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/* sinf.c
*
* Circular sine
*
*
*
* SYNOPSIS:
*
* float x, y, sinf();
*
* y = sinf( x );
*
*
*
* DESCRIPTION:
*
* Range reduction is into intervals of pi/4. The reduction
* error is nearly eliminated by contriving an extended precision
* modular arithmetic.
*
* Two polynomial approximating functions are employed.
* Between 0 and pi/4 the sine is approximated by
* x + x**3 P(x**2).
* Between pi/4 and pi/2 the cosine is represented as
* 1 - x**2 Q(x**2).
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -4096,+4096 100,000 1.2e-7 3.0e-8
* IEEE -8192,+8192 100,000 3.0e-7 3.0e-8
*
* ERROR MESSAGES:
*
* message condition value returned
* sin total loss x > 2^24 0.0
*
* Partial loss of accuracy begins to occur at x = 2^13
* = 8192. Results may be meaningless for x >= 2^24
* The routine as implemented flags a TLOSS error
* for x >= 2^24 and returns 0.0.
*/
/* cosf.c
*
* Circular cosine
*
*
*
* SYNOPSIS:
*
* float x, y, cosf();
*
* y = cosf( x );
*
*
*
* DESCRIPTION:
*
* Range reduction is into intervals of pi/4. The reduction
* error is nearly eliminated by contriving an extended precision
* modular arithmetic.
*
* Two polynomial approximating functions are employed.
* Between 0 and pi/4 the cosine is approximated by
* 1 - x**2 Q(x**2).
* Between pi/4 and pi/2 the sine is represented as
* x + x**3 P(x**2).
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -8192,+8192 100,000 3.0e-7 3.0e-8
*/
/*
Cephes Math Library Release 2.2: June, 1992
Copyright 1985, 1987, 1988, 1992 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
/* Single precision circular sine
* test interval: [-pi/4, +pi/4]
* trials: 10000
* peak relative error: 6.8e-8
* rms relative error: 2.6e-8
*/
#include <math.h>
static float FOPI = 1.27323954473516;
extern float PIO4F;
/* Note, these constants are for a 32-bit significand: */
/*
static float DP1 = 0.7853851318359375;
static float DP2 = 1.30315311253070831298828125e-5;
static float DP3 = 3.03855025325309630e-11;
static float lossth = 65536.;
*/
/* These are for a 24-bit significand: */
static float DP1 = 0.78515625;
static float DP2 = 2.4187564849853515625e-4;
static float DP3 = 3.77489497744594108e-8;
static float lossth = 8192.;
static float T24M1 = 16777215.;
static float sincof[] = {
-1.9515295891E-4,
8.3321608736E-3,
-1.6666654611E-1
};
static float coscof[] = {
2.443315711809948E-005,
-1.388731625493765E-003,
4.166664568298827E-002
};
float sinf( float xx )
{
float *p;
float x, y, z;
register unsigned long j;
register int sign;
sign = 1;
x = xx;
if( xx < 0 )
{
sign = -1;
x = -xx;
}
if( x > T24M1 )
{
mtherr( "sinf", TLOSS );
return(0.0);
}
j = FOPI * x; /* integer part of x/(PI/4) */
y = j;
/* map zeros to origin */
if( j & 1 )
{
j += 1;
y += 1.0;
}
j &= 7; /* octant modulo 360 degrees */
/* reflect in x axis */
if( j > 3)
{
sign = -sign;
j -= 4;
}
if( x > lossth )
{
mtherr( "sinf", PLOSS );
x = x - y * PIO4F;
}
else
{
/* Extended precision modular arithmetic */
x = ((x - y * DP1) - y * DP2) - y * DP3;
}
/*einits();*/
z = x * x;
if( (j==1) || (j==2) )
{
/* measured relative error in +/- pi/4 is 7.8e-8 */
/*
y = (( 2.443315711809948E-005 * z
- 1.388731625493765E-003) * z
+ 4.166664568298827E-002) * z * z;
*/
p = coscof;
y = *p++;
y = y * z + *p++;
y = y * z + *p++;
y *= z * z;
y -= 0.5 * z;
y += 1.0;
}
else
{
/* Theoretical relative error = 3.8e-9 in [-pi/4, +pi/4] */
/*
y = ((-1.9515295891E-4 * z
+ 8.3321608736E-3) * z
- 1.6666654611E-1) * z * x;
y += x;
*/
p = sincof;
y = *p++;
y = y * z + *p++;
y = y * z + *p++;
y *= z * x;
y += x;
}
/*einitd();*/
if(sign < 0)
y = -y;
return( y);
}
/* Single precision circular cosine
* test interval: [-pi/4, +pi/4]
* trials: 10000
* peak relative error: 8.3e-8
* rms relative error: 2.2e-8
*/
float cosf( float xx )
{
float x, y, z;
int j, sign;
/* make argument positive */
sign = 1;
x = xx;
if( x < 0 )
x = -x;
if( x > T24M1 )
{
mtherr( "cosf", TLOSS );
return(0.0);
}
j = FOPI * x; /* integer part of x/PIO4 */
y = j;
/* integer and fractional part modulo one octant */
if( j & 1 ) /* map zeros to origin */
{
j += 1;
y += 1.0;
}
j &= 7;
if( j > 3)
{
j -=4;
sign = -sign;
}
if( j > 1 )
sign = -sign;
if( x > lossth )
{
mtherr( "cosf", PLOSS );
x = x - y * PIO4F;
}
else
/* Extended precision modular arithmetic */
x = ((x - y * DP1) - y * DP2) - y * DP3;
z = x * x;
if( (j==1) || (j==2) )
{
y = (((-1.9515295891E-4 * z
+ 8.3321608736E-3) * z
- 1.6666654611E-1) * z * x)
+ x;
}
else
{
y = (( 2.443315711809948E-005 * z
- 1.388731625493765E-003) * z
+ 4.166664568298827E-002) * z * z;
y -= 0.5 * z;
y += 1.0;
}
if(sign < 0)
y = -y;
return( y );
}
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