3502 lines
76 KiB
ArmAsm
3502 lines
76 KiB
ArmAsm
.file "sincos.s"
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// Copyright (C) 2000, 2001, Intel Corporation
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// All rights reserved.
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//
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// Contributed 2/2/2000 by John Harrison, Ted Kubaska, Bob Norin, Shane Story,
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// and Ping Tak Peter Tang of the Computational Software Lab, 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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//
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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://developer.intel.com/opensource.
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//
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// History
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//==============================================================
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// 2/02/00 Initial revision
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// 4/02/00 Unwind support added.
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// 6/16/00 Updated tables to enforce symmetry
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// 8/31/00 Saved 2 cycles in main path, and 9 in other paths.
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// 9/20/00 The updated tables regressed to an old version, so reinstated them
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// 10/18/00 Changed one table entry to ensure symmetry
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// 1/03/01 Improved speed, fixed flag settings for small arguments.
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// API
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//==============================================================
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// double sin( double x);
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// double cos( double x);
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//
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// Overview of operation
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//==============================================================
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//
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// Step 1
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// ======
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// Reduce x to region -1/2*pi/2^k ===== 0 ===== +1/2*pi/2^k where k=4
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// divide x by pi/2^k.
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// Multiply by 2^k/pi.
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// nfloat = Round result to integer (round-to-nearest)
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//
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// r = x - nfloat * pi/2^k
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// Do this as (x - nfloat * HIGH(pi/2^k)) - nfloat * LOW(pi/2^k) for increased accuracy.
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// pi/2^k is stored as two numbers that when added make pi/2^k.
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// pi/2^k = HIGH(pi/2^k) + LOW(pi/2^k)
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//
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// x = (nfloat * pi/2^k) + r
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// r is small enough that we can use a polynomial approximation
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// and is referred to as the reduced argument.
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//
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// Step 3
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// ======
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// Take the unreduced part and remove the multiples of 2pi.
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// So nfloat = nfloat (with lower k+1 bits cleared) + lower k+1 bits
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//
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// nfloat (with lower k+1 bits cleared) is a multiple of 2^(k+1)
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// N * 2^(k+1)
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// nfloat * pi/2^k = N * 2^(k+1) * pi/2^k + (lower k+1 bits) * pi/2^k
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// nfloat * pi/2^k = N * 2 * pi + (lower k+1 bits) * pi/2^k
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// nfloat * pi/2^k = N2pi + M * pi/2^k
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//
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//
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// Sin(x) = Sin((nfloat * pi/2^k) + r)
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// = Sin(nfloat * pi/2^k) * Cos(r) + Cos(nfloat * pi/2^k) * Sin(r)
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//
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// Sin(nfloat * pi/2^k) = Sin(N2pi + Mpi/2^k)
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// = Sin(N2pi)Cos(Mpi/2^k) + Cos(N2pi)Sin(Mpi/2^k)
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// = Sin(Mpi/2^k)
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//
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// Cos(nfloat * pi/2^k) = Cos(N2pi + Mpi/2^k)
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// = Cos(N2pi)Cos(Mpi/2^k) + Sin(N2pi)Sin(Mpi/2^k)
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// = Cos(Mpi/2^k)
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//
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// Sin(x) = Sin(Mpi/2^k) Cos(r) + Cos(Mpi/2^k) Sin(r)
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//
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//
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// Step 4
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// ======
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// 0 <= M < 2^(k+1)
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// There are 2^(k+1) Sin entries in a table.
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// There are 2^(k+1) Cos entries in a table.
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//
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// Get Sin(Mpi/2^k) and Cos(Mpi/2^k) by table lookup.
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//
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//
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// Step 5
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// ======
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// Calculate Cos(r) and Sin(r) by polynomial approximation.
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//
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// Cos(r) = 1 + r^2 q1 + r^4 q2 + r^6 q3 + ... = Series for Cos
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// Sin(r) = r + r^3 p1 + r^5 p2 + r^7 p3 + ... = Series for Sin
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//
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// and the coefficients q1, q2, ... and p1, p2, ... are stored in a table
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//
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//
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// Calculate
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// Sin(x) = Sin(Mpi/2^k) Cos(r) + Cos(Mpi/2^k) Sin(r)
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//
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// as follows
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//
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// Sm = Sin(Mpi/2^k) and Cm = Cos(Mpi/2^k)
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// rsq = r*r
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//
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//
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// P = p1 + r^2p2 + r^4p3 + r^6p4
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// Q = q1 + r^2q2 + r^4q3 + r^6q4
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//
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// rcub = r * rsq
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// Sin(r) = r + rcub * P
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// = r + r^3p1 + r^5p2 + r^7p3 + r^9p4 + ... = Sin(r)
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//
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// The coefficients are not exactly these values, but almost.
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//
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// p1 = -1/6 = -1/3!
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// p2 = 1/120 = 1/5!
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// p3 = -1/5040 = -1/7!
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// p4 = 1/362889 = 1/9!
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//
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// P = r + rcub * P
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//
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// Answer = Sm Cos(r) + Cm P
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//
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// Cos(r) = 1 + rsq Q
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// Cos(r) = 1 + r^2 Q
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// Cos(r) = 1 + r^2 (q1 + r^2q2 + r^4q3 + r^6q4)
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// Cos(r) = 1 + r^2q1 + r^4q2 + r^6q3 + r^8q4 + ...
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//
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// Sm Cos(r) = Sm(1 + rsq Q)
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// Sm Cos(r) = Sm + Sm rsq Q
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// Sm Cos(r) = Sm + s_rsq Q
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// Q = Sm + s_rsq Q
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//
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// Then,
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//
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// Answer = Q + Cm P
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#include "libm_support.h"
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// Registers used
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//==============================================================
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// general input registers:
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// r14 -> r19
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// r32 -> r45
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// predicate registers used:
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// p6 -> p14
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// floating-point registers used
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// f9 -> f15
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// f32 -> f61
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// Assembly macros
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//==============================================================
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sind_NORM_f8 = f9
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sind_W = f10
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sind_int_Nfloat = f11
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sind_Nfloat = f12
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sind_r = f13
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sind_rsq = f14
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sind_rcub = f15
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sind_Inv_Pi_by_16 = f32
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sind_Pi_by_16_hi = f33
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sind_Pi_by_16_lo = f34
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sind_Inv_Pi_by_64 = f35
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sind_Pi_by_64_hi = f36
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sind_Pi_by_64_lo = f37
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sind_Sm = f38
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sind_Cm = f39
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sind_P1 = f40
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sind_Q1 = f41
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sind_P2 = f42
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sind_Q2 = f43
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sind_P3 = f44
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sind_Q3 = f45
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sind_P4 = f46
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sind_Q4 = f47
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sind_P_temp1 = f48
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sind_P_temp2 = f49
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sind_Q_temp1 = f50
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sind_Q_temp2 = f51
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sind_P = f52
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sind_Q = f53
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sind_srsq = f54
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sind_SIG_INV_PI_BY_16_2TO61 = f55
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sind_RSHF_2TO61 = f56
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sind_RSHF = f57
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sind_2TOM61 = f58
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sind_NFLOAT = f59
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sind_W_2TO61_RSH = f60
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fp_tmp = f61
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/////////////////////////////////////////////////////////////
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sind_AD_1 = r33
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sind_AD_2 = r34
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sind_exp_limit = r35
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sind_r_signexp = r36
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sind_AD_beta_table = r37
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sind_r_sincos = r38
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sind_r_exp = r39
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sind_r_17_ones = r40
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sind_GR_sig_inv_pi_by_16 = r14
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sind_GR_rshf_2to61 = r15
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sind_GR_rshf = r16
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sind_GR_exp_2tom61 = r17
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sind_GR_n = r18
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sind_GR_m = r19
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sind_GR_32m = r19
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gr_tmp = r41
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GR_SAVE_PFS = r41
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GR_SAVE_B0 = r42
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GR_SAVE_GP = r43
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#ifdef _LIBC
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.rodata
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#else
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.data
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#endif
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.align 16
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double_sind_pi:
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ASM_TYPE_DIRECTIVE(double_sind_pi,@object)
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// data8 0xA2F9836E4E44152A, 0x00004001 // 16/pi (significand loaded w/ setf)
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// c90fdaa22168c234
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data8 0xC90FDAA22168C234, 0x00003FFC // pi/16 hi
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// c4c6628b80dc1cd1 29024e088a
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data8 0xC4C6628B80DC1CD1, 0x00003FBC // pi/16 lo
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ASM_SIZE_DIRECTIVE(double_sind_pi)
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double_sind_pq_k4:
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ASM_TYPE_DIRECTIVE(double_sind_pq_k4,@object)
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data8 0x3EC71C963717C63A // P4
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data8 0x3EF9FFBA8F191AE6 // Q4
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data8 0xBF2A01A00F4E11A8 // P3
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data8 0xBF56C16C05AC77BF // Q3
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data8 0x3F8111111110F167 // P2
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data8 0x3FA555555554DD45 // Q2
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data8 0xBFC5555555555555 // P1
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data8 0xBFDFFFFFFFFFFFFC // Q1
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ASM_SIZE_DIRECTIVE(double_sind_pq_k4)
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double_sin_cos_beta_k4:
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ASM_TYPE_DIRECTIVE(double_sin_cos_beta_k4,@object)
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data8 0x0000000000000000 , 0x00000000 // sin( 0 pi/16) S0
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data8 0x8000000000000000 , 0x00003fff // cos( 0 pi/16) C0
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data8 0xc7c5c1e34d3055b3 , 0x00003ffc // sin( 1 pi/16) S1
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data8 0xfb14be7fbae58157 , 0x00003ffe // cos( 1 pi/16) C1
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data8 0xc3ef1535754b168e , 0x00003ffd // sin( 2 pi/16) S2
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data8 0xec835e79946a3146 , 0x00003ffe // cos( 2 pi/16) C2
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data8 0x8e39d9cd73464364 , 0x00003ffe // sin( 3 pi/16) S3
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data8 0xd4db3148750d181a , 0x00003ffe // cos( 3 pi/16) C3
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data8 0xb504f333f9de6484 , 0x00003ffe // sin( 4 pi/16) S4
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data8 0xb504f333f9de6484 , 0x00003ffe // cos( 4 pi/16) C4
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data8 0xd4db3148750d181a , 0x00003ffe // sin( 5 pi/16) C3
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data8 0x8e39d9cd73464364 , 0x00003ffe // cos( 5 pi/16) S3
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data8 0xec835e79946a3146 , 0x00003ffe // sin( 6 pi/16) C2
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data8 0xc3ef1535754b168e , 0x00003ffd // cos( 6 pi/16) S2
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data8 0xfb14be7fbae58157 , 0x00003ffe // sin( 7 pi/16) C1
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data8 0xc7c5c1e34d3055b3 , 0x00003ffc // cos( 7 pi/16) S1
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data8 0x8000000000000000 , 0x00003fff // sin( 8 pi/16) C0
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data8 0x0000000000000000 , 0x00000000 // cos( 8 pi/16) S0
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data8 0xfb14be7fbae58157 , 0x00003ffe // sin( 9 pi/16) C1
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data8 0xc7c5c1e34d3055b3 , 0x0000bffc // cos( 9 pi/16) -S1
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data8 0xec835e79946a3146 , 0x00003ffe // sin(10 pi/16) C2
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data8 0xc3ef1535754b168e , 0x0000bffd // cos(10 pi/16) -S2
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data8 0xd4db3148750d181a , 0x00003ffe // sin(11 pi/16) C3
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data8 0x8e39d9cd73464364 , 0x0000bffe // cos(11 pi/16) -S3
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data8 0xb504f333f9de6484 , 0x00003ffe // sin(12 pi/16) S4
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data8 0xb504f333f9de6484 , 0x0000bffe // cos(12 pi/16) -S4
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data8 0x8e39d9cd73464364 , 0x00003ffe // sin(13 pi/16) S3
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data8 0xd4db3148750d181a , 0x0000bffe // cos(13 pi/16) -C3
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data8 0xc3ef1535754b168e , 0x00003ffd // sin(14 pi/16) S2
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data8 0xec835e79946a3146 , 0x0000bffe // cos(14 pi/16) -C2
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data8 0xc7c5c1e34d3055b3 , 0x00003ffc // sin(15 pi/16) S1
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data8 0xfb14be7fbae58157 , 0x0000bffe // cos(15 pi/16) -C1
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data8 0x0000000000000000 , 0x00000000 // sin(16 pi/16) S0
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data8 0x8000000000000000 , 0x0000bfff // cos(16 pi/16) -C0
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data8 0xc7c5c1e34d3055b3 , 0x0000bffc // sin(17 pi/16) -S1
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data8 0xfb14be7fbae58157 , 0x0000bffe // cos(17 pi/16) -C1
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data8 0xc3ef1535754b168e , 0x0000bffd // sin(18 pi/16) -S2
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data8 0xec835e79946a3146 , 0x0000bffe // cos(18 pi/16) -C2
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data8 0x8e39d9cd73464364 , 0x0000bffe // sin(19 pi/16) -S3
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data8 0xd4db3148750d181a , 0x0000bffe // cos(19 pi/16) -C3
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data8 0xb504f333f9de6484 , 0x0000bffe // sin(20 pi/16) -S4
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data8 0xb504f333f9de6484 , 0x0000bffe // cos(20 pi/16) -S4
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data8 0xd4db3148750d181a , 0x0000bffe // sin(21 pi/16) -C3
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data8 0x8e39d9cd73464364 , 0x0000bffe // cos(21 pi/16) -S3
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data8 0xec835e79946a3146 , 0x0000bffe // sin(22 pi/16) -C2
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data8 0xc3ef1535754b168e , 0x0000bffd // cos(22 pi/16) -S2
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data8 0xfb14be7fbae58157 , 0x0000bffe // sin(23 pi/16) -C1
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data8 0xc7c5c1e34d3055b3 , 0x0000bffc // cos(23 pi/16) -S1
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data8 0x8000000000000000 , 0x0000bfff // sin(24 pi/16) -C0
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data8 0x0000000000000000 , 0x00000000 // cos(24 pi/16) S0
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data8 0xfb14be7fbae58157 , 0x0000bffe // sin(25 pi/16) -C1
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data8 0xc7c5c1e34d3055b3 , 0x00003ffc // cos(25 pi/16) S1
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data8 0xec835e79946a3146 , 0x0000bffe // sin(26 pi/16) -C2
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data8 0xc3ef1535754b168e , 0x00003ffd // cos(26 pi/16) S2
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data8 0xd4db3148750d181a , 0x0000bffe // sin(27 pi/16) -C3
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data8 0x8e39d9cd73464364 , 0x00003ffe // cos(27 pi/16) S3
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data8 0xb504f333f9de6484 , 0x0000bffe // sin(28 pi/16) -S4
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data8 0xb504f333f9de6484 , 0x00003ffe // cos(28 pi/16) S4
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data8 0x8e39d9cd73464364 , 0x0000bffe // sin(29 pi/16) -S3
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data8 0xd4db3148750d181a , 0x00003ffe // cos(29 pi/16) C3
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data8 0xc3ef1535754b168e , 0x0000bffd // sin(30 pi/16) -S2
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data8 0xec835e79946a3146 , 0x00003ffe // cos(30 pi/16) C2
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data8 0xc7c5c1e34d3055b3 , 0x0000bffc // sin(31 pi/16) -S1
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data8 0xfb14be7fbae58157 , 0x00003ffe // cos(31 pi/16) C1
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data8 0x0000000000000000 , 0x00000000 // sin(32 pi/16) S0
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data8 0x8000000000000000 , 0x00003fff // cos(32 pi/16) C0
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ASM_SIZE_DIRECTIVE(double_sin_cos_beta_k4)
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.align 32
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.global sin#
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.global cos#
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#ifdef _LIBC
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.global __sin#
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.global __cos#
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#endif
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////////////////////////////////////////////////////////
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// There are two entry points: sin and cos
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// If from sin, p8 is true
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// If from cos, p9 is true
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.section .text
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.proc sin#
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#ifdef _LIBC
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.proc __sin#
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#endif
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.align 32
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sin:
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#ifdef _LIBC
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__sin:
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#endif
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{ .mlx
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alloc r32=ar.pfs,1,13,0,0
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movl sind_GR_sig_inv_pi_by_16 = 0xA2F9836E4E44152A // significand of 16/pi
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}
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{ .mlx
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addl sind_AD_1 = @ltoff(double_sind_pi), gp
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movl sind_GR_rshf_2to61 = 0x47b8000000000000 // 1.1000 2^(63+63-2)
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}
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;;
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{ .mfi
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ld8 sind_AD_1 = [sind_AD_1]
|
|
fnorm sind_NORM_f8 = f8
|
|
cmp.eq p8,p9 = r0, r0
|
|
}
|
|
{ .mib
|
|
mov sind_GR_exp_2tom61 = 0xffff-61 // exponent of scaling factor 2^-61
|
|
mov sind_r_sincos = 0x0
|
|
br.cond.sptk L(SIND_SINCOS)
|
|
}
|
|
;;
|
|
|
|
.endp sin
|
|
ASM_SIZE_DIRECTIVE(sin)
|
|
|
|
|
|
.section .text
|
|
.proc cos#
|
|
#ifdef _LIBC
|
|
.proc __cos#
|
|
#endif
|
|
.align 32
|
|
cos:
|
|
#ifdef _LIBC
|
|
__cos:
|
|
#endif
|
|
|
|
{ .mlx
|
|
alloc r32=ar.pfs,1,13,0,0
|
|
movl sind_GR_sig_inv_pi_by_16 = 0xA2F9836E4E44152A // significand of 16/pi
|
|
}
|
|
{ .mlx
|
|
addl sind_AD_1 = @ltoff(double_sind_pi), gp
|
|
movl sind_GR_rshf_2to61 = 0x47b8000000000000 // 1.1000 2^(63+63-2)
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
ld8 sind_AD_1 = [sind_AD_1]
|
|
fnorm.s1 sind_NORM_f8 = f8
|
|
cmp.eq p9,p8 = r0, r0
|
|
}
|
|
{ .mib
|
|
mov sind_GR_exp_2tom61 = 0xffff-61 // exponent of scaling factor 2^-61
|
|
mov sind_r_sincos = 0x8
|
|
br.cond.sptk L(SIND_SINCOS)
|
|
}
|
|
;;
|
|
|
|
|
|
////////////////////////////////////////////////////////
|
|
// All entry points end up here.
|
|
// If from sin, sind_r_sincos is 0 and p8 is true
|
|
// If from cos, sind_r_sincos is 8 = 2^(k-1) and p9 is true
|
|
// We add sind_r_sincos to N
|
|
|
|
L(SIND_SINCOS):
|
|
|
|
|
|
// Form two constants we need
|
|
// 16/pi * 2^-2 * 2^63, scaled by 2^61 since we just loaded the significand
|
|
// 1.1000...000 * 2^(63+63-2) to right shift int(W) into the low significand
|
|
// fcmp used to set denormal, and invalid on snans
|
|
{ .mfi
|
|
setf.sig sind_SIG_INV_PI_BY_16_2TO61 = sind_GR_sig_inv_pi_by_16
|
|
fcmp.eq.s0 p12,p0=f8,f0
|
|
mov sind_r_17_ones = 0x1ffff
|
|
}
|
|
{ .mlx
|
|
setf.d sind_RSHF_2TO61 = sind_GR_rshf_2to61
|
|
movl sind_GR_rshf = 0x43e8000000000000 // 1.1000 2^63 for right shift
|
|
}
|
|
;;
|
|
|
|
// Form another constant
|
|
// 2^-61 for scaling Nfloat
|
|
// 0x10009 is register_bias + 10.
|
|
// So if f8 > 2^10 = Gamma, go to DBX
|
|
{ .mfi
|
|
setf.exp sind_2TOM61 = sind_GR_exp_2tom61
|
|
fclass.m p13,p0 = f8, 0x23 // Test for x inf
|
|
mov sind_exp_limit = 0x10009
|
|
}
|
|
;;
|
|
|
|
// Load the two pieces of pi/16
|
|
// Form another constant
|
|
// 1.1000...000 * 2^63, the right shift constant
|
|
{ .mmf
|
|
ldfe sind_Pi_by_16_hi = [sind_AD_1],16
|
|
setf.d sind_RSHF = sind_GR_rshf
|
|
fclass.m p14,p0 = f8, 0xc3 // Test for x nan
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
ldfe sind_Pi_by_16_lo = [sind_AD_1],16
|
|
(p13) frcpa.s0 f8,p12=f0,f0 // force qnan indef for x=inf
|
|
addl gr_tmp = -1,r0
|
|
}
|
|
{ .mfb
|
|
addl sind_AD_beta_table = @ltoff(double_sin_cos_beta_k4), gp
|
|
nop.f 999
|
|
(p13) br.ret.spnt b0 ;; // Exit for x=inf
|
|
}
|
|
|
|
// Start loading P, Q coefficients
|
|
// SIN(0)
|
|
{ .mfi
|
|
ldfpd sind_P4,sind_Q4 = [sind_AD_1],16
|
|
(p8) fclass.m.unc p6,p0 = f8, 0x07 // Test for sin(0)
|
|
nop.i 999
|
|
}
|
|
{ .mfb
|
|
addl sind_AD_beta_table = @ltoff(double_sin_cos_beta_k4), gp
|
|
(p14) fma.d f8=f8,f1,f0 // qnan for x=nan
|
|
(p14) br.ret.spnt b0 ;; // Exit for x=nan
|
|
}
|
|
|
|
|
|
// COS(0)
|
|
{ .mfi
|
|
getf.exp sind_r_signexp = f8
|
|
(p9) fclass.m.unc p7,p0 = f8, 0x07 // Test for sin(0)
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
ld8 sind_AD_beta_table = [sind_AD_beta_table]
|
|
nop.f 999
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mmb
|
|
ldfpd sind_P3,sind_Q3 = [sind_AD_1],16
|
|
setf.sig fp_tmp = gr_tmp // Create constant such that fmpy sets inexact
|
|
(p6) br.ret.spnt b0 ;;
|
|
}
|
|
|
|
{ .mfb
|
|
and sind_r_exp = sind_r_17_ones, sind_r_signexp
|
|
(p7) fmerge.s f8 = f1,f1
|
|
(p7) br.ret.spnt b0 ;;
|
|
}
|
|
|
|
// p10 is true if we must call routines to handle larger arguments
|
|
// p10 is true if f8 exp is > 0x10009
|
|
|
|
{ .mfi
|
|
ldfpd sind_P2,sind_Q2 = [sind_AD_1],16
|
|
nop.f 999
|
|
cmp.ge p10,p0 = sind_r_exp,sind_exp_limit
|
|
}
|
|
;;
|
|
|
|
// sind_W = x * sind_Inv_Pi_by_16
|
|
// Multiply x by scaled 16/pi and add large const to shift integer part of W to
|
|
// rightmost bits of significand
|
|
{ .mfi
|
|
ldfpd sind_P1,sind_Q1 = [sind_AD_1]
|
|
fma.s1 sind_W_2TO61_RSH = sind_NORM_f8,sind_SIG_INV_PI_BY_16_2TO61,sind_RSHF_2TO61
|
|
nop.i 999
|
|
}
|
|
{ .mbb
|
|
(p10) cmp.ne.unc p11,p12=sind_r_sincos,r0 // p11 call __libm_cos_double_dbx
|
|
// p12 call __libm_sin_double_dbx
|
|
(p11) br.cond.spnt L(COSD_DBX)
|
|
(p12) br.cond.spnt L(SIND_DBX)
|
|
}
|
|
;;
|
|
|
|
|
|
// sind_NFLOAT = Round_Int_Nearest(sind_W)
|
|
// This is done by scaling back by 2^-61 and subtracting the shift constant
|
|
{ .mfi
|
|
nop.m 999
|
|
fms.s1 sind_NFLOAT = sind_W_2TO61_RSH,sind_2TOM61,sind_RSHF
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
|
|
// get N = (int)sind_int_Nfloat
|
|
{ .mfi
|
|
getf.sig sind_GR_n = sind_W_2TO61_RSH
|
|
nop.f 999
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
// Add 2^(k-1) (which is in sind_r_sincos) to N
|
|
// sind_r = -sind_Nfloat * sind_Pi_by_16_hi + x
|
|
// sind_r = sind_r -sind_Nfloat * sind_Pi_by_16_lo
|
|
{ .mfi
|
|
add sind_GR_n = sind_GR_n, sind_r_sincos
|
|
fnma.s1 sind_r = sind_NFLOAT, sind_Pi_by_16_hi, sind_NORM_f8
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
|
|
// Get M (least k+1 bits of N)
|
|
{ .mmi
|
|
and sind_GR_m = 0x1f,sind_GR_n ;;
|
|
nop.m 999
|
|
shl sind_GR_32m = sind_GR_m,5 ;;
|
|
}
|
|
|
|
// Add 32*M to address of sin_cos_beta table
|
|
{ .mmi
|
|
add sind_AD_2 = sind_GR_32m, sind_AD_beta_table
|
|
nop.m 999
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
ldfe sind_Sm = [sind_AD_2],16
|
|
(p8) fclass.m.unc p10,p0=f8,0x0b // If sin, note denormal input to set uflow
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
ldfe sind_Cm = [sind_AD_2]
|
|
fnma.s1 sind_r = sind_NFLOAT, sind_Pi_by_16_lo, sind_r
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
// get rsq
|
|
{ .mfi
|
|
nop.m 999
|
|
fma.s1 sind_rsq = sind_r, sind_r, f0
|
|
nop.i 999
|
|
}
|
|
{ .mfi
|
|
nop.m 999
|
|
fmpy.s0 fp_tmp = fp_tmp,fp_tmp // fmpy forces inexact flag
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
// form P and Q series
|
|
{ .mfi
|
|
nop.m 999
|
|
fma.s1 sind_P_temp1 = sind_rsq, sind_P4, sind_P3
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
fma.s1 sind_Q_temp1 = sind_rsq, sind_Q4, sind_Q3
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
// get rcube and sm*rsq
|
|
{ .mfi
|
|
nop.m 999
|
|
fmpy.s1 sind_srsq = sind_Sm,sind_rsq
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
fmpy.s1 sind_rcub = sind_r, sind_rsq
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
fma.s1 sind_Q_temp2 = sind_rsq, sind_Q_temp1, sind_Q2
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
fma.s1 sind_P_temp2 = sind_rsq, sind_P_temp1, sind_P2
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
fma.s1 sind_Q = sind_rsq, sind_Q_temp2, sind_Q1
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
fma.s1 sind_P = sind_rsq, sind_P_temp2, sind_P1
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
// Get final P and Q
|
|
{ .mfi
|
|
nop.m 999
|
|
fma.s1 sind_Q = sind_srsq,sind_Q, sind_Sm
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
fma.s1 sind_P = sind_rcub,sind_P, sind_r
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
// If sin(denormal), force inexact to be set
|
|
{ .mfi
|
|
nop.m 999
|
|
(p10) fmpy.d.s0 fp_tmp = f8,f8
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
// Final calculation
|
|
{ .mfb
|
|
nop.m 999
|
|
fma.d f8 = sind_Cm, sind_P, sind_Q
|
|
br.ret.sptk b0 ;;
|
|
}
|
|
.endp cos#
|
|
ASM_SIZE_DIRECTIVE(cos#)
|
|
|
|
|
|
|
|
.proc __libm_callout_1s
|
|
__libm_callout_1s:
|
|
L(SIND_DBX):
|
|
.prologue
|
|
{ .mfi
|
|
nop.m 0
|
|
nop.f 0
|
|
.save ar.pfs,GR_SAVE_PFS
|
|
mov GR_SAVE_PFS=ar.pfs
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
mov GR_SAVE_GP=gp
|
|
nop.f 0
|
|
.save b0, GR_SAVE_B0
|
|
mov GR_SAVE_B0=b0
|
|
}
|
|
|
|
.body
|
|
{ .mib
|
|
nop.m 999
|
|
nop.i 999
|
|
br.call.sptk.many b0=__libm_sin_double_dbx# ;;
|
|
}
|
|
;;
|
|
|
|
|
|
{ .mfi
|
|
mov gp = GR_SAVE_GP
|
|
nop.f 999
|
|
mov b0 = GR_SAVE_B0
|
|
}
|
|
;;
|
|
|
|
{ .mib
|
|
nop.m 999
|
|
mov ar.pfs = GR_SAVE_PFS
|
|
br.ret.sptk b0 ;;
|
|
}
|
|
.endp __libm_callout_1s
|
|
ASM_SIZE_DIRECTIVE(__libm_callout_1s)
|
|
|
|
|
|
.proc __libm_callout_1c
|
|
__libm_callout_1c:
|
|
L(COSD_DBX):
|
|
.prologue
|
|
{ .mfi
|
|
nop.m 0
|
|
nop.f 0
|
|
.save ar.pfs,GR_SAVE_PFS
|
|
mov GR_SAVE_PFS=ar.pfs
|
|
}
|
|
;;
|
|
|
|
{ .mfi
|
|
mov GR_SAVE_GP=gp
|
|
nop.f 0
|
|
.save b0, GR_SAVE_B0
|
|
mov GR_SAVE_B0=b0
|
|
}
|
|
|
|
.body
|
|
{ .mib
|
|
nop.m 999
|
|
nop.i 999
|
|
br.call.sptk.many b0=__libm_cos_double_dbx# ;;
|
|
}
|
|
;;
|
|
|
|
|
|
{ .mfi
|
|
mov gp = GR_SAVE_GP
|
|
nop.f 999
|
|
mov b0 = GR_SAVE_B0
|
|
}
|
|
;;
|
|
|
|
{ .mib
|
|
nop.m 999
|
|
mov ar.pfs = GR_SAVE_PFS
|
|
br.ret.sptk b0 ;;
|
|
}
|
|
.endp __libm_callout_1c
|
|
ASM_SIZE_DIRECTIVE(__libm_callout_1c)
|
|
|
|
|
|
// ====================================================================
|
|
// ====================================================================
|
|
|
|
// These functions calculate the sin and cos for inputs
|
|
// greater than 2^10
|
|
// __libm_sin_double_dbx# and __libm_cos_double_dbx#
|
|
|
|
// *********************************************************************
|
|
// *********************************************************************
|
|
//
|
|
// Function: Combined sin(x) and cos(x), where
|
|
//
|
|
// sin(x) = sine(x), for double precision x values
|
|
// cos(x) = cosine(x), for double precision x values
|
|
//
|
|
// *********************************************************************
|
|
//
|
|
// Accuracy: Within .7 ulps for 80-bit floating point values
|
|
// Very accurate for double precision values
|
|
//
|
|
// *********************************************************************
|
|
//
|
|
// Resources Used:
|
|
//
|
|
// Floating-Point Registers: f8 (Input and Return Value)
|
|
// f32-f99
|
|
//
|
|
// General Purpose Registers:
|
|
// r32-r43
|
|
// r44-r45 (Used to pass arguments to pi_by_2 reduce routine)
|
|
//
|
|
// Predicate Registers: p6-p13
|
|
//
|
|
// *********************************************************************
|
|
//
|
|
// IEEE Special Conditions:
|
|
//
|
|
// Denormal fault raised on denormal inputs
|
|
// Overflow exceptions do not occur
|
|
// Underflow exceptions raised when appropriate for sin
|
|
// (No specialized error handling for this routine)
|
|
// Inexact raised when appropriate by algorithm
|
|
//
|
|
// sin(SNaN) = QNaN
|
|
// sin(QNaN) = QNaN
|
|
// sin(inf) = QNaN
|
|
// sin(+/-0) = +/-0
|
|
// cos(inf) = QNaN
|
|
// cos(SNaN) = QNaN
|
|
// cos(QNaN) = QNaN
|
|
// cos(0) = 1
|
|
//
|
|
// *********************************************************************
|
|
//
|
|
// Mathematical Description
|
|
// ========================
|
|
//
|
|
// The computation of FSIN and FCOS is best handled in one piece of
|
|
// code. The main reason is that given any argument Arg, computation
|
|
// of trigonometric functions first calculate N and an approximation
|
|
// to alpha where
|
|
//
|
|
// Arg = N pi/2 + alpha, |alpha| <= pi/4.
|
|
//
|
|
// Since
|
|
//
|
|
// cos( Arg ) = sin( (N+1) pi/2 + alpha ),
|
|
//
|
|
// therefore, the code for computing sine will produce cosine as long
|
|
// as 1 is added to N immediately after the argument reduction
|
|
// process.
|
|
//
|
|
// Let M = N if sine
|
|
// N+1 if cosine.
|
|
//
|
|
// Now, given
|
|
//
|
|
// Arg = M pi/2 + alpha, |alpha| <= pi/4,
|
|
//
|
|
// let I = M mod 4, or I be the two lsb of M when M is represented
|
|
// as 2's complement. I = [i_0 i_1]. Then
|
|
//
|
|
// sin( Arg ) = (-1)^i_0 sin( alpha ) if i_1 = 0,
|
|
// = (-1)^i_0 cos( alpha ) if i_1 = 1.
|
|
//
|
|
// For example:
|
|
// if M = -1, I = 11
|
|
// sin ((-pi/2 + alpha) = (-1) cos (alpha)
|
|
// if M = 0, I = 00
|
|
// sin (alpha) = sin (alpha)
|
|
// if M = 1, I = 01
|
|
// sin (pi/2 + alpha) = cos (alpha)
|
|
// if M = 2, I = 10
|
|
// sin (pi + alpha) = (-1) sin (alpha)
|
|
// if M = 3, I = 11
|
|
// sin ((3/2)pi + alpha) = (-1) cos (alpha)
|
|
//
|
|
// The value of alpha is obtained by argument reduction and
|
|
// represented by two working precision numbers r and c where
|
|
//
|
|
// alpha = r + c accurately.
|
|
//
|
|
// The reduction method is described in a previous write up.
|
|
// The argument reduction scheme identifies 4 cases. For Cases 2
|
|
// and 4, because |alpha| is small, sin(r+c) and cos(r+c) can be
|
|
// computed very easily by 2 or 3 terms of the Taylor series
|
|
// expansion as follows:
|
|
//
|
|
// Case 2:
|
|
// -------
|
|
//
|
|
// sin(r + c) = r + c - r^3/6 accurately
|
|
// cos(r + c) = 1 - 2^(-67) accurately
|
|
//
|
|
// Case 4:
|
|
// -------
|
|
//
|
|
// sin(r + c) = r + c - r^3/6 + r^5/120 accurately
|
|
// cos(r + c) = 1 - r^2/2 + r^4/24 accurately
|
|
//
|
|
// The only cases left are Cases 1 and 3 of the argument reduction
|
|
// procedure. These two cases will be merged since after the
|
|
// argument is reduced in either cases, we have the reduced argument
|
|
// represented as r + c and that the magnitude |r + c| is not small
|
|
// enough to allow the usage of a very short approximation.
|
|
//
|
|
// The required calculation is either
|
|
//
|
|
// sin(r + c) = sin(r) + correction, or
|
|
// cos(r + c) = cos(r) + correction.
|
|
//
|
|
// Specifically,
|
|
//
|
|
// sin(r + c) = sin(r) + c sin'(r) + O(c^2)
|
|
// = sin(r) + c cos (r) + O(c^2)
|
|
// = sin(r) + c(1 - r^2/2) accurately.
|
|
// Similarly,
|
|
//
|
|
// cos(r + c) = cos(r) - c sin(r) + O(c^2)
|
|
// = cos(r) - c(r - r^3/6) accurately.
|
|
//
|
|
// We therefore concentrate on accurately calculating sin(r) and
|
|
// cos(r) for a working-precision number r, |r| <= pi/4 to within
|
|
// 0.1% or so.
|
|
//
|
|
// The greatest challenge of this task is that the second terms of
|
|
// the Taylor series
|
|
//
|
|
// r - r^3/3! + r^r/5! - ...
|
|
//
|
|
// and
|
|
//
|
|
// 1 - r^2/2! + r^4/4! - ...
|
|
//
|
|
// are not very small when |r| is close to pi/4 and the rounding
|
|
// errors will be a concern if simple polynomial accumulation is
|
|
// used. When |r| < 2^-3, however, the second terms will be small
|
|
// enough (6 bits or so of right shift) that a normal Horner
|
|
// recurrence suffices. Hence there are two cases that we consider
|
|
// in the accurate computation of sin(r) and cos(r), |r| <= pi/4.
|
|
//
|
|
// Case small_r: |r| < 2^(-3)
|
|
// --------------------------
|
|
//
|
|
// Since Arg = M pi/4 + r + c accurately, and M mod 4 is [i_0 i_1],
|
|
// we have
|
|
//
|
|
// sin(Arg) = (-1)^i_0 * sin(r + c) if i_1 = 0
|
|
// = (-1)^i_0 * cos(r + c) if i_1 = 1
|
|
//
|
|
// can be accurately approximated by
|
|
//
|
|
// sin(Arg) = (-1)^i_0 * [sin(r) + c] if i_1 = 0
|
|
// = (-1)^i_0 * [cos(r) - c*r] if i_1 = 1
|
|
//
|
|
// because |r| is small and thus the second terms in the correction
|
|
// are unneccessary.
|
|
//
|
|
// Finally, sin(r) and cos(r) are approximated by polynomials of
|
|
// moderate lengths.
|
|
//
|
|
// sin(r) = r + S_1 r^3 + S_2 r^5 + ... + S_5 r^11
|
|
// cos(r) = 1 + C_1 r^2 + C_2 r^4 + ... + C_5 r^10
|
|
//
|
|
// We can make use of predicates to selectively calculate
|
|
// sin(r) or cos(r) based on i_1.
|
|
//
|
|
// Case normal_r: 2^(-3) <= |r| <= pi/4
|
|
// ------------------------------------
|
|
//
|
|
// This case is more likely than the previous one if one considers
|
|
// r to be uniformly distributed in [-pi/4 pi/4]. Again,
|
|
//
|
|
// sin(Arg) = (-1)^i_0 * sin(r + c) if i_1 = 0
|
|
// = (-1)^i_0 * cos(r + c) if i_1 = 1.
|
|
//
|
|
// Because |r| is now larger, we need one extra term in the
|
|
// correction. sin(Arg) can be accurately approximated by
|
|
//
|
|
// sin(Arg) = (-1)^i_0 * [sin(r) + c(1-r^2/2)] if i_1 = 0
|
|
// = (-1)^i_0 * [cos(r) - c*r*(1 - r^2/6)] i_1 = 1.
|
|
//
|
|
// Finally, sin(r) and cos(r) are approximated by polynomials of
|
|
// moderate lengths.
|
|
//
|
|
// sin(r) = r + PP_1_hi r^3 + PP_1_lo r^3 +
|
|
// PP_2 r^5 + ... + PP_8 r^17
|
|
//
|
|
// cos(r) = 1 + QQ_1 r^2 + QQ_2 r^4 + ... + QQ_8 r^16
|
|
//
|
|
// where PP_1_hi is only about 16 bits long and QQ_1 is -1/2.
|
|
// The crux in accurate computation is to calculate
|
|
//
|
|
// r + PP_1_hi r^3 or 1 + QQ_1 r^2
|
|
//
|
|
// accurately as two pieces: U_hi and U_lo. The way to achieve this
|
|
// is to obtain r_hi as a 10 sig. bit number that approximates r to
|
|
// roughly 8 bits or so of accuracy. (One convenient way is
|
|
//
|
|
// r_hi := frcpa( frcpa( r ) ).)
|
|
//
|
|
// This way,
|
|
//
|
|
// r + PP_1_hi r^3 = r + PP_1_hi r_hi^3 +
|
|
// PP_1_hi (r^3 - r_hi^3)
|
|
// = [r + PP_1_hi r_hi^3] +
|
|
// [PP_1_hi (r - r_hi)
|
|
// (r^2 + r_hi r + r_hi^2) ]
|
|
// = U_hi + U_lo
|
|
//
|
|
// Since r_hi is only 10 bit long and PP_1_hi is only 16 bit long,
|
|
// PP_1_hi * r_hi^3 is only at most 46 bit long and thus computed
|
|
// exactly. Furthermore, r and PP_1_hi r_hi^3 are of opposite sign
|
|
// and that there is no more than 8 bit shift off between r and
|
|
// PP_1_hi * r_hi^3. Hence the sum, U_hi, is representable and thus
|
|
// calculated without any error. Finally, the fact that
|
|
//
|
|
// |U_lo| <= 2^(-8) |U_hi|
|
|
//
|
|
// says that U_hi + U_lo is approximating r + PP_1_hi r^3 to roughly
|
|
// 8 extra bits of accuracy.
|
|
//
|
|
// Similarly,
|
|
//
|
|
// 1 + QQ_1 r^2 = [1 + QQ_1 r_hi^2] +
|
|
// [QQ_1 (r - r_hi)(r + r_hi)]
|
|
// = U_hi + U_lo.
|
|
//
|
|
// Summarizing, we calculate r_hi = frcpa( frcpa( r ) ).
|
|
//
|
|
// If i_1 = 0, then
|
|
//
|
|
// U_hi := r + PP_1_hi * r_hi^3
|
|
// U_lo := PP_1_hi * (r - r_hi) * (r^2 + r*r_hi + r_hi^2)
|
|
// poly := PP_1_lo r^3 + PP_2 r^5 + ... + PP_8 r^17
|
|
// correction := c * ( 1 + C_1 r^2 )
|
|
//
|
|
// Else ...i_1 = 1
|
|
//
|
|
// U_hi := 1 + QQ_1 * r_hi * r_hi
|
|
// U_lo := QQ_1 * (r - r_hi) * (r + r_hi)
|
|
// poly := QQ_2 * r^4 + QQ_3 * r^6 + ... + QQ_8 r^16
|
|
// correction := -c * r * (1 + S_1 * r^2)
|
|
//
|
|
// End
|
|
//
|
|
// Finally,
|
|
//
|
|
// V := poly + ( U_lo + correction )
|
|
//
|
|
// / U_hi + V if i_0 = 0
|
|
// result := |
|
|
// \ (-U_hi) - V if i_0 = 1
|
|
//
|
|
// It is important that in the last step, negation of U_hi is
|
|
// performed prior to the subtraction which is to be performed in
|
|
// the user-set rounding mode.
|
|
//
|
|
//
|
|
// Algorithmic Description
|
|
// =======================
|
|
//
|
|
// The argument reduction algorithm is tightly integrated into FSIN
|
|
// and FCOS which share the same code. The following is complete and
|
|
// self-contained. The argument reduction description given
|
|
// previously is repeated below.
|
|
//
|
|
//
|
|
// Step 0. Initialization.
|
|
//
|
|
// If FSIN is invoked, set N_inc := 0; else if FCOS is invoked,
|
|
// set N_inc := 1.
|
|
//
|
|
// Step 1. Check for exceptional and special cases.
|
|
//
|
|
// * If Arg is +-0, +-inf, NaN, NaT, go to Step 10 for special
|
|
// handling.
|
|
// * If |Arg| < 2^24, go to Step 2 for reduction of moderate
|
|
// arguments. This is the most likely case.
|
|
// * If |Arg| < 2^63, go to Step 8 for pre-reduction of large
|
|
// arguments.
|
|
// * If |Arg| >= 2^63, go to Step 10 for special handling.
|
|
//
|
|
// Step 2. Reduction of moderate arguments.
|
|
//
|
|
// If |Arg| < pi/4 ...quick branch
|
|
// N_fix := N_inc (integer)
|
|
// r := Arg
|
|
// c := 0.0
|
|
// Branch to Step 4, Case_1_complete
|
|
// Else ...cf. argument reduction
|
|
// N := Arg * two_by_PI (fp)
|
|
// N_fix := fcvt.fx( N ) (int)
|
|
// N := fcvt.xf( N_fix )
|
|
// N_fix := N_fix + N_inc
|
|
// s := Arg - N * P_1 (first piece of pi/2)
|
|
// w := -N * P_2 (second piece of pi/2)
|
|
//
|
|
// If |s| >= 2^(-33)
|
|
// go to Step 3, Case_1_reduce
|
|
// Else
|
|
// go to Step 7, Case_2_reduce
|
|
// Endif
|
|
// Endif
|
|
//
|
|
// Step 3. Case_1_reduce.
|
|
//
|
|
// r := s + w
|
|
// c := (s - r) + w ...observe order
|
|
//
|
|
// Step 4. Case_1_complete
|
|
//
|
|
// ...At this point, the reduced argument alpha is
|
|
// ...accurately represented as r + c.
|
|
// If |r| < 2^(-3), go to Step 6, small_r.
|
|
//
|
|
// Step 5. Normal_r.
|
|
//
|
|
// Let [i_0 i_1] by the 2 lsb of N_fix.
|
|
// FR_rsq := r * r
|
|
// r_hi := frcpa( frcpa( r ) )
|
|
// r_lo := r - r_hi
|
|
//
|
|
// If i_1 = 0, then
|
|
// poly := r*FR_rsq*(PP_1_lo + FR_rsq*(PP_2 + ... FR_rsq*PP_8))
|
|
// U_hi := r + PP_1_hi*r_hi*r_hi*r_hi ...any order
|
|
// U_lo := PP_1_hi*r_lo*(r*r + r*r_hi + r_hi*r_hi)
|
|
// correction := c + c*C_1*FR_rsq ...any order
|
|
// Else
|
|
// poly := FR_rsq*FR_rsq*(QQ_2 + FR_rsq*(QQ_3 + ... + FR_rsq*QQ_8))
|
|
// U_hi := 1 + QQ_1 * r_hi * r_hi ...any order
|
|
// U_lo := QQ_1 * r_lo * (r + r_hi)
|
|
// correction := -c*(r + S_1*FR_rsq*r) ...any order
|
|
// Endif
|
|
//
|
|
// V := poly + (U_lo + correction) ...observe order
|
|
//
|
|
// result := (i_0 == 0? 1.0 : -1.0)
|
|
//
|
|
// Last instruction in user-set rounding mode
|
|
//
|
|
// result := (i_0 == 0? result*U_hi + V :
|
|
// result*U_hi - V)
|
|
//
|
|
// Return
|
|
//
|
|
// Step 6. Small_r.
|
|
//
|
|
// ...Use flush to zero mode without causing exception
|
|
// Let [i_0 i_1] be the two lsb of N_fix.
|
|
//
|
|
// FR_rsq := r * r
|
|
//
|
|
// If i_1 = 0 then
|
|
// z := FR_rsq*FR_rsq; z := FR_rsq*z *r
|
|
// poly_lo := S_3 + FR_rsq*(S_4 + FR_rsq*S_5)
|
|
// poly_hi := r*FR_rsq*(S_1 + FR_rsq*S_2)
|
|
// correction := c
|
|
// result := r
|
|
// Else
|
|
// z := FR_rsq*FR_rsq; z := FR_rsq*z
|
|
// poly_lo := C_3 + FR_rsq*(C_4 + FR_rsq*C_5)
|
|
// poly_hi := FR_rsq*(C_1 + FR_rsq*C_2)
|
|
// correction := -c*r
|
|
// result := 1
|
|
// Endif
|
|
//
|
|
// poly := poly_hi + (z * poly_lo + correction)
|
|
//
|
|
// If i_0 = 1, result := -result
|
|
//
|
|
// Last operation. Perform in user-set rounding mode
|
|
//
|
|
// result := (i_0 == 0? result + poly :
|
|
// result - poly )
|
|
// Return
|
|
//
|
|
// Step 7. Case_2_reduce.
|
|
//
|
|
// ...Refer to the write up for argument reduction for
|
|
// ...rationale. The reduction algorithm below is taken from
|
|
// ...argument reduction description and integrated this.
|
|
//
|
|
// w := N*P_3
|
|
// U_1 := N*P_2 + w ...FMA
|
|
// U_2 := (N*P_2 - U_1) + w ...2 FMA
|
|
// ...U_1 + U_2 is N*(P_2+P_3) accurately
|
|
//
|
|
// r := s - U_1
|
|
// c := ( (s - r) - U_1 ) - U_2
|
|
//
|
|
// ...The mathematical sum r + c approximates the reduced
|
|
// ...argument accurately. Note that although compared to
|
|
// ...Case 1, this case requires much more work to reduce
|
|
// ...the argument, the subsequent calculation needed for
|
|
// ...any of the trigonometric function is very little because
|
|
// ...|alpha| < 1.01*2^(-33) and thus two terms of the
|
|
// ...Taylor series expansion suffices.
|
|
//
|
|
// If i_1 = 0 then
|
|
// poly := c + S_1 * r * r * r ...any order
|
|
// result := r
|
|
// Else
|
|
// poly := -2^(-67)
|
|
// result := 1.0
|
|
// Endif
|
|
//
|
|
// If i_0 = 1, result := -result
|
|
//
|
|
// Last operation. Perform in user-set rounding mode
|
|
//
|
|
// result := (i_0 == 0? result + poly :
|
|
// result - poly )
|
|
//
|
|
// Return
|
|
//
|
|
//
|
|
// Step 8. Pre-reduction of large arguments.
|
|
//
|
|
// ...Again, the following reduction procedure was described
|
|
// ...in the separate write up for argument reduction, which
|
|
// ...is tightly integrated here.
|
|
|
|
// N_0 := Arg * Inv_P_0
|
|
// N_0_fix := fcvt.fx( N_0 )
|
|
// N_0 := fcvt.xf( N_0_fix)
|
|
|
|
// Arg' := Arg - N_0 * P_0
|
|
// w := N_0 * d_1
|
|
// N := Arg' * two_by_PI
|
|
// N_fix := fcvt.fx( N )
|
|
// N := fcvt.xf( N_fix )
|
|
// N_fix := N_fix + N_inc
|
|
//
|
|
// s := Arg' - N * P_1
|
|
// w := w - N * P_2
|
|
//
|
|
// If |s| >= 2^(-14)
|
|
// go to Step 3
|
|
// Else
|
|
// go to Step 9
|
|
// Endif
|
|
//
|
|
// Step 9. Case_4_reduce.
|
|
//
|
|
// ...first obtain N_0*d_1 and -N*P_2 accurately
|
|
// U_hi := N_0 * d_1 V_hi := -N*P_2
|
|
// U_lo := N_0 * d_1 - U_hi V_lo := -N*P_2 - U_hi ...FMAs
|
|
//
|
|
// ...compute the contribution from N_0*d_1 and -N*P_3
|
|
// w := -N*P_3
|
|
// w := w + N_0*d_2
|
|
// t := U_lo + V_lo + w ...any order
|
|
//
|
|
// ...at this point, the mathematical value
|
|
// ...s + U_hi + V_hi + t approximates the true reduced argument
|
|
// ...accurately. Just need to compute this accurately.
|
|
//
|
|
// ...Calculate U_hi + V_hi accurately:
|
|
// A := U_hi + V_hi
|
|
// if |U_hi| >= |V_hi| then
|
|
// a := (U_hi - A) + V_hi
|
|
// else
|
|
// a := (V_hi - A) + U_hi
|
|
// endif
|
|
// ...order in computing "a" must be observed. This branch is
|
|
// ...best implemented by predicates.
|
|
// ...A + a is U_hi + V_hi accurately. Moreover, "a" is
|
|
// ...much smaller than A: |a| <= (1/2)ulp(A).
|
|
//
|
|
// ...Just need to calculate s + A + a + t
|
|
// C_hi := s + A t := t + a
|
|
// C_lo := (s - C_hi) + A
|
|
// C_lo := C_lo + t
|
|
//
|
|
// ...Final steps for reduction
|
|
// r := C_hi + C_lo
|
|
// c := (C_hi - r) + C_lo
|
|
//
|
|
// ...At this point, we have r and c
|
|
// ...And all we need is a couple of terms of the corresponding
|
|
// ...Taylor series.
|
|
//
|
|
// If i_1 = 0
|
|
// poly := c + r*FR_rsq*(S_1 + FR_rsq*S_2)
|
|
// result := r
|
|
// Else
|
|
// poly := FR_rsq*(C_1 + FR_rsq*C_2)
|
|
// result := 1
|
|
// Endif
|
|
//
|
|
// If i_0 = 1, result := -result
|
|
//
|
|
// Last operation. Perform in user-set rounding mode
|
|
//
|
|
// result := (i_0 == 0? result + poly :
|
|
// result - poly )
|
|
// Return
|
|
//
|
|
// Large Arguments: For arguments above 2**63, a Payne-Hanek
|
|
// style argument reduction is used and pi_by_2 reduce is called.
|
|
//
|
|
|
|
|
|
#ifdef _LIBC
|
|
.rodata
|
|
#else
|
|
.data
|
|
#endif
|
|
.align 64
|
|
|
|
FSINCOS_CONSTANTS:
|
|
ASM_TYPE_DIRECTIVE(FSINCOS_CONSTANTS,@object)
|
|
data4 0x4B800000, 0xCB800000, 0x00000000,0x00000000 // two**24, -two**24
|
|
data4 0x4E44152A, 0xA2F9836E, 0x00003FFE,0x00000000 // Inv_pi_by_2
|
|
data4 0xCE81B9F1, 0xC84D32B0, 0x00004016,0x00000000 // P_0
|
|
data4 0x2168C235, 0xC90FDAA2, 0x00003FFF,0x00000000 // P_1
|
|
data4 0xFC8F8CBB, 0xECE675D1, 0x0000BFBD,0x00000000 // P_2
|
|
data4 0xACC19C60, 0xB7ED8FBB, 0x0000BF7C,0x00000000 // P_3
|
|
data4 0x5F000000, 0xDF000000, 0x00000000,0x00000000 // two_to_63, -two_to_63
|
|
data4 0x6EC6B45A, 0xA397E504, 0x00003FE7,0x00000000 // Inv_P_0
|
|
data4 0xDBD171A1, 0x8D848E89, 0x0000BFBF,0x00000000 // d_1
|
|
data4 0x18A66F8E, 0xD5394C36, 0x0000BF7C,0x00000000 // d_2
|
|
data4 0x2168C234, 0xC90FDAA2, 0x00003FFE,0x00000000 // pi_by_4
|
|
data4 0x2168C234, 0xC90FDAA2, 0x0000BFFE,0x00000000 // neg_pi_by_4
|
|
data4 0x3E000000, 0xBE000000, 0x00000000,0x00000000 // two**-3, -two**-3
|
|
data4 0x2F000000, 0xAF000000, 0x9E000000,0x00000000 // two**-33, -two**-33, -two**-67
|
|
data4 0xA21C0BC9, 0xCC8ABEBC, 0x00003FCE,0x00000000 // PP_8
|
|
data4 0x720221DA, 0xD7468A05, 0x0000BFD6,0x00000000 // PP_7
|
|
data4 0x640AD517, 0xB092382F, 0x00003FDE,0x00000000 // PP_6
|
|
data4 0xD1EB75A4, 0xD7322B47, 0x0000BFE5,0x00000000 // PP_5
|
|
data4 0xFFFFFFFE, 0xFFFFFFFF, 0x0000BFFD,0x00000000 // C_1
|
|
data4 0x00000000, 0xAAAA0000, 0x0000BFFC,0x00000000 // PP_1_hi
|
|
data4 0xBAF69EEA, 0xB8EF1D2A, 0x00003FEC,0x00000000 // PP_4
|
|
data4 0x0D03BB69, 0xD00D00D0, 0x0000BFF2,0x00000000 // PP_3
|
|
data4 0x88888962, 0x88888888, 0x00003FF8,0x00000000 // PP_2
|
|
data4 0xAAAB0000, 0xAAAAAAAA, 0x0000BFEC,0x00000000 // PP_1_lo
|
|
data4 0xC2B0FE52, 0xD56232EF, 0x00003FD2,0x00000000 // QQ_8
|
|
data4 0x2B48DCA6, 0xC9C99ABA, 0x0000BFDA,0x00000000 // QQ_7
|
|
data4 0x9C716658, 0x8F76C650, 0x00003FE2,0x00000000 // QQ_6
|
|
data4 0xFDA8D0FC, 0x93F27DBA, 0x0000BFE9,0x00000000 // QQ_5
|
|
data4 0xAAAAAAAA, 0xAAAAAAAA, 0x0000BFFC,0x00000000 // S_1
|
|
data4 0x00000000, 0x80000000, 0x0000BFFE,0x00000000 // QQ_1
|
|
data4 0x0C6E5041, 0xD00D00D0, 0x00003FEF,0x00000000 // QQ_4
|
|
data4 0x0B607F60, 0xB60B60B6, 0x0000BFF5,0x00000000 // QQ_3
|
|
data4 0xAAAAAA9B, 0xAAAAAAAA, 0x00003FFA,0x00000000 // QQ_2
|
|
data4 0xFFFFFFFE, 0xFFFFFFFF, 0x0000BFFD,0x00000000 // C_1
|
|
data4 0xAAAA719F, 0xAAAAAAAA, 0x00003FFA,0x00000000 // C_2
|
|
data4 0x0356F994, 0xB60B60B6, 0x0000BFF5,0x00000000 // C_3
|
|
data4 0xB2385EA9, 0xD00CFFD5, 0x00003FEF,0x00000000 // C_4
|
|
data4 0x292A14CD, 0x93E4BD18, 0x0000BFE9,0x00000000 // C_5
|
|
data4 0xAAAAAAAA, 0xAAAAAAAA, 0x0000BFFC,0x00000000 // S_1
|
|
data4 0x888868DB, 0x88888888, 0x00003FF8,0x00000000 // S_2
|
|
data4 0x055EFD4B, 0xD00D00D0, 0x0000BFF2,0x00000000 // S_3
|
|
data4 0x839730B9, 0xB8EF1C5D, 0x00003FEC,0x00000000 // S_4
|
|
data4 0xE5B3F492, 0xD71EA3A4, 0x0000BFE5,0x00000000 // S_5
|
|
data4 0x38800000, 0xB8800000, 0x00000000 // two**-14, -two**-14
|
|
ASM_SIZE_DIRECTIVE(FSINCOS_CONSTANTS)
|
|
|
|
FR_Input_X = f8
|
|
FR_Neg_Two_to_M3 = f32
|
|
FR_Two_to_63 = f32
|
|
FR_Two_to_24 = f33
|
|
FR_Pi_by_4 = f33
|
|
FR_Two_to_M14 = f34
|
|
FR_Two_to_M33 = f35
|
|
FR_Neg_Two_to_24 = f36
|
|
FR_Neg_Pi_by_4 = f36
|
|
FR_Neg_Two_to_M14 = f37
|
|
FR_Neg_Two_to_M33 = f38
|
|
FR_Neg_Two_to_M67 = f39
|
|
FR_Inv_pi_by_2 = f40
|
|
FR_N_float = f41
|
|
FR_N_fix = f42
|
|
FR_P_1 = f43
|
|
FR_P_2 = f44
|
|
FR_P_3 = f45
|
|
FR_s = f46
|
|
FR_w = f47
|
|
FR_c = f48
|
|
FR_r = f49
|
|
FR_Z = f50
|
|
FR_A = f51
|
|
FR_a = f52
|
|
FR_t = f53
|
|
FR_U_1 = f54
|
|
FR_U_2 = f55
|
|
FR_C_1 = f56
|
|
FR_C_2 = f57
|
|
FR_C_3 = f58
|
|
FR_C_4 = f59
|
|
FR_C_5 = f60
|
|
FR_S_1 = f61
|
|
FR_S_2 = f62
|
|
FR_S_3 = f63
|
|
FR_S_4 = f64
|
|
FR_S_5 = f65
|
|
FR_poly_hi = f66
|
|
FR_poly_lo = f67
|
|
FR_r_hi = f68
|
|
FR_r_lo = f69
|
|
FR_rsq = f70
|
|
FR_r_cubed = f71
|
|
FR_C_hi = f72
|
|
FR_N_0 = f73
|
|
FR_d_1 = f74
|
|
FR_V = f75
|
|
FR_V_hi = f75
|
|
FR_V_lo = f76
|
|
FR_U_hi = f77
|
|
FR_U_lo = f78
|
|
FR_U_hiabs = f79
|
|
FR_V_hiabs = f80
|
|
FR_PP_8 = f81
|
|
FR_QQ_8 = f81
|
|
FR_PP_7 = f82
|
|
FR_QQ_7 = f82
|
|
FR_PP_6 = f83
|
|
FR_QQ_6 = f83
|
|
FR_PP_5 = f84
|
|
FR_QQ_5 = f84
|
|
FR_PP_4 = f85
|
|
FR_QQ_4 = f85
|
|
FR_PP_3 = f86
|
|
FR_QQ_3 = f86
|
|
FR_PP_2 = f87
|
|
FR_QQ_2 = f87
|
|
FR_QQ_1 = f88
|
|
FR_N_0_fix = f89
|
|
FR_Inv_P_0 = f90
|
|
FR_corr = f91
|
|
FR_poly = f92
|
|
FR_d_2 = f93
|
|
FR_Two_to_M3 = f94
|
|
FR_Neg_Two_to_63 = f94
|
|
FR_P_0 = f95
|
|
FR_C_lo = f96
|
|
FR_PP_1 = f97
|
|
FR_PP_1_lo = f98
|
|
FR_ArgPrime = f99
|
|
|
|
GR_Table_Base = r32
|
|
GR_Table_Base1 = r33
|
|
GR_i_0 = r34
|
|
GR_i_1 = r35
|
|
GR_N_Inc = r36
|
|
GR_Sin_or_Cos = r37
|
|
|
|
GR_SAVE_B0 = r39
|
|
GR_SAVE_GP = r40
|
|
GR_SAVE_PFS = r41
|
|
|
|
.section .text
|
|
.proc __libm_sin_double_dbx#
|
|
.align 64
|
|
__libm_sin_double_dbx:
|
|
|
|
{ .mlx
|
|
alloc GR_Table_Base = ar.pfs,0,12,2,0
|
|
movl GR_Sin_or_Cos = 0x0 ;;
|
|
}
|
|
|
|
{ .mmi
|
|
nop.m 999
|
|
addl GR_Table_Base = @ltoff(FSINCOS_CONSTANTS#), gp
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
{ .mmi
|
|
ld8 GR_Table_Base = [GR_Table_Base]
|
|
nop.m 999
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
|
|
{ .mib
|
|
nop.m 999
|
|
nop.i 999
|
|
br.cond.sptk L(SINCOS_CONTINUE) ;;
|
|
}
|
|
|
|
.endp __libm_sin_double_dbx#
|
|
ASM_SIZE_DIRECTIVE(__libm_sin_double_dbx)
|
|
|
|
.section .text
|
|
.proc __libm_cos_double_dbx#
|
|
__libm_cos_double_dbx:
|
|
|
|
{ .mlx
|
|
alloc GR_Table_Base= ar.pfs,0,12,2,0
|
|
movl GR_Sin_or_Cos = 0x1 ;;
|
|
}
|
|
|
|
{ .mmi
|
|
nop.m 999
|
|
addl GR_Table_Base = @ltoff(FSINCOS_CONSTANTS#), gp
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
{ .mmi
|
|
ld8 GR_Table_Base = [GR_Table_Base]
|
|
nop.m 999
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
//
|
|
// Load Table Address
|
|
//
|
|
L(SINCOS_CONTINUE):
|
|
|
|
{ .mmi
|
|
add GR_Table_Base1 = 96, GR_Table_Base
|
|
ldfs FR_Two_to_24 = [GR_Table_Base], 4
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
{ .mmi
|
|
nop.m 999
|
|
//
|
|
// Load 2**24, load 2**63.
|
|
//
|
|
ldfs FR_Neg_Two_to_24 = [GR_Table_Base], 12
|
|
mov r41 = ar.pfs ;;
|
|
}
|
|
|
|
{ .mfi
|
|
ldfs FR_Two_to_63 = [GR_Table_Base1], 4
|
|
//
|
|
// Check for unnormals - unsupported operands. We do not want
|
|
// to generate denormal exception
|
|
// Check for NatVals, QNaNs, SNaNs, +/-Infs
|
|
// Check for EM unsupporteds
|
|
// Check for Zero
|
|
//
|
|
fclass.m.unc p6, p8 = FR_Input_X, 0x1E3
|
|
mov r40 = gp ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
fclass.nm.unc p8, p0 = FR_Input_X, 0x1FF
|
|
// GR_Sin_or_Cos denotes
|
|
mov r39 = b0
|
|
}
|
|
|
|
{ .mfb
|
|
ldfs FR_Neg_Two_to_63 = [GR_Table_Base1], 12
|
|
fclass.m.unc p10, p0 = FR_Input_X, 0x007
|
|
(p6) br.cond.spnt L(SINCOS_SPECIAL) ;;
|
|
}
|
|
|
|
{ .mib
|
|
nop.m 999
|
|
nop.i 999
|
|
(p8) br.cond.spnt L(SINCOS_SPECIAL) ;;
|
|
}
|
|
|
|
{ .mib
|
|
nop.m 999
|
|
nop.i 999
|
|
//
|
|
// Branch if +/- NaN, Inf.
|
|
// Load -2**24, load -2**63.
|
|
//
|
|
(p10) br.cond.spnt L(SINCOS_ZERO) ;;
|
|
}
|
|
|
|
{ .mmb
|
|
ldfe FR_Inv_pi_by_2 = [GR_Table_Base], 16
|
|
ldfe FR_Inv_P_0 = [GR_Table_Base1], 16
|
|
nop.b 999 ;;
|
|
}
|
|
|
|
{ .mmb
|
|
nop.m 999
|
|
ldfe FR_d_1 = [GR_Table_Base1], 16
|
|
nop.b 999 ;;
|
|
}
|
|
//
|
|
// Raise possible denormal operand flag with useful fcmp
|
|
// Is x <= -2**63
|
|
// Load Inv_P_0 for pre-reduction
|
|
// Load Inv_pi_by_2
|
|
//
|
|
|
|
{ .mmb
|
|
ldfe FR_P_0 = [GR_Table_Base], 16
|
|
ldfe FR_d_2 = [GR_Table_Base1], 16
|
|
nop.b 999 ;;
|
|
}
|
|
//
|
|
// Load P_0
|
|
// Load d_1
|
|
// Is x >= 2**63
|
|
// Is x <= -2**24?
|
|
//
|
|
|
|
{ .mmi
|
|
ldfe FR_P_1 = [GR_Table_Base], 16 ;;
|
|
//
|
|
// Load P_1
|
|
// Load d_2
|
|
// Is x >= 2**24?
|
|
//
|
|
ldfe FR_P_2 = [GR_Table_Base], 16
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mmf
|
|
nop.m 999
|
|
ldfe FR_P_3 = [GR_Table_Base], 16
|
|
fcmp.le.unc.s1 p7, p8 = FR_Input_X, FR_Neg_Two_to_24
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// Branch if +/- zero.
|
|
// Decide about the paths to take:
|
|
// If -2**24 < FR_Input_X < 2**24 - CASE 1 OR 2
|
|
// OTHERWISE - CASE 3 OR 4
|
|
//
|
|
fcmp.le.unc.s0 p10, p11 = FR_Input_X, FR_Neg_Two_to_63
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p8) fcmp.ge.s1 p7, p0 = FR_Input_X, FR_Two_to_24
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
ldfe FR_Pi_by_4 = [GR_Table_Base1], 16
|
|
(p11) fcmp.ge.s1 p10, p0 = FR_Input_X, FR_Two_to_63
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mmi
|
|
ldfe FR_Neg_Pi_by_4 = [GR_Table_Base1], 16 ;;
|
|
ldfs FR_Two_to_M3 = [GR_Table_Base1], 4
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mib
|
|
ldfs FR_Neg_Two_to_M3 = [GR_Table_Base1], 12
|
|
nop.i 999
|
|
//
|
|
// Load P_2
|
|
// Load P_3
|
|
// Load pi_by_4
|
|
// Load neg_pi_by_4
|
|
// Load 2**(-3)
|
|
// Load -2**(-3).
|
|
//
|
|
(p10) br.cond.spnt L(SINCOS_ARG_TOO_LARGE) ;;
|
|
}
|
|
|
|
{ .mib
|
|
nop.m 999
|
|
nop.i 999
|
|
//
|
|
// Branch out if x >= 2**63. Use Payne-Hanek Reduction
|
|
//
|
|
(p7) br.cond.spnt L(SINCOS_LARGER_ARG) ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// Branch if Arg <= -2**24 or Arg >= 2**24 and use pre-reduction.
|
|
//
|
|
fma.s1 FR_N_float = FR_Input_X, FR_Inv_pi_by_2, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
fcmp.lt.unc.s1 p6, p7 = FR_Input_X, FR_Pi_by_4
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// Select the case when |Arg| < pi/4
|
|
// Else Select the case when |Arg| >= pi/4
|
|
//
|
|
fcvt.fx.s1 FR_N_fix = FR_N_float
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N = Arg * 2/pi
|
|
// Check if Arg < pi/4
|
|
//
|
|
(p6) fcmp.gt.s1 p6, p7 = FR_Input_X, FR_Neg_Pi_by_4
|
|
nop.i 999 ;;
|
|
}
|
|
//
|
|
// Case 2: Convert integer N_fix back to normalized floating-point value.
|
|
// Case 1: p8 is only affected when p6 is set
|
|
//
|
|
|
|
{ .mfi
|
|
(p7) ldfs FR_Two_to_M33 = [GR_Table_Base1], 4
|
|
//
|
|
// Grab the integer part of N and call it N_fix
|
|
//
|
|
(p6) fmerge.se FR_r = FR_Input_X, FR_Input_X
|
|
// If |x| < pi/4, r = x and c = 0
|
|
// lf |x| < pi/4, is x < 2**(-3).
|
|
// r = Arg
|
|
// c = 0
|
|
(p6) mov GR_N_Inc = GR_Sin_or_Cos ;;
|
|
}
|
|
|
|
{ .mmf
|
|
nop.m 999
|
|
(p7) ldfs FR_Neg_Two_to_M33 = [GR_Table_Base1], 4
|
|
(p6) fmerge.se FR_c = f0, f0
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p6) fcmp.lt.unc.s1 p8, p9 = FR_Input_X, FR_Two_to_M3
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// lf |x| < pi/4, is -2**(-3)< x < 2**(-3) - set p8.
|
|
// If |x| >= pi/4,
|
|
// Create the right N for |x| < pi/4 and otherwise
|
|
// Case 2: Place integer part of N in GP register
|
|
//
|
|
(p7) fcvt.xf FR_N_float = FR_N_fix
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mmf
|
|
nop.m 999
|
|
(p7) getf.sig GR_N_Inc = FR_N_fix
|
|
(p8) fcmp.gt.s1 p8, p0 = FR_Input_X, FR_Neg_Two_to_M3 ;;
|
|
}
|
|
|
|
{ .mib
|
|
nop.m 999
|
|
nop.i 999
|
|
//
|
|
// Load 2**(-33), -2**(-33)
|
|
//
|
|
(p8) br.cond.spnt L(SINCOS_SMALL_R) ;;
|
|
}
|
|
|
|
{ .mib
|
|
nop.m 999
|
|
nop.i 999
|
|
(p6) br.cond.sptk L(SINCOS_NORMAL_R) ;;
|
|
}
|
|
//
|
|
// if |x| < pi/4, branch based on |x| < 2**(-3) or otherwise.
|
|
//
|
|
//
|
|
// In this branch, |x| >= pi/4.
|
|
//
|
|
|
|
{ .mfi
|
|
ldfs FR_Neg_Two_to_M67 = [GR_Table_Base1], 8
|
|
//
|
|
// Load -2**(-67)
|
|
//
|
|
fnma.s1 FR_s = FR_N_float, FR_P_1, FR_Input_X
|
|
//
|
|
// w = N * P_2
|
|
// s = -N * P_1 + Arg
|
|
//
|
|
add GR_N_Inc = GR_N_Inc, GR_Sin_or_Cos
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
fma.s1 FR_w = FR_N_float, FR_P_2, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// Adjust N_fix by N_inc to determine whether sine or
|
|
// cosine is being calculated
|
|
//
|
|
fcmp.lt.unc.s1 p7, p6 = FR_s, FR_Two_to_M33
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p7) fcmp.gt.s1 p7, p6 = FR_s, FR_Neg_Two_to_M33
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
// Remember x >= pi/4.
|
|
// Is s <= -2**(-33) or s >= 2**(-33) (p6)
|
|
// or -2**(-33) < s < 2**(-33) (p7)
|
|
(p6) fms.s1 FR_r = FR_s, f1, FR_w
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p7) fma.s1 FR_w = FR_N_float, FR_P_3, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p7) fma.s1 FR_U_1 = FR_N_float, FR_P_2, FR_w
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p6) fms.s1 FR_c = FR_s, f1, FR_r
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// For big s: r = s - w: No futher reduction is necessary
|
|
// For small s: w = N * P_3 (change sign) More reduction
|
|
//
|
|
(p6) fcmp.lt.unc.s1 p8, p9 = FR_r, FR_Two_to_M3
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p8) fcmp.gt.s1 p8, p9 = FR_r, FR_Neg_Two_to_M3
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p7) fms.s1 FR_r = FR_s, f1, FR_U_1
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfb
|
|
nop.m 999
|
|
//
|
|
// For big s: Is |r| < 2**(-3)?
|
|
// For big s: c = S - r
|
|
// For small s: U_1 = N * P_2 + w
|
|
//
|
|
// If p8 is set, prepare to branch to Small_R.
|
|
// If p9 is set, prepare to branch to Normal_R.
|
|
// For big s, r is complete here.
|
|
//
|
|
(p6) fms.s1 FR_c = FR_c, f1, FR_w
|
|
//
|
|
// For big s: c = c + w (w has not been negated.)
|
|
// For small s: r = S - U_1
|
|
//
|
|
(p8) br.cond.spnt L(SINCOS_SMALL_R) ;;
|
|
}
|
|
|
|
{ .mib
|
|
nop.m 999
|
|
nop.i 999
|
|
(p9) br.cond.sptk L(SINCOS_NORMAL_R) ;;
|
|
}
|
|
|
|
{ .mfi
|
|
(p7) add GR_Table_Base1 = 224, GR_Table_Base1
|
|
//
|
|
// Branch to SINCOS_SMALL_R or SINCOS_NORMAL_R
|
|
//
|
|
(p7) fms.s1 FR_U_2 = FR_N_float, FR_P_2, FR_U_1
|
|
//
|
|
// c = S - U_1
|
|
// r = S_1 * r
|
|
//
|
|
//
|
|
(p7) extr.u GR_i_1 = GR_N_Inc, 0, 1
|
|
}
|
|
|
|
{ .mmi
|
|
nop.m 999 ;;
|
|
//
|
|
// Get [i_0,i_1] - two lsb of N_fix_gr.
|
|
// Do dummy fmpy so inexact is always set.
|
|
//
|
|
(p7) cmp.eq.unc p9, p10 = 0x0, GR_i_1
|
|
(p7) extr.u GR_i_0 = GR_N_Inc, 1, 1 ;;
|
|
}
|
|
//
|
|
// For small s: U_2 = N * P_2 - U_1
|
|
// S_1 stored constant - grab the one stored with the
|
|
// coefficients.
|
|
//
|
|
|
|
{ .mfi
|
|
(p7) ldfe FR_S_1 = [GR_Table_Base1], 16
|
|
//
|
|
// Check if i_1 and i_0 != 0
|
|
//
|
|
(p10) fma.s1 FR_poly = f0, f1, FR_Neg_Two_to_M67
|
|
(p7) cmp.eq.unc p11, p12 = 0x0, GR_i_0 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p7) fms.s1 FR_s = FR_s, f1, FR_r
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// S = S - r
|
|
// U_2 = U_2 + w
|
|
// load S_1
|
|
//
|
|
(p7) fma.s1 FR_rsq = FR_r, FR_r, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p7) fma.s1 FR_U_2 = FR_U_2, f1, FR_w
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p7) fmerge.se FR_Input_X = FR_r, FR_r
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p10) fma.s1 FR_Input_X = f0, f1, f1
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// FR_rsq = r * r
|
|
// Save r as the result.
|
|
//
|
|
(p7) fms.s1 FR_c = FR_s, f1, FR_U_1
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if ( i_1 ==0) poly = c + S_1*r*r*r
|
|
// else Result = 1
|
|
//
|
|
(p12) fnma.s1 FR_Input_X = FR_Input_X, f1, f0
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p7) fma.s1 FR_r = FR_S_1, FR_r, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p7) fma.d.s0 FR_S_1 = FR_S_1, FR_S_1, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// If i_1 != 0, poly = 2**(-67)
|
|
//
|
|
(p7) fms.s1 FR_c = FR_c, f1, FR_U_2
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// c = c - U_2
|
|
//
|
|
(p9) fma.s1 FR_poly = FR_r, FR_rsq, FR_c
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// i_0 != 0, so Result = -Result
|
|
//
|
|
(p11) fma.d.s0 FR_Input_X = FR_Input_X, f1, FR_poly
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfb
|
|
nop.m 999
|
|
(p12) fms.d.s0 FR_Input_X = FR_Input_X, f1, FR_poly
|
|
//
|
|
// if (i_0 == 0), Result = Result + poly
|
|
// else Result = Result - poly
|
|
//
|
|
br.ret.sptk b0 ;;
|
|
}
|
|
L(SINCOS_LARGER_ARG):
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
fma.s1 FR_N_0 = FR_Input_X, FR_Inv_P_0, f0
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
// This path for argument > 2*24
|
|
// Adjust table_ptr1 to beginning of table.
|
|
//
|
|
|
|
{ .mmi
|
|
nop.m 999
|
|
addl GR_Table_Base = @ltoff(FSINCOS_CONSTANTS#), gp
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
{ .mmi
|
|
ld8 GR_Table_Base = [GR_Table_Base]
|
|
nop.m 999
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
|
|
//
|
|
// Point to 2*-14
|
|
// N_0 = Arg * Inv_P_0
|
|
//
|
|
|
|
{ .mmi
|
|
add GR_Table_Base = 688, GR_Table_Base ;;
|
|
ldfs FR_Two_to_M14 = [GR_Table_Base], 4
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
ldfs FR_Neg_Two_to_M14 = [GR_Table_Base], 0
|
|
nop.f 999
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// Load values 2**(-14) and -2**(-14)
|
|
//
|
|
fcvt.fx.s1 FR_N_0_fix = FR_N_0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N_0_fix = integer part of N_0
|
|
//
|
|
fcvt.xf FR_N_0 = FR_N_0_fix
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// Make N_0 the integer part
|
|
//
|
|
fnma.s1 FR_ArgPrime = FR_N_0, FR_P_0, FR_Input_X
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
fma.s1 FR_w = FR_N_0, FR_d_1, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// Arg' = -N_0 * P_0 + Arg
|
|
// w = N_0 * d_1
|
|
//
|
|
fma.s1 FR_N_float = FR_ArgPrime, FR_Inv_pi_by_2, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N = A' * 2/pi
|
|
//
|
|
fcvt.fx.s1 FR_N_fix = FR_N_float
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N_fix is the integer part
|
|
//
|
|
fcvt.xf FR_N_float = FR_N_fix
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
getf.sig GR_N_Inc = FR_N_fix
|
|
nop.f 999
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mii
|
|
nop.m 999
|
|
nop.i 999 ;;
|
|
add GR_N_Inc = GR_N_Inc, GR_Sin_or_Cos ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// N is the integer part of the reduced-reduced argument.
|
|
// Put the integer in a GP register
|
|
//
|
|
fnma.s1 FR_s = FR_N_float, FR_P_1, FR_ArgPrime
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
fnma.s1 FR_w = FR_N_float, FR_P_2, FR_w
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// s = -N*P_1 + Arg'
|
|
// w = -N*P_2 + w
|
|
// N_fix_gr = N_fix_gr + N_inc
|
|
//
|
|
fcmp.lt.unc.s1 p9, p8 = FR_s, FR_Two_to_M14
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p9) fcmp.gt.s1 p9, p8 = FR_s, FR_Neg_Two_to_M14
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// For |s| > 2**(-14) r = S + w (r complete)
|
|
// Else U_hi = N_0 * d_1
|
|
//
|
|
(p9) fma.s1 FR_V_hi = FR_N_float, FR_P_2, f0
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p9) fma.s1 FR_U_hi = FR_N_0, FR_d_1, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// Either S <= -2**(-14) or S >= 2**(-14)
|
|
// or -2**(-14) < s < 2**(-14)
|
|
//
|
|
(p8) fma.s1 FR_r = FR_s, f1, FR_w
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p9) fma.s1 FR_w = FR_N_float, FR_P_3, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// We need abs of both U_hi and V_hi - don't
|
|
// worry about switched sign of V_hi.
|
|
//
|
|
(p9) fms.s1 FR_A = FR_U_hi, f1, FR_V_hi
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// Big s: finish up c = (S - r) + w (c complete)
|
|
// Case 4: A = U_hi + V_hi
|
|
// Note: Worry about switched sign of V_hi, so subtract instead of add.
|
|
//
|
|
(p9) fnma.s1 FR_V_lo = FR_N_float, FR_P_2, FR_V_hi
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p9) fms.s1 FR_U_lo = FR_N_0, FR_d_1, FR_U_hi
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p9) fmerge.s FR_V_hiabs = f0, FR_V_hi
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
// For big s: c = S - r
|
|
// For small s do more work: U_lo = N_0 * d_1 - U_hi
|
|
//
|
|
(p9) fmerge.s FR_U_hiabs = f0, FR_U_hi
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// For big s: Is |r| < 2**(-3)
|
|
// For big s: if p12 set, prepare to branch to Small_R.
|
|
// For big s: If p13 set, prepare to branch to Normal_R.
|
|
//
|
|
(p8) fms.s1 FR_c = FR_s, f1, FR_r
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// For small S: V_hi = N * P_2
|
|
// w = N * P_3
|
|
// Note the product does not include the (-) as in the writeup
|
|
// so (-) missing for V_hi and w.
|
|
//
|
|
(p8) fcmp.lt.unc.s1 p12, p13 = FR_r, FR_Two_to_M3
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fcmp.gt.s1 p12, p13 = FR_r, FR_Neg_Two_to_M3
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p8) fma.s1 FR_c = FR_c, f1, FR_w
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfb
|
|
nop.m 999
|
|
(p9) fms.s1 FR_w = FR_N_0, FR_d_2, FR_w
|
|
(p12) br.cond.spnt L(SINCOS_SMALL_R) ;;
|
|
}
|
|
|
|
{ .mib
|
|
nop.m 999
|
|
nop.i 999
|
|
(p13) br.cond.sptk L(SINCOS_NORMAL_R) ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// Big s: Vector off when |r| < 2**(-3). Recall that p8 will be true.
|
|
// The remaining stuff is for Case 4.
|
|
// Small s: V_lo = N * P_2 + U_hi (U_hi is in place of V_hi in writeup)
|
|
// Note: the (-) is still missing for V_lo.
|
|
// Small s: w = w + N_0 * d_2
|
|
// Note: the (-) is now incorporated in w.
|
|
//
|
|
(p9) fcmp.ge.unc.s1 p10, p11 = FR_U_hiabs, FR_V_hiabs
|
|
extr.u GR_i_1 = GR_N_Inc, 0, 1 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// C_hi = S + A
|
|
//
|
|
(p9) fma.s1 FR_t = FR_U_lo, f1, FR_V_lo
|
|
extr.u GR_i_0 = GR_N_Inc, 1, 1 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// t = U_lo + V_lo
|
|
//
|
|
//
|
|
(p10) fms.s1 FR_a = FR_U_hi, f1, FR_A
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p11) fma.s1 FR_a = FR_V_hi, f1, FR_A
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
{ .mmi
|
|
nop.m 999
|
|
addl GR_Table_Base = @ltoff(FSINCOS_CONSTANTS#), gp
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
{ .mmi
|
|
ld8 GR_Table_Base = [GR_Table_Base]
|
|
nop.m 999
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
|
|
{ .mfi
|
|
add GR_Table_Base = 528, GR_Table_Base
|
|
//
|
|
// Is U_hiabs >= V_hiabs?
|
|
//
|
|
(p9) fma.s1 FR_C_hi = FR_s, f1, FR_A
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mmi
|
|
ldfe FR_C_1 = [GR_Table_Base], 16 ;;
|
|
ldfe FR_C_2 = [GR_Table_Base], 64
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mmf
|
|
nop.m 999
|
|
//
|
|
// c = c + C_lo finished.
|
|
// Load C_2
|
|
//
|
|
ldfe FR_S_1 = [GR_Table_Base], 16
|
|
//
|
|
// C_lo = S - C_hi
|
|
//
|
|
fma.s1 FR_t = FR_t, f1, FR_w ;;
|
|
}
|
|
//
|
|
// r and c have been computed.
|
|
// Make sure ftz mode is set - should be automatic when using wre
|
|
// |r| < 2**(-3)
|
|
// Get [i_0,i_1] - two lsb of N_fix.
|
|
// Load S_1
|
|
//
|
|
|
|
{ .mfi
|
|
ldfe FR_S_2 = [GR_Table_Base], 64
|
|
//
|
|
// t = t + w
|
|
//
|
|
(p10) fms.s1 FR_a = FR_a, f1, FR_V_hi
|
|
cmp.eq.unc p9, p10 = 0x0, GR_i_0
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// For larger u than v: a = U_hi - A
|
|
// Else a = V_hi - A (do an add to account for missing (-) on V_hi
|
|
//
|
|
fms.s1 FR_C_lo = FR_s, f1, FR_C_hi
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p11) fms.s1 FR_a = FR_U_hi, f1, FR_a
|
|
cmp.eq.unc p11, p12 = 0x0, GR_i_1
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// If u > v: a = (U_hi - A) + V_hi
|
|
// Else a = (V_hi - A) + U_hi
|
|
// In each case account for negative missing from V_hi.
|
|
//
|
|
fma.s1 FR_C_lo = FR_C_lo, f1, FR_A
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// C_lo = (S - C_hi) + A
|
|
//
|
|
fma.s1 FR_t = FR_t, f1, FR_a
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// t = t + a
|
|
//
|
|
fma.s1 FR_C_lo = FR_C_lo, f1, FR_t
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// C_lo = C_lo + t
|
|
// Adjust Table_Base to beginning of table
|
|
//
|
|
fma.s1 FR_r = FR_C_hi, f1, FR_C_lo
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// Load S_2
|
|
//
|
|
fma.s1 FR_rsq = FR_r, FR_r, f0
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// Table_Base points to C_1
|
|
// r = C_hi + C_lo
|
|
//
|
|
fms.s1 FR_c = FR_C_hi, f1, FR_r
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if i_1 ==0: poly = S_2 * FR_rsq + S_1
|
|
// else poly = C_2 * FR_rsq + C_1
|
|
//
|
|
(p11) fma.s1 FR_Input_X = f0, f1, FR_r
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fma.s1 FR_Input_X = f0, f1, f1
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// Compute r_cube = FR_rsq * r
|
|
//
|
|
(p11) fma.s1 FR_poly = FR_rsq, FR_S_2, FR_S_1
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fma.s1 FR_poly = FR_rsq, FR_C_2, FR_C_1
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// Compute FR_rsq = r * r
|
|
// Is i_1 == 0 ?
|
|
//
|
|
fma.s1 FR_r_cubed = FR_rsq, FR_r, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// c = C_hi - r
|
|
// Load C_1
|
|
//
|
|
fma.s1 FR_c = FR_c, f1, FR_C_lo
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if i_1 ==0: poly = r_cube * poly + c
|
|
// else poly = FR_rsq * poly
|
|
//
|
|
(p10) fms.s1 FR_Input_X = f0, f1, FR_Input_X
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if i_1 ==0: Result = r
|
|
// else Result = 1.0
|
|
//
|
|
(p11) fma.s1 FR_poly = FR_r_cubed, FR_poly, FR_c
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fma.s1 FR_poly = FR_rsq, FR_poly, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if i_0 !=0: Result = -Result
|
|
//
|
|
(p9) fma.d.s0 FR_Input_X = FR_Input_X, f1, FR_poly
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfb
|
|
nop.m 999
|
|
(p10) fms.d.s0 FR_Input_X = FR_Input_X, f1, FR_poly
|
|
//
|
|
// if i_0 == 0: Result = Result + poly
|
|
// else Result = Result - poly
|
|
//
|
|
br.ret.sptk b0 ;;
|
|
}
|
|
L(SINCOS_SMALL_R):
|
|
|
|
{ .mii
|
|
nop.m 999
|
|
extr.u GR_i_1 = GR_N_Inc, 0, 1 ;;
|
|
//
|
|
//
|
|
// Compare both i_1 and i_0 with 0.
|
|
// if i_1 == 0, set p9.
|
|
// if i_0 == 0, set p11.
|
|
//
|
|
cmp.eq.unc p9, p10 = 0x0, GR_i_1 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
fma.s1 FR_rsq = FR_r, FR_r, f0
|
|
extr.u GR_i_0 = GR_N_Inc, 1, 1 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// Z = Z * FR_rsq
|
|
//
|
|
(p10) fnma.s1 FR_c = FR_c, FR_r, f0
|
|
cmp.eq.unc p11, p12 = 0x0, GR_i_0
|
|
}
|
|
;;
|
|
|
|
// ******************************************************************
|
|
// ******************************************************************
|
|
// ******************************************************************
|
|
// r and c have been computed.
|
|
// We know whether this is the sine or cosine routine.
|
|
// Make sure ftz mode is set - should be automatic when using wre
|
|
// |r| < 2**(-3)
|
|
//
|
|
// Set table_ptr1 to beginning of constant table.
|
|
// Get [i_0,i_1] - two lsb of N_fix_gr.
|
|
//
|
|
|
|
{ .mmi
|
|
nop.m 999
|
|
addl GR_Table_Base = @ltoff(FSINCOS_CONSTANTS#), gp
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
{ .mmi
|
|
ld8 GR_Table_Base = [GR_Table_Base]
|
|
nop.m 999
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
|
|
//
|
|
// Set table_ptr1 to point to S_5.
|
|
// Set table_ptr1 to point to C_5.
|
|
// Compute FR_rsq = r * r
|
|
//
|
|
|
|
{ .mfi
|
|
(p9) add GR_Table_Base = 672, GR_Table_Base
|
|
(p10) fmerge.s FR_r = f1, f1
|
|
(p10) add GR_Table_Base = 592, GR_Table_Base ;;
|
|
}
|
|
//
|
|
// Set table_ptr1 to point to S_5.
|
|
// Set table_ptr1 to point to C_5.
|
|
//
|
|
|
|
{ .mmi
|
|
(p9) ldfe FR_S_5 = [GR_Table_Base], -16 ;;
|
|
//
|
|
// if (i_1 == 0) load S_5
|
|
// if (i_1 != 0) load C_5
|
|
//
|
|
(p9) ldfe FR_S_4 = [GR_Table_Base], -16
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mmf
|
|
(p10) ldfe FR_C_5 = [GR_Table_Base], -16
|
|
//
|
|
// Z = FR_rsq * FR_rsq
|
|
//
|
|
(p9) ldfe FR_S_3 = [GR_Table_Base], -16
|
|
//
|
|
// Compute FR_rsq = r * r
|
|
// if (i_1 == 0) load S_4
|
|
// if (i_1 != 0) load C_4
|
|
//
|
|
fma.s1 FR_Z = FR_rsq, FR_rsq, f0 ;;
|
|
}
|
|
//
|
|
// if (i_1 == 0) load S_3
|
|
// if (i_1 != 0) load C_3
|
|
//
|
|
|
|
{ .mmi
|
|
(p9) ldfe FR_S_2 = [GR_Table_Base], -16 ;;
|
|
//
|
|
// if (i_1 == 0) load S_2
|
|
// if (i_1 != 0) load C_2
|
|
//
|
|
(p9) ldfe FR_S_1 = [GR_Table_Base], -16
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mmi
|
|
(p10) ldfe FR_C_4 = [GR_Table_Base], -16 ;;
|
|
(p10) ldfe FR_C_3 = [GR_Table_Base], -16
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mmi
|
|
(p10) ldfe FR_C_2 = [GR_Table_Base], -16 ;;
|
|
(p10) ldfe FR_C_1 = [GR_Table_Base], -16
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if (i_1 != 0):
|
|
// poly_lo = FR_rsq * C_5 + C_4
|
|
// poly_hi = FR_rsq * C_2 + C_1
|
|
//
|
|
(p9) fma.s1 FR_Z = FR_Z, FR_r, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if (i_1 == 0) load S_1
|
|
// if (i_1 != 0) load C_1
|
|
//
|
|
(p9) fma.s1 FR_poly_lo = FR_rsq, FR_S_5, FR_S_4
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// c = -c * r
|
|
// dummy fmpy's to flag inexact.
|
|
//
|
|
(p9) fma.d.s0 FR_S_4 = FR_S_4, FR_S_4, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// poly_lo = FR_rsq * poly_lo + C_3
|
|
// poly_hi = FR_rsq * poly_hi
|
|
//
|
|
fma.s1 FR_Z = FR_Z, FR_rsq, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p9) fma.s1 FR_poly_hi = FR_rsq, FR_S_2, FR_S_1
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if (i_1 == 0):
|
|
// poly_lo = FR_rsq * S_5 + S_4
|
|
// poly_hi = FR_rsq * S_2 + S_1
|
|
//
|
|
(p10) fma.s1 FR_poly_lo = FR_rsq, FR_C_5, FR_C_4
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if (i_1 == 0):
|
|
// Z = Z * r for only one of the small r cases - not there
|
|
// in original implementation notes.
|
|
//
|
|
(p9) fma.s1 FR_poly_lo = FR_rsq, FR_poly_lo, FR_S_3
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p10) fma.s1 FR_poly_hi = FR_rsq, FR_C_2, FR_C_1
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p10) fma.d.s0 FR_C_1 = FR_C_1, FR_C_1, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p9) fma.s1 FR_poly_hi = FR_poly_hi, FR_rsq, f0
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// poly_lo = FR_rsq * poly_lo + S_3
|
|
// poly_hi = FR_rsq * poly_hi
|
|
//
|
|
(p10) fma.s1 FR_poly_lo = FR_rsq, FR_poly_lo, FR_C_3
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p10) fma.s1 FR_poly_hi = FR_poly_hi, FR_rsq, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if (i_1 == 0): dummy fmpy's to flag inexact
|
|
// r = 1
|
|
//
|
|
(p9) fma.s1 FR_poly_hi = FR_r, FR_poly_hi, f0
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// poly_hi = r * poly_hi
|
|
//
|
|
fma.s1 FR_poly = FR_Z, FR_poly_lo, FR_c
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fms.s1 FR_r = f0, f1, FR_r
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// poly_hi = Z * poly_lo + c
|
|
// if i_0 == 1: r = -r
|
|
//
|
|
fma.s1 FR_poly = FR_poly, f1, FR_poly_hi
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fms.d.s0 FR_Input_X = FR_r, f1, FR_poly
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfb
|
|
nop.m 999
|
|
//
|
|
// poly = poly + poly_hi
|
|
//
|
|
(p11) fma.d.s0 FR_Input_X = FR_r, f1, FR_poly
|
|
//
|
|
// if (i_0 == 0) Result = r + poly
|
|
// if (i_0 != 0) Result = r - poly
|
|
//
|
|
br.ret.sptk b0 ;;
|
|
}
|
|
L(SINCOS_NORMAL_R):
|
|
|
|
{ .mii
|
|
nop.m 999
|
|
extr.u GR_i_1 = GR_N_Inc, 0, 1 ;;
|
|
//
|
|
// Set table_ptr1 and table_ptr2 to base address of
|
|
// constant table.
|
|
cmp.eq.unc p9, p10 = 0x0, GR_i_1 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
fma.s1 FR_rsq = FR_r, FR_r, f0
|
|
extr.u GR_i_0 = GR_N_Inc, 1, 1 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
frcpa.s1 FR_r_hi, p6 = f1, FR_r
|
|
cmp.eq.unc p11, p12 = 0x0, GR_i_0
|
|
}
|
|
;;
|
|
|
|
// ******************************************************************
|
|
// ******************************************************************
|
|
// ******************************************************************
|
|
//
|
|
// r and c have been computed.
|
|
// We known whether this is the sine or cosine routine.
|
|
// Make sure ftz mode is set - should be automatic when using wre
|
|
// Get [i_0,i_1] - two lsb of N_fix_gr alone.
|
|
//
|
|
|
|
{ .mmi
|
|
nop.m 999
|
|
addl GR_Table_Base = @ltoff(FSINCOS_CONSTANTS#), gp
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
{ .mmi
|
|
ld8 GR_Table_Base = [GR_Table_Base]
|
|
nop.m 999
|
|
nop.i 999
|
|
}
|
|
;;
|
|
|
|
|
|
{ .mfi
|
|
(p10) add GR_Table_Base = 384, GR_Table_Base
|
|
(p12) fms.s1 FR_Input_X = f0, f1, f1
|
|
(p9) add GR_Table_Base = 224, GR_Table_Base ;;
|
|
}
|
|
|
|
{ .mmf
|
|
nop.m 999
|
|
(p10) ldfe FR_QQ_8 = [GR_Table_Base], 16
|
|
//
|
|
// if (i_1==0) poly = poly * FR_rsq + PP_1_lo
|
|
// else poly = FR_rsq * poly
|
|
//
|
|
(p11) fma.s1 FR_Input_X = f0, f1, f1 ;;
|
|
}
|
|
|
|
{ .mmf
|
|
(p10) ldfe FR_QQ_7 = [GR_Table_Base], 16
|
|
//
|
|
// Adjust table pointers based on i_0
|
|
// Compute rsq = r * r
|
|
//
|
|
(p9) ldfe FR_PP_8 = [GR_Table_Base], 16
|
|
fma.s1 FR_r_cubed = FR_r, FR_rsq, f0 ;;
|
|
}
|
|
|
|
{ .mmf
|
|
(p9) ldfe FR_PP_7 = [GR_Table_Base], 16
|
|
(p10) ldfe FR_QQ_6 = [GR_Table_Base], 16
|
|
//
|
|
// Load PP_8 and QQ_8; PP_7 and QQ_7
|
|
//
|
|
frcpa.s1 FR_r_hi, p6 = f1, FR_r_hi ;;
|
|
}
|
|
//
|
|
// if (i_1==0) poly = PP_7 + FR_rsq * PP_8.
|
|
// else poly = QQ_7 + FR_rsq * QQ_8.
|
|
//
|
|
|
|
{ .mmb
|
|
(p9) ldfe FR_PP_6 = [GR_Table_Base], 16
|
|
(p10) ldfe FR_QQ_5 = [GR_Table_Base], 16
|
|
nop.b 999 ;;
|
|
}
|
|
|
|
{ .mmb
|
|
(p9) ldfe FR_PP_5 = [GR_Table_Base], 16
|
|
(p10) ldfe FR_S_1 = [GR_Table_Base], 16
|
|
nop.b 999 ;;
|
|
}
|
|
|
|
{ .mmb
|
|
(p10) ldfe FR_QQ_1 = [GR_Table_Base], 16
|
|
(p9) ldfe FR_C_1 = [GR_Table_Base], 16
|
|
nop.b 999 ;;
|
|
}
|
|
|
|
{ .mmi
|
|
(p10) ldfe FR_QQ_4 = [GR_Table_Base], 16 ;;
|
|
(p9) ldfe FR_PP_1 = [GR_Table_Base], 16
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mmf
|
|
(p10) ldfe FR_QQ_3 = [GR_Table_Base], 16
|
|
//
|
|
// if (i_1=0) corr = corr + c*c
|
|
// else corr = corr * c
|
|
//
|
|
(p9) ldfe FR_PP_4 = [GR_Table_Base], 16
|
|
(p10) fma.s1 FR_poly = FR_rsq, FR_QQ_8, FR_QQ_7 ;;
|
|
}
|
|
//
|
|
// if (i_1=0) poly = rsq * poly + PP_5
|
|
// else poly = rsq * poly + QQ_5
|
|
// Load PP_4 or QQ_4
|
|
//
|
|
|
|
{ .mmf
|
|
(p9) ldfe FR_PP_3 = [GR_Table_Base], 16
|
|
(p10) ldfe FR_QQ_2 = [GR_Table_Base], 16
|
|
//
|
|
// r_hi = frcpa(frcpa(r)).
|
|
// r_cube = r * FR_rsq.
|
|
//
|
|
(p9) fma.s1 FR_poly = FR_rsq, FR_PP_8, FR_PP_7 ;;
|
|
}
|
|
//
|
|
// Do dummy multiplies so inexact is always set.
|
|
//
|
|
|
|
{ .mfi
|
|
(p9) ldfe FR_PP_2 = [GR_Table_Base], 16
|
|
//
|
|
// r_lo = r - r_hi
|
|
//
|
|
(p9) fma.s1 FR_U_lo = FR_r_hi, FR_r_hi, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mmf
|
|
nop.m 999
|
|
(p9) ldfe FR_PP_1_lo = [GR_Table_Base], 16
|
|
(p10) fma.s1 FR_corr = FR_S_1, FR_r_cubed, FR_r
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p10) fma.s1 FR_poly = FR_rsq, FR_poly, FR_QQ_6
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if (i_1=0) U_lo = r_hi * r_hi
|
|
// else U_lo = r_hi + r
|
|
//
|
|
(p9) fma.s1 FR_corr = FR_C_1, FR_rsq, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if (i_1=0) corr = C_1 * rsq
|
|
// else corr = S_1 * r_cubed + r
|
|
//
|
|
(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_6
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p10) fma.s1 FR_U_lo = FR_r_hi, f1, FR_r
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if (i_1=0) U_hi = r_hi + U_hi
|
|
// else U_hi = QQ_1 * U_hi + 1
|
|
//
|
|
(p9) fma.s1 FR_U_lo = FR_r, FR_r_hi, FR_U_lo
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// U_hi = r_hi * r_hi
|
|
//
|
|
fms.s1 FR_r_lo = FR_r, f1, FR_r_hi
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// Load PP_1, PP_6, PP_5, and C_1
|
|
// Load QQ_1, QQ_6, QQ_5, and S_1
|
|
//
|
|
fma.s1 FR_U_hi = FR_r_hi, FR_r_hi, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p10) fma.s1 FR_poly = FR_rsq, FR_poly, FR_QQ_5
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p10) fnma.s1 FR_corr = FR_corr, FR_c, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if (i_1=0) U_lo = r * r_hi + U_lo
|
|
// else U_lo = r_lo * U_lo
|
|
//
|
|
(p9) fma.s1 FR_corr = FR_corr, FR_c, FR_c
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_5
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if (i_1 =0) U_hi = r + U_hi
|
|
// if (i_1 =0) U_lo = r_lo * U_lo
|
|
//
|
|
//
|
|
(p9) fma.d.s0 FR_PP_5 = FR_PP_5, FR_PP_4, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p9) fma.s1 FR_U_lo = FR_r, FR_r, FR_U_lo
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p10) fma.s1 FR_U_lo = FR_r_lo, FR_U_lo, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if (i_1=0) poly = poly * rsq + PP_6
|
|
// else poly = poly * rsq + QQ_6
|
|
//
|
|
(p9) fma.s1 FR_U_hi = FR_r_hi, FR_U_hi, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p10) fma.s1 FR_poly = FR_rsq, FR_poly, FR_QQ_4
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p10) fma.s1 FR_U_hi = FR_QQ_1, FR_U_hi, f1
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p10) fma.d.s0 FR_QQ_5 = FR_QQ_5, FR_QQ_5, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if (i_1!=0) U_hi = PP_1 * U_hi
|
|
// if (i_1!=0) U_lo = r * r + U_lo
|
|
// Load PP_3 or QQ_3
|
|
//
|
|
(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_4
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p9) fma.s1 FR_U_lo = FR_r_lo, FR_U_lo, f0
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p10) fma.s1 FR_U_lo = FR_QQ_1,FR_U_lo, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p9) fma.s1 FR_U_hi = FR_PP_1, FR_U_hi, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p10) fma.s1 FR_poly = FR_rsq, FR_poly, FR_QQ_3
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// Load PP_2, QQ_2
|
|
//
|
|
(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_3
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if (i_1==0) poly = FR_rsq * poly + PP_3
|
|
// else poly = FR_rsq * poly + QQ_3
|
|
// Load PP_1_lo
|
|
//
|
|
(p9) fma.s1 FR_U_lo = FR_PP_1, FR_U_lo, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if (i_1 =0) poly = poly * rsq + pp_r4
|
|
// else poly = poly * rsq + qq_r4
|
|
//
|
|
(p9) fma.s1 FR_U_hi = FR_r, f1, FR_U_hi
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p10) fma.s1 FR_poly = FR_rsq, FR_poly, FR_QQ_2
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if (i_1==0) U_lo = PP_1_hi * U_lo
|
|
// else U_lo = QQ_1 * U_lo
|
|
//
|
|
(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_2
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if (i_0==0) Result = 1
|
|
// else Result = -1
|
|
//
|
|
fma.s1 FR_V = FR_U_lo, f1, FR_corr
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p10) fma.s1 FR_poly = FR_rsq, FR_poly, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if (i_1==0) poly = FR_rsq * poly + PP_2
|
|
// else poly = FR_rsq * poly + QQ_2
|
|
//
|
|
(p9) fma.s1 FR_poly = FR_rsq, FR_poly, FR_PP_1_lo
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p10) fma.s1 FR_poly = FR_rsq, FR_poly, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// V = U_lo + corr
|
|
//
|
|
(p9) fma.s1 FR_poly = FR_r_cubed, FR_poly, f0
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
//
|
|
// if (i_1==0) poly = r_cube * poly
|
|
// else poly = FR_rsq * poly
|
|
//
|
|
fma.s1 FR_V = FR_poly, f1, FR_V
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p12) fms.d.s0 FR_Input_X = FR_Input_X, FR_U_hi, FR_V
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfb
|
|
nop.m 999
|
|
//
|
|
// V = V + poly
|
|
//
|
|
(p11) fma.d.s0 FR_Input_X = FR_Input_X, FR_U_hi, FR_V
|
|
//
|
|
// if (i_0==0) Result = Result * U_hi + V
|
|
// else Result = Result * U_hi - V
|
|
//
|
|
br.ret.sptk b0 ;;
|
|
}
|
|
|
|
//
|
|
// If cosine, FR_Input_X = 1
|
|
// If sine, FR_Input_X = +/-Zero (Input FR_Input_X)
|
|
// Results are exact, no exceptions
|
|
//
|
|
L(SINCOS_ZERO):
|
|
|
|
{ .mmb
|
|
cmp.eq.unc p6, p7 = 0x1, GR_Sin_or_Cos
|
|
nop.m 999
|
|
nop.b 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p7) fmerge.s FR_Input_X = FR_Input_X, FR_Input_X
|
|
nop.i 999
|
|
}
|
|
|
|
{ .mfb
|
|
nop.m 999
|
|
(p6) fmerge.s FR_Input_X = f1, f1
|
|
br.ret.sptk b0 ;;
|
|
}
|
|
|
|
L(SINCOS_SPECIAL):
|
|
|
|
//
|
|
// Path for Arg = +/- QNaN, SNaN, Inf
|
|
// Invalid can be raised. SNaNs
|
|
// become QNaNs
|
|
//
|
|
|
|
{ .mfb
|
|
nop.m 999
|
|
fmpy.d.s0 FR_Input_X = FR_Input_X, f0
|
|
br.ret.sptk b0 ;;
|
|
}
|
|
.endp __libm_cos_double_dbx#
|
|
ASM_SIZE_DIRECTIVE(__libm_cos_double_dbx#)
|
|
|
|
|
|
|
|
//
|
|
// Call int pi_by_2_reduce(double* x, double *y)
|
|
// for |arguments| >= 2**63
|
|
// Address to save r and c as double
|
|
//
|
|
//
|
|
// psp sp+64
|
|
// sp+48 -> f0 c
|
|
// r45 sp+32 -> f0 r
|
|
// r44 -> sp+16 -> InputX
|
|
// sp sp -> scratch provided to callee
|
|
|
|
|
|
|
|
.proc __libm_callout_2
|
|
__libm_callout_2:
|
|
L(SINCOS_ARG_TOO_LARGE):
|
|
|
|
.prologue
|
|
{ .mfi
|
|
add r45=-32,sp // Parameter: r address
|
|
nop.f 0
|
|
.save ar.pfs,GR_SAVE_PFS
|
|
mov GR_SAVE_PFS=ar.pfs // Save ar.pfs
|
|
}
|
|
{ .mfi
|
|
.fframe 64
|
|
add sp=-64,sp // Create new stack
|
|
nop.f 0
|
|
mov GR_SAVE_GP=gp // Save gp
|
|
};;
|
|
{ .mmi
|
|
stfe [r45] = f0,16 // Clear Parameter r on stack
|
|
add r44 = 16,sp // Parameter x address
|
|
.save b0, GR_SAVE_B0
|
|
mov GR_SAVE_B0=b0 // Save b0
|
|
};;
|
|
.body
|
|
{ .mib
|
|
stfe [r45] = f0,-16 // Clear Parameter c on stack
|
|
nop.i 0
|
|
nop.b 0
|
|
}
|
|
{ .mib
|
|
stfe [r44] = FR_Input_X // Store Parameter x on stack
|
|
nop.i 0
|
|
br.call.sptk b0=__libm_pi_by_2_reduce# ;;
|
|
};;
|
|
|
|
|
|
{ .mii
|
|
ldfe FR_Input_X =[r44],16
|
|
//
|
|
// Get r and c off stack
|
|
//
|
|
adds GR_Table_Base1 = -16, GR_Table_Base1
|
|
//
|
|
// Get r and c off stack
|
|
//
|
|
add GR_N_Inc = GR_Sin_or_Cos,r8 ;;
|
|
}
|
|
{ .mmb
|
|
ldfe FR_r =[r45],16
|
|
//
|
|
// Get X off the stack
|
|
// Readjust Table ptr
|
|
//
|
|
ldfs FR_Two_to_M3 = [GR_Table_Base1],4
|
|
nop.b 999 ;;
|
|
}
|
|
{ .mmb
|
|
ldfs FR_Neg_Two_to_M3 = [GR_Table_Base1],0
|
|
ldfe FR_c =[r45]
|
|
nop.b 999 ;;
|
|
}
|
|
|
|
{ .mfi
|
|
.restore sp
|
|
add sp = 64,sp // Restore stack pointer
|
|
fcmp.lt.unc.s1 p6, p0 = FR_r, FR_Two_to_M3
|
|
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
|
|
nop.b 0
|
|
};;
|
|
|
|
|
|
{ .mfi
|
|
nop.m 999
|
|
(p6) fcmp.gt.unc.s1 p6, p0 = FR_r, FR_Neg_Two_to_M3
|
|
nop.i 999 ;;
|
|
}
|
|
|
|
{ .mib
|
|
nop.m 999
|
|
nop.i 999
|
|
(p6) br.cond.spnt L(SINCOS_SMALL_R) ;;
|
|
}
|
|
|
|
{ .mib
|
|
nop.m 999
|
|
nop.i 999
|
|
br.cond.sptk L(SINCOS_NORMAL_R) ;;
|
|
}
|
|
|
|
.endp __libm_callout_2
|
|
ASM_SIZE_DIRECTIVE(__libm_callout_2)
|
|
|
|
.type __libm_pi_by_2_reduce#,@function
|
|
.global __libm_pi_by_2_reduce#
|
|
|
|
|
|
.type __libm_sin_double_dbx#,@function
|
|
.global __libm_sin_double_dbx#
|
|
.type __libm_cos_double_dbx#,@function
|
|
.global __libm_cos_double_dbx#
|