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