fix: evaluate both tails without cancellation in stats/base/dists/cauchy/logcdf

[Abhist17]

Description

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This pull request:

• evaluates both tails of stats/base/dists/cauchy/logcdf without cancellation (JS, factory and C).

ln( 0.5 + atan(z)/pi ) cancels left of the median: the sum loses everything once atan(z)/pi is within an ULP of -0.5, so the function returns -Infinity from about x = x0 - 1e16*gamma on, and is off by hundreds of ULP well before that.

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

logcdf( -1.0e16, 0.0, 1.0 );
// develop: -Infinity
// this PR: -37.98609137375413

logcdf( -1.0e30, 0.0, 1.0 );
// develop: -Infinity
// this PR: -70.22228267567077

• Below z = -1: 0.5 + atan(z)/pi equals atan2( gamma, x0-x )/pi exactly, with no subtraction.
• Above z = 1: the result is ln of a number approaching 1, so it uses log1p( -atan2( gamma, x-x0 )/pi ).
• In between, the original expression is kept. Applied there too, the identities were less accurate than the sum (atan2 is close to pi/2 and picks up the rounding of 1/pi): on large_gamma the first version was further from exact than develop on 325 points; gated, it's 2. x = x0 still gives ln(0.5); ±Infinity are unchanged.

The fixtures had to change. They were generated with Distributions.jl's logcdf, which evaluates the same formula in Float64. Against mpmath, on the 1,046 points where this PR and the fixtures disagree, the fixtures (and develop) are up to 782 ULP off, and this PR is within 2. develop only passed because it reproduces the reference's own error. gen now evaluates the textbook formula in 2048-bit BigFloat, as in #15772 and #15774. All three existing fixtures were regenerated with that, plus a new left_tail.json with x from 1e6 to 1e296 scale units below x0.

With the new fixtures, every point is within 2 ULP (the existing tolerances of 6–12 ULP are unchanged; the new left-tail blocks use 2). test.logcdf.js / test.factory.js / test.native.js pass 4,014 / 4,018 / 4,014. On develop, 1,211 assertions in test.logcdf.js fail.

Related Issues

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Coverage Report

Package	Statements	Branches	Functions	Lines
stats/base/dists/cauchy/logcdf	$\\color{green}307/307$ $\\color{green}+100.00\\%$	$\\color{green}27/27$ $\\color{green}+100.00\\%$	$\\color{green}4/4$ $\\color{green}+100.00\\%$	$\\color{green}307/307$ $\\color{green}+100.00\\%$

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

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