  We revisit the problem of tolerant distribution testing. That is, given
samples from an unknown distribution $p$ over $\{1, \dots, n\}$, is it
$\varepsilon_1$-close to or $\varepsilon_2$-far from a reference distribution
$q$ (in total variation distance)? Despite significant interest over the past
decade, this problem is well understood only in the extreme cases. In the
noiseless setting (i.e., $\varepsilon_1 = 0$) the sample complexity is
$\Theta(\sqrt{n})$, strongly sublinear in the domain size. At the other end of
the spectrum, when $\varepsilon_1 = \varepsilon_2/2$, the sample complexity
jumps to the barely sublinear $\Theta(n/\log n)$. However, very little is known
about the intermediate regime. We fully characterize the price of tolerance in
distribution testing as a function of $n$, $\varepsilon_1$, $\varepsilon_2$, up
to a single $\log n$ factor. Specifically, we show the sample complexity to be
\[\tilde \Theta\left(\frac{\sqrt{n}}{\varepsilon_2^{2}} + \frac{n}{\log n}
\cdot \max
\left\{\frac{\varepsilon_1}{\varepsilon_2^2},\left(\frac{\varepsilon_1}{\varepsilon_2^2}\right)^{\!\!2}\right\}\right),\]
providing a smooth tradeoff between the two previously known cases. We also
provide a similar characterization for the problem of tolerant equivalence
testing, where both $p$ and $q$ are unknown. Surprisingly, in both cases, the
main quantity dictating the sample complexity is the ratio
$\varepsilon_1/\varepsilon_2^2$, and not the more intuitive
$\varepsilon_1/\varepsilon_2$. Of particular technical interest is our lower
bound framework, which involves novel approximation-theoretic tools required to
handle the asymmetry between $\varepsilon_1$ and $\varepsilon_2$, a challenge
absent from previous works.
