DDE
|
|
data4 0x22EECA07, 0x00000000
|
|
data8 0x3FFE1F1935204180
|
|
data4 0x24191079, 0x00000000
|
|
data8 0x3FFD8FCA54F7E60A
|
|
data4 0x248D3058, 0x00000000
|
|
LOCAL_OBJECT_END(tanl_table_cm2)
|
|
|
|
LOCAL_OBJECT_START(tanl_table_cm1)
|
|
//
|
|
// Entries C_hi double-precision memory format
|
|
// Index = 0,1,...,19 B = 2^(-1)*(1+Index/32+1/64)
|
|
// Entries C_lo single-precision memory format
|
|
// Index = 0,1,...,19 B = 2^(-1)*(1+Index/32+1/64)
|
|
//
|
|
data8 0x3FFCC06A79F6FADE
|
|
data4 0x239C7886, 0x00000000
|
|
data8 0x3FFBB91F891662A6
|
|
data4 0x250BD191, 0x00000000
|
|
data8 0x3FFABFB6529F155D
|
|
data4 0x256CC3E6, 0x00000000
|
|
data8 0x3FF9D3002E964AE9
|
|
data4 0x250843E3, 0x00000000
|
|
data8 0x3FF8F1EF89DCB383
|
|
data4 0x2277C87E, 0x00000000
|
|
data8 0x3FF81B937C87DBD6
|
|
data4 0x256DA6CF, 0x00000000
|
|
data8 0x3FF74F141042EDE4
|
|
data4 0x2573D28A, 0x00000000
|
|
data8 0x3FF68BAF1784B360
|
|
data4 0x242E489A, 0x00000000
|
|
data8 0x3FF5D0B57C923C4C
|
|
data4 0x2532D940, 0x00000000
|
|
data8 0x3FF51D88F418EF20
|
|
data4 0x253C7DD6, 0x00000000
|
|
data8 0x3FF4719A02F88DAE
|
|
data4 0x23DB59BF, 0x00000000
|
|
data8 0x3FF3CC6649DA0788
|
|
data4 0x252B4756, 0x00000000
|
|
data8 0x3FF32D770B980DB8
|
|
data4 0x23FE585F, 0x00000000
|
|
data8 0x3FF2945FE56C987A
|
|
data4 0x25378A63, 0x00000000
|
|
data8 0x3FF200BDB16523F6
|
|
data4 0x247BB2E0, 0x00000000
|
|
data8 0x3FF172358CE27778
|
|
data4 0x24446538, 0x00000000
|
|
data8 0x3FF0E873FDEFE692
|
|
data4 0x2514638F, 0x00000000
|
|
data8 0x3FF0632C33154062
|
|
data4 0x24A7FC27, 0x00000000
|
|
data8 0x3FEFC42EB3EF115F
|
|
data4 0x248FD0FE, 0x00000000
|
|
data8 0x3FEEC9E8135D26F6
|
|
data4 0x2385C719, 0x00000000
|
|
LOCAL_OBJECT_END(tanl_table_cm1)
|
|
|
|
LOCAL_OBJECT_START(tanl_table_scim2)
|
|
//
|
|
// Entries SC_inv in Swapped IEEE format (extended)
|
|
// Index = 0,1,...,31 B = 2^(-2)*(1+Index/32+1/64)
|
|
//
|
|
data8 0x839D6D4A1BF30C9E, 0x00004001
|
|
data8 0x80092804554B0EB0, 0x00004001
|
|
data8 0xF959F94CA1CF0DE9, 0x00004000
|
|
data8 0xF3086BA077378677, 0x00004000
|
|
data8 0xED154515CCD4723C, 0x00004000
|
|
data8 0xE77909441C27CF25, 0x00004000
|
|
data8 0xE22D037D8DDACB88, 0x00004000
|
|
data8 0xDD2B2D8A89C73522, 0x00004000
|
|
data8 0xD86E1A23BB2C1171, 0x00004000
|
|
data8 0xD3F0E288DFF5E0F9, 0x00004000
|
|
data8 0xCFAF16B1283BEBD5, 0x00004000
|
|
data8 0xCBA4AFAA0D88DD53, 0x00004000
|
|
data8 0xC7CE03CCCA67C43D, 0x00004000
|
|
data8 0xC427BC820CA0DDB0, 0x00004000
|
|
data8 0xC0AECD57F13D8CAB, 0x00004000
|
|
data8 0xBD606C3871ECE6B1, 0x00004000
|
|
data8 0xBA3A0A96A44C4929, 0x00004000
|
|
data8 0xB7394F6FE5CCCEC1, 0x00004000
|
|
data8 0xB45C12039637D8BC, 0x00004000
|
|
data8 0xB1A0552892CB051B, 0x00004000
|
|
data8 0xAF04432B6BA2FFD0, 0x00004000
|
|
data8 0xAC862A237221235F, 0x00004000
|
|
data8 0xAA2478AF5F00A9D1, 0x00004000
|
|
data8 0xA7DDBB0C81E082BF, 0x00004000
|
|
data8 0xA5B0987D45684FEE, 0x00004000
|
|
data8 0xA39BD0F5627A8F53, 0x00004000
|
|
data8 0xA19E3B036EC5C8B0, 0x00004000
|
|
data8 0x9FB6C1F091CD7C66, 0x00004000
|
|
data8 0x9DE464101FA3DF8A, 0x00004000
|
|
data8 0x9C263139A8F6B888, 0x00004000
|
|
data8 0x9A7B4968C27B0450, 0x00004000
|
|
data8 0x98E2DB7E5EE614EE, 0x00004000
|
|
LOCAL_OBJECT_END(tanl_table_scim2)
|
|
|
|
LOCAL_OBJECT_START(tanl_table_scim1)
|
|
//
|
|
// Entries SC_inv in Swapped IEEE format (extended)
|
|
// Index = 0,1,...,19 B = 2^(-1)*(1+Index/32+1/64)
|
|
//
|
|
data8 0x969F335C13B2B5BA, 0x00004000
|
|
data8 0x93D446D9D4C0F548, 0x00004000
|
|
data8 0x9147094F61B798AF, 0x00004000
|
|
data8 0x8EF317CC758787AC, 0x00004000
|
|
data8 0x8CD498B3B99EEFDB, 0x00004000
|
|
data8 0x8AE82A7DDFF8BC37, 0x00004000
|
|
data8 0x892AD546E3C55D42, 0x00004000
|
|
data8 0x8799FEA9D15573C1, 0x00004000
|
|
data8 0x86335F88435A4B4C, 0x00004000
|
|
data8 0x84F4FB6E3E93A87B, 0x00004000
|
|
data8 0x83DD195280A382FB, 0x00004000
|
|
data8 0x82EA3D7FA4CB8C9E, 0x00004000
|
|
data8 0x821B247C6861D0A8, 0x00004000
|
|
data8 0x816EBED163E8D244, 0x00004000
|
|
data8 0x80E42D9127E4CFC6, 0x00004000
|
|
data8 0x807ABF8D28E64AFD, 0x00004000
|
|
data8 0x8031EF26863B4FD8, 0x00004000
|
|
data8 0x800960ADAE8C11FD, 0x00004000
|
|
data8 0x8000E1475FDBEC21, 0x00004000
|
|
data8 0x80186650A07791FA, 0x00004000
|
|
LOCAL_OBJECT_END(tanl_table_scim1)
|
|
|
|
Arg = f8
|
|
Save_Norm_Arg = f8 // For input to reduction routine
|
|
Result = f8
|
|
r = f8 // For output from reduction routine
|
|
c = f9 // For output from reduction routine
|
|
U_2 = f10
|
|
rsq = f11
|
|
C_hi = f12
|
|
C_lo = f13
|
|
T_hi = f14
|
|
T_lo = f15
|
|
|
|
d_1 = f33
|
|
N_0 = f34
|
|
tail = f35
|
|
tanx = f36
|
|
Cx = f37
|
|
Sx = f38
|
|
sgn_r = f39
|
|
CORR = f40
|
|
P = f41
|
|
D = f42
|
|
ArgPrime = f43
|
|
P_0 = f44
|
|
|
|
P2_1 = f45
|
|
P2_2 = f46
|
|
P2_3 = f47
|
|
|
|
P1_1 = f45
|
|
P1_2 = f46
|
|
P1_3 = f47
|
|
|
|
P1_4 = f48
|
|
P1_5 = f49
|
|
P1_6 = f50
|
|
P1_7 = f51
|
|
P1_8 = f52
|
|
P1_9 = f53
|
|
|
|
x = f56
|
|
xsq = f57
|
|
Tx = f58
|
|
Tx1 = f59
|
|
Set = f60
|
|
poly1 = f61
|
|
poly2 = f62
|
|
Poly = f63
|
|
Poly1 = f64
|
|
Poly2 = f65
|
|
r_to_the_8 = f66
|
|
B = f67
|
|
SC_inv = f68
|
|
Pos_r = f69
|
|
N_0_fix = f70
|
|
d_2 = f71
|
|
PI_BY_4 = f72
|
|
TWO_TO_NEG14 = f74
|
|
TWO_TO_NEG33 = f75
|
|
NEGTWO_TO_NEG14 = f76
|
|
NEGTWO_TO_NEG33 = f77
|
|
two_by_PI = f78
|
|
N = f79
|
|
N_fix = f80
|
|
P_1 = f81
|
|
P_2 = f82
|
|
P_3 = f83
|
|
s_val = f84
|
|
w = f85
|
|
B_mask1 = f86
|
|
B_mask2 = f87
|
|
w2 = f88
|
|
A = f89
|
|
a = f90
|
|
t = f91
|
|
U_1 = f92
|
|
NEGTWO_TO_NEG2 = f93
|
|
TWO_TO_NEG2 = f94
|
|
Q1_1 = f95
|
|
Q1_2 = f96
|
|
Q1_3 = f97
|
|
Q1_4 = f98
|
|
Q1_5 = f99
|
|
Q1_6 = f100
|
|
Q1_7 = f101
|
|
Q1_8 = f102
|
|
S_hi = f103
|
|
S_lo = f104
|
|
V_hi = f105
|
|
V_lo = f106
|
|
U_hi = f107
|
|
U_lo = f108
|
|
U_hiabs = f109
|
|
V_hiabs = f110
|
|
V = f111
|
|
Inv_P_0 = f112
|
|
|
|
FR_inv_pi_2to63 = f113
|
|
FR_rshf_2to64 = f114
|
|
FR_2tom64 = f115
|
|
FR_rshf = f116
|
|
Norm_Arg = f117
|
|
Abs_Arg = f118
|
|
TWO_TO_NEG65 = f119
|
|
fp_tmp = f120
|
|
mOne = f121
|
|
|
|
GR_sig_inv_pi = r14
|
|
GR_rshf_2to64 = r15
|
|
GR_exp_2tom64 = r16
|
|
GR_rshf = r17
|
|
GR_exp_2_to_63 = r18
|
|
GR_exp_2_to_24 = r19
|
|
GR_signexp_x = r20
|
|
GR_exp_x = r21
|
|
GR_exp_mask = r22
|
|
GR_exp_2tom14 = r23
|
|
GR_exp_m2tom14 = r24
|
|
GR_exp_2tom33 = r25
|
|
GR_exp_m2tom33 = r26
|
|
|
|
GR_SAVE_B0 = r33
|
|
GR_SAVE_GP = r34
|
|
GR_SAVE_PFS = r35
|
|
table_base = r36
|
|
table_ptr1 = r37
|
|
table_ptr2 = r38
|
|
table_ptr3 = r39
|
|
lookup = r40
|
|
N_fix_gr = r41
|
|
GR_exp_2tom2 = r42
|
|
GR_exp_2tom65 = r43
|
|
exp_r = r44
|
|
sig_r = r45
|
|
bmask1 = r46
|
|
table_offset = r47
|
|
bmask2 = r48
|
|
gr_tmp = r49
|
|
cot_flag = r50
|
|
|
|
GR_SAVE_B0 = r51
|
|
GR_SAVE_PFS = r52
|
|
GR_SAVE_GP = r53
|
|
GR_Parameter_X = r54
|
|
GR_Parameter_Y = r55
|
|
GR_Parameter_RESULT = r56
|
|
GR_Parameter_Tag = r57
|
|
|
|
|
|
.section .text
|
|
.global __libm_tanl#
|
|
.global __libm_cotl#
|
|
|
|
.proc __libm_cotl#
|
|
__libm_cotl:
|
|
.endp __libm_cotl#
|
|
LOCAL_LIBM_ENTRY(cotl)
|
|
|
|
{ .mlx
|
|
alloc r32 = ar.pfs, 0,22,4,0
|
|
movl GR_sig_inv_pi = 0xa2f9836e4e44152a // significand of 1/pi
|
|
}
|
|
{ .mlx
|
|
mov GR_exp_mask = 0x1ffff // Exponent mask
|
|
movl GR_rshf_2to64 = 0x47e8000000000000 // 1.1000 2^(63+64)
|
|
}
|
|
;;
|
|
|
|
// Check for NatVals, Infs , NaNs, and Zeros
|
|
{ .mfi
|
|
getf.exp GR_signexp_x = Arg // Get sign and exponent of x
|
|
fclass.m p6,p0 = Arg, 0x1E7 // Test for natval, nan, inf, zero
|
|
mov cot_flag = 0x1
|
|
}
|
|
{ .mfb
|
|
addl table_base = @ltoff(TANL_BASE_CONSTANTS), gp // Pointer to table ptr
|
|
fnorm.s1 Norm_Arg = Arg // Normalize x
|
|
br.cond.sptk COMMON_PATH
|
|
};;
|
|
|
|
LOCAL_LIBM_END(cotl)
|
|
|
|
.proc __libm_tanl#
|
|
__libm_tanl:
|
|
.endp __libm_tanl#
|
|
GLOBAL_IEEE754_ENTRY(tanl)
|
|
|
|
{ .mlx
|
|
alloc r32 = ar.pfs, 0,22,4,0
|
|
movl GR_sig_inv_pi = 0xa2f9836e4e44152a // significand of 1/pi
|
|
}
|
|
{ .mlx
|
|
mov GR_exp_mask = 0x1ffff // Exponent mask
|
|
movl GR_rshf_2to64 = 0x47e8000000000000 // 1.1000 2^(63+64)
|
|
}
|
|
;;
|
|
|
|
// Check for NatVals, Infs , NaNs, and Zeros
|
|
{ .mfi
|
|
getf.exp GR_signexp_x = Arg // Get sign and exponent of x
|
|
fclass.m p6,p0 = Arg, 0x1E7 // Test for natval, nan, inf, zero
|
|
mov cot_flag = 0x0
|
|
}
|
|
{ .mfi
|
|
addl table_base = @ltoff(TANL_BASE_CONSTANTS), gp // Pointer to table ptr
|
|
fnorm.s1 Norm_Arg = Arg // Normalize x
|
|
nop.i 0
|
|
};;
|
|
|
|
// Common path for both tanl and cotl
|
|
COMMON_PATH:
|
|
{ .mfi
|
|
setf.sig FR_inv_pi_2to63 = GR_sig_inv_pi // Form 1/pi * 2^63
|
|
fclass.m p9, p0 = Arg, 0x0b // Test x denormal
|
|
mov GR_exp_2tom64 = 0xffff - 64 // Scaling constant to compute N
|
|
}
|
|
{ .mlx
|
|
setf.d FR_rshf_2to64 = GR_rshf_2to64 // Form const 1.1000 * 2^(63+64)
|
|
movl GR_rshf = 0x43e8000000000000 // Form const 1.1000 * 2^63
|
|
}
|
|
;;
|
|
|
|
// Check for everything - if false, then must be pseudo-zero or pseudo-nan.
|
|
// Branch out to deal with special values.
|
|
{ .mfi
|
|
addl gr_tmp = -1,r0
|
|
fclass.nm p7,p0 = Arg, 0x1FF // Test x unsupported
|
|
mov GR_exp_2_to_63 = 0xffff + 63 // Exponent of 2^63
|
|
}
|
|
{ .mfb
|
|
ld8 table_base = [table_base] // Get pointer to constant table
|
|
fms.s1 mOne = f0, f0, f1
|
|
(p6) br.cond.spnt TANL_SPECIAL // Branch if x natval, nan, inf, zero
|
|
}
|
|
;;
|
|
|
|
{ .mmb
|
|
setf.sig fp_tmp = gr_tmp // Make a constant so fmpy produces inexact
|
|
mov GR_exp_2_to_24 = 0xffff + 24 // Exponent of 2^24
|
|
(p9) br.cond.spnt TANL_DENORMAL // Branch if x denormal
|
|
}
|
|
;;
|
|
|
|
TANL_COMMON:
|
|
// Return to here if x denormal
|
|
//
|
|
// Do fcmp to generate Denormal exception
|
|
// - can't do FNORM (will generate Underflow when U is unmasked!)
|
|
// Branch out to deal with unsupporteds values.
|
|
{ .mfi
|
|
setf.exp FR_2tom64 = GR_exp_2tom64 // Form 2^-64 for scaling N_float
|
|
fcmp.eq.s0 p0, p6 = Arg, f1 // Dummy to flag denormals
|
|
add table_ptr1 = 0, table_base // Point to tanl_table_1
|
|
}
|
|
{ .mib
|
|
setf.d FR_rshf = GR_rshf // Form right shift const 1.1000 * 2^63
|
|
add table_ptr2 = 80, table_base // Point to tanl_table_2
|
|
(p7) br.cond.spnt TANL_UNSUPPORTED // Branch if x unsupported type
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
and GR_exp_x = GR_exp_mask, GR_signexp_x // Get exponent of x
|
|
fmpy.s1 Save_Norm_Arg = Norm_Arg, f1 // Save x if large arg reduction
|
|
dep.z bmask1 = 0x7c, 56, 8 // Form mask to get 5 msb of r
|
|
// bmask1 = 0x7c00000000000000
|
|
}
|
|
;;
|
|
|
|
//
|
|
// Decide about the paths to take:
|
|
// Set PR_6 if |Arg| >= 2**63
|
|
// Set PR_9 if |Arg| < 2**24 - CASE 1 OR 2
|
|
// OTHERWISE Set PR_8 - CASE 3 OR 4
|
|
//
|
|
// Branch out if the magnitude of the input argument is >= 2^63
|
|
// - do this branch before the next.
|
|
{ .mfi
|
|
ldfe two_by_PI = [table_ptr1],16 // Load 2/pi
|
|
nop.f 999
|
|
dep.z bmask2 = 0x41, 57, 7 // Form mask to OR to produce B
|
|
// bmask2 = 0x8200000000000000
|
|
}
|
|
{ .mib
|
|
ldfe PI_BY_4 = [table_ptr2],16 // Load pi/4
|
|
cmp.ge p6,p0 = GR_exp_x, GR_exp_2_to_63 // Is |x| >= 2^63
|
|
(p6) br.cond.spnt TANL_ARG_TOO_LARGE // Branch if |x| >= 2^63
|
|
}
|
|
;;
|
|
|
|
{ .mmi
|
|
ldfe P_0 = [table_ptr1],16 // Load P_0
|
|
ldfe Inv_P_0 = [table_ptr2],16 // Load Inv_P_0
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
ldfe P_1 = [table_ptr1],16 // Load P_1
|
|
fmerge.s Abs_Arg = f0, Norm_Arg // Get |x|
|
|
mov GR_exp_m2tom33 = 0x2ffff - 33 // Form signexp of -2^-33
|
|
}
|
|
{ .mfi
|
|
ldfe d_1 = [table_ptr2],16 // Load d_1 for 2^24 <= |x| < 2^63
|
|
nop.f 999
|
|
mov GR_exp_2tom33 = 0xffff - 33 // Form signexp of 2^-33
|
|
}
|
|
;;
|
|
|
|
{ .mmi
|
|
ldfe P_2 = [table_ptr1],16 // Load P_2
|
|
ldfe d_2 = [table_ptr2],16 // Load d_2 for 2^24 <= |x| < 2^63
|
|
cmp.ge p8,p0 = GR_exp_x, GR_exp_2_to_24 // Is |x| >= 2^24
|
|
}
|
|
;;
|
|
|
|
// Use special scaling to right shift so N=Arg * 2/pi is in rightmost bits
|
|
// Branch to Cases 3 or 4 if Arg <= -2**24 or Arg >= 2**24
|
|
{ .mfb
|
|
ldfe P_3 = [table_ptr1],16 // Load P_3
|
|
fma.s1 N_fix = Norm_Arg, FR_inv_pi_2to63, FR_rshf_2to64
|
|
(p8) br.cond.spnt TANL_LARGER_ARG // Branch if 2^24 <= |x| < 2^63
|
|
}
|
|
;;
|
|
|
|
// Here if 0 < |x| < 2^24
|
|
// ARGUMENT REDUCTION CODE - CASE 1 and 2
|
|
//
|
|
{ .mmf
|
|
setf.exp TWO_TO_NEG33 = GR_exp_2tom33 // Form 2^-33
|
|
setf.exp NEGTWO_TO_NEG33 = GR_exp_m2tom33 // Form -2^-33
|
|
fmerge.s r = Norm_Arg,Norm_Arg // Assume r=x, ok if |x| < pi/4
|
|
}
|
|
;;
|
|
|
|
//
|
|
// If |Arg| < pi/4, set PR_8, else pi/4 <=|Arg| < 2^24 - set PR_9.
|
|
//
|
|
// Case 2: Convert integer N_fix back to normalized floating-point value.
|
|
{ .mfi
|
|
getf.sig sig_r = Norm_Arg // Get sig_r if 1/4 <= |x| < pi/4
|
|
fcmp.lt.s1 p8,p9= Abs_Arg,PI_BY_4 // Test |x| < pi/4
|
|
mov GR_exp_2tom2 = 0xffff - 2 // Form signexp of 2^-2
|
|
}
|
|
{ .mfi
|
|
ldfps TWO_TO_NEG2, NEGTWO_TO_NEG2 = [table_ptr2] // Load 2^-2, -2^-2
|
|
fms.s1 N = N_fix, FR_2tom64, FR_rshf // Use scaling to get N floated
|
|
mov N_fix_gr = r0 // Assume N=0, ok if |x| < pi/4
|
|
}
|
|
;;
|
|
|
|
//
|
|
// Case 1: Is |r| < 2**(-2).
|
|
// Arg is the same as r in this case.
|
|
// r = Arg
|
|
// c = 0
|
|
//
|
|
// Case 2: Place integer part of N in GP register.
|
|
{ .mfi
|
|
(p9) getf.sig N_fix_gr = N_fix
|
|
fmerge.s c = f0, f0 // Assume c=0, ok if |x| < pi/4
|
|
cmp.lt p10, p0 = GR_exp_x, GR_exp_2tom2 // Test if |x| < 1/4
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
setf.sig B_mask1 = bmask1 // Form mask to get 5 msb of r
|
|
nop.f 999
|
|
mov exp_r = GR_exp_x // Get exp_r if 1/4 <= |x| < pi/4
|
|
}
|
|
{ .mbb
|
|
setf.sig B_mask2 = bmask2 // Form mask to form B from r
|
|
(p10) br.cond.spnt TANL_SMALL_R // Branch if 0 < |x| < 1/4
|
|
(p8) br.cond.spnt TANL_NORMAL_R // Branch if 1/4 <= |x| < pi/4
|
|
}
|
|
;;
|
|
|
|
// Here if pi/4 <= |x| < 2^24
|
|
//
|
|
// Case 1: PR_3 is only affected when PR_1 is set.
|
|
//
|
|
//
|
|
// Case 2: w = N * P_2
|
|
// Case 2: s_val = -N * P_1 + Arg
|
|
//
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
fnma.s1 s_val = N, P_1, Norm_Arg
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
fmpy.s1 w = N, P_2 // w = N * P_2 for |s| >= 2^-33
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
// Case 2_reduce: w = N * P_3 (change sign)
|
|
{ .mfi
|
|
nop.m 999
|
|
fmpy.s1 w2 = N, P_3 // w = N * P_3 for |s| < 2^-33
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
// Case 1_reduce: r = s + w (change sign)
|
|
{ .mfi
|
|
nop.m 999
|
|
fsub.s1 r = s_val, w // r = s_val - w for |s| >= 2^-33
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
// Case 2_reduce: U_1 = N * P_2 + w
|
|
{ .mfi
|
|
nop.m 999
|
|
fma.s1 U_1 = N, P_2, w2 // U_1 = N * P_2 + w for |s| < 2^-33
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
//
|
|
// Decide between case_1 and case_2 reduce:
|
|
// Case 1_reduce: |s| >= 2**(-33)
|
|
// Case 2_reduce: |s| < 2**(-33)
|
|
//
|
|
{ .mfi
|
|
nop.m 999
|
|
fcmp.lt.s1 p9, p8 = s_val, TWO_TO_NEG33
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p9) fcmp.gt.s1 p9, p8 = s_val, NEGTWO_TO_NEG33
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
// Case 1_reduce: c = s - r
|
|
{ .mfi
|
|
nop.m 999
|
|
fsub.s1 c = s_val, r // c = s_val - r for |s| >= 2^-33
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
// Case 2_reduce: r is complete here - continue to calculate c .
|
|
// r = s - U_1
|
|
{ .mfi
|
|
nop.m 999
|
|
(p9) fsub.s1 r = s_val, U_1
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p9) fms.s1 U_2 = N, P_2, U_1
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
//
|
|
// Case 1_reduce: Is |r| < 2**(-2), if so set PR_10
|
|
// else set PR_13.
|
|
//
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
fand B = B_mask1, r
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p8) fcmp.lt.unc.s1 p10, p13 = r, TWO_TO_NEG2
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
(p8) getf.sig sig_r = r // Get signif of r if |s| >= 2^-33
|
|
nop.f 999
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
(p8) getf.exp exp_r = r // Extract signexp of r if |s| >= 2^-33
|
|
(p10) fcmp.gt.s1 p10, p13 = r, NEGTWO_TO_NEG2
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
// Case 1_reduce: c is complete here.
|
|
// Case 1: Branch to SMALL_R or NORMAL_R.
|
|
// c = c + w (w has not been negated.)
|
|
{ .mfi
|
|
nop.m 999
|
|
(p8) fsub.s1 c = c, w // c = c - w for |s| >= 2^-33
|
|
nop.i 999
|
|
}
|
|
{ .mbb
|
|
nop.m 999
|
|
(p10) br.cond.spnt TANL_SMALL_R // Branch if pi/4 < |x| < 2^24 and |r|<1/4
|
|
(p13) br.cond.sptk TANL_NORMAL_R_A // Branch if pi/4 < |x| < 2^24 and |r|>=1/4
|
|
}
|
|
;;
|
|
|
|
|
|
// Here if pi/4 < |x| < 2^24 and |s| < 2^-33
|
|
//
|
|
// Is i_1 = lsb of N_fix_gr even or odd?
|
|
// if i_1 == 0, set p11, else set p12.
|
|
//
|
|
{ .mfi
|
|
nop.m 999
|
|
fsub.s1 s_val = s_val, r
|
|
add N_fix_gr = N_fix_gr, cot_flag // N = N + 1 (for cotl)
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// Case 2_reduce:
|
|
// U_2 = N * P_2 - U_1
|
|
// Not needed until later.
|
|
//
|
|
fadd.s1 U_2 = U_2, w2
|
|
//
|
|
// Case 2_reduce:
|
|
// s = s - r
|
|
// U_2 = U_2 + w
|
|
//
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
//
|
|
// Case 2_reduce:
|
|
// c = c - U_2
|
|
// c is complete here
|
|
// Argument reduction ends here.
|
|
//
|
|
{ .mfi
|
|
nop.m 999
|
|
fmpy.s1 rsq = r, r
|
|
tbit.z p11, p12 = N_fix_gr, 0 ;; // Set p11 if N even, p12 if odd
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) frcpa.s1 S_hi,p0 = f1, r
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
fsub.s1 c = s_val, U_1
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
{ .mmi
|
|
add table_ptr1 = 160, table_base ;; // Point to tanl_table_p1
|
|
ldfe P1_1 = [table_ptr1],144
|
|
nop.i 999 ;;
|
|
}
|
|
//
|
|
// Load P1_1 and point to Q1_1 .
|
|
//
|
|
{ .mfi
|
|
ldfe Q1_1 = [table_ptr1]
|
|
//
|
|
// N even: rsq = r * Z
|
|
// N odd: S_hi = frcpa(r)
|
|
//
|
|
(p12) fmerge.ns S_hi = S_hi, S_hi
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// Case 2_reduce:
|
|
// c = s - U_1
|
|
//
|
|
(p9) fsub.s1 c = c, U_2
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fma.s1 poly1 = S_hi, r, f1
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N odd: Change sign of S_hi
|
|
//
|
|
(p11) fmpy.s1 rsq = rsq, P1_1
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fma.s1 S_hi = S_hi, poly1, S_hi
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: rsq = rsq * P1_1
|
|
// N odd: poly1 = 1.0 + S_hi * r 16 bits partial account for necessary
|
|
//
|
|
(p11) fma.s1 Poly = r, rsq, c
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: Poly = c + r * rsq
|
|
// N odd: S_hi = S_hi + S_hi*poly1 16 bits account for necessary
|
|
//
|
|
(p12) fma.s1 poly1 = S_hi, r, f1
|
|
(p11) tbit.z.unc p14, p15 = cot_flag, 0 ;; // p14=1 for tanl; p15=1 for cotl
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: Result = Poly + r
|
|
// N odd: poly1 = 1.0 + S_hi * r 32 bits partial
|
|
//
|
|
(p14) fadd.s0 Result = r, Poly // for tanl
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p15) fms.s0 Result = r, mOne, Poly // for cotl
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fma.s1 S_hi = S_hi, poly1, S_hi
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: Result1 = Result + r
|
|
// N odd: S_hi = S_hi * poly1 + S_hi 32 bits
|
|
//
|
|
(p12) fma.s1 poly1 = S_hi, r, f1
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N odd: poly1 = S_hi * r + 1.0 64 bits partial
|
|
//
|
|
(p12) fma.s1 S_hi = S_hi, poly1, S_hi
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N odd: poly1 = S_hi * poly + 1.0 64 bits
|
|
//
|
|
(p12) fma.s1 poly1 = S_hi, r, f1
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N odd: poly1 = S_hi * r + 1.0
|
|
//
|
|
(p12) fma.s1 poly1 = S_hi, c, poly1
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N odd: poly1 = S_hi * c + poly1
|
|
//
|
|
(p12) fmpy.s1 S_lo = S_hi, poly1
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N odd: S_lo = S_hi * poly1
|
|
//
|
|
(p12) fma.s1 S_lo = Q1_1, r, S_lo
|
|
(p12) tbit.z.unc p14, p15 = cot_flag, 0 // p14=1 for tanl; p15=1 for cotl
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N odd: Result = S_hi + S_lo
|
|
//
|
|
fmpy.s0 fp_tmp = fp_tmp, fp_tmp // Dummy mult to set inexact
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N odd: S_lo = S_lo + Q1_1 * r
|
|
//
|
|
(p14) fadd.s0 Result = S_hi, S_lo // for tanl
|
|
nop.i 999
|
|
}
|
|
{ .mfb
|
|
nop.m 999
|
|
(p15) fms.s0 Result = S_hi, mOne, S_lo // for cotl
|
|
br.ret.sptk b0 ;; // Exit for pi/4 <= |x| < 2^24 and |s| < 2^-33
|
|
}
|
|
|
|
|
|
TANL_LARGER_ARG:
|
|
// Here if 2^24 <= |x| < 2^63
|
|
//
|
|
// ARGUMENT REDUCTION CODE - CASE 3 and 4
|
|
//
|
|
|
|
{ .mmf
|
|
mov GR_exp_2tom14 = 0xffff - 14 // Form signexp of 2^-14
|
|
mov GR_exp_m2tom14 = 0x2ffff - 14 // Form signexp of -2^-14
|
|
fmpy.s1 N_0 = Norm_Arg, Inv_P_0
|
|
}
|
|
;;
|
|
|
|
{ .mmi
|
|
setf.exp TWO_TO_NEG14 = GR_exp_2tom14 // Form 2^-14
|
|
setf.exp NEGTWO_TO_NEG14 = GR_exp_m2tom14// Form -2^-14
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
|
|
//
|
|
// Adjust table_ptr1 to beginning of table.
|
|
// N_0 = Arg * Inv_P_0
|
|
//
|
|
{ .mmi
|
|
add table_ptr2 = 144, table_base ;; // Point to 2^-2
|
|
ldfps TWO_TO_NEG2, NEGTWO_TO_NEG2 = [table_ptr2]
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
//
|
|
// N_0_fix = integer part of N_0 .
|
|
//
|
|
//
|
|
// Make N_0 the integer part.
|
|
//
|
|
{ .mfi
|
|
nop.m 999
|
|
fcvt.fx.s1 N_0_fix = N_0
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
setf.sig B_mask1 = bmask1 // Form mask to get 5 msb of r
|
|
fcvt.xf N_0 = N_0_fix
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
setf.sig B_mask2 = bmask2 // Form mask to form B from r
|
|
fnma.s1 ArgPrime = N_0, P_0, Norm_Arg
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
fmpy.s1 w = N_0, d_1
|
|
nop.i 999 ;;
|
|
}
|
|
//
|
|
// ArgPrime = -N_0 * P_0 + Arg
|
|
// w = N_0 * d_1
|
|
//
|
|
//
|
|
// N = ArgPrime * 2/pi
|
|
//
|
|
// fcvt.fx.s1 N_fix = N
|
|
// Use special scaling to right shift so N=Arg * 2/pi is in rightmost bits
|
|
// Branch to Cases 3 or 4 if Arg <= -2**24 or Arg >= 2**24
|
|
{ .mfi
|
|
nop.m 999
|
|
fma.s1 N_fix = ArgPrime, FR_inv_pi_2to63, FR_rshf_2to64
|
|
|
|
nop.i 999 ;;
|
|
}
|
|
// Convert integer N_fix back to normalized floating-point value.
|
|
{ .mfi
|
|
nop.m 999
|
|
fms.s1 N = N_fix, FR_2tom64, FR_rshf // Use scaling to get N floated
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
//
|
|
// N is the integer part of the reduced-reduced argument.
|
|
// Put the integer in a GP register.
|
|
//
|
|
{ .mfi
|
|
getf.sig N_fix_gr = N_fix
|
|
nop.f 999
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
//
|
|
// s_val = -N*P_1 + ArgPrime
|
|
// w = -N*P_2 + w
|
|
//
|
|
{ .mfi
|
|
nop.m 999
|
|
fnma.s1 s_val = N, P_1, ArgPrime
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
fnma.s1 w = N, P_2, w
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
// Case 4: V_hi = N * P_2
|
|
// Case 4: U_hi = N_0 * d_1
|
|
{ .mfi
|
|
nop.m 999
|
|
fmpy.s1 V_hi = N, P_2 // V_hi = N * P_2 for |s| < 2^-14
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
fmpy.s1 U_hi = N_0, d_1 // U_hi = N_0 * d_1 for |s| < 2^-14
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
// Case 3: r = s_val + w (Z complete)
|
|
// Case 4: w = N * P_3
|
|
{ .mfi
|
|
nop.m 999
|
|
fadd.s1 r = s_val, w // r = s_val + w for |s| >= 2^-14
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
fmpy.s1 w2 = N, P_3 // w = N * P_3 for |s| < 2^-14
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
// Case 4: A = U_hi + V_hi
|
|
// Note: Worry about switched sign of V_hi, so subtract instead of add.
|
|
// Case 4: V_lo = -N * P_2 - V_hi (U_hi is in place of V_hi in writeup)
|
|
// Note: the (-) is still missing for V_hi.
|
|
{ .mfi
|
|
nop.m 999
|
|
fsub.s1 A = U_hi, V_hi // A = U_hi - V_hi for |s| < 2^-14
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
fnma.s1 V_lo = N, P_2, V_hi // V_lo = V_hi - N * P_2 for |s| < 2^-14
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
// Decide between case 3 and 4:
|
|
// Case 3: |s| >= 2**(-14) Set p10
|
|
// Case 4: |s| < 2**(-14) Set p11
|
|
//
|
|
// Case 4: U_lo = N_0 * d_1 - U_hi
|
|
{ .mfi
|
|
nop.m 999
|
|
fms.s1 U_lo = N_0, d_1, U_hi // U_lo = N_0*d_1 - U_hi for |s| < 2^-14
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
fcmp.lt.s1 p11, p10 = s_val, TWO_TO_NEG14
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
// Case 4: We need abs of both U_hi and V_hi - dont
|
|
// worry about switched sign of V_hi.
|
|
{ .mfi
|
|
nop.m 999
|
|
fabs V_hiabs = V_hi // |V_hi| for |s| < 2^-14
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p11) fcmp.gt.s1 p11, p10 = s_val, NEGTWO_TO_NEG14
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
// Case 3: c = s_val - r
|
|
{ .mfi
|
|
nop.m 999
|
|
fabs U_hiabs = U_hi // |U_hi| for |s| < 2^-14
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
fsub.s1 c = s_val, r // c = s_val - r for |s| >= 2^-14
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
// For Case 3, |s| >= 2^-14, determine if |r| < 1/4
|
|
//
|
|
// Case 4: C_hi = s_val + A
|
|
//
|
|
{ .mfi
|
|
nop.m 999
|
|
(p11) fadd.s1 C_hi = s_val, A // C_hi = s_val + A for |s| < 2^-14
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p10) fcmp.lt.unc.s1 p14, p15 = r, TWO_TO_NEG2
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
getf.sig sig_r = r // Get signif of r if |s| >= 2^-33
|
|
fand B = B_mask1, r
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
// Case 4: t = U_lo + V_lo
|
|
{ .mfi
|
|
getf.exp exp_r = r // Extract signexp of r if |s| >= 2^-33
|
|
(p11) fadd.s1 t = U_lo, V_lo // t = U_lo + V_lo for |s| < 2^-14
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p14) fcmp.gt.s1 p14, p15 = r, NEGTWO_TO_NEG2
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
// Case 3: c = (s - r) + w (c complete)
|
|
{ .mfi
|
|
nop.m 999
|
|
(p10) fadd.s1 c = c, w // c = c + w for |s| >= 2^-14
|
|
nop.i 999
|
|
}
|
|
{ .mbb
|
|
nop.m 999
|
|
(p14) br.cond.spnt TANL_SMALL_R // Branch if 2^24 <= |x| < 2^63 and |r|< 1/4
|
|
(p15) br.cond.sptk TANL_NORMAL_R_A // Branch if 2^24 <= |x| < 2^63 and |r|>=1/4
|
|
}
|
|
;;
|
|
|
|
|
|
// Here if 2^24 <= |x| < 2^63 and |s| < 2^-14 >>>>>>> Case 4.
|
|
//
|
|
// Case 4: Set P_12 if U_hiabs >= V_hiabs
|
|
// Case 4: w = w + N_0 * d_2
|
|
// Note: the (-) is now incorporated in w .
|
|
{ .mfi
|
|
add table_ptr1 = 160, table_base // Point to tanl_table_p1
|
|
fcmp.ge.unc.s1 p12, p13 = U_hiabs, V_hiabs
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
fms.s1 w2 = N_0, d_2, w2
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
// Case 4: C_lo = s_val - C_hi
|
|
{ .mfi
|
|
ldfe P1_1 = [table_ptr1], 16 // Load P1_1
|
|
fsub.s1 C_lo = s_val, C_hi
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
//
|
|
// Case 4: a = U_hi - A
|
|
// a = V_hi - A (do an add to account for missing (-) on V_hi
|
|
//
|
|
{ .mfi
|
|
ldfe P1_2 = [table_ptr1], 128 // Load P1_2
|
|
(p12) fsub.s1 a = U_hi, A
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p13) fadd.s1 a = V_hi, A
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
// Case 4: t = U_lo + V_lo + w
|
|
{ .mfi
|
|
ldfe Q1_1 = [table_ptr1], 16 // Load Q1_1
|
|
fadd.s1 t = t, w2
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
// Case 4: a = (U_hi - A) + V_hi
|
|
// a = (V_hi - A) + U_hi
|
|
// In each case account for negative missing form V_hi .
|
|
//
|
|
{ .mfi
|
|
ldfe Q1_2 = [table_ptr1], 16 // Load Q1_2
|
|
(p12) fsub.s1 a = a, V_hi
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p13) fsub.s1 a = U_hi, a
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
//
|
|
// Case 4: C_lo = (s_val - C_hi) + A
|
|
//
|
|
{ .mfi
|
|
nop.m 999
|
|
fadd.s1 C_lo = C_lo, A
|
|
nop.i 999 ;;
|
|
}
|
|
//
|
|
// Case 4: t = t + a
|
|
//
|
|
{ .mfi
|
|
nop.m 999
|
|
fadd.s1 t = t, a
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
// Case 4: C_lo = C_lo + t
|
|
// Case 4: r = C_hi + C_lo
|
|
{ .mfi
|
|
nop.m 999
|
|
fadd.s1 C_lo = C_lo, t
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
fadd.s1 r = C_hi, C_lo
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
//
|
|
// Case 4: c = C_hi - r
|
|
//
|
|
{ .mfi
|
|
nop.m 999
|
|
fsub.s1 c = C_hi, r
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
fmpy.s1 rsq = r, r
|
|
add N_fix_gr = N_fix_gr, cot_flag // N = N + 1 (for cotl)
|
|
}
|
|
;;
|
|
|
|
// Case 4: c = c + C_lo finished.
|
|
//
|
|
// Is i_1 = lsb of N_fix_gr even or odd?
|
|
// if i_1 == 0, set PR_11, else set PR_12.
|
|
//
|
|
{ .mfi
|
|
nop.m 999
|
|
fadd.s1 c = c , C_lo
|
|
tbit.z p11, p12 = N_fix_gr, 0
|
|
}
|
|
;;
|
|
|
|
// r and c have been computed.
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) frcpa.s1 S_hi, p0 = f1, r
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N odd: Change sign of S_hi
|
|
//
|
|
(p11) fma.s1 Poly = rsq, P1_2, P1_1
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fma.s1 P = rsq, Q1_2, Q1_1
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N odd: Result = S_hi + S_lo (User supplied rounding mode for C1)
|
|
//
|
|
fmpy.s0 fp_tmp = fp_tmp, fp_tmp // Dummy mult to set inexact
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: rsq = r * r
|
|
// N odd: S_hi = frcpa(r)
|
|
//
|
|
(p12) fmerge.ns S_hi = S_hi, S_hi
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: rsq = rsq * P1_2 + P1_1
|
|
// N odd: poly1 = 1.0 + S_hi * r 16 bits partial account for necessary
|
|
//
|
|
(p11) fmpy.s1 Poly = rsq, Poly
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fma.s1 poly1 = S_hi, r,f1
|
|
(p11) tbit.z.unc p14, p15 = cot_flag, 0 // p14=1 for tanl; p15=1 for cotl
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: Poly = Poly * rsq
|
|
// N odd: S_hi = S_hi + S_hi*poly1 16 bits account for necessary
|
|
//
|
|
(p11) fma.s1 Poly = r, Poly, c
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fma.s1 S_hi = S_hi, poly1, S_hi
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N odd: S_hi = S_hi * poly1 + S_hi 32 bits
|
|
//
|
|
(p14) fadd.s0 Result = r, Poly // for tanl
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
.pred.rel "mutex",p15,p12
|
|
{ .mfi
|
|
nop.m 999
|
|
(p15) fms.s0 Result = r, mOne, Poly // for cotl
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fma.s1 poly1 = S_hi, r, f1
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: Poly = Poly * r + c
|
|
// N odd: poly1 = 1.0 + S_hi * r 32 bits partial
|
|
//
|
|
(p12) fma.s1 S_hi = S_hi, poly1, S_hi
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fma.s1 poly1 = S_hi, r, f1
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: Result = Poly + r (Rounding mode S0)
|
|
// N odd: poly1 = S_hi * r + 1.0 64 bits partial
|
|
//
|
|
(p12) fma.s1 S_hi = S_hi, poly1, S_hi
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N odd: poly1 = S_hi * poly + S_hi 64 bits
|
|
//
|
|
(p12) fma.s1 poly1 = S_hi, r, f1
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N odd: poly1 = S_hi * r + 1.0
|
|
//
|
|
(p12) fma.s1 poly1 = S_hi, c, poly1
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N odd: poly1 = S_hi * c + poly1
|
|
//
|
|
(p12) fmpy.s1 S_lo = S_hi, poly1
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N odd: S_lo = S_hi * poly1
|
|
//
|
|
(p12) fma.s1 S_lo = P, r, S_lo
|
|
(p12) tbit.z.unc p14, p15 = cot_flag, 0 ;; // p14=1 for tanl; p15=1 for cotl
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p14) fadd.s0 Result = S_hi, S_lo // for tanl
|
|
nop.i 999
|
|
}
|
|
{ .mfb
|
|
nop.m 999
|
|
//
|
|
// N odd: S_lo = S_lo + r * P
|
|
//
|
|
(p15) fms.s0 Result = S_hi, mOne, S_lo // for cotl
|
|
br.ret.sptk b0 ;; // Exit for 2^24 <= |x| < 2^63 and |s| < 2^-14
|
|
}
|
|
|
|
|
|
TANL_SMALL_R:
|
|
// Here if |r| < 1/4
|
|
// r and c have been computed.
|
|
// *****************************************************************
|
|
// *****************************************************************
|
|
// *****************************************************************
|
|
// N odd: S_hi = frcpa(r)
|
|
// Get [i_1] - lsb of N_fix_gr. Set p11 if N even, p12 if N odd.
|
|
// N even: rsq = r * r
|
|
{ .mfi
|
|
add table_ptr1 = 160, table_base // Point to tanl_table_p1
|
|
frcpa.s1 S_hi, p0 = f1, r // S_hi for N odd
|
|
add N_fix_gr = N_fix_gr, cot_flag // N = N + 1 (for cotl)
|
|
}
|
|
{ .mfi
|
|
add table_ptr2 = 400, table_base // Point to Q1_7
|
|
fmpy.s1 rsq = r, r
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
{ .mmi
|
|
ldfe P1_1 = [table_ptr1], 16
|
|
;;
|
|
ldfe P1_2 = [table_ptr1], 16
|
|
tbit.z p11, p12 = N_fix_gr, 0
|
|
}
|
|
;;
|
|
|
|
|
|
{ .mfi
|
|
ldfe P1_3 = [table_ptr1], 96
|
|
nop.f 999
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
(p11) ldfe P1_9 = [table_ptr1], -16
|
|
(p12) fmerge.ns S_hi = S_hi, S_hi
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p11) fmpy.s1 r_to_the_8 = rsq, rsq
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
//
|
|
// N even: Poly2 = P1_7 + Poly2 * rsq
|
|
// N odd: poly2 = Q1_5 + poly2 * rsq
|
|
//
|
|
{ .mfi
|
|
(p11) ldfe P1_8 = [table_ptr1], -16
|
|
(p11) fadd.s1 CORR = rsq, f1
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
//
|
|
// N even: Poly1 = P1_2 + P1_3 * rsq
|
|
// N odd: poly1 = 1.0 + S_hi * r
|
|
// 16 bits partial account for necessary (-1)
|
|
//
|
|
{ .mmi
|
|
(p11) ldfe P1_7 = [table_ptr1], -16
|
|
;;
|
|
(p11) ldfe P1_6 = [table_ptr1], -16
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
//
|
|
// N even: Poly1 = P1_1 + Poly1 * rsq
|
|
// N odd: S_hi = S_hi + S_hi * poly1) 16 bits account for necessary
|
|
//
|
|
//
|
|
// N even: Poly2 = P1_5 + Poly2 * rsq
|
|
// N odd: poly2 = Q1_3 + poly2 * rsq
|
|
//
|
|
{ .mfi
|
|
(p11) ldfe P1_5 = [table_ptr1], -16
|
|
(p11) fmpy.s1 r_to_the_8 = r_to_the_8, r_to_the_8
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fma.s1 poly1 = S_hi, r, f1
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
//
|
|
// N even: Poly1 = Poly1 * rsq
|
|
// N odd: poly1 = 1.0 + S_hi * r 32 bits partial
|
|
//
|
|
|
|
//
|
|
// N even: CORR = CORR * c
|
|
// N odd: S_hi = S_hi * poly1 + S_hi 32 bits
|
|
//
|
|
|
|
//
|
|
// N even: Poly2 = P1_6 + Poly2 * rsq
|
|
// N odd: poly2 = Q1_4 + poly2 * rsq
|
|
//
|
|
|
|
{ .mmf
|
|
(p11) ldfe P1_4 = [table_ptr1], -16
|
|
nop.m 999
|
|
(p11) fmpy.s1 CORR = CORR, c
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p11) fma.s1 Poly1 = P1_3, rsq, P1_2
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
(p12) ldfe Q1_7 = [table_ptr2], -16
|
|
(p12) fma.s1 S_hi = S_hi, poly1, S_hi
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
(p12) ldfe Q1_6 = [table_ptr2], -16
|
|
(p11) fma.s1 Poly2 = P1_9, rsq, P1_8
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mmi
|
|
(p12) ldfe Q1_5 = [table_ptr2], -16 ;;
|
|
(p12) ldfe Q1_4 = [table_ptr2], -16
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
(p12) ldfe Q1_3 = [table_ptr2], -16
|
|
//
|
|
// N even: Poly2 = P1_8 + P1_9 * rsq
|
|
// N odd: poly2 = Q1_6 + Q1_7 * rsq
|
|
//
|
|
(p11) fma.s1 Poly1 = Poly1, rsq, P1_1
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
(p12) ldfe Q1_2 = [table_ptr2], -16
|
|
(p12) fma.s1 poly1 = S_hi, r, f1
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
(p12) ldfe Q1_1 = [table_ptr2], -16
|
|
(p11) fma.s1 Poly2 = Poly2, rsq, P1_7
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: CORR = rsq + 1
|
|
// N even: r_to_the_8 = rsq * rsq
|
|
//
|
|
(p11) fmpy.s1 Poly1 = Poly1, rsq
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fma.s1 S_hi = S_hi, poly1, S_hi
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fma.s1 poly2 = Q1_7, rsq, Q1_6
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p11) fma.s1 Poly2 = Poly2, rsq, P1_6
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fma.s1 poly1 = S_hi, r, f1
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fma.s1 poly2 = poly2, rsq, Q1_5
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p11) fma.s1 Poly2= Poly2, rsq, P1_5
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fma.s1 S_hi = S_hi, poly1, S_hi
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fma.s1 poly2 = poly2, rsq, Q1_4
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: r_to_the_8 = r_to_the_8 * r_to_the_8
|
|
// N odd: poly1 = S_hi * r + 1.0 64 bits partial
|
|
//
|
|
(p11) fma.s1 Poly2 = Poly2, rsq, P1_4
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: Poly = CORR + Poly * r
|
|
// N odd: P = Q1_1 + poly2 * rsq
|
|
//
|
|
(p12) fma.s1 poly1 = S_hi, r, f1
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fma.s1 poly2 = poly2, rsq, Q1_3
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: Poly2 = P1_4 + Poly2 * rsq
|
|
// N odd: poly2 = Q1_2 + poly2 * rsq
|
|
//
|
|
(p11) fma.s1 Poly = Poly2, r_to_the_8, Poly1
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fma.s1 poly1 = S_hi, c, poly1
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fma.s1 poly2 = poly2, rsq, Q1_2
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: Poly = Poly1 + Poly2 * r_to_the_8
|
|
// N odd: S_hi = S_hi * poly1 + S_hi 64 bits
|
|
//
|
|
(p11) fma.s1 Poly = Poly, r, CORR
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: Result = r + Poly (User supplied rounding mode)
|
|
// N odd: poly1 = S_hi * c + poly1
|
|
//
|
|
(p12) fmpy.s1 S_lo = S_hi, poly1
|
|
(p11) tbit.z.unc p14, p15 = cot_flag, 0 // p14=1 for tanl; p15=1 for cotl
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fma.s1 P = poly2, rsq, Q1_1
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N odd: poly1 = S_hi * r + 1.0
|
|
//
|
|
//
|
|
// N odd: S_lo = S_hi * poly1
|
|
//
|
|
(p14) fadd.s0 Result = Poly, r // for tanl
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p15) fms.s0 Result = Poly, mOne, r // for cotl
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N odd: S_lo = Q1_1 * c + S_lo
|
|
//
|
|
(p12) fma.s1 S_lo = Q1_1, c, S_lo
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
fmpy.s0 fp_tmp = fp_tmp, fp_tmp // Dummy mult to set inexact
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N odd: Result = S_lo + r * P
|
|
//
|
|
(p12) fma.s1 Result = P, r, S_lo
|
|
(p12) tbit.z.unc p14, p15 = cot_flag, 0 ;; // p14=1 for tanl; p15=1 for cotl
|
|
}
|
|
|
|
//
|
|
// N odd: Result = Result + S_hi (user supplied rounding mode)
|
|
//
|
|
{ .mfi
|
|
nop.m 999
|
|
(p14) fadd.s0 Result = Result, S_hi // for tanl
|
|
nop.i 999
|
|
}
|
|
{ .mfb
|
|
nop.m 999
|
|
(p15) fms.s0 Result = Result, mOne, S_hi // for cotl
|
|
br.ret.sptk b0 ;; // Exit |r| < 1/4 path
|
|
}
|
|
|
|
|
|
TANL_NORMAL_R:
|
|
// Here if 1/4 <= |x| < pi/4 or if |x| >= 2^63 and |r| >= 1/4
|
|
// *******************************************************************
|
|
// *******************************************************************
|
|
// *******************************************************************
|
|
//
|
|
// r and c have been computed.
|
|
//
|
|
{ .mfi
|
|
nop.m 999
|
|
fand B = B_mask1, r
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
TANL_NORMAL_R_A:
|
|
// Enter here if pi/4 <= |x| < 2^63 and |r| >= 1/4
|
|
// Get the 5 bits or r for the lookup. 1.xxxxx ....
|
|
{ .mmi
|
|
add table_ptr1 = 416, table_base // Point to tanl_table_p2
|
|
mov GR_exp_2tom65 = 0xffff - 65 // Scaling constant for B
|
|
extr.u lookup = sig_r, 58, 5
|
|
}
|
|
;;
|
|
|
|
{ .mmi
|
|
ldfe P2_1 = [table_ptr1], 16
|
|
setf.exp TWO_TO_NEG65 = GR_exp_2tom65 // 2^-65 for scaling B if exp_r=-2
|
|
add N_fix_gr = N_fix_gr, cot_flag // N = N + 1 (for cotl)
|
|
}
|
|
;;
|
|
|
|
.pred.rel "mutex",p11,p12
|
|
// B = 2^63 * 1.xxxxx 100...0
|
|
{ .mfi
|
|
ldfe P2_2 = [table_ptr1], 16
|
|
for B = B_mask2, B
|
|
mov table_offset = 512 // Assume table offset is 512
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
ldfe P2_3 = [table_ptr1], 16
|
|
fmerge.s Pos_r = f1, r
|
|
tbit.nz p8,p9 = exp_r, 0
|
|
}
|
|
;;
|
|
|
|
// Is B = 2** -2 or B= 2** -1? If 2**-1, then
|
|
// we want an offset of 512 for table addressing.
|
|
{ .mii
|
|
add table_ptr2 = 1296, table_base // Point to tanl_table_cm2
|
|
(p9) shladd table_offset = lookup, 4, table_offset
|
|
(p8) shladd table_offset = lookup, 4, r0
|
|
}
|
|
;;
|
|
|
|
{ .mmi
|
|
add table_ptr1 = table_ptr1, table_offset // Point to T_hi
|
|
add table_ptr2 = table_ptr2, table_offset // Point to C_hi
|
|
add table_ptr3 = 2128, table_base // Point to tanl_table_scim2
|
|
}
|
|
;;
|
|
|
|
{ .mmi
|
|
ldfd T_hi = [table_ptr1], 8 // Load T_hi
|
|
;;
|
|
ldfd C_hi = [table_ptr2], 8 // Load C_hi
|
|
add table_ptr3 = table_ptr3, table_offset // Point to SC_inv
|
|
}
|
|
;;
|
|
|
|
//
|
|
// x = |r| - B
|
|
//
|
|
// Convert B so it has the same exponent as Pos_r before subtracting
|
|
{ .mfi
|
|
ldfs T_lo = [table_ptr1] // Load T_lo
|
|
(p9) fnma.s1 x = B, FR_2tom64, Pos_r
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p8) fnma.s1 x = B, TWO_TO_NEG65, Pos_r
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
ldfs C_lo = [table_ptr2] // Load C_lo
|
|
nop.f 999
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
ldfe SC_inv = [table_ptr3] // Load SC_inv
|
|
fmerge.s sgn_r = r, f1
|
|
tbit.z p11, p12 = N_fix_gr, 0 // p11 if N even, p12 if odd
|
|
|
|
}
|
|
;;
|
|
|
|
//
|
|
// xsq = x * x
|
|
// N even: Tx = T_hi * x
|
|
//
|
|
// N even: Tx1 = Tx + 1
|
|
// N odd: Cx1 = 1 - Cx
|
|
//
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
fmpy.s1 xsq = x, x
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p11) fmpy.s1 Tx = T_hi, x
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
//
|
|
// N odd: Cx = C_hi * x
|
|
//
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fmpy.s1 Cx = C_hi, x
|
|
nop.i 999
|
|
}
|
|
;;
|
|
//
|
|
// N even and odd: P = P2_3 + P2_2 * xsq
|
|
//
|
|
{ .mfi
|
|
nop.m 999
|
|
fma.s1 P = P2_3, xsq, P2_2
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p11) fadd.s1 Tx1 = Tx, f1
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: D = C_hi - tanx
|
|
// N odd: D = T_hi + tanx
|
|
//
|
|
(p11) fmpy.s1 CORR = SC_inv, T_hi
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
fmpy.s1 Sx = SC_inv, x
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fmpy.s1 CORR = SC_inv, C_hi
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fsub.s1 V_hi = f1, Cx
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
fma.s1 P = P, xsq, P2_1
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even and odd: P = P2_1 + P * xsq
|
|
//
|
|
(p11) fma.s1 V_hi = Tx, Tx1, f1
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: Result = sgn_r * tail + T_hi (user rounding mode for C1)
|
|
// N odd: Result = sgn_r * tail + C_hi (user rounding mode for C1)
|
|
//
|
|
fmpy.s0 fp_tmp = fp_tmp, fp_tmp // Dummy mult to set inexact
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
fmpy.s1 CORR = CORR, c
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fnma.s1 V_hi = Cx,V_hi,f1
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: V_hi = Tx * Tx1 + 1
|
|
// N odd: Cx1 = 1 - Cx * Cx1
|
|
//
|
|
fmpy.s1 P = P, xsq
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even and odd: P = P * xsq
|
|
//
|
|
(p11) fmpy.s1 V_hi = V_hi, T_hi
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even and odd: tail = P * tail + V_lo
|
|
//
|
|
(p11) fmpy.s1 T_hi = sgn_r, T_hi
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
fmpy.s1 CORR = CORR, sgn_r
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fmpy.s1 V_hi = V_hi,C_hi
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: V_hi = T_hi * V_hi
|
|
// N odd: V_hi = C_hi * V_hi
|
|
//
|
|
fma.s1 tanx = P, x, x
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fnmpy.s1 C_hi = sgn_r, C_hi
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: V_lo = 1 - V_hi + C_hi
|
|
// N odd: V_lo = 1 - V_hi + T_hi
|
|
//
|
|
(p11) fadd.s1 CORR = CORR, T_lo
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fsub.s1 CORR = CORR, C_lo
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even and odd: tanx = x + x * P
|
|
// N even and odd: Sx = SC_inv * x
|
|
//
|
|
(p11) fsub.s1 D = C_hi, tanx
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fadd.s1 D = T_hi, tanx
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N odd: CORR = SC_inv * C_hi
|
|
// N even: CORR = SC_inv * T_hi
|
|
//
|
|
fnma.s1 D = V_hi, D, f1
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even and odd: D = 1 - V_hi * D
|
|
// N even and odd: CORR = CORR * c
|
|
//
|
|
fma.s1 V_hi = V_hi, D, V_hi
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even and odd: V_hi = V_hi + V_hi * D
|
|
// N even and odd: CORR = sgn_r * CORR
|
|
//
|
|
(p11) fnma.s1 V_lo = V_hi, C_hi, f1
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fnma.s1 V_lo = V_hi, T_hi, f1
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: CORR = COOR + T_lo
|
|
// N odd: CORR = CORR - C_lo
|
|
//
|
|
(p11) fma.s1 V_lo = tanx, V_hi, V_lo
|
|
tbit.nz p15, p0 = cot_flag, 0 // p15=1 if we compute cotl
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fnma.s1 V_lo = tanx, V_hi, V_lo
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p15) fms.s1 T_hi = f0, f0, T_hi // to correct result's sign for cotl
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p15) fms.s1 C_hi = f0, f0, C_hi // to correct result's sign for cotl
|
|
nop.i 999
|
|
};;
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p15) fms.s1 sgn_r = f0, f0, sgn_r // to correct result's sign for cotl
|
|
nop.i 999
|
|
};;
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: V_lo = V_lo + V_hi * tanx
|
|
// N odd: V_lo = V_lo - V_hi * tanx
|
|
//
|
|
(p11) fnma.s1 V_lo = C_lo, V_hi, V_lo
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fnma.s1 V_lo = T_lo, V_hi, V_lo
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: V_lo = V_lo - V_hi * C_lo
|
|
// N odd: V_lo = V_lo - V_hi * T_lo
|
|
//
|
|
fmpy.s1 V_lo = V_hi, V_lo
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even and odd: V_lo = V_lo * V_hi
|
|
//
|
|
fadd.s1 tail = V_hi, V_lo
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even and odd: tail = V_hi + V_lo
|
|
//
|
|
fma.s1 tail = tail, P, V_lo
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even: T_hi = sgn_r * T_hi
|
|
// N odd : C_hi = -sgn_r * C_hi
|
|
//
|
|
fma.s1 tail = tail, Sx, CORR
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even and odd: tail = Sx * tail + CORR
|
|
//
|
|
fma.s1 tail = V_hi, Sx, tail
|
|
nop.i 999 ;;
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N even an odd: tail = Sx * V_hi + tail
|
|
//
|
|
(p11) fma.s0 Result = sgn_r, tail, T_hi
|
|
nop.i 999
|
|
}
|
|
{ .mfb
|
|
nop.m 999
|
|
(p12) fma.s0 Result = sgn_r, tail, C_hi
|
|
br.ret.sptk b0 ;; // Exit for 1/4 <= |r| < pi/4
|
|
}
|
|
|
|
TANL_DENORMAL:
|
|
// Here if x denormal
|
|
{ .mfb
|
|
getf.exp GR_signexp_x = Norm_Arg // Get sign and exponent of x
|
|
nop.f 999
|
|
br.cond.sptk TANL_COMMON // Return to common code
|
|
}
|
|
;;
|
|
|
|
|
|
TANL_SPECIAL:
|
|
TANL_UNSUPPORTED:
|
|
//
|
|
// Code for NaNs, Unsupporteds, Infs, or +/- zero ?
|
|
// Invalid raised for Infs and SNaNs.
|
|
//
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
fmerge.s f10 = f8, f8 // Save input for error call
|
|
tbit.nz p6, p7 = cot_flag, 0 // p6=1 if we compute cotl
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p6) fclass.m p6, p7 = f8, 0x7 // Test for zero (cotl only)
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
.pred.rel "mutex", p6, p7
|
|
{ .mfi
|
|
(p6) mov GR_Parameter_Tag = 225 // (cotl)
|
|
(p6) frcpa.s0 f8, p0 = f1, f8 // cotl(+-0) = +-Inf
|
|
nop.i 999
|
|
}
|
|
{ .mfb
|
|
nop.m 999
|
|
(p7) fmpy.s0 f8 = f8, f0
|
|
(p7) br.ret.sptk b0
|
|
}
|
|
;;
|
|
|
|
GLOBAL_IEEE754_END(tanl)
|
|
|
|
LOCAL_LIBM_ENTRY(__libm_error_region)
|
|
.prologue
|
|
|
|
// (1)
|
|
{ .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
|
|
};;
|
|
|
|
// (2)
|
|
{ .mmi
|
|
stfe [GR_Parameter_Y] = f1,16 // 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
|
|
// (3)
|
|
{ .mib
|
|
stfe [GR_Parameter_X] = f10 // STORE Parameter 1 on stack
|
|
add GR_Parameter_RESULT = 0,GR_Parameter_Y // Parameter 3 address
|
|
nop.b 0
|
|
}
|
|
{ .mib
|
|
stfe [GR_Parameter_Y] = f8 // 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
|
|
};;
|
|
|
|
// (4)
|
|
{ .mmi
|
|
ldfe 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#
|
|
|
|
|
|
// *******************************************************************
|
|
// *******************************************************************
|
|
// *******************************************************************
|
|
//
|
|
// Special Code to handle very large argument case.
|
|
// Call int __libm_pi_by_2_reduce(x,r,c) for |arguments| >= 2**63
|
|
// The interface is custom:
|
|
// On input:
|
|
// (Arg or x) is in f8
|
|
// On output:
|
|
// r is in f8
|
|
// c is in f9
|
|
// N is in r8
|
|
// We know also that __libm_pi_by_2_reduce preserves f10-15, f71-127. We
|
|
// use this to eliminate save/restore of key fp registers in this calling
|
|
// function.
|
|
//
|
|
// *******************************************************************
|
|
// *******************************************************************
|
|
// *******************************************************************
|
|
|
|
LOCAL_LIBM_ENTRY(__libm_callout)
|
|
TANL_ARG_TOO_LARGE:
|
|
.prologue
|
|
{ .mfi
|
|
add table_ptr2 = 144, table_base // Point to 2^-2
|
|
nop.f 999
|
|
.save ar.pfs,GR_SAVE_PFS
|
|
mov GR_SAVE_PFS=ar.pfs // Save ar.pfs
|
|
}
|
|
;;
|
|
|
|
// Load 2^-2, -2^-2
|
|
{ .mmi
|
|
ldfps TWO_TO_NEG2, NEGTWO_TO_NEG2 = [table_ptr2]
|
|
setf.sig B_mask1 = bmask1 // Form mask to get 5 msb of r
|
|
.save b0, GR_SAVE_B0
|
|
mov GR_SAVE_B0=b0 // Save b0
|
|
};;
|
|
|
|
.body
|
|
//
|
|
// Call argument reduction with x in f8
|
|
// Returns with N in r8, r in f8, c in f9
|
|
// Assumes f71-127 are preserved across the call
|
|
//
|
|
{ .mib
|
|
setf.sig B_mask2 = bmask2 // Form mask to form B from r
|
|
mov GR_SAVE_GP=gp // Save gp
|
|
br.call.sptk b0=__libm_pi_by_2_reduce#
|
|
}
|
|
;;
|
|
|
|
//
|
|
// Is |r| < 2**(-2)
|
|
//
|
|
{ .mfi
|
|
getf.sig sig_r = r // Extract significand of r
|
|
fcmp.lt.s1 p6, p0 = r, TWO_TO_NEG2
|
|
mov gp = GR_SAVE_GP // Restore gp
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
getf.exp exp_r = r // Extract signexp of r
|
|
nop.f 999
|
|
mov b0 = GR_SAVE_B0 // Restore return address
|
|
}
|
|
;;
|
|
|
|
//
|
|
// Get N_fix_gr
|
|
//
|
|
{ .mfi
|
|
mov N_fix_gr = r8
|
|
(p6) fcmp.gt.unc.s1 p6, p0 = r, NEGTWO_TO_NEG2
|
|
mov ar.pfs = GR_SAVE_PFS // Restore pfs
|
|
}
|
|
;;
|
|
|
|
{ .mbb
|
|
nop.m 999
|
|
(p6) br.cond.spnt TANL_SMALL_R // Branch if |r| < 1/4
|
|
br.cond.sptk TANL_NORMAL_R // Branch if 1/4 <= |r| < pi/4
|
|
}
|
|
;;
|
|
|
|
LOCAL_LIBM_END(__libm_callout)
|
|
|
|
.type __libm_pi_by_2_reduce#,@function
|
|
.global __libm_pi_by_2_reduce#
|