From 537a3d2fc4d90579ab29d0b4d73284688c466f5d Mon Sep 17 00:00:00 2001 From: Abhist17 Date: Thu, 8 Oct 2026 03:02:39 +0530 Subject: [PATCH] fix: evaluate the left tail without cancellation in `stats/base/dists/cauchy/cdf` `0.5 + atan(z)/pi` cancels below the median: `cdf( -1e16, 0, 1 )` returned 0 (about 3.18e-17) and `cdf( -1e8, 0, 1 )` had 8 correct digits. Below `z = -1` use the equivalent `atan2( gamma, x0-x )/pi`; nearer the median the sum doesn't cancel and stays as it was. Fixtures are now evaluated in 2048-bit BigFloat (the Float64 references were up to 12,729 ULP off), with a new far-left-tail fixture. --- .../base/dists/cauchy/cdf/lib/factory.js | 4 ++++ .../stats/base/dists/cauchy/cdf/lib/main.js | 4 ++++ .../stats/base/dists/cauchy/cdf/src/main.c | 4 ++++ .../cdf/test/fixtures/julia/large_gamma.json | 2 +- .../cdf/test/fixtures/julia/left_tail.json | 1 + .../test/fixtures/julia/negative_median.json | 2 +- .../test/fixtures/julia/positive_median.json | 2 +- .../cauchy/cdf/test/fixtures/julia/runner.jl | 14 ++++++++---- .../base/dists/cauchy/cdf/test/test.cdf.js | 20 +++++++++++++++++ .../dists/cauchy/cdf/test/test.factory.js | 22 +++++++++++++++++++ .../base/dists/cauchy/cdf/test/test.native.js | 20 +++++++++++++++++ 11 files changed, 88 insertions(+), 7 deletions(-) create mode 100644 lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/fixtures/julia/left_tail.json diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/lib/factory.js b/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/lib/factory.js index 6d92be313e5f..40cf8183273f 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/lib/factory.js +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/lib/factory.js @@ -73,6 +73,10 @@ function factory( x0, gamma ) { if ( isnan( x ) ) { return NaN; } + // Below `z = -1`, `0.5 + atan(z)/pi` cancels more and more as `z` goes to `-infinity`; `atan2( gamma, x0-x )/pi` is the same quantity evaluated without the subtraction. Nearer the median the sum does not cancel and is the more accurate of the two... + if ( x-x0 < -gamma ) { + return ONE_OVER_PI * atan2( gamma, x0-x ); + } return ( ONE_OVER_PI * atan2( x-x0, gamma ) ) + 0.5; } } diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/lib/main.js b/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/lib/main.js index 374df5b57d69..776ba9524f85 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/lib/main.js +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/lib/main.js @@ -72,6 +72,10 @@ function cdf( x, x0, gamma ) { ) { return NaN; } + // Below `z = -1`, `0.5 + atan(z)/pi` cancels more and more as `z` goes to `-infinity`; `atan2( gamma, x0-x )/pi` is the same quantity evaluated without the subtraction. Nearer the median the sum does not cancel and is the more accurate of the two... + if ( x-x0 < -gamma ) { + return ONE_OVER_PI * atan2( gamma, x0-x ); + } return ( ONE_OVER_PI * atan2( x-x0, gamma ) ) + 0.5; } diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/src/main.c b/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/src/main.c index 4625b9dd03fb..041692ba0e1b 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/src/main.c +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/src/main.c @@ -47,5 +47,9 @@ double stdlib_base_dists_cauchy_cdf( const double x, const double x0, const doub if ( stdlib_base_is_infinite( x ) ) { return ( x < 0.0 ) ? 0.0 : 1.0; } + // Below `z = -1`, `0.5 + atan(z)/pi` cancels more and more as `z` goes to `-infinity`; `atan2( gamma, x0-x )/pi` is the same quantity evaluated without the subtraction. Nearer the median the sum does not cancel and is the more accurate of the two... + if ( x-x0 < -gamma ) { + return ONE_OVER_PI * stdlib_base_atan2( gamma, x0-x ); + } return ( ONE_OVER_PI * stdlib_base_atan2( x-x0, gamma ) ) + 0.5; } diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/fixtures/julia/large_gamma.json b/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/fixtures/julia/large_gamma.json index 04c19cb3fe7a..393f200edcc8 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/fixtures/julia/large_gamma.json +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/fixtures/julia/large_gamma.json @@ -1 +1 @@ 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diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/fixtures/julia/left_tail.json b/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/fixtures/julia/left_tail.json new file mode 100644 index 000000000000..2848f17b3dca --- /dev/null +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/fixtures/julia/left_tail.json @@ -0,0 +1 @@ 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diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/fixtures/julia/negative_median.json b/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/fixtures/julia/negative_median.json index d2925921011d..3382af5e42bf 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/fixtures/julia/negative_median.json +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/fixtures/julia/negative_median.json @@ -1 +1 @@ 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diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/fixtures/julia/positive_median.json b/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/fixtures/julia/positive_median.json index 2d25f8aededb..63f774055f04 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/fixtures/julia/positive_median.json +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/fixtures/julia/positive_median.json @@ -1 +1 @@ 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diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/fixtures/julia/runner.jl b/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/fixtures/julia/runner.jl index 360e763960fc..6ba221b2070c 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/fixtures/julia/runner.jl +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/fixtures/julia/runner.jl @@ -16,7 +16,6 @@ # See the License for the specific language governing permissions and # limitations under the License. -import Distributions: cdf, Cauchy import JSON """ @@ -41,9 +40,9 @@ julia> gen( x, x0, gamma, \"data.json\" ); ``` """ function gen( x, x0, gamma, name ) - z = Array{Float64}( undef, length(x) ); - for i in eachindex(x) - z[ i ] = cdf( Cauchy( x0[i], gamma[i] ), x[i] ); + # The CDF is evaluated in extended precision, as `0.5 + atan(z)/pi` cancels in double precision left of the median... + z = setprecision( BigFloat, 2048 ) do + Float64.( 0.5 .+ ( atan.( ( BigFloat.( x ) .- BigFloat.( x0 ) ) ./ BigFloat.( gamma ) ) ./ BigFloat( pi ) ) ); end # Store data to be written to file as a collection: @@ -87,3 +86,10 @@ x = rand( 1000 ) .* 2.0; x0 = rand( 1000 ) .* 1.0; gamma = rand( 1000 ) .* 50.0; gen( x, x0, gamma, "large_gamma.json" ); + +# Far left tail, many orders of magnitude below the median: +z = .-exp10.( ( rand( 1000 ) .* 290.0 ) .+ 6.0 ); +x0 = ( rand( 1000 ) .* 20.0 ) .- 10.0; +gamma = ( rand( 1000 ) .* 10.0 ) .+ 0.1; +x = ( z .* gamma ) .+ x0; +gen( x, x0, gamma, "left_tail.json" ); diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/test.cdf.js b/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/test.cdf.js index 3ae2d57872bd..d856600385a3 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/test.cdf.js +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/test.cdf.js @@ -32,6 +32,7 @@ var cdf = require( './../lib' ); var largeGamma = require( './fixtures/julia/large_gamma.json' ); var negativeMedian = require( './fixtures/julia/negative_median.json' ); +var leftTail = require( './fixtures/julia/left_tail.json' ); var positiveMedian = require( './fixtures/julia/positive_median.json' ); @@ -148,3 +149,22 @@ tape( 'the function evaluates the cdf for `x` given `x0` and `gamma` (`x0 > 0`)' } t.end(); }); + +tape( 'the function evaluates the cdf for `x` given `x0` and `gamma` far into the left tail', function test( t ) { + var expected; + var gamma; + var x0; + var x; + var y; + var i; + + expected = leftTail.expected; + x = leftTail.x; + x0 = leftTail.x0; + gamma = leftTail.gamma; + for ( i = 0; i < x.length; i++ ) { + y = cdf( x[i], x0[i], gamma[i] ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' ); + } + t.end(); +}); diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/test.factory.js b/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/test.factory.js index 36863315f58a..36f54801eef2 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/test.factory.js +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/test.factory.js @@ -32,6 +32,7 @@ var factory = require( './../lib/factory.js' ); var largeGamma = require( './fixtures/julia/large_gamma.json' ); var negativeMedian = require( './fixtures/julia/negative_median.json' ); +var leftTail = require( './fixtures/julia/left_tail.json' ); var positiveMedian = require( './fixtures/julia/positive_median.json' ); @@ -197,3 +198,24 @@ tape( 'the created function evaluates the cdf for `x` given `x0` and `gamma` (`x } t.end(); }); + +tape( 'the created function evaluates the cdf for `x` given `x0` and `gamma` far into the left tail', function test( t ) { + var expected; + var gamma; + var cdf; + var x0; + var x; + var y; + var i; + + expected = leftTail.expected; + x = leftTail.x; + x0 = leftTail.x0; + gamma = leftTail.gamma; + for ( i = 0; i < x.length; i++ ) { + cdf = factory( x0[i], gamma[i] ); + y = cdf( x[i] ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' ); + } + t.end(); +}); diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/test.native.js b/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/test.native.js index b283b4175ffc..f685000ede27 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/test.native.js +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/cdf/test/test.native.js @@ -33,6 +33,7 @@ var NINF = require( '@stdlib/constants/float64/ninf' ); var largeGamma = require( './fixtures/julia/large_gamma.json' ); var negativeMedian = require( './fixtures/julia/negative_median.json' ); +var leftTail = require( './fixtures/julia/left_tail.json' ); var positiveMedian = require( './fixtures/julia/positive_median.json' ); @@ -157,3 +158,22 @@ tape( 'the function evaluates the cdf for `x` given `x0` and `gamma` (`x0 > 0`)' } t.end(); }); + +tape( 'the function evaluates the cdf for `x` given `x0` and `gamma` far into the left tail', opts, function test( t ) { + var expected; + var gamma; + var x0; + var x; + var y; + var i; + + expected = leftTail.expected; + x = leftTail.x; + x0 = leftTail.x0; + gamma = leftTail.gamma; + for ( i = 0; i < x.length; i++ ) { + y = cdf( x[i], x0[i], gamma[i] ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' ); + } + t.end(); +});