fix: avoid cancellation for large negative m in math/base/special/ellipk

lib/node_modules/@stdlib/math/base/special/ellipk/lib/main.js

@@ -137,14 +137,18 @@ function ellipk( m ) {
 	var x;
 
 	x = m;
+	td = 1.0 - m;
 	if ( m < 0.0 ) {
 		x = m / ( m - 1.0 );
+
+		// Form `1-x` as `1/(1-m)` rather than by subtracting `x` from one, which cancels as `m` grows more negative and gives `x = 1` once `m < -2^53`:
+		td = 1.0 / ( 1.0 - m );
 		FLG = true;
 	}
 	if ( x === 0.0 ) {
 		return HALF_PI;
 	}
-	if ( x === 1.0 ) {
+	if ( td === 0.0 && !FLG ) {
 		return PINF;
 	}
 	if ( x > 1.0 ) {
@@ -171,7 +175,6 @@ function ellipk( m ) {
 	} else if ( x < 0.9 ) {
 		t = poly10( x - 0.875 );
 	} else {
-		td = 1.0 - x;
 		qd = poly11( td );
 		kdm = poly12( td - 0.05 );
 		t = -ln( qd ) * ( kdm * ONE_DIV_PI );

lib/node_modules/@stdlib/math/base/special/ellipk/src/main.c

@@ -348,14 +348,18 @@ double stdlib_base_ellipk( const double m ) {
 
 	FLG = 0;
 	x = m;
+	td = 1.0 - m;
 	if ( m < 0.0 ) {
 		x = m / ( m - 1.0 );
+
+		// Form `1-x` as `1/(1-m)` rather than by subtracting `x` from one, which cancels as `m` grows more negative and gives `x = 1` once `m < -2^53`:
+		td = 1.0 / ( 1.0 - m );
 		FLG += 1;
 	}
 	if ( x == 0.0 ) {
 		return STDLIB_CONSTANT_FLOAT64_HALF_PI;
 	}
-	if ( x == 1.0 ) {
+	if ( td == 0.0 && FLG == 0 ) {
 		return STDLIB_CONSTANT_FLOAT64_PINF;
 	}
 	if ( x > 1.0 ) {
@@ -382,7 +386,6 @@ double stdlib_base_ellipk( const double m ) {
 	} else if ( x < 0.9 ) {
 		t = poly_p10( x - 0.875 );
 	} else {
-		td = 1.0 - x;
 		qd = poly_p11( td );
 		kdm = poly_p12( td - 0.05 );
 		t = -stdlib_base_ln( qd ) * ( kdm * ONE_DIV_PI );

lib/node_modules/@stdlib/math/base/special/ellipk/test/fixtures/cpp/runner.cpp

@@ -330,8 +330,6 @@ int main( void ) {
 	rand_array_f64( x, len, -1.0, 0.0);
 	generate( x, len, "medium_negative.json" );
 
-	rand_array_f64( x, len, -1.0e10, -1.0);
-	generate( x, len, "large_negative.json" );
 
 	// Free allocated memory:
 	free( x );

lib/node_modules/@stdlib/math/base/special/ellipk/test/fixtures/julia/REQUIRE

@@ -0,0 +1,2 @@
+julia 1.5
+JSON 0.21

lib/node_modules/@stdlib/math/base/special/ellipk/test/fixtures/julia/runner.jl

@@ -0,0 +1,96 @@
+#!/usr/bin/env julia
+#
+# @license Apache-2.0
+#
+# Copyright (c) 2026 The Stdlib Authors.
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#    http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+import JSON
+
+"""
+	ellipk( m )
+
+Evaluate the complete elliptic integral of the first kind in extended precision.
+
+# Arguments
+
+* `m`: parameter
+
+# Examples
+
+``` julia
+julia> y = ellipk( BigFloat( -1.0 ) );
+```
+"""
+function ellipk( m )
+	# Use `K(m) = π / ( 2 * agm( 1, sqrt(1-m) ) )`, which holds for all `m < 1` without transforming a negative `m` into a parameter close to one:
+	a = BigFloat( 1 );
+	b = sqrt( 1 - m );
+	while abs( a - b ) > eps( a ) * 4
+		a, b = ( a + b ) / 2, sqrt( a * b );
+	end
+	return BigFloat( pi ) / ( a + b );
+end
+
+"""
+	gen( x, name )
+
+Generate fixture data and write to file.
+
+# Arguments
+
+* `x`: input values
+* `name::AbstractString`: output filename
+
+# Examples
+
+``` julia
+julia> x = -rand( 1000 ) .* 1.0e10;
+julia> gen( x, "data.json" );
+```
+"""
+function gen( x, name )
+	y = setprecision( BigFloat, 2048 ) do
+		Float64.( ellipk.( BigFloat.( x ) ) );
+	end
+
+	# Store data to be written to file as a collection:
+	data = Dict([
+		("x", x),
+		("expected", y)
+	]);
+
+	# Based on the script directory, create an output filepath:
+	filepath = joinpath( dir, name );
+
+	# Write the data to the output filepath as JSON:
+	outfile = open( filepath, "w" );
+	write( outfile, JSON.json(data) );
+	write( outfile, "\n" );
+	close( outfile );
+end
+
+# Get the filename:
+file = @__FILE__;
+
+# Extract the directory in which this file resides:
+dir = dirname( file );
+
+# Large negative values:
+x = -( rand( 4000 ) .* ( 1.0e10 - 1.0 ) .+ 1.0 );
+gen( x, "large_negative.json" );
+
+# Huge negative values (below `-2^53`, `m/(m-1)` rounds to one):
+x = -exp10.( range( 10.0, stop = 300.0, length = 1000 ) );
+gen( x, "huge_negative.json" );

lib/node_modules/@stdlib/math/base/special/ellipk/test/test.js

@@ -34,7 +34,8 @@
 var mediumPositive = require( './fixtures/cpp/medium_positive.json' );
 var closeToUnity = require( './fixtures/cpp/close_to_unity.json' );
 var mediumNegative = require( './fixtures/cpp/medium_negative.json' );
-var largeNegative = require( './fixtures/cpp/large_negative.json' );
+var largeNegative = require( './fixtures/julia/large_negative.json' );
+var hugeNegative = require( './fixtures/julia/huge_negative.json' );
 
 
 // TESTS //
@@ -72,7 +73,7 @@
 
 	for ( i = 0; i < x.length; i++ ) {
 		y = ellipk( x[i] );
 		t.strictEqual( isAlmostSameValue( y, expected[ i ], 12065 ), true, 'returns expected value' ); // FIXME: investigate the reason for this large discrepancy
 	}
 	t.end();
 });
@@ -104,7 +105,23 @@
 
 	for ( i = 0; i < x.length; i++ ) {
 		y = ellipk( x[i] );
-		t.strictEqual( isAlmostSameValue( y, expected[ i ], 204773163 ), true, 'returns expected value' ); // FIXME: investigate the reason for this large discrepancy
+		t.strictEqual( isAlmostSameValue( y, expected[ i ], 5 ), true, 'returns expected value' );
+	}
+	t.end();
+});
+
+tape( 'the function evaluates the complete elliptic integral of the first kind (huge negative values)', function test( t ) {
+	var expected;
+	var x;
+	var y;
+	var i;
+
+	expected = hugeNegative.expected;
+	x = hugeNegative.x;
+
+	for ( i = 0; i < x.length; i++ ) {
+		y = ellipk( x[i] );
+		t.strictEqual( isAlmostSameValue( y, expected[ i ], 4 ), true, 'returns expected value' );
 	}
 	t.end();
 });

lib/node_modules/@stdlib/math/base/special/ellipk/test/test.native.js

@@ -35,7 +35,8 @@
 var mediumPositive = require( './fixtures/cpp/medium_positive.json' );
 var closeToUnity = require( './fixtures/cpp/close_to_unity.json' );
 var mediumNegative = require( './fixtures/cpp/medium_negative.json' );
-var largeNegative = require( './fixtures/cpp/large_negative.json' );
+var largeNegative = require( './fixtures/julia/large_negative.json' );
+var hugeNegative = require( './fixtures/julia/huge_negative.json' );
 
 
 // VARIABLES //
@@ -81,7 +82,7 @@
 
 	for ( i = 0; i < x.length; i++ ) {
 		y = ellipk( x[i] );
 		t.strictEqual( isAlmostSameValue( y, expected[ i ], 12065 ), true, 'returns expected value' ); // FIXME: investigate the reason for this large discrepancy
 	}
 	t.end();
 });
@@ -113,7 +114,23 @@
 
 	for ( i = 0; i < x.length; i++ ) {
 		y = ellipk( x[i] );
-		t.strictEqual( isAlmostSameValue( y, expected[ i ], 204773163 ), true, 'returns expected value' ); // FIXME: investigate the reason for this large discrepancy
+		t.strictEqual( isAlmostSameValue( y, expected[ i ], 5 ), true, 'returns expected value' );
+	}
+	t.end();
+});
+
+tape( 'the function evaluates the complete elliptic integral of the first kind (huge negative values)', opts, function test( t ) {
+	var expected;
+	var x;
+	var y;
+	var i;
+
+	expected = hugeNegative.expected;
+	x = hugeNegative.x;
+
+	for ( i = 0; i < x.length; i++ ) {
+		y = ellipk( x[i] );
+		t.strictEqual( isAlmostSameValue( y, expected[ i ], 4 ), true, 'returns expected value' );
 	}
 	t.end();
 });