fix: evaluate the left tail without cancellation in stats/base/dists/cauchy/cdf

[Abhist17]

Description

What is the purpose of this pull request?

This pull request:

• evaluates the left tail of stats/base/dists/cauchy/cdf without cancellation (JS, factory and C).

0.5 + atan(z)/pi cancels below the median, so the CDF loses digits steadily and reaches 0 from about x = x0 - 1e16*gamma:

var cdf = require( '@stdlib/stats/base/dists/cauchy/cdf' );

cdf( -1.0e8, 0.0, 1.0 );
// develop: 3.1830988400649574e-9   (8 correct digits)
// this PR: 3.183098861837907e-9

cdf( -1.0e16, 0.0, 1.0 );
// develop: 0
// this PR: 3.183098861837907e-17

Below z = -1 it uses the identity 0.5 + atan(z)/pi = atan2( gamma, x0-x )/pi, which has no subtraction. Above that the sum doesn't cancel. I first applied the identity everywhere left of the median, but near the median it's less accurate than the sum (atan2 is close to pi/2 there and picks up the rounding of 1/pi), so it's gated at z = -1.

Fixtures. They came from Distributions.jl's Float64 cdf, which uses the same formula. Against the regenerated 2048-bit BigFloat reference, develop is up to 12,729 ULP off on the existing fixtures and this PR is within 3. Point by point, this PR is closer to the exact value on 1,867 of the 2,000 negative_median and positive_median points, and further on 37, by at most an ULP. gen now evaluates in BigFloat (as in #15772 and #15966); all three fixtures were regenerated, plus a new left_tail.json. Tolerances are unchanged, and the new blocks use 2 ULP. test.cdf.js / test.factory.js / test.native.js pass 4,014 / 4,018 / 4,014.

Same fix as #15966 for cauchy/logcdf.

Related Issues

Does this pull request have any related issues?

None. Related to #15966 (cauchy/logcdf).

Questions

Any questions for reviewers of this pull request?

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Coverage Report

Package	Statements	Branches	Functions	Lines
stats/base/dists/cauchy/cdf	$\\color{green}295/295$ $\\color{green}+100.00\\%$	$\\color{green}23/23$ $\\color{green}+100.00\\%$	$\\color{green}4/4$ $\\color{green}+100.00\\%$	$\\color{green}295/295$ $\\color{green}+100.00\\%$

The above coverage report was generated for the changes in this PR.