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/* Compute cosine of argument.
Copyright (C) 2017 Free Software Foundation, Inc.
This file is part of the GNU C Library.
The GNU C Library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
The GNU C Library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with the GNU C Library; if not, see
<http://www.gnu.org/licenses/>. */
#include <errno.h>
#include <math.h>
#include <math_private.h>
#include <libm-alias-float.h>
#ifndef COSF
# define COSF_FUNC __cosf
#else
# define COSF_FUNC COSF
#endif
/* Chebyshev constants for cos, range -PI/4 - PI/4. */
static const double C0 = -0x1.ffffffffe98aep-2;
static const double C1 = 0x1.55555545c50c7p-5;
static const double C2 = -0x1.6c16b348b6874p-10;
static const double C3 = 0x1.a00eb9ac43ccp-16;
static const double C4 = -0x1.23c97dd8844d7p-22;
/* Chebyshev constants for sin, range -PI/4 - PI/4. */
static const double S0 = -0x1.5555555551cd9p-3;
static const double S1 = 0x1.1111110c2688bp-7;
static const double S2 = -0x1.a019f8b4bd1f9p-13;
static const double S3 = 0x1.71d7264e6b5b4p-19;
static const double S4 = -0x1.a947e1674b58ap-26;
/* Chebyshev constants for cos, range 2^-27 - 2^-5. */
static const double CC0 = -0x1.fffffff5cc6fdp-2;
static const double CC1 = 0x1.55514b178dac5p-5;
/* PI/2 with 98 bits of accuracy. */
static const double PI_2_hi = 0x1.921fb544p+0;
static const double PI_2_lo = 0x1.0b4611a626332p-34;
static const double inv_PI_4 = 0x1.45f306dc9c883p+0; /* 4/PI. */
#define FLOAT_EXPONENT_SHIFT 23
#define FLOAT_EXPONENT_BIAS 127
static const double pio2_table[] = {
0 * M_PI_2,
1 * M_PI_2,
2 * M_PI_2,
3 * M_PI_2,
4 * M_PI_2,
5 * M_PI_2
};
static const double invpio4_table[] = {
0x0p+0,
0x1.45f306cp+0,
0x1.c9c882ap-28,
0x1.4fe13a8p-58,
0x1.f47d4dp-85,
0x1.bb81b6cp-112,
0x1.4acc9ep-142,
0x1.0e4107cp-169
};
static const double ones[] = { 1.0, -1.0 };
/* Compute the cosine value using Chebyshev polynomials where
THETA is the range reduced absolute value of the input
and it is less than Pi/4,
N is calculated as trunc(|x|/(Pi/4)) + 1 and it is used to decide
whether a sine or cosine approximation is more accurate and
the sign of the result. */
static inline float
reduced (double theta, unsigned int n)
{
double sign, cx;
const double theta2 = theta * theta;
/* Determine positive or negative primary interval. */
n += 2;
sign = ones[(n >> 2) & 1];
/* Are we in the primary interval of sin or cos? */
if ((n & 2) == 0)
{
/* Here cosf() is calculated using sin Chebyshev polynomial:
x+x^3*(S0+x^2*(S1+x^2*(S2+x^2*(S3+x^2*S4)))). */
cx = S3 + theta2 * S4;
cx = S2 + theta2 * cx;
cx = S1 + theta2 * cx;
cx = S0 + theta2 * cx;
cx = theta + theta * theta2 * cx;
}
else
{
/* Here cosf() is calculated using cos Chebyshev polynomial:
1.0+x^2*(C0+x^2*(C1+x^2*(C2+x^2*(C3+x^2*C4)))). */
cx = C3 + theta2 * C4;
cx = C2 + theta2 * cx;
cx = C1 + theta2 * cx;
cx = C0 + theta2 * cx;
cx = 1. + theta2 * cx;
}
return sign * cx;
}
float
COSF_FUNC (float x)
{
double theta = x;
double abstheta = fabs (theta);
if (isless (abstheta, M_PI_4))
{
double cx;
if (abstheta >= 0x1p-5)
{
const double theta2 = theta * theta;
/* Chebyshev polynomial of the form for cos:
* 1 + x^2 (C0 + x^2 (C1 + x^2 (C2 + x^2 (C3 + x^2 * C4)))). */
cx = C3 + theta2 * C4;
cx = C2 + theta2 * cx;
cx = C1 + theta2 * cx;
cx = C0 + theta2 * cx;
cx = 1. + theta2 * cx;
return cx;
}
else if (abstheta >= 0x1p-27)
{
/* A simpler Chebyshev approximation is close enough for this range:
* 1 + x^2 (CC0 + x^3 * CC1). */
const double theta2 = theta * theta;
cx = CC0 + theta * theta2 * CC1;
cx = 1.0 + theta2 * cx;
return cx;
}
else
{
/* For small enough |theta|, this is close enough. */
return 1.0 - abstheta;
}
}
else /* |theta| >= Pi/4. */
{
if (isless (abstheta, 9 * M_PI_4))
{
/* There are cases where FE_UPWARD rounding mode can
produce a result of abstheta * inv_PI_4 == 9,
where abstheta < 9pi/4, so the domain for
pio2_table must go to 5 (9 / 2 + 1). */
unsigned int n = (abstheta * inv_PI_4) + 1;
theta = abstheta - pio2_table[n / 2];
return reduced (theta, n);
}
else if (isless (abstheta, INFINITY))
{
if (abstheta < 0x1p+23)
{
unsigned int n = ((unsigned int) (abstheta * inv_PI_4)) + 1;
double x = n / 2;
theta = (abstheta - x * PI_2_hi) - x * PI_2_lo;
/* Argument reduction needed. */
return reduced (theta, n);
}
else /* |theta| >= 2^23. */
{
x = fabsf (x);
int exponent;
GET_FLOAT_WORD (exponent, x);
exponent = (exponent >> FLOAT_EXPONENT_SHIFT)
- FLOAT_EXPONENT_BIAS;
exponent += 3;
exponent /= 28;
double a = invpio4_table[exponent] * x;
double b = invpio4_table[exponent + 1] * x;
double c = invpio4_table[exponent + 2] * x;
double d = invpio4_table[exponent + 3] * x;
uint64_t l = a;
l &= ~0x7;
a -= l;
double e = a + b;
l = e;
e = a - l;
if (l & 1)
{
e -= 1.0;
e += b;
e += c;
e += d;
e *= M_PI_4;
return reduced (e, l + 1);
}
else
{
e += b;
e += c;
e += d;
if (e <= 1.0)
{
e *= M_PI_4;
return reduced (e, l + 1);
}
else
{
l++;
e -= 2.0;
e *= M_PI_4;
return reduced (e, l + 1);
}
}
}
}
else
{
int32_t ix;
GET_FLOAT_WORD (ix, abstheta);
/* cos(Inf or NaN) is NaN. */
if (ix == 0x7f800000) /* Inf. */
__set_errno (EDOM);
return x - x;
}
}
}
#ifndef COSF
libm_alias_float (__cos, cos)
#endif
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