From 3efabc48c97eef59ed36de01e426e8d8f6f63810 Mon Sep 17 00:00:00 2001 From: Abhist17 Date: Thu, 8 Oct 2026 03:00:13 +0530 Subject: [PATCH 1/2] fix: evaluate both tails without cancellation in `stats/base/dists/cauchy/logcdf` `ln( 0.5 + atan(z)/pi )` cancels left of the median and returns -Infinity from about `x = x0 - 1e16*gamma`; `logcdf( -1e30, 0, 1 )` is about -70.22. Use the equivalent `atan2( gamma, x0-x )/pi` there, and `log1p( -atan2( gamma, x-x0 )/pi )` right of the median, where the result is `ln` of a number close to 1. The fixtures were generated with Distributions.jl's Float64 logcdf, which uses the same formula and is up to 782 ULP off; they are now evaluated in 2048-bit BigFloat, plus a new far-left-tail fixture. All four fixtures are within 2 ULP; develop fails 1211 of the assertions. --- .../base/dists/cauchy/logcdf/lib/factory.js | 9 ++++++++ .../base/dists/cauchy/logcdf/lib/main.js | 9 ++++++++ .../base/dists/cauchy/logcdf/manifest.json | 3 +++ .../stats/base/dists/cauchy/logcdf/src/main.c | 9 ++++++++ .../test/fixtures/julia/large_gamma.json | 2 +- .../logcdf/test/fixtures/julia/left_tail.json | 1 + .../test/fixtures/julia/negative_median.json | 2 +- .../test/fixtures/julia/positive_median.json | 2 +- .../logcdf/test/fixtures/julia/runner.jl | 14 ++++++++---- .../dists/cauchy/logcdf/test/test.factory.js | 22 +++++++++++++++++++ .../dists/cauchy/logcdf/test/test.logcdf.js | 20 +++++++++++++++++ .../dists/cauchy/logcdf/test/test.native.js | 20 +++++++++++++++++ 12 files changed, 106 insertions(+), 7 deletions(-) create mode 100644 lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/fixtures/julia/left_tail.json diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/lib/factory.js b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/lib/factory.js index d67c1a72767b..8cfe524f42a3 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/lib/factory.js +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/lib/factory.js @@ -24,6 +24,7 @@ var constantFunction = require( '@stdlib/utils/constant-function' ); var isnan = require( '@stdlib/math/base/assert/is-nan' ); var atan2 = require( '@stdlib/math/base/special/atan2' ); var ln = require( '@stdlib/math/base/special/ln' ); +var log1p = require( '@stdlib/math/base/special/log1p' ); // VARIABLES // @@ -74,6 +75,14 @@ function factory( x0, gamma ) { if ( isnan( x ) ) { return NaN; } + // Left of the median, `0.5 + atan(z)/pi` cancels as `z` goes to `-infinity`; `atan2( gamma, x0-x )/pi` is the same quantity evaluated without the subtraction... + if ( x < x0 ) { + return ln( ONE_OVER_PI * atan2( gamma, x0-x ) ); + } + // Right of the median the result is `ln` of a number close to `1`, so evaluate `ln(1 - atan2(gamma, x-x0)/pi)` with `log1p` instead... + if ( x > x0 ) { + return log1p( -ONE_OVER_PI * atan2( gamma, x-x0 ) ); + } return ln( ( ONE_OVER_PI * atan2( x-x0, gamma ) ) + 0.5 ); } } diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/lib/main.js b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/lib/main.js index 783a38c57c56..f2a088222579 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/lib/main.js +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/lib/main.js @@ -23,6 +23,7 @@ var isnan = require( '@stdlib/math/base/assert/is-nan' ); var atan2 = require( '@stdlib/math/base/special/atan2' ); var ln = require( '@stdlib/math/base/special/ln' ); +var log1p = require( '@stdlib/math/base/special/log1p' ); // VARIABLES // @@ -73,6 +74,14 @@ function logcdf( x, x0, gamma ) { ) { return NaN; } + // Left of the median, `0.5 + atan(z)/pi` cancels as `z` goes to `-infinity`; `atan2( gamma, x0-x )/pi` is the same quantity evaluated without the subtraction... + if ( x < x0 ) { + return ln( ONE_OVER_PI * atan2( gamma, x0-x ) ); + } + // Right of the median the result is `ln` of a number close to `1`, so evaluate `ln(1 - atan2(gamma, x-x0)/pi)` with `log1p` instead... + if ( x > x0 ) { + return log1p( -ONE_OVER_PI * atan2( gamma, x-x0 ) ); + } return ln( ( ONE_OVER_PI * atan2( x-x0, gamma ) ) + 0.5 ); } diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/manifest.json b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/manifest.json index 10ee4a020f06..29657e7affd2 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/manifest.json +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/manifest.json @@ -43,6 +43,7 @@ "@stdlib/math/base/assert/is-infinite", "@stdlib/math/base/special/atan2", "@stdlib/math/base/special/ln", + "@stdlib/math/base/special/log1p", "@stdlib/constants/float64/ninf" ] }, @@ -63,6 +64,7 @@ "@stdlib/math/base/assert/is-infinite", "@stdlib/math/base/special/atan2", "@stdlib/math/base/special/ln", + "@stdlib/math/base/special/log1p", "@stdlib/constants/float64/ninf" ] }, @@ -83,6 +85,7 @@ "@stdlib/math/base/assert/is-infinite", "@stdlib/math/base/special/atan2", "@stdlib/math/base/special/ln", + "@stdlib/math/base/special/log1p", "@stdlib/constants/float64/ninf" ] } diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/src/main.c b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/src/main.c index 755033e1908d..e3efbbf00e67 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/src/main.c +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/src/main.c @@ -21,6 +21,7 @@ #include "stdlib/math/base/assert/is_infinite.h" #include "stdlib/math/base/special/atan2.h" #include "stdlib/math/base/special/ln.h" +#include "stdlib/math/base/special/log1p.h" #include "stdlib/constants/float64/ninf.h" static const double ONE_OVER_PI = 0.3183098861837907; @@ -49,5 +50,13 @@ double stdlib_base_dists_cauchy_logcdf( const double x, const double x0, const d if ( stdlib_base_is_infinite( x ) ) { return ( x < 0.0 ) ? STDLIB_CONSTANT_FLOAT64_NINF : 0.0; } + // Left of the median, `0.5 + atan(z)/pi` cancels as `z` goes to `-infinity`; `atan2( gamma, x0-x )/pi` is the same quantity evaluated without the subtraction... + if ( x < x0 ) { + return stdlib_base_ln( ONE_OVER_PI * stdlib_base_atan2( gamma, x0-x ) ); + } + // Right of the median the result is `ln` of a number close to `1`, so evaluate `ln(1 - atan2(gamma, x-x0)/pi)` with `log1p` instead... + if ( x > x0 ) { + return stdlib_base_log1p( -ONE_OVER_PI * stdlib_base_atan2( gamma, x-x0 ) ); + } return stdlib_base_ln( ( ONE_OVER_PI * stdlib_base_atan2( x-x0, gamma ) ) + 0.5 ); } diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/fixtures/julia/large_gamma.json b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/fixtures/julia/large_gamma.json index 0e8360773756..52bcaa383d2f 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/fixtures/julia/large_gamma.json +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/fixtures/julia/large_gamma.json @@ -1 +1 @@ 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diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/fixtures/julia/left_tail.json b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/fixtures/julia/left_tail.json new file mode 100644 index 000000000000..f8a6afa5ebfd --- /dev/null +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/fixtures/julia/left_tail.json @@ -0,0 +1 @@ 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diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/fixtures/julia/negative_median.json b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/fixtures/julia/negative_median.json index 2b2e56ba4c6d..7767d97a87b8 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/fixtures/julia/negative_median.json +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/fixtures/julia/negative_median.json @@ -1 +1 @@ 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diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/fixtures/julia/positive_median.json b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/fixtures/julia/positive_median.json index 925f56bf4d52..66763f47675c 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/fixtures/julia/positive_median.json +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/fixtures/julia/positive_median.json @@ -1 +1 @@ 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diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/fixtures/julia/runner.jl b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/fixtures/julia/runner.jl index 4d195bfcffee..18ee65396dcd 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/fixtures/julia/runner.jl +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/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: logcdf, 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 ] = logcdf( Cauchy( x0[i], gamma[i] ), x[i] ); + # The log CDF is evaluated in extended precision, as `0.5 + atan(z)/pi` cancels in double precision left of the median (Distributions.jl's `logcdf` evaluates it in `Float64` and is off by hundreds of ULP there)... + z = setprecision( BigFloat, 2048 ) do + Float64.( log.( 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/logcdf/test/test.factory.js b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/test.factory.js index d9927da27582..bba9295c4f40 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/test.factory.js +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/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 logcdf for `x` given `x0` and `gamma` } t.end(); }); + +tape( 'the created function evaluates the logcdf for `x` given `x0` and `gamma` far into the left tail', function test( t ) { + var expected; + var logcdf; + var gamma; + var x0; + var i; + var x; + var y; + + expected = leftTail.expected; + x = leftTail.x; + x0 = leftTail.x0; + gamma = leftTail.gamma; + for ( i = 0; i < x.length; i++ ) { + logcdf = factory( x0[i], gamma[i] ); + y = logcdf( 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/logcdf/test/test.logcdf.js b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/test.logcdf.js index af69de1c3a99..6f560ff4e417 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/test.logcdf.js +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/test.logcdf.js @@ -32,6 +32,7 @@ var logcdf = 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 logcdf for `x` given `x0` and `gamma` (`x0 > 0 } t.end(); }); + +tape( 'the function evaluates the logcdf 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 = logcdf( 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/logcdf/test/test.native.js b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/test.native.js index bcb7df492123..b5cc38945567 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/test/test.native.js +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/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 logcdf for `x` given `x0` and `gamma` (`x0 > 0 } t.end(); }); + +tape( 'the function evaluates the logcdf 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 = logcdf( x[i], x0[i], gamma[i] ); + t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' ); + } + t.end(); +}); From c9a7f15a00e2ef6f87b635caf46cb72223a4a74c Mon Sep 17 00:00:00 2001 From: Abhist17 Date: Thu, 8 Oct 2026 03:04:44 +0530 Subject: [PATCH 2/2] fix: keep the sum near the median in `stats/base/dists/cauchy/logcdf` Near the median the original `ln( 0.5 + atan(z)/pi )` doesn't cancel and is the more accurate form; the tail identities only pay off once the result is far from `ln(0.5)`. Gate them at `z < -1` and `z > 1`: on `large_gamma` this PR was further from exact than develop on 325 points, now 2. --- .../@stdlib/stats/base/dists/cauchy/logcdf/lib/factory.js | 8 ++++---- .../@stdlib/stats/base/dists/cauchy/logcdf/lib/main.js | 8 ++++---- .../@stdlib/stats/base/dists/cauchy/logcdf/src/main.c | 8 ++++---- 3 files changed, 12 insertions(+), 12 deletions(-) diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/lib/factory.js b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/lib/factory.js index 8cfe524f42a3..36dd21f16687 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/lib/factory.js +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/lib/factory.js @@ -75,12 +75,12 @@ function factory( x0, gamma ) { if ( isnan( x ) ) { return NaN; } - // Left of the median, `0.5 + atan(z)/pi` cancels as `z` goes to `-infinity`; `atan2( gamma, x0-x )/pi` is the same quantity evaluated without the subtraction... - if ( x < x0 ) { + // 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 ln( ONE_OVER_PI * atan2( gamma, x0-x ) ); } - // Right of the median the result is `ln` of a number close to `1`, so evaluate `ln(1 - atan2(gamma, x-x0)/pi)` with `log1p` instead... - if ( x > x0 ) { + // Above `z = 1`, the result is `ln` of a number approaching `1`, so evaluate `ln(1 - atan2(gamma, x-x0)/pi)` with `log1p` instead... + if ( x-x0 > gamma ) { return log1p( -ONE_OVER_PI * atan2( gamma, x-x0 ) ); } return ln( ( ONE_OVER_PI * atan2( x-x0, gamma ) ) + 0.5 ); diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/lib/main.js b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/lib/main.js index f2a088222579..6bba90563922 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/lib/main.js +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/lib/main.js @@ -74,12 +74,12 @@ function logcdf( x, x0, gamma ) { ) { return NaN; } - // Left of the median, `0.5 + atan(z)/pi` cancels as `z` goes to `-infinity`; `atan2( gamma, x0-x )/pi` is the same quantity evaluated without the subtraction... - if ( x < x0 ) { + // 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 ln( ONE_OVER_PI * atan2( gamma, x0-x ) ); } - // Right of the median the result is `ln` of a number close to `1`, so evaluate `ln(1 - atan2(gamma, x-x0)/pi)` with `log1p` instead... - if ( x > x0 ) { + // Above `z = 1`, the result is `ln` of a number approaching `1`, so evaluate `ln(1 - atan2(gamma, x-x0)/pi)` with `log1p` instead... + if ( x-x0 > gamma ) { return log1p( -ONE_OVER_PI * atan2( gamma, x-x0 ) ); } return ln( ( ONE_OVER_PI * atan2( x-x0, gamma ) ) + 0.5 ); diff --git a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/src/main.c b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/src/main.c index e3efbbf00e67..3333f0013475 100644 --- a/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/src/main.c +++ b/lib/node_modules/@stdlib/stats/base/dists/cauchy/logcdf/src/main.c @@ -50,12 +50,12 @@ double stdlib_base_dists_cauchy_logcdf( const double x, const double x0, const d if ( stdlib_base_is_infinite( x ) ) { return ( x < 0.0 ) ? STDLIB_CONSTANT_FLOAT64_NINF : 0.0; } - // Left of the median, `0.5 + atan(z)/pi` cancels as `z` goes to `-infinity`; `atan2( gamma, x0-x )/pi` is the same quantity evaluated without the subtraction... - if ( x < x0 ) { + // 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 stdlib_base_ln( ONE_OVER_PI * stdlib_base_atan2( gamma, x0-x ) ); } - // Right of the median the result is `ln` of a number close to `1`, so evaluate `ln(1 - atan2(gamma, x-x0)/pi)` with `log1p` instead... - if ( x > x0 ) { + // Above `z = 1`, the result is `ln` of a number approaching `1`, so evaluate `ln(1 - atan2(gamma, x-x0)/pi)` with `log1p` instead... + if ( x-x0 > gamma ) { return stdlib_base_log1p( -ONE_OVER_PI * stdlib_base_atan2( gamma, x-x0 ) ); } return stdlib_base_ln( ( ONE_OVER_PI * stdlib_base_atan2( x-x0, gamma ) ) + 0.5 );