282 lines
10 KiB
C
282 lines
10 KiB
C
/*
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* Copyright (c) 1985, 1993
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* The Regents of the University of California. All rights reserved.
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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
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* are met:
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* 1. 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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* 2. 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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* 3. All advertising materials mentioning features or use of this software
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* must display the following acknowledgement:
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* This product includes software developed by the University of
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* California, Berkeley and its contributors.
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* 4. Neither the name of the University nor the names of its contributors
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* may be used to endorse or promote products derived from this software
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* without specific prior written permission.
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*
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* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
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* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
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* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
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* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
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* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
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* SUCH DAMAGE.
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*/
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#ifndef lint
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static char sccsid[] = "@(#)atan2.c 8.1 (Berkeley) 6/4/93";
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#endif /* not lint */
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/* ATAN2(Y,X)
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* RETURN ARG (X+iY)
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* DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
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* CODED IN C BY K.C. NG, 1/8/85;
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* REVISED BY K.C. NG on 2/7/85, 2/13/85, 3/7/85, 3/30/85, 6/29/85.
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*
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* Required system supported functions :
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* copysign(x,y)
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* scalb(x,y)
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* logb(x)
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*
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* Method :
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* 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x).
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* 2. Reduce x to positive by (if x and y are unexceptional):
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* ARG (x+iy) = arctan(y/x) ... if x > 0,
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* ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0,
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* 3. According to the integer k=4t+0.25 truncated , t=y/x, the argument
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* is further reduced to one of the following intervals and the
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* arctangent of y/x is evaluated by the corresponding formula:
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*
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* [0,7/16] atan(y/x) = t - t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
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* [7/16,11/16] atan(y/x) = atan(1/2) + atan( (y-x/2)/(x+y/2) )
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* [11/16.19/16] atan(y/x) = atan( 1 ) + atan( (y-x)/(x+y) )
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* [19/16,39/16] atan(y/x) = atan(3/2) + atan( (y-1.5x)/(x+1.5y) )
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* [39/16,INF] atan(y/x) = atan(INF) + atan( -x/y )
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*
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* Special cases:
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* Notations: atan2(y,x) == ARG (x+iy) == ARG(x,y).
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*
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* ARG( NAN , (anything) ) is NaN;
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* ARG( (anything), NaN ) is NaN;
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* ARG(+(anything but NaN), +-0) is +-0 ;
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* ARG(-(anything but NaN), +-0) is +-PI ;
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* ARG( 0, +-(anything but 0 and NaN) ) is +-PI/2;
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* ARG( +INF,+-(anything but INF and NaN) ) is +-0 ;
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* ARG( -INF,+-(anything but INF and NaN) ) is +-PI;
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* ARG( +INF,+-INF ) is +-PI/4 ;
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* ARG( -INF,+-INF ) is +-3PI/4;
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* ARG( (anything but,0,NaN, and INF),+-INF ) is +-PI/2;
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*
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* Accuracy:
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* atan2(y,x) returns (PI/pi) * the exact ARG (x+iy) nearly rounded,
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* where
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*
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* in decimal:
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* pi = 3.141592653589793 23846264338327 .....
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* 53 bits PI = 3.141592653589793 115997963 ..... ,
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* 56 bits PI = 3.141592653589793 227020265 ..... ,
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*
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* in hexadecimal:
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* pi = 3.243F6A8885A308D313198A2E....
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* 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps
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* 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps
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*
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* In a test run with 356,000 random argument on [-1,1] * [-1,1] on a
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* VAX, the maximum observed error was 1.41 ulps (units of the last place)
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* compared with (PI/pi)*(the exact ARG(x+iy)).
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*
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* Note:
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* We use machine PI (the true pi rounded) in place of the actual
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* value of pi for all the trig and inverse trig functions. In general,
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* if trig is one of sin, cos, tan, then computed trig(y) returns the
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* exact trig(y*pi/PI) nearly rounded; correspondingly, computed arctrig
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* returns the exact arctrig(y)*PI/pi nearly rounded. These guarantee the
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* trig functions have period PI, and trig(arctrig(x)) returns x for
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* all critical values x.
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*
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* Constants:
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* The hexadecimal values are the intended ones for the following constants.
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* The decimal values may be used, provided that the compiler will convert
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* from decimal to binary accurately enough to produce the hexadecimal values
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* shown.
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*/
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#include "mathimpl.h"
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vc(athfhi, 4.6364760900080611433E-1 ,6338,3fed,da7b,2b0d, -1, .ED63382B0DDA7B)
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vc(athflo, 1.9338828231967579916E-19 ,5005,2164,92c0,9cfe, -62, .E450059CFE92C0)
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vc(PIo4, 7.8539816339744830676E-1 ,0fda,4049,68c2,a221, 0, .C90FDAA22168C2)
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vc(at1fhi, 9.8279372324732906796E-1 ,985e,407b,b4d9,940f, 0, .FB985E940FB4D9)
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vc(at1flo,-3.5540295636764633916E-18 ,1edc,a383,eaea,34d6, -57,-.831EDC34D6EAEA)
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vc(PIo2, 1.5707963267948966135E0 ,0fda,40c9,68c2,a221, 1, .C90FDAA22168C2)
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vc(PI, 3.1415926535897932270E0 ,0fda,4149,68c2,a221, 2, .C90FDAA22168C2)
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vc(a1, 3.3333333333333473730E-1 ,aaaa,3faa,ab75,aaaa, -1, .AAAAAAAAAAAB75)
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vc(a2, -2.0000000000017730678E-1 ,cccc,bf4c,946e,cccd, -2,-.CCCCCCCCCD946E)
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vc(a3, 1.4285714286694640301E-1 ,4924,3f12,4262,9274, -2, .92492492744262)
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vc(a4, -1.1111111135032672795E-1 ,8e38,bee3,6292,ebc6, -3,-.E38E38EBC66292)
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vc(a5, 9.0909091380563043783E-2 ,2e8b,3eba,d70c,b31b, -3, .BA2E8BB31BD70C)
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vc(a6, -7.6922954286089459397E-2 ,89c8,be9d,7f18,27c3, -3,-.9D89C827C37F18)
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vc(a7, 6.6663180891693915586E-2 ,86b4,3e88,9e58,ae37, -3, .8886B4AE379E58)
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vc(a8, -5.8772703698290408927E-2 ,bba5,be70,a942,8481, -4,-.F0BBA58481A942)
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vc(a9, 5.2170707402812969804E-2 ,b0f3,3e55,13ab,a1ab, -4, .D5B0F3A1AB13AB)
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vc(a10, -4.4895863157820361210E-2 ,e4b9,be37,048f,7fd1, -4,-.B7E4B97FD1048F)
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vc(a11, 3.3006147437343875094E-2 ,3174,3e07,2d87,3cf7, -4, .8731743CF72D87)
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vc(a12, -1.4614844866464185439E-2 ,731a,bd6f,76d9,2f34, -6,-.EF731A2F3476D9)
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ic(athfhi, 4.6364760900080609352E-1 , -2, 1.DAC670561BB4F)
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ic(athflo, 4.6249969567426939759E-18 , -58, 1.5543B8F253271)
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ic(PIo4, 7.8539816339744827900E-1 , -1, 1.921FB54442D18)
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ic(at1fhi, 9.8279372324732905408E-1 , -1, 1.F730BD281F69B)
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ic(at1flo,-2.4407677060164810007E-17 , -56, -1.C23DFEFEAE6B5)
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ic(PIo2, 1.5707963267948965580E0 , 0, 1.921FB54442D18)
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ic(PI, 3.1415926535897931160E0 , 1, 1.921FB54442D18)
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ic(a1, 3.3333333333333942106E-1 , -2, 1.55555555555C3)
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ic(a2, -1.9999999999979536924E-1 , -3, -1.9999999997CCD)
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ic(a3, 1.4285714278004377209E-1 , -3, 1.24924921EC1D7)
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ic(a4, -1.1111110579344973814E-1 , -4, -1.C71C7059AF280)
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ic(a5, 9.0908906105474668324E-2 , -4, 1.745CE5AA35DB2)
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ic(a6, -7.6919217767468239799E-2 , -4, -1.3B0FA54BEC400)
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ic(a7, 6.6614695906082474486E-2 , -4, 1.10DA924597FFF)
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ic(a8, -5.8358371008508623523E-2 , -5, -1.DE125FDDBD793)
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ic(a9, 4.9850617156082015213E-2 , -5, 1.9860524BDD807)
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ic(a10, -3.6700606902093604877E-2 , -5, -1.2CA6C04C6937A)
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ic(a11, 1.6438029044759730479E-2 , -6, 1.0D52174A1BB54)
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#ifdef vccast
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#define athfhi vccast(athfhi)
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#define athflo vccast(athflo)
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#define PIo4 vccast(PIo4)
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#define at1fhi vccast(at1fhi)
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#define at1flo vccast(at1flo)
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#define PIo2 vccast(PIo2)
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#define PI vccast(PI)
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#define a1 vccast(a1)
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#define a2 vccast(a2)
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#define a3 vccast(a3)
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#define a4 vccast(a4)
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#define a5 vccast(a5)
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#define a6 vccast(a6)
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#define a7 vccast(a7)
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#define a8 vccast(a8)
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#define a9 vccast(a9)
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#define a10 vccast(a10)
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#define a11 vccast(a11)
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#define a12 vccast(a12)
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#endif
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double atan2(y,x)
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double y,x;
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{
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static const double zero=0, one=1, small=1.0E-9, big=1.0E18;
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double t,z,signy,signx,hi,lo;
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int k,m;
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#if !defined(vax)&&!defined(tahoe)
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/* if x or y is NAN */
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if(x!=x) return(x); if(y!=y) return(y);
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#endif /* !defined(vax)&&!defined(tahoe) */
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/* copy down the sign of y and x */
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signy = copysign(one,y) ;
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signx = copysign(one,x) ;
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/* if x is 1.0, goto begin */
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if(x==1) { y=copysign(y,one); t=y; if(finite(t)) goto begin;}
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/* when y = 0 */
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if(y==zero) return((signx==one)?y:copysign(PI,signy));
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/* when x = 0 */
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if(x==zero) return(copysign(PIo2,signy));
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/* when x is INF */
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if(!finite(x))
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if(!finite(y))
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return(copysign((signx==one)?PIo4:3*PIo4,signy));
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else
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return(copysign((signx==one)?zero:PI,signy));
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/* when y is INF */
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if(!finite(y)) return(copysign(PIo2,signy));
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/* compute y/x */
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x=copysign(x,one);
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y=copysign(y,one);
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if((m=(k=logb(y))-logb(x)) > 60) t=big+big;
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else if(m < -80 ) t=y/x;
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else { t = y/x ; y = scalb(y,-k); x=scalb(x,-k); }
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/* begin argument reduction */
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begin:
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if (t < 2.4375) {
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/* truncate 4(t+1/16) to integer for branching */
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k = 4 * (t+0.0625);
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switch (k) {
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/* t is in [0,7/16] */
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case 0:
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case 1:
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if (t < small)
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{ big + small ; /* raise inexact flag */
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return (copysign((signx>zero)?t:PI-t,signy)); }
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hi = zero; lo = zero; break;
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/* t is in [7/16,11/16] */
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case 2:
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hi = athfhi; lo = athflo;
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z = x+x;
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t = ( (y+y) - x ) / ( z + y ); break;
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/* t is in [11/16,19/16] */
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case 3:
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case 4:
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hi = PIo4; lo = zero;
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t = ( y - x ) / ( x + y ); break;
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/* t is in [19/16,39/16] */
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default:
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hi = at1fhi; lo = at1flo;
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z = y-x; y=y+y+y; t = x+x;
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t = ( (z+z)-x ) / ( t + y ); break;
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}
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}
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/* end of if (t < 2.4375) */
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else
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{
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hi = PIo2; lo = zero;
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/* t is in [2.4375, big] */
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if (t <= big) t = - x / y;
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/* t is in [big, INF] */
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else
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{ big+small; /* raise inexact flag */
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t = zero; }
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}
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/* end of argument reduction */
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/* compute atan(t) for t in [-.4375, .4375] */
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z = t*t;
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#if defined(vax)||defined(tahoe)
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z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+
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z*(a9+z*(a10+z*(a11+z*a12))))))))))));
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#else /* defined(vax)||defined(tahoe) */
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z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+
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z*(a9+z*(a10+z*a11)))))))))));
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#endif /* defined(vax)||defined(tahoe) */
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z = lo - z; z += t; z += hi;
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return(copysign((signx>zero)?z:PI-z,signy));
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}
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