Function primitive_float_csc_rational 

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

Computes $\csc x$, the cosecant of a Rational, returning the result as a primitive float.

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

Special cases:

• $f(0)=\infty$

Overflow is possible: a Rational within about $2^{-129}$ of a nonzero multiple of $\pi$ has a cosecant beyond the largest f32, and one within about $2^{-1025}$ of one beyond the largest f64; so does any Rational small enough that its reciprocal alone leaves the range, and $0$ itself, whose cosecant is $\infty$. Underflow is not possible, since $|\csc x| \geq 1$.

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::csc::primitive_float_csc_rational;
use malachite_q::Rational;

assert_eq!(
    NiceFloat(primitive_float_csc_rational::<f64>(&Rational::ZERO)),
    NiceFloat(f64::INFINITY)
);
assert_eq!(
    NiceFloat(primitive_float_csc_rational::<f64>(
        &Rational::from_unsigneds(1u8, 3)
    )),
    NiceFloat(3.0562842545795195)
);
assert_eq!(
    NiceFloat(primitive_float_csc_rational::<f32>(
        &Rational::from_unsigneds(1u8, 3)
    )),
    NiceFloat(3.0562842)
);
assert_eq!(
    NiceFloat(primitive_float_csc_rational::<f64>(&Rational::from(10000))),
    NiceFloat(-3.2720972452826818)
);