4.3BSD/usr/src/usr.lib/libm/IEEE/trig.c
/*
* Copyright (c) 1985 Regents of the University of California.
*
* Use and reproduction of this software are granted in accordance with
* the terms and conditions specified in the Berkeley Software License
* Agreement (in particular, this entails acknowledgement of the programs'
* source, and inclusion of this notice) with the additional understanding
* that all recipients should regard themselves as participants in an
* ongoing research project and hence should feel obligated to report
* their experiences (good or bad) with these elementary function codes,
* using "sendbug 4bsd-bugs@BERKELEY", to the authors.
*/
#ifndef lint
static char sccsid[] = "@(#)trig.c 1.2 (Berkeley) 8/22/85";
#endif not lint
/* SIN(X), COS(X), TAN(X)
* RETURN THE SINE, COSINE, AND TANGENT OF X RESPECTIVELY
* DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
* CODED IN C BY K.C. NG, 1/8/85;
* REVISED BY W. Kahan and K.C. NG, 8/17/85.
*
* Required system supported functions:
* copysign(x,y)
* finite(x)
* drem(x,p)
*
* Static kernel functions:
* sin__S(z) ....sin__S(x*x) return (sin(x)-x)/x
* cos__C(z) ....cos__C(x*x) return cos(x)-1-x*x/2
*
* Method.
* Let S and C denote the polynomial approximations to sin and cos
* respectively on [-PI/4, +PI/4].
*
* SIN and COS:
* 1. Reduce the argument into [-PI , +PI] by the remainder function.
* 2. For x in (-PI,+PI), there are three cases:
* case 1: |x| < PI/4
* case 2: PI/4 <= |x| < 3PI/4
* case 3: 3PI/4 <= |x|.
* SIN and COS of x are computed by:
*
* sin(x) cos(x) remark
* ----------------------------------------------------------
* case 1 S(x) C(x)
* case 2 sign(x)*C(y) S(y) y=PI/2-|x|
* case 3 S(y) -C(y) y=sign(x)*(PI-|x|)
* ----------------------------------------------------------
*
* TAN:
* 1. Reduce the argument into [-PI/2 , +PI/2] by the remainder function.
* 2. For x in (-PI/2,+PI/2), there are two cases:
* case 1: |x| < PI/4
* case 2: PI/4 <= |x| < PI/2
* TAN of x is computed by:
*
* tan (x) remark
* ----------------------------------------------------------
* case 1 S(x)/C(x)
* case 2 C(y)/S(y) y=sign(x)*(PI/2-|x|)
* ----------------------------------------------------------
*
* Notes:
* 1. S(y) and C(y) were computed by:
* S(y) = y+y*sin__S(y*y)
* C(y) = 1-(y*y/2-cos__C(x*x)) ... if y*y/2 < thresh,
* = 0.5-((y*y/2-0.5)-cos__C(x*x)) ... if y*y/2 >= thresh.
* where
* thresh = 0.5*(acos(3/4)**2)
*
* 2. For better accuracy, we use the following formula for S/C for tan
* (k=0): let ss=sin__S(y*y), and cc=cos__C(y*y), then
*
* y+y*ss (y*y/2-cc)+ss
* S(y)/C(y) = -------- = y + y * ---------------.
* C C
*
*
* Special cases:
* Let trig be any of sin, cos, or tan.
* trig(+-INF) is NaN, with signals;
* trig(NaN) is that NaN;
* trig(n*PI/2) is exact for any integer n, provided n*PI is
* representable; otherwise, trig(x) is inexact.
*
* Accuracy:
* trig(x) returns the exact trig(x*pi/PI) nearly rounded, where
*
* Decimal:
* pi = 3.141592653589793 23846264338327 .....
* 53 bits PI = 3.141592653589793 115997963 ..... ,
* 56 bits PI = 3.141592653589793 227020265 ..... ,
*
* Hexadecimal:
* pi = 3.243F6A8885A308D313198A2E....
* 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps
* 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps
*
* In a test run with 1,024,000 random arguments on a VAX, the maximum
* observed errors (compared with the exact trig(x*pi/PI)) were
* tan(x) : 2.09 ulps (around 4.716340404662354)
* sin(x) : .861 ulps
* cos(x) : .857 ulps
*
* Constants:
* The hexadecimal values are the intended ones for the following constants.
* The decimal values may be used, provided that the compiler will convert
* from decimal to binary accurately enough to produce the hexadecimal values
* shown.
*/
#ifdef VAX
/*thresh = 2.6117239648121182150E-1 , Hex 2^ -1 * .85B8636B026EA0 */
/*PIo4 = 7.8539816339744830676E-1 , Hex 2^ 0 * .C90FDAA22168C2 */
/*PIo2 = 1.5707963267948966135E0 , Hex 2^ 1 * .C90FDAA22168C2 */
/*PI3o4 = 2.3561944901923449203E0 , Hex 2^ 2 * .96CBE3F9990E92 */
/*PI = 3.1415926535897932270E0 , Hex 2^ 2 * .C90FDAA22168C2 */
/*PI2 = 6.2831853071795864540E0 ; Hex 2^ 3 * .C90FDAA22168C2 */
static long threshx[] = { 0xb8633f85, 0x6ea06b02};
#define thresh (*(double*)threshx)
static long PIo4x[] = { 0x0fda4049, 0x68c2a221};
#define PIo4 (*(double*)PIo4x)
static long PIo2x[] = { 0x0fda40c9, 0x68c2a221};
#define PIo2 (*(double*)PIo2x)
static long PI3o4x[] = { 0xcbe34116, 0x0e92f999};
#define PI3o4 (*(double*)PI3o4x)
static long PIx[] = { 0x0fda4149, 0x68c2a221};
#define PI (*(double*)PIx)
static long PI2x[] = { 0x0fda41c9, 0x68c2a221};
#define PI2 (*(double*)PI2x)
#else /* IEEE double */
static double
thresh = 2.6117239648121182150E-1 , /*Hex 2^ -2 * 1.0B70C6D604DD4 */
PIo4 = 7.8539816339744827900E-1 , /*Hex 2^ -1 * 1.921FB54442D18 */
PIo2 = 1.5707963267948965580E0 , /*Hex 2^ 0 * 1.921FB54442D18 */
PI3o4 = 2.3561944901923448370E0 , /*Hex 2^ 1 * 1.2D97C7F3321D2 */
PI = 3.1415926535897931160E0 , /*Hex 2^ 1 * 1.921FB54442D18 */
PI2 = 6.2831853071795862320E0 ; /*Hex 2^ 2 * 1.921FB54442D18 */
#endif
static double zero=0, one=1, negone= -1, half=1.0/2.0,
small=1E-10, /* 1+small**2==1; better values for small:
small = 1.5E-9 for VAX D
= 1.2E-8 for IEEE Double
= 2.8E-10 for IEEE Extended */
big=1E20; /* big = 1/(small**2) */
double tan(x)
double x;
{
double copysign(),drem(),cos__C(),sin__S(),a,z,ss,cc,c;
int finite(),k;
/* tan(NaN) and tan(INF) must be NaN */
if(!finite(x)) return(x-x);
x=drem(x,PI); /* reduce x into [-PI/2, PI/2] */
a=copysign(x,one); /* ... = abs(x) */
if ( a >= PIo4 ) {k=1; x = copysign( PIo2 - a , x ); }
else { k=0; if(a < small ) { big + a; return(x); }}
z = x*x;
cc = cos__C(z);
ss = sin__S(z);
z = z*half ; /* Next get c = cos(x) accurately */
c = (z >= thresh )? half-((z-half)-cc) : one-(z-cc);
if (k==0) return ( x + (x*(z-(cc-ss)))/c ); /* sin/cos */
return( c/(x+x*ss) ); /* ... cos/sin */
}
double sin(x)
double x;
{
double copysign(),drem(),sin__S(),cos__C(),a,c,z;
int finite();
/* sin(NaN) and sin(INF) must be NaN */
if(!finite(x)) return(x-x);
x=drem(x,PI2); /* reduce x into [-PI, PI] */
a=copysign(x,one);
if( a >= PIo4 ) {
if( a >= PI3o4 ) /* .. in [3PI/4, PI ] */
x=copysign((a=PI-a),x);
else { /* .. in [PI/4, 3PI/4] */
a=PIo2-a; /* return sign(x)*C(PI/2-|x|) */
z=a*a;
c=cos__C(z);
z=z*half;
a=(z>=thresh)?half-((z-half)-c):one-(z-c);
return(copysign(a,x));
}
}
/* return S(x) */
if( a < small) { big + a; return(x);}
return(x+x*sin__S(x*x));
}
double cos(x)
double x;
{
double copysign(),drem(),sin__S(),cos__C(),a,c,z,s=1.0;
int finite();
/* cos(NaN) and cos(INF) must be NaN */
if(!finite(x)) return(x-x);
x=drem(x,PI2); /* reduce x into [-PI, PI] */
a=copysign(x,one);
if ( a >= PIo4 ) {
if ( a >= PI3o4 ) /* .. in [3PI/4, PI ] */
{ a=PI-a; s= negone; }
else /* .. in [PI/4, 3PI/4] */
/* return S(PI/2-|x|) */
{ a=PIo2-a; return(a+a*sin__S(a*a));}
}
/* return s*C(a) */
if( a < small) { big + a; return(s);}
z=a*a;
c=cos__C(z);
z=z*half;
a=(z>=thresh)?half-((z-half)-c):one-(z-c);
return(copysign(a,s));
}
/* sin__S(x*x)
* DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
* STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X)
* CODED IN C BY K.C. NG, 1/21/85;
* REVISED BY K.C. NG on 8/13/85.
*
* sin(x*k) - x
* RETURN --------------- on [-PI/4,PI/4] , where k=pi/PI, PI is the rounded
* x
* value of pi in machine precision:
*
* Decimal:
* pi = 3.141592653589793 23846264338327 .....
* 53 bits PI = 3.141592653589793 115997963 ..... ,
* 56 bits PI = 3.141592653589793 227020265 ..... ,
*
* Hexadecimal:
* pi = 3.243F6A8885A308D313198A2E....
* 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18
* 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2
*
* Method:
* 1. Let z=x*x. Create a polynomial approximation to
* (sin(k*x)-x)/x = z*(S0 + S1*z^1 + ... + S5*z^5).
* Then
* sin__S(x*x) = z*(S0 + S1*z^1 + ... + S5*z^5)
*
* The coefficient S's are obtained by a special Remez algorithm.
*
* Accuracy:
* In the absence of rounding error, the approximation has absolute error
* less than 2**(-61.11) for VAX D FORMAT, 2**(-57.45) for IEEE DOUBLE.
*
* Constants:
* The hexadecimal values are the intended ones for the following constants.
* The decimal values may be used, provided that the compiler will convert
* from decimal to binary accurately enough to produce the hexadecimal values
* shown.
*
*/
#ifdef VAX
/*S0 = -1.6666666666666646660E-1 , Hex 2^ -2 * -.AAAAAAAAAAAA71 */
/*S1 = 8.3333333333297230413E-3 , Hex 2^ -6 * .8888888888477F */
/*S2 = -1.9841269838362403710E-4 , Hex 2^-12 * -.D00D00CF8A1057 */
/*S3 = 2.7557318019967078930E-6 , Hex 2^-18 * .B8EF1CA326BEDC */
/*S4 = -2.5051841873876551398E-8 , Hex 2^-25 * -.D73195374CE1D3 */
/*S5 = 1.6028995389845827653E-10 , Hex 2^-32 * .B03D9C6D26CCCC */
/*S6 = -6.2723499671769283121E-13 ; Hex 2^-40 * -.B08D0B7561EA82 */
static long S0x[] = { 0xaaaabf2a, 0xaa71aaaa};
#define S0 (*(double*)S0x)
static long S1x[] = { 0x88883d08, 0x477f8888};
#define S1 (*(double*)S1x)
static long S2x[] = { 0x0d00ba50, 0x1057cf8a};
#define S2 (*(double*)S2x)
static long S3x[] = { 0xef1c3738, 0xbedca326};
#define S3 (*(double*)S3x)
static long S4x[] = { 0x3195b3d7, 0xe1d3374c};
#define S4 (*(double*)S4x)
static long S5x[] = { 0x3d9c3030, 0xcccc6d26};
#define S5 (*(double*)S5x)
static long S6x[] = { 0x8d0bac30, 0xea827561};
#define S6 (*(double*)S6x)
#else /* IEEE double */
static double
S0 = -1.6666666666666463126E-1 , /*Hex 2^ -3 * -1.555555555550C */
S1 = 8.3333333332992771264E-3 , /*Hex 2^ -7 * 1.111111110C461 */
S2 = -1.9841269816180999116E-4 , /*Hex 2^-13 * -1.A01A019746345 */
S3 = 2.7557309793219876880E-6 , /*Hex 2^-19 * 1.71DE3209CDCD9 */
S4 = -2.5050225177523807003E-8 , /*Hex 2^-26 * -1.AE5C0E319A4EF */
S5 = 1.5868926979889205164E-10 ; /*Hex 2^-33 * 1.5CF61DF672B13 */
#endif
static double sin__S(z)
double z;
{
#ifdef VAX
return(z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*(S5+z*S6)))))));
#else /* IEEE double */
return(z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*S5))))));
#endif
}
/* cos__C(x*x)
* DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS)
* STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X)
* CODED IN C BY K.C. NG, 1/21/85;
* REVISED BY K.C. NG on 8/13/85.
*
* x*x
* RETURN cos(k*x) - 1 + ----- on [-PI/4,PI/4], where k = pi/PI,
* 2
* PI is the rounded value of pi in machine precision :
*
* Decimal:
* pi = 3.141592653589793 23846264338327 .....
* 53 bits PI = 3.141592653589793 115997963 ..... ,
* 56 bits PI = 3.141592653589793 227020265 ..... ,
*
* Hexadecimal:
* pi = 3.243F6A8885A308D313198A2E....
* 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18
* 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2
*
*
* Method:
* 1. Let z=x*x. Create a polynomial approximation to
* cos(k*x)-1+z/2 = z*z*(C0 + C1*z^1 + ... + C5*z^5)
* then
* cos__C(z) = z*z*(C0 + C1*z^1 + ... + C5*z^5)
*
* The coefficient C's are obtained by a special Remez algorithm.
*
* Accuracy:
* In the absence of rounding error, the approximation has absolute error
* less than 2**(-64) for VAX D FORMAT, 2**(-58.3) for IEEE DOUBLE.
*
*
* Constants:
* The hexadecimal values are the intended ones for the following constants.
* The decimal values may be used, provided that the compiler will convert
* from decimal to binary accurately enough to produce the hexadecimal values
* shown.
*
*/
#ifdef VAX
/*C0 = 4.1666666666666504759E-2 , Hex 2^ -4 * .AAAAAAAAAAA9F0 */
/*C1 = -1.3888888888865302059E-3 , Hex 2^ -9 * -.B60B60B60A0CCA */
/*C2 = 2.4801587285601038265E-5 , Hex 2^-15 * .D00D00CDCD098F */
/*C3 = -2.7557313470902390219E-7 , Hex 2^-21 * -.93F27BB593E805 */
/*C4 = 2.0875623401082232009E-9 , Hex 2^-28 * .8F74C8FA1E3FF0 */
/*C5 = -1.1355178117642986178E-11 ; Hex 2^-36 * -.C7C32D0A5C5A63 */
static long C0x[] = { 0xaaaa3e2a, 0xa9f0aaaa};
#define C0 (*(double*)C0x)
static long C1x[] = { 0x0b60bbb6, 0x0ccab60a};
#define C1 (*(double*)C1x)
static long C2x[] = { 0x0d0038d0, 0x098fcdcd};
#define C2 (*(double*)C2x)
static long C3x[] = { 0xf27bb593, 0xe805b593};
#define C3 (*(double*)C3x)
static long C4x[] = { 0x74c8320f, 0x3ff0fa1e};
#define C4 (*(double*)C4x)
static long C5x[] = { 0xc32dae47, 0x5a630a5c};
#define C5 (*(double*)C5x)
#else /* IEEE double */
static double
C0 = 4.1666666666666504759E-2 , /*Hex 2^ -5 * 1.555555555553E */
C1 = -1.3888888888865301516E-3 , /*Hex 2^-10 * -1.6C16C16C14199 */
C2 = 2.4801587269650015769E-5 , /*Hex 2^-16 * 1.A01A01971CAEB */
C3 = -2.7557304623183959811E-7 , /*Hex 2^-22 * -1.27E4F1314AD1A */
C4 = 2.0873958177697780076E-9 , /*Hex 2^-29 * 1.1EE3B60DDDC8C */
C5 = -1.1250289076471311557E-11 ; /*Hex 2^-37 * -1.8BD5986B2A52E */
#endif
static double cos__C(z)
double z;
{
return(z*z*(C0+z*(C1+z*(C2+z*(C3+z*(C4+z*C5))))));
}