Function primitive_float_sin_with_period_rational 

pub fn primitive_float_sin_with_period_rational<T>(x: &Rational, u: u64) -> T
where
    Float: PartialOrd<T>,
    for<'a> T: ExactFrom<&'a Float> + RoundingFrom<&'a Float> + PrimitiveFloat,

Computes $\sin(2\pi x/u)$, the sine of a Rational measured in $u$ths of a turn (so that u = 360 is degrees), returning the result as a primitive float.

$$ f(x,u) = \sin(2\pi x/u)+\varepsilon. $$

• If $u=0$, $\varepsilon$ may be ignored or assumed to be 0.
• If $u\neq 0$, then $|\varepsilon| < 2^{\lfloor\log_2 |\sin(2\pi x/u)|\rfloor-p}$, where $p$ is the precision of the output (24 if T is a f32 and 53 if T is a f64).

Special cases:

• $f(x,0)=\text{NaN}$
• $f(0,u)=0$
• If $x/u$ is a multiple of $1/2$, the result is exactly $0.0$ with the sign of $x$ (following IEEE 754-2019’s sinPi, so that the function is odd); if it is an odd multiple of $1/4$, the result is exactly $1$ or $-1$; and if it is $\pm1/12$ or $\pm5/12$ modulo $1$, the result is exactly $1/2$ or $-1/2$.

Overflow is not possible, since the result lies in $[-1, 1]$. The result underflows, to a subnormal or to zero, only when $2\pi x/u$ does, for a tiny $x/u$; a Rational close enough to a half turn, without being one, for its sine to be subnormal would need a denominator of more than 100 bits, in which case the result is still correctly rounded.

Worst-case complexity

$T(m) = O(m (\log m)^2 \log\log m)$

$M(m) = O(m \log m)$

where $T$ is time, $M$ is additional memory, and $m$ is x.significant_bits(): the fraction of a turn is reduced modulo 1 exactly, so the magnitude of $x$ does not drive the cost.

Examples

use malachite_base::num::basic::traits::Zero;
use malachite_base::num::float::NiceFloat;
use malachite_float::float::arithmetic::sin::primitive_float_sin_with_period_rational;
use malachite_q::Rational;

assert!(primitive_float_sin_with_period_rational::<f64>(&Rational::ZERO, 0).is_nan());
assert_eq!(
    NiceFloat(primitive_float_sin_with_period_rational::<f64>(
        &Rational::ZERO,
        360
    )),
    NiceFloat(0.0)
);
// a twelfth of a turn is exactly 1/2
assert_eq!(
    NiceFloat(primitive_float_sin_with_period_rational::<f64>(
        &Rational::from_unsigneds(1u8, 12),
        1
    )),
    NiceFloat(0.5)
);
assert_eq!(
    NiceFloat(primitive_float_sin_with_period_rational::<f32>(
        &Rational::from_unsigneds(1u8, 7),
        1
    )),
    NiceFloat(0.7818315)
);
assert_eq!(
    NiceFloat(primitive_float_sin_with_period_rational::<f64>(
        &Rational::from_unsigneds(1u8, 7),
        1
    )),
    NiceFloat(0.7818314824680298)
);