359 lines
8.8 KiB
C
359 lines
8.8 KiB
C
/* sincos.c
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*
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* Circular sine and cosine of argument in degrees
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* Table lookup and interpolation algorithm
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*
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*
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*
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* SYNOPSIS:
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*
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* double x, sine, cosine, flg, sincos();
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*
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* sincos( x, &sine, &cosine, flg );
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*
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*
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*
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* DESCRIPTION:
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*
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* Returns both the sine and the cosine of the argument x.
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* Several different compile time options and minimax
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* approximations are supplied to permit tailoring the
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* tradeoff between computation speed and accuracy.
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*
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* Since range reduction is time consuming, the reduction
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* of x modulo 360 degrees is also made optional.
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*
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* sin(i) is internally tabulated for 0 <= i <= 90 degrees.
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* Approximation polynomials, ranging from linear interpolation
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* to cubics in (x-i)**2, compute the sine and cosine
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* of the residual x-i which is between -0.5 and +0.5 degree.
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* In the case of the high accuracy options, the residual
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* and the tabulated values are combined using the trigonometry
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* formulas for sin(A+B) and cos(A+B).
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*
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* Compile time options are supplied for 5, 11, or 17 decimal
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* relative accuracy (ACC5, ACC11, ACC17 respectively).
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* A subroutine flag argument "flg" chooses betwen this
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* accuracy and table lookup only (peak absolute error
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* = 0.0087).
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*
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* If the argument flg = 1, then the tabulated value is
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* returned for the nearest whole number of degrees. The
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* approximation polynomials are not computed. At
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* x = 0.5 deg, the absolute error is then sin(0.5) = 0.0087.
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*
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* An intermediate speed and precision can be obtained using
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* the compile time option LINTERP and flg = 1. This yields
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* a linear interpolation using a slope estimated from the sine
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* or cosine at the nearest integer argument. The peak absolute
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* error with this option is 3.8e-5. Relative error at small
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* angles is about 1e-5.
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*
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* If flg = 0, then the approximation polynomials are computed
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* and applied.
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*
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*
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*
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* SPEED:
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*
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* Relative speed comparisons follow for 6MHz IBM AT clone
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* and Microsoft C version 4.0. These figures include
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* software overhead of do loop and function calls.
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* Since system hardware and software vary widely, the
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* numbers should be taken as representative only.
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*
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* flg=0 flg=0 flg=1 flg=1
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* ACC11 ACC5 LINTERP Lookup only
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* In-line 8087 (/FPi)
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* sin(), cos() 1.0 1.0 1.0 1.0
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*
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* In-line 8087 (/FPi)
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* sincos() 1.1 1.4 1.9 3.0
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*
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* Software (/FPa)
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* sin(), cos() 0.19 0.19 0.19 0.19
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*
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* Software (/FPa)
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* sincos() 0.39 0.50 0.73 1.7
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*
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*
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*
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* ACCURACY:
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*
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* The accurate approximations are designed with a relative error
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* criterion. The absolute error is greatest at x = 0.5 degree.
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* It decreases from a local maximum at i+0.5 degrees to full
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* machine precision at each integer i degrees. With the
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* ACC5 option, the relative error of 6.3e-6 is equivalent to
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* an absolute angular error of 0.01 arc second in the argument
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* at x = i+0.5 degrees. For small angles < 0.5 deg, the ACC5
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* accuracy is 6.3e-6 (.00063%) of reading; i.e., the absolute
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* error decreases in proportion to the argument. This is true
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* for both the sine and cosine approximations, since the latter
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* is for the function 1 - cos(x).
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*
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* If absolute error is of most concern, use the compile time
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* option ABSERR to obtain an absolute error of 2.7e-8 for ACC5
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* precision. This is about half the absolute error of the
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* relative precision option. In this case the relative error
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* for small angles will increase to 9.5e-6 -- a reasonable
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* tradeoff.
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*/
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#include "mconf.h"
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/* Define one of the following to be 1:
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*/
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#define ACC5 1
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#define ACC11 0
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#define ACC17 0
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/* Option for linear interpolation when flg = 1
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*/
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#define LINTERP 1
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/* Option for absolute error criterion
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*/
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#define ABSERR 1
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/* Option to include modulo 360 function:
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*/
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#define MOD360 1
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/*
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Cephes Math Library Release 2.1
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Copyright 1987 by Stephen L. Moshier
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Direct inquiries to 30 Frost Street, Cambridge, MA 02140
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*/
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/* Table of sin(i degrees)
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* for 0 <= i <= 90
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*/
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const static double sintbl[92] = {
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0.00000000000000000000E0,
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1.74524064372835128194E-2,
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3.48994967025009716460E-2,
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5.23359562429438327221E-2,
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6.97564737441253007760E-2,
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8.71557427476581735581E-2,
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1.04528463267653471400E-1,
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1.21869343405147481113E-1,
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1.39173100960065444112E-1,
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1.56434465040230869010E-1,
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1.73648177666930348852E-1,
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1.90808995376544812405E-1,
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2.07911690817759337102E-1,
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2.24951054343864998051E-1,
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2.41921895599667722560E-1,
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2.58819045102520762349E-1,
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2.75637355816999185650E-1,
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2.92371704722736728097E-1,
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3.09016994374947424102E-1,
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3.25568154457156668714E-1,
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3.42020143325668733044E-1,
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3.58367949545300273484E-1,
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3.74606593415912035415E-1,
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3.90731128489273755062E-1,
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4.06736643075800207754E-1,
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4.22618261740699436187E-1,
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4.38371146789077417453E-1,
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4.53990499739546791560E-1,
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4.69471562785890775959E-1,
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4.84809620246337029075E-1,
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5.00000000000000000000E-1,
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5.15038074910054210082E-1,
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5.29919264233204954047E-1,
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5.44639035015027082224E-1,
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5.59192903470746830160E-1,
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5.73576436351046096108E-1,
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5.87785252292473129169E-1,
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6.01815023152048279918E-1,
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6.15661475325658279669E-1,
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6.29320391049837452706E-1,
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6.42787609686539326323E-1,
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6.56059028990507284782E-1,
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6.69130606358858213826E-1,
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6.81998360062498500442E-1,
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6.94658370458997286656E-1,
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7.07106781186547524401E-1,
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7.19339800338651139356E-1,
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7.31353701619170483288E-1,
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7.43144825477394235015E-1,
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7.54709580222771997943E-1,
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7.66044443118978035202E-1,
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7.77145961456970879980E-1,
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7.88010753606721956694E-1,
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7.98635510047292846284E-1,
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8.09016994374947424102E-1,
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8.19152044288991789684E-1,
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8.29037572555041692006E-1,
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8.38670567945424029638E-1,
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8.48048096156425970386E-1,
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8.57167300702112287465E-1,
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8.66025403784438646764E-1,
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8.74619707139395800285E-1,
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8.82947592858926942032E-1,
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8.91006524188367862360E-1,
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8.98794046299166992782E-1,
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9.06307787036649963243E-1,
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9.13545457642600895502E-1,
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9.20504853452440327397E-1,
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9.27183854566787400806E-1,
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9.33580426497201748990E-1,
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9.39692620785908384054E-1,
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9.45518575599316810348E-1,
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9.51056516295153572116E-1,
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9.56304755963035481339E-1,
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9.61261695938318861916E-1,
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9.65925826289068286750E-1,
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9.70295726275996472306E-1,
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9.74370064785235228540E-1,
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9.78147600733805637929E-1,
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9.81627183447663953497E-1,
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9.84807753012208059367E-1,
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9.87688340595137726190E-1,
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9.90268068741570315084E-1,
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9.92546151641322034980E-1,
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9.94521895368273336923E-1,
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9.96194698091745532295E-1,
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9.97564050259824247613E-1,
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9.98629534754573873784E-1,
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9.99390827019095730006E-1,
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9.99847695156391239157E-1,
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1.00000000000000000000E0,
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9.99847695156391239157E-1,
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};
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#ifdef ANSIPROT
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double floor ( double );
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#else
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double floor();
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#endif
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int sincos(x, s, c, flg)
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double x;
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double *s, *c;
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int flg;
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{
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int ix, ssign, csign, xsign;
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double y, z, sx, sz, cx, cz;
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/* Make argument nonnegative.
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*/
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xsign = 1;
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if( x < 0.0 )
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{
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xsign = -1;
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x = -x;
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}
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#if MOD360
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x = x - 360.0 * floor( x/360.0 );
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#endif
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/* Find nearest integer to x.
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* Note there should be a domain error test here,
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* but this is omitted to gain speed.
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*/
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ix = x + 0.5;
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z = x - ix; /* the residual */
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/* Look up the sine and cosine of the integer.
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*/
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if( ix <= 180 )
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{
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ssign = 1;
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csign = 1;
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}
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else
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{
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ssign = -1;
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csign = -1;
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ix -= 180;
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}
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if( ix > 90 )
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{
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csign = -csign;
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ix = 180 - ix;
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}
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sx = sintbl[ix];
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if( ssign < 0 )
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sx = -sx;
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cx = sintbl[ 90-ix ];
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if( csign < 0 )
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cx = -cx;
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/* If the flag argument is set, then just return
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* the tabulated values for arg to the nearest whole degree.
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*/
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if( flg )
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{
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#if LINTERP
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y = sx + 1.74531263774940077459e-2 * z * cx;
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cx -= 1.74531263774940077459e-2 * z * sx;
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sx = y;
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#endif
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if( xsign < 0 )
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sx = -sx;
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*s = sx; /* sine */
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*c = cx; /* cosine */
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return 0;
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}
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/* Find sine and cosine
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* of the residual angle between -0.5 and +0.5 degree.
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*/
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#if ACC5
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#if ABSERR
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/* absolute error = 2.769e-8: */
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sz = 1.74531263774940077459e-2 * z;
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/* absolute error = 4.146e-11: */
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cz = 1.0 - 1.52307909153324666207e-4 * z * z;
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#else
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/* relative error = 6.346e-6: */
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sz = 1.74531817576426662296e-2 * z;
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/* relative error = 3.173e-6: */
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cz = 1.0 - 1.52308226602566149927e-4 * z * z;
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#endif
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#else
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y = z * z;
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#endif
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#if ACC11
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sz = ( -8.86092781698004819918e-7 * y
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+ 1.74532925198378577601e-2 ) * z;
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cz = 1.0 - ( -3.86631403698859047896e-9 * y
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+ 1.52308709893047593702e-4 ) * y;
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#endif
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#if ACC17
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sz = (( 1.34959795251974073996e-11 * y
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- 8.86096155697856783296e-7 ) * y
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+ 1.74532925199432957214e-2 ) * z;
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cz = 1.0 - (( 3.92582397764340914444e-14 * y
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- 3.86632385155548605680e-9 ) * y
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+ 1.52308709893354299569e-4 ) * y;
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#endif
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/* Combine the tabulated part and the calculated part
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* by trigonometry.
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*/
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y = sx * cz + cx * sz;
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if( xsign < 0 )
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y = - y;
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*s = y; /* sine */
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*c = cx * cz - sx * sz; /* cosine */
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return 0;
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}
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