/* NFS 20-Nov-85 14:34:56 from trig.c */
/* sin(), cos(), tan(), atan(), atan2(), asin(), acos(), sinh(), cosh() */
/* tanh(), asinh(), acosh(), atanh() */
asm (" export Libm");
mesa double LibmSupport←copysign(), LibmSupport←scalb(), LibmSupport←logb();
mesa double LibmSupport←drem(), Libm←sqrt(), Libm←expm1(), LibmSupport←expE();
mesa double Libm←exp(), Libm←log1p();
mesa int LibmSupport←finite();
#define copysign(x,y) (LibmSupport←copysign(x,y))
#define scalb(x,n) (LibmSupport←scalb(x,n))
#define logb(x) (LibmSupport←logb(x))
#define finite(x) (LibmSupport←finite(x))
#define drem(x,p) (LibmSupport←drem(x,p))
#define sqrt(x) (Libm←sqrt(x))
#define expm1(x) (Libm←expm1(x))
#define exp←←E(x,c) (LibmSupport←expE(x,c))
#define exp(x) (Libm←exp(x))
#define log1p(x) (Libm←log1p(x))
/* 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.
*/
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 */
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) */
asm (" exportproc ←tan, Libm");
double tan(x)
double x;
{
double LibmSupport←copysign(),
DoubleReal←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 */
}
asm (" exportproc ←sin, Libm");
double sin(x)
double x;
{
double LibmSupport←copysign(),
DoubleReal←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));
}
asm (" exportproc ←cos, Libm");
double cos(x)
double x;
{
double LibmSupport←copysign(),
DoubleReal←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.
*
*/
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 */
static double sin←←S(z)
double z;
{
return(z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*S5))))));
}
/* 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.
*
*/
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 */
static double cos←←C(z)
double z;
{
return(z*z*(C0+z*(C1+z*(C2+z*(C3+z*(C4+z*C5))))));
}
/* ATAN(X)
* RETURNS ARC TANGENT OF X
* DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits)
* CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85.
*
* Required kernel function:
* atan2(y,x)
*
* Method:
* atan(x) = atan2(x,1.0).
*
* Special case:
* if x is NaN, return x itself.
*
* Accuracy:
* 1) If atan2() uses machine PI, then
*
* atan(x) returns (PI/pi) * (the exact arc tangent of x) nearly rounded;
* and PI is the exact pi rounded to machine precision (see atan2 for
* details):
*
* in decimal:
* pi = 3.141592653589793 23846264338327 .....
* 53 bits PI = 3.141592653589793 115997963 ..... ,
* 56 bits PI = 3.141592653589793 227020265 ..... ,
*
* in 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 more than 200,000 random arguments on a VAX, the
* maximum observed error in ulps (units in the last place) was
* 0.86 ulps. (comparing against (PI/pi)*(exact atan(x))).
*
* 2) If atan2() uses true pi, then
*
* atan(x) returns the exact atan(x) with error below about 2 ulps.
*
* In a test run with more than 1,024,000 random arguments on a VAX, the
* maximum observed error in ulps (units in the last place) was
* 0.85 ulps.
*/
asm (" exportproc ←atan, Libm");
double atan(x)
double x;
{
double atan2(),one=1.0;
return(atan2(x,one));
}
/* ATAN2(Y,X)
* RETURN ARG (X+iY)
* DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
* CODED IN C BY K.C. NG, 1/8/85;
* REVISED BY K.C. NG on 2/7/85, 2/13/85, 3/7/85, 3/30/85, 6/29/85.
*
* Required system supported functions :
* copysign(x,y)
* scalb(x,y)
* logb(x)
*
* Method :
* 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x).
* 2. Reduce x to positive by (if x and y are unexceptional):
* ARG (x+iy) = arctan(y/x) ... if x > 0,
* ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0,
* 3. According to the integer k=4t+0.25 truncated , t=y/x, the argument
* is further reduced to one of the following intervals and the
* arctangent of y/x is evaluated by the corresponding formula:
*
* [0,7/16] atan(y/x) = t - t↑3*(a1+t↑2*(a2+...(a10+t↑2*a11)...)
* [7/16,11/16] atan(y/x) = atan(1/2) + atan( (y-x/2)/(x+y/2) )
* [11/16.19/16] atan(y/x) = atan( 1 ) + atan( (y-x)/(x+y) )
* [19/16,39/16] atan(y/x) = atan(3/2) + atan( (y-1.5x)/(x+1.5y) )
* [39/16,INF] atan(y/x) = atan(INF) + atan( -x/y )
*
* Special cases:
* Notations: atan2(y,x) == ARG (x+iy) == ARG(x,y).
*
* ARG( NAN , (anything) ) is NaN;
* ARG( (anything), NaN ) is NaN;
* ARG(+(anything but NaN), +-0) is +-0 ;
* ARG(-(anything but NaN), +-0) is +-PI ;
* ARG( 0, +-(anything but 0 and NaN) ) is +-PI/2;
* ARG( +INF,+-(anything but INF and NaN) ) is +-0 ;
* ARG( -INF,+-(anything but INF and NaN) ) is +-PI;
* ARG( +INF,+-INF ) is +-PI/4 ;
* ARG( -INF,+-INF ) is +-3PI/4;
* ARG( (anything but,0,NaN, and INF),+-INF ) is +-PI/2;
*
* Accuracy:
* atan2(y,x) returns (PI/pi) * the exact ARG (x+iy) nearly rounded,
* where
*
* in decimal:
* pi = 3.141592653589793 23846264338327 .....
* 53 bits PI = 3.141592653589793 115997963 ..... ,
* 56 bits PI = 3.141592653589793 227020265 ..... ,
*
* in 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 356,000 random argument on [-1,1] * [-1,1] on a
* VAX, the maximum observed error was 1.41 ulps (units of the last place)
* compared with (PI/pi)*(the exact ARG(x+iy)).
*
* Note:
* We use machine PI (the true pi rounded) in place of the actual
* value of pi for all the trig and inverse trig functions. In general,
* if trig is one of sin, cos, tan, then computed trig(y) returns the
* exact trig(y*pi/PI) nearly rounded; correspondingly, computed arctrig
* returns the exact arctrig(y)*PI/pi nearly rounded. These guarantee the
* trig functions have period PI, and trig(arctrig(x)) returns x for
* all critical values x.
*
* 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.
*/
static double
athfhi = 4.6364760900080609352E-1 , /*Hex 2↑ -2 * 1.DAC670561BB4F */
athflo = 4.6249969567426939759E-18 , /*Hex 2↑-58 * 1.5543B8F253271 */
at1fhi = 9.8279372324732905408E-1 , /*Hex 2↑ -1 * 1.F730BD281F69B */
at1flo = -2.4407677060164810007E-17 , /*Hex 2↑-56 * -1.C23DFEFEAE6B5 */
a1 = 3.3333333333333942106E-1 , /*Hex 2↑ -2 * 1.55555555555C3 */
a2 = -1.9999999999979536924E-1 , /*Hex 2↑ -3 * -1.9999999997CCD */
a3 = 1.4285714278004377209E-1 , /*Hex 2↑ -3 * 1.24924921EC1D7 */
a4 = -1.1111110579344973814E-1 , /*Hex 2↑ -4 * -1.C71C7059AF280 */
a5 = 9.0908906105474668324E-2 , /*Hex 2↑ -4 * 1.745CE5AA35DB2 */
a6 = -7.6919217767468239799E-2 , /*Hex 2↑ -4 * -1.3B0FA54BEC400 */
a7 = 6.6614695906082474486E-2 , /*Hex 2↑ -4 * 1.10DA924597FFF */
a8 = -5.8358371008508623523E-2 , /*Hex 2↑ -5 * -1.DE125FDDBD793 */
a9 = 4.9850617156082015213E-2 , /*Hex 2↑ -5 * 1.9860524BDD807 */
a10 = -3.6700606902093604877E-2 , /*Hex 2↑ -5 * -1.2CA6C04C6937A */
a11 = 1.6438029044759730479E-2 ; /*Hex 2↑ -6 * 1.0D52174A1BB54 */
asm (" exportproc ←atan2, Libm");
double atan2(y,x)
double y,x;
{
static double zero=0, one=1, small=1.0E-9, big=1.0E18;
double LibmSupport←copysign(),
LibmSupport←logb(),
LibmSupport←scalb(),t,z,signy,signx,hi,lo;
int LibmSupport←finite(), k,m;
/* if x or y is NAN */
if(x!=x) return(x); if(y!=y) return(y);
/* copy down the sign of y and x */
signy = copysign(one,y) ;
signx = copysign(one,x) ;
/* if x is 1.0, goto begin */
if(x==1) { y=copysign(y,one); t=y; if(finite(t)) goto begin;}
/* when y = 0 */
if(y==zero) return((signx==one)?y:copysign(PI,signy));
/* when x = 0 */
if(x==zero) return(copysign(PIo2,signy));
/* when x is INF */
if(!finite(x))
if(!finite(y))
return(copysign((signx==one)?PIo4:3*PIo4,signy));
else
return(copysign((signx==one)?zero:PI,signy));
/* when y is INF */
if(!finite(y)) return(copysign(PIo2,signy));
/* compute y/x */
x=copysign(x,one);
y=copysign(y,one);
if((m=(k=logb(y))-logb(x)) > 60) t=big+big;
else if(m < -80 ) t=y/x;
else { t = y/x ; y = scalb(y,-k); x=scalb(x,-k); }
/* begin argument reduction */
begin:
if (t < 2.4375) {
/* truncate 4(t+1/16) to integer for branching */
k = 4 * (t+0.0625);
switch (k) {
/* t is in [0,7/16] */
case 0:
case 1:
if (t < small)
{ big + small ; /* raise inexact flag */
return (copysign((signx>zero)?t:PI-t,signy)); }
hi = zero; lo = zero; break;
/* t is in [7/16,11/16] */
case 2:
hi = athfhi; lo = athflo;
z = x+x;
t = ( (y+y) - x ) / ( z + y ); break;
/* t is in [11/16,19/16] */
case 3:
case 4:
hi = PIo4; lo = zero;
t = ( y - x ) / ( x + y ); break;
/* t is in [19/16,39/16] */
default:
hi = at1fhi; lo = at1flo;
z = y-x; y=y+y+y; t = x+x;
t = ( (z+z)-x ) / ( t + y ); break;
}
}
/* end of if (t < 2.4375) */
else
{
hi = PIo2; lo = zero;
/* t is in [2.4375, big] */
if (t <= big) t = - x / y;
/* t is in [big, INF] */
else
{ big+small; /* raise inexact flag */
t = zero; }
}
/* end of argument reduction */
/* compute atan(t) for t in [-.4375, .4375] */
z = t*t;
z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+
z*(a9+z*(a10+z*a11)))))))))));
z = lo - z; z += t; z += hi;
return(copysign((signx>zero)?z:PI-z,signy));
}
/* ASIN(X)
* RETURNS ARC SINE OF X
* DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits)
* CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85.
*
* Required system supported functions:
* copysign(x,y)
* sqrt(x)
*
* Required kernel function:
* atan2(y,x)
*
* Method :
* asin(x) = atan2(x,sqrt(1-x*x)); for better accuracy, 1-x*x is
* computed as follows
* 1-x*x if x < 0.5,
* 2*(1-|x|)-(1-|x|)*(1-|x|) if x >= 0.5.
*
* Special cases:
* if x is NaN, return x itself;
* if |x|>1, return NaN.
*
* Accuracy:
* 1) If atan2() uses machine PI, then
*
* asin(x) returns (PI/pi) * (the exact arc sine of x) nearly rounded;
* and PI is the exact pi rounded to machine precision (see atan2 for
* details):
*
* in decimal:
* pi = 3.141592653589793 23846264338327 .....
* 53 bits PI = 3.141592653589793 115997963 ..... ,
* 56 bits PI = 3.141592653589793 227020265 ..... ,
*
* in 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 more than 200,000 random arguments on a VAX, the
* maximum observed error in ulps (units in the last place) was
* 2.06 ulps. (comparing against (PI/pi)*(exact asin(x)));
*
* 2) If atan2() uses true pi, then
*
* asin(x) returns the exact asin(x) with error below about 2 ulps.
*
* In a test run with more than 1,024,000 random arguments on a VAX, the
* maximum observed error in ulps (units in the last place) was
* 1.99 ulps.
*/
asm (" exportproc ←asin, Libm");
double asin(x)
double x;
{
double s,t,
LibmSupport←copysign(),
Libm←atan2(),
sqrt(),
one=1.0;
if(x!=x) return(x); /* x is NaN */
s=copysign(x,one);
if(s <= 0.5)
return(atan2(x,sqrt(one-x*x)));
else
{ t=one-s; s=t+t; return(atan2(x,sqrt(s-t*t))); }
}
/* ACOS(X)
* RETURNS ARC COS OF X
* DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits)
* CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85.
*
* Required system supported functions:
* copysign(x,y)
* sqrt(x)
*
* Required kernel function:
* atan2(y,x)
*
* Method :
* ←←←←←←←←
* / 1 - x
* acos(x) = 2*atan2( / -------- , 1 ) .
* \/ 1 + x
*
* Special cases:
* if x is NaN, return x itself;
* if |x|>1, return NaN.
*
* Accuracy:
* 1) If atan2() uses machine PI, then
*
* acos(x) returns (PI/pi) * (the exact arc cosine of x) nearly rounded;
* and PI is the exact pi rounded to machine precision (see atan2 for
* details):
*
* in decimal:
* pi = 3.141592653589793 23846264338327 .....
* 53 bits PI = 3.141592653589793 115997963 ..... ,
* 56 bits PI = 3.141592653589793 227020265 ..... ,
*
* in 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 more than 200,000 random arguments on a VAX, the
* maximum observed error in ulps (units in the last place) was
* 2.07 ulps. (comparing against (PI/pi)*(exact acos(x)));
*
* 2) If atan2() uses true pi, then
*
* acos(x) returns the exact acos(x) with error below about 2 ulps.
*
* In a test run with more than 1,024,000 random arguments on a VAX, the
* maximum observed error in ulps (units in the last place) was
* 2.15 ulps.
*/
asm (" exportproc ←acos, Libm");
double acos(x)
double x;
{
double t,
LibmSupport←copysign(),
Libm←atan2(),
sqrt(),
one=1.0;
if(x!=x) return(x);
if( x != -1.0)
t=atan2(sqrt((one-x)/(one+x)),one);
else
t=atan2(one,0.0); /* t = PI/2 */
return(t+t);
}
/* SINH(X)
* RETURN THE HYPERBOLIC SINE OF X
* DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
* CODED IN C BY K.C. NG, 1/8/85;
* REVISED BY K.C. NG on 2/8/85, 3/7/85, 3/24/85, 4/16/85.
*
* Required system supported functions :
* copysign(x,y)
* scalb(x,N)
*
* Required kernel functions:
* expm1(x) ...return exp(x)-1
*
* Method :
* 1. reduce x to non-negative by sinh(-x) = - sinh(x).
* 2.
*
* expm1(x) + expm1(x)/(expm1(x)+1)
* 0 <= x <= lnovfl : sinh(x) := --------------------------------
* 2
* lnovfl <= x <= lnovfl+ln2 : sinh(x) := expm1(x)/2 (avoid overflow)
* lnovfl+ln2 < x < INF : overflow to INF
*
*
* Special cases:
* sinh(x) is x if x is +INF, -INF, or NaN.
* only sinh(0)=0 is exact for finite argument.
*
* Accuracy:
* sinh(x) returns the exact hyperbolic sine of x nearly rounded. In
* a test run with 1,024,000 random arguments on a VAX, the maximum
* observed error was 1.93 ulps (units in the last place).
*
* 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.
*/
double static
mln2hi = 7.0978271289338397310E2 , /*Hex 2↑ 10 * 1.62E42FEFA39EF */
mln2lo = 2.3747039373786107478E-14 , /*Hex 2↑-45 * 1.ABC9E3B39803F */
lnovfl = 7.0978271289338397310E2 ; /*Hex 2↑ 9 * 1.62E42FEFA39EF */
static max = 1023 ;
asm (" exportproc ←sinh, Libm");
double sinh(x)
double x;
{
static double one=1.0, half=1.0/2.0 ;
double expm1(), t, LibmSupport←scalb(), LibmSupport←copysign(), sign;
if(x!=x) return(x); /* x is NaN */
sign=copysign(one,x);
x=copysign(x,one);
if(x<lnovfl)
{t=expm1(x); return(copysign((t+t/(one+t))*half,sign));}
else if(x <= lnovfl+0.7)
/* subtract x by ln(2↑(max+1)) and return 2↑max*exp(x)
to avoid unnecessary overflow */
return(copysign(scalb(one+expm1((x-mln2hi)-mln2lo),max),sign));
else /* sinh(+-INF) = +-INF, sinh(+-big no.) overflow to +-INF */
return( expm1(x)*sign );
}
/* TANH(X)
* RETURN THE HYPERBOLIC TANGENT OF X
* DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS)
* CODED IN C BY K.C. NG, 1/8/85;
* REVISED BY K.C. NG on 2/8/85, 2/11/85, 3/7/85, 3/24/85.
*
* Required system supported functions :
* copysign(x,y)
* finite(x)
*
* Required kernel function:
* expm1(x) ...exp(x)-1
*
* Method :
* 1. reduce x to non-negative by tanh(-x) = - tanh(x).
* 2.
* 0 < x <= 1.e-10 : tanh(x) := x
* -expm1(-2x)
* 1.e-10 < x <= 1 : tanh(x) := --------------
* expm1(-2x) + 2
* 2
* 1 <= x <= 22.0 : tanh(x) := 1 - ---------------
* expm1(2x) + 2
* 22.0 < x <= INF : tanh(x) := 1.
*
* Note: 22 was chosen so that fl(1.0+2/(expm1(2*22)+2)) == 1.
*
* Special cases:
* tanh(NaN) is NaN;
* only tanh(0)=0 is exact for finite argument.
*
* Accuracy:
* tanh(x) returns the exact hyperbolic tangent of x nealy rounded.
* In a test run with 1,024,000 random arguments on a VAX, the maximum
* observed error was 2.22 ulps (units in the last place).
*/
asm (" exportproc ←tanh, Libm");
double tanh(x)
double x;
{
static double one=1.0, two=2.0, small = 1.0e-10, big = 1.0e10;
double expm1(), t, LibmSupport←copysign(), sign;
int finite();
if(x!=x) return(x); /* x is NaN */
sign=copysign(one,x);
x=copysign(x,one);
if(x < 22.0)
if( x > one )
return(copysign(one-two/(expm1(x+x)+two),sign));
else if ( x > small )
{t= -expm1(-(x+x)); return(copysign(t/(two-t),sign));}
else /* raise the INEXACT flag for non-zero x */
{big+x; return(copysign(x,sign));}
else if(finite(x))
return (sign+1.0E-37); /* raise the INEXACT flag */
else
return(sign); /* x is +- INF */
}
/* COSH(X)
* RETURN THE HYPERBOLIC COSINE OF X
* DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
* CODED IN C BY K.C. NG, 1/8/85;
* REVISED BY K.C. NG on 2/8/85, 2/23/85, 3/7/85, 3/29/85, 4/16/85.
*
* Required system supported functions :
* copysign(x,y)
* scalb(x,N)
*
* Required kernel function:
* exp(x)
* exp←←E(x,c) ...return exp(x+c)-1-x for |x|<0.3465
*
* Method :
* 1. Replace x by |x|.
* 2.
* [ exp(x) - 1 ]↑2
* 0 <= x <= 0.3465 : cosh(x) := 1 + -------------------
* 2*exp(x)
*
* exp(x) + 1/exp(x)
* 0.3465 <= x <= 22 : cosh(x) := -------------------
* 2
* 22 <= x <= lnovfl : cosh(x) := exp(x)/2
* lnovfl <= x <= lnovfl+log(2)
* : cosh(x) := exp(x)/2 (avoid overflow)
* log(2)+lnovfl < x < INF: overflow to INF
*
* Note: .3465 is a number near one half of ln2.
*
* Special cases:
* cosh(x) is x if x is +INF, -INF, or NaN.
* only cosh(0)=1 is exact for finite x.
*
* Accuracy:
* cosh(x) returns the exact hyperbolic cosine of x nearly rounded.
* In a test run with 768,000 random arguments on a VAX, the maximum
* observed error was 1.23 ulps (units in the last place).
*
* 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.
*/
asm (" exportproc ←cosh, Libm");
double cosh(x)
double x;
{
static double half=1.0/2.0,one=1.0, small=1.0E-18; /* fl(1+small)==1 */
double Libmsupport←scalb(),LibmSupport←copysign(),exp(),
LibmSupport←exp←←E(),t;
if(x!=x) return(x); /* x is NaN */
if((x=copysign(x,one)) <= 22)
if(x<0.3465)
if(x<small) return(one+x);
else {t=x+exp←←E(x,0.0);x=t+t; return(one+t*t/(2.0+x)); }
else /* for x lies in [0.3465,22] */
{ t=exp(x); return((t+one/t)*half); }
if( lnovfl <= x && x <= (lnovfl+0.7))
/* for x lies in [lnovfl, lnovfl+ln2], decrease x by ln(2↑(max+1))
* and return 2↑max*exp(x) to avoid unnecessary overflow
*/
return(scalb(exp((x-mln2hi)-mln2lo), max));
else
return(exp(x)*half); /* for large x, cosh(x)=exp(x)/2 */
}
/* ASINH(X)
* RETURN THE INVERSE HYPERBOLIC SINE OF X
* DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
* CODED IN C BY K.C. NG, 2/16/85;
* REVISED BY K.C. NG on 3/7/85, 3/24/85, 4/16/85.
*
* Required system supported functions :
* copysign(x,y)
* sqrt(x)
*
* Required kernel function:
* log1p(x) ...return log(1+x)
*
* Method :
* Based on
* asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ]
* we have
* asinh(x) := x if 1+x*x=1,
* := sign(x)*(log1p(x)+ln2)) if sqrt(1+x*x)=x, else
* := sign(x)*log1p(|x| + |x|/(1/|x| + sqrt(1+(1/|x|)↑2)) )
*
* Accuracy:
* asinh(x) returns the exact inverse hyperbolic sine of x nearly rounded.
* In a test run with 52,000 random arguments on a VAX, the maximum
* observed error was 1.58 ulps (units in the last place).
*
* 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.
*/
static double
ln2hi = 6.9314718036912381649E-1 , /*Hex 2↑ -1 * 1.62E42FEE00000 */
ln2lo = 1.9082149292705877000E-10 ; /*Hex 2↑-33 * 1.A39EF35793C76 */
asm (" exportproc ←asinh, Libm");
double asinh(x)
double x;
{
double LibmSupport←copysign(),log1p(),Libm←sqrt(),t,s;
static double small=1.0E-10, /* fl(1+small*small) == 1 */
big =1.0E20, /* fl(1+big) == big */
one =1.0 ;
if(x!=x) return(x); /* x is NaN */
if((t=copysign(x,one))>small)
if(t<big) {
s=one/t; return(copysign(log1p(t+t/(s+sqrt(one+s*s))),x)); }
else /* if |x| > big */
{s=log1p(t)+ln2lo; return(copysign(s+ln2hi,x));}
else /* if |x| < small */
return(x);
}
/* ACOSH(X)
* RETURN THE INVERSE HYPERBOLIC COSINE OF X
* DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS)
* CODED IN C BY K.C. NG, 2/16/85;
* REVISED BY K.C. NG on 3/6/85, 3/24/85, 4/16/85, 8/17/85.
*
* Required system supported functions :
* sqrt(x)
*
* Required kernel function:
* log1p(x) ...return log(1+x)
*
* Method :
* Based on
* acosh(x) = log [ x + sqrt(x*x-1) ]
* we have
* acosh(x) := log1p(x)+ln2, if (x > 1.0E20); else
* acosh(x) := log1p( sqrt(x-1) * (sqrt(x-1) + sqrt(x+1)) ) .
* These formulae avoid the over/underflow complication.
*
* Special cases:
* acosh(x) is NaN with signal if x<1.
* acosh(NaN) is NaN without signal.
*
* Accuracy:
* acosh(x) returns the exact inverse hyperbolic cosine of x nearly
* rounded. In a test run with 512,000 random arguments on a VAX, the
* maximum observed error was 3.30 ulps (units of the last place) at
* x=1.0070493753568216 .
*
* 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.
*/
asm (" exportproc ←acosh, Libm");
double acosh(x)
double x;
{
double log1p(),sqrt(),t,big=1.E20; /* big+1==big */
if(x!=x) return(x); /* x is NaN */
/* return log1p(x) + log(2) if x is large */
if(x>big) {t=log1p(x)+ln2lo; return(t+ln2hi);}
t=sqrt(x-1.0);
return(log1p(t*(t+sqrt(x+1.0))));
}
/* ATANH(X)
* RETURN THE HYPERBOLIC ARC TANGENT OF X
* DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
* CODED IN C BY K.C. NG, 1/8/85;
* REVISED BY K.C. NG on 2/7/85, 3/7/85, 8/18/85.
*
* Required kernel function:
* log1p(x) ...return log(1+x)
*
* Method :
* Return
* 1 2x x
* atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------)
* 2 1 - x 1 - x
*
* Special cases:
* atanh(x) is NaN if |x| > 1 with signal;
* atanh(NaN) is that NaN with no signal;
* atanh(+-1) is +-INF with signal.
*
* Accuracy:
* atanh(x) returns the exact hyperbolic arc tangent of x nearly rounded.
* In a test run with 512,000 random arguments on a VAX, the maximum
* observed error was 1.87 ulps (units in the last place) at
* x= -3.8962076028810414000e-03.
*/
asm (" exportproc ←atanh, Libm");
double atanh(x)
double x;
{
double LibmSupport←copysign(),log1p(),z;
z = copysign(0.5,x);
x = copysign(x,1.0);
x = x/(1.0-x);
return( z*log1p(x+x) );
}