Function primitive_float_sin_rational 

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

Computes $\sin x$, the sine of a Rational, returning the result as a primitive float.

$$ f(x) = \sin x+\varepsilon, $$ where $|\varepsilon| < 2^{\lfloor\log_2 |\sin x|\rfloor-p}$, and $p$ is the precision of the output (24 if T is a f32 and 53 if T is a f64).

Special cases:

• $f(0)=0$

Overflow is not possible, since the result lies in $[-1, 1]$. The result underflows, to a subnormal or to zero, when $x$ is tiny, since $\sin x$ is then very close to $x$; a Rational close enough to a nonzero multiple of $\pi$ 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

$T(m, e) = O((m+e) (\log (m+e))^2 \log\log (m+e))$

$M(m, e) = O((m+e) \log (m+e))$

where $T$ is time, $M$ is additional memory, $m$ is x.significant_bits(), and $e$ is x.floor_log_base_2_abs() (taken as 0 when it is negative or $x = 0$): for $|x| \geq 3$ the argument is reduced modulo $2\pi$, which needs $\pi$ to about $e$ bits.

Examples

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

assert_eq!(
    NiceFloat(primitive_float_sin_rational::<f64>(&Rational::ZERO)),
    NiceFloat(0.0)
);
assert_eq!(
    NiceFloat(primitive_float_sin_rational::<f64>(
        &Rational::from_unsigneds(1u8, 3)
    )),
    NiceFloat(0.32719469679615226)
);
assert_eq!(
    NiceFloat(primitive_float_sin_rational::<f32>(
        &Rational::from_unsigneds(1u8, 3)
    )),
    NiceFloat(0.3271947)
);
assert_eq!(
    NiceFloat(primitive_float_sin_rational::<f64>(&Rational::from(10000))),
    NiceFloat(-0.30561438888825215)
);