dd_real cos(const dd_real &a) {
if (a.is_zero()) {
return 1.0;
}
// approximately reduce modulo 2*pi
dd_real z = nint(a / dd_real::_2pi);
dd_real r = a - z * dd_real::_2pi;
// approximately reduce modulo pi/2 and then modulo pi/16
dd_real t;
double q = std::floor(r.x[0] / dd_real::_pi2.x[0] + 0.5);
t = r - dd_real::_pi2 * q;
int j = static_cast<int>(q);
q = std::floor(t.x[0] / _pi16.x[0] + 0.5);
t -= _pi16 * q;
int k = static_cast<int>(q);
int abs_k = std::abs(k);
if (j < -2 || j > 2) {
dd_real::error("(dd_real::cos): Cannot reduce modulo pi/2.");
return dd_real::_nan;
}
if (abs_k > 4) {
dd_real::error("(dd_real::cos): Cannot reduce modulo pi/16.");
return dd_real::_nan;
}
if (k == 0) {
switch (j) {
case 0:
return cos_taylor(t);
case 1:
return -sin_taylor(t);
case -1:
return sin_taylor(t);
default:
return -cos_taylor(t);
}
}
dd_real sin_t, cos_t;
sincos_taylor(t, sin_t, cos_t);
dd_real u = cos_table[abs_k-1];
dd_real v = sin_table[abs_k-1];
if (j == 0) {
if (k > 0) {
r = u * cos_t - v * sin_t;
} else {
r = u * cos_t + v * sin_t;
}
} else if (j == 1) {
if (k > 0) {
r = - u * sin_t - v * cos_t;
} else {
r = v * cos_t - u * sin_t;
}
} else if (j == -1) {
if (k > 0) {
r = u * sin_t + v * cos_t;
} else {
r = u * sin_t - v * cos_t;
}
} else {
if (k > 0) {
r = v * sin_t - u * cos_t;
} else {
r = - u * cos_t - v * sin_t;
}
}
return r;
}
void sincos(const dd_real &a, dd_real &sin_a, dd_real &cos_a) {
if (a.is_zero()) {
sin_a = 0.0;
cos_a = 1.0;
return;
}
// approximately reduce modulo 2*pi
dd_real z = nint(a / dd_real::_2pi);
dd_real r = a - dd_real::_2pi * z;
// approximately reduce module pi/2 and pi/16
dd_real t;
double q = std::floor(r.x[0] / dd_real::_pi2.x[0] + 0.5);
t = r - dd_real::_pi2 * q;
int j = static_cast<int>(q);
int abs_j = std::abs(j);
q = std::floor(t.x[0] / _pi16.x[0] + 0.5);
t -= _pi16 * q;
int k = static_cast<int>(q);
int abs_k = std::abs(k);
if (abs_j > 2) {
dd_real::error("(dd_real::sincos): Cannot reduce modulo pi/2.");
cos_a = sin_a = dd_real::_nan;
return;
}
if (abs_k > 4) {
dd_real::error("(dd_real::sincos): Cannot reduce modulo pi/16.");
cos_a = sin_a = dd_real::_nan;
return;
}
dd_real sin_t, cos_t;
dd_real s, c;
sincos_taylor(t, sin_t, cos_t);
if (abs_k == 0) {
s = sin_t;
c = cos_t;
} else {
dd_real u = cos_table[abs_k-1];
dd_real v = sin_table[abs_k-1];
if (k > 0) {
s = u * sin_t + v * cos_t;
c = u * cos_t - v * sin_t;
} else {
s = u * sin_t - v * cos_t;
c = u * cos_t + v * sin_t;
}
}
if (abs_j == 0) {
sin_a = s;
cos_a = c;
} else if (j == 1) {
sin_a = c;
cos_a = -s;
} else if (j == -1) {
sin_a = -c;
cos_a = s;
} else {
sin_a = -s;
cos_a = -c;
}
}
dd_real sin(const dd_real &a) {
/* Strategy. To compute sin(x), we choose integers a, b so that
x = s + a * (pi/2) + b * (pi/16)
and |s| <= pi/32. Using the fact that
sin(pi/16) = 0.5 * sqrt(2 - sqrt(2 + sqrt(2)))
we can compute sin(x) from sin(s), cos(s). This greatly
increases the convergence of the sine Taylor series. */
if (a.is_zero()) {
return 0.0;
}
// approximately reduce modulo 2*pi
dd_real z = nint(a / dd_real::_2pi);
dd_real r = a - dd_real::_2pi * z;
// approximately reduce modulo pi/2 and then modulo pi/16.
dd_real t;
double q = std::floor(r.x[0] / dd_real::_pi2.x[0] + 0.5);
t = r - dd_real::_pi2 * q;
int j = static_cast<int>(q);
q = std::floor(t.x[0] / _pi16.x[0] + 0.5);
t -= _pi16 * q;
int k = static_cast<int>(q);
int abs_k = std::abs(k);
if (j < -2 || j > 2) {
dd_real::error("(dd_real::sin): Cannot reduce modulo pi/2.");
return dd_real::_nan;
}
if (abs_k > 4) {
dd_real::error("(dd_real::sin): Cannot reduce modulo pi/16.");
return dd_real::_nan;
}
if (k == 0) {
switch (j) {
case 0:
return sin_taylor(t);
case 1:
return cos_taylor(t);
case -1:
return -cos_taylor(t);
default:
return -sin_taylor(t);
}
}
dd_real u = cos_table[abs_k-1];
dd_real v = sin_table[abs_k-1];
dd_real sin_t, cos_t;
sincos_taylor(t, sin_t, cos_t);
if (j == 0) {
if (k > 0) {
r = u * sin_t + v * cos_t;
} else {
r = u * sin_t - v * cos_t;
}
} else if (j == 1) {
if (k > 0) {
r = u * cos_t - v * sin_t;
} else {
r = u * cos_t + v * sin_t;
}
} else if (j == -1) {
if (k > 0) {
r = v * sin_t - u * cos_t;
} else if (k < 0) {
r = -u * cos_t - v * sin_t;
}
} else {
if (k > 0) {
r = -u * sin_t - v * cos_t;
} else {
r = v * cos_t - u * sin_t;
}
}
return r;
}