fix: avoid cancellation near multiples of 2π in math/base/special/cosm1

lib/node_modules/@stdlib/math/base/special/cosm1/lib/main.js

@@ -34,7 +34,7 @@
 
 // MODULES //
 
-var cos = require( '@stdlib/math/base/special/cos' );
+var sin = require( '@stdlib/math/base/special/sin' );
 var polyval = require( './polyval_p.js' );
 
 
@@ -73,8 +73,11 @@ var PIO4 = 7.85398163397448309616e-1; // 4/π
 */
 function cosm1( x ) {
 	var x2;
+	var s;
 	if ( x < -PIO4 || x > PIO4 ) {
-		return cos( x ) - 1.0;
+		// Outside `[-π/4,π/4]`, `cos(x) - 1` cancels where `cos(x)` is close to one (near multiples of 2π). Use the half-angle identity `cos(x) - 1 = -2 sin^2(x/2)` instead, which has no cancellation:
+		s = sin( x / 2.0 );
+		return -2.0 * s * s;
 	}
 	x2 = x * x;
 	return ( -0.5*x2 ) + ( x2*x2*polyval( x2 ) );

lib/node_modules/@stdlib/math/base/special/cosm1/manifest.json

@@ -37,7 +37,7 @@
       "libpath": [],
       "dependencies": [
         "@stdlib/math/base/napi/unary",
-        "@stdlib/math/base/special/cos"
+        "@stdlib/math/base/special/sin"
       ]
     },
     {
@@ -51,7 +51,7 @@
       "libraries": [],
       "libpath": [],
       "dependencies": [
-        "@stdlib/math/base/special/cos"
+        "@stdlib/math/base/special/sin"
       ]
     },
     {
@@ -65,7 +65,7 @@
       "libraries": [],
       "libpath": [],
       "dependencies": [
-        "@stdlib/math/base/special/cos"
+        "@stdlib/math/base/special/sin"
       ]
     }
   ]

lib/node_modules/@stdlib/math/base/special/cosm1/src/main.c

@@ -31,7 +31,7 @@
 */
 
 #include "stdlib/math/base/special/cosm1.h"
-#include "stdlib/math/base/special/cos.h"
+#include "stdlib/math/base/special/sin.h"
 
 static const double PIO4 = 7.85398163397448309616e-1; // 4/π
 
@@ -71,8 +71,11 @@ static double polyval_p( const double x ) {
 */
 double stdlib_base_cosm1( const double x ) {
 	double x2;
+	double s;
 	if ( x < -PIO4 || x > PIO4 ) {
-		return stdlib_base_cos( x ) - 1.0;
+		// Outside `[-π/4,π/4]`, `cos(x) - 1` cancels where `cos(x)` is close to one (near multiples of 2π). Use the half-angle identity `cos(x) - 1 = -2 sin^2(x/2)` instead, which has no cancellation:
+		s = stdlib_base_sin( x / 2.0 );
+		return -2.0 * s * s;
 	}
 	x2 = x * x;
 	return ( -0.5 * x2 ) + ( x2 * x2 * polyval_p( x2 ) );

lib/node_modules/@stdlib/math/base/special/cosm1/test/fixtures/julia/REQUIRE

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

lib/node_modules/@stdlib/math/base/special/cosm1/test/fixtures/julia/runner.jl

@@ -0,0 +1,82 @@
+#!/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
+
+"""
+	gen( domain, name )
+
+Generate fixture data and write to file.
+
+The cosine minus one is evaluated in extended precision, as `cos(x) - 1` cancels near multiples of 2π in `Float64`.
+
+# Arguments
+
+* `domain`: domain
+* `name::AbstractString`: output filename
+
+# Examples
+
+``` julia
+julia> x = range( 1.0, stop = 10.0, length = 2001 );
+julia> gen( x, "data.json" );
+```
+"""
+function gen( domain, name )
+	x = collect( domain );
+	y = setprecision( BigFloat, 256 ) do
+		Float64.( cos.( BigFloat.( x ) ) .- 1 );
+	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 );
+
+# Values outside `[-π/4,π/4]`:
+x = range( pi/4, stop = 4*pi, length = 1000 );
+gen( x, "positive.json" );
+
+x = range( -4*pi, stop = -pi/4, length = 1000 );
+gen( x, "negative.json" );
+
+# Values close to multiples of 2π:
+x = Float64[];
+for k in 1:10
+	for d in exp10.( range( -8.0, stop = -1.0, length = 50 ) )
+		push!( x, 2*pi*k + d, 2*pi*k - d, -2*pi*k + d, -2*pi*k - d );
+	end
+end
+gen( x, "near_two_pi.json" );

lib/node_modules/@stdlib/math/base/special/cosm1/test/test.js

@@ -21,25 +21,19 @@
 // MODULES //
 
 var tape = require( 'tape' );
-var linspace = require( '@stdlib/array/base/linspace' );
 var isnan = require( '@stdlib/math/base/assert/is-nan' );
-var cos = require( '@stdlib/math/base/special/cos' );
 var PINF = require( '@stdlib/constants/float64/pinf' );
 var NINF = require( '@stdlib/constants/float64/ninf' );
-var PI = require( '@stdlib/constants/float64/pi' );
 var isAlmostSameValue = require( '@stdlib/assert/is-almost-same-value' );
 var cosm1 = require( './../lib' );
 
 
-// VARIABLES //
-
-var NPIO4 = -7.85398163397448309616e-1; // -π/4
-var PIO4 = 7.85398163397448309616e-1; // π/4
-
-
 // FIXTURES //
 
 var data = require( './fixtures/python/data.json' );
+var positive = require( './fixtures/julia/positive.json' );
+var negative = require( './fixtures/julia/negative.json' );
+var nearTwoPi = require( './fixtures/julia/near_two_pi.json' );
 
 
 // TESTS //
@@ -67,28 +61,48 @@ tape( 'the function computes the cosine minus one more accurately inside the int
 	t.end();
 });
 
-tape( 'the function computes `cos(x) - 1.0` outside the interval [-π/4,π/4]', function test( t ) {
+tape( 'the function computes the cosine minus one outside the interval [-π/4,π/4] (positive values)', function test( t ) {
 	var expected;
 	var x;
 	var y;
 	var i;
 
-	x = linspace( PIO4, 4.0*PI, 100 );
-
+	x = positive.x;
+	expected = positive.expected;
 	for ( i = 0; i < x.length; i++ ) {
 		y = cosm1( x[i] );
-		expected = cos( x[i] ) - 1;
-		t.strictEqual( isAlmostSameValue( y, expected, 1 ), true, 'returns expected value' );
+		t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' );
 	}
+	t.end();
+});
 
-	x = linspace( -4.0*PI, NPIO4, 100 );
+tape( 'the function computes the cosine minus one outside the interval [-π/4,π/4] (negative values)', function test( t ) {
+	var expected;
+	var x;
+	var y;
+	var i;
 
+	x = negative.x;
+	expected = negative.expected;
 	for ( i = 0; i < x.length; i++ ) {
 		y = cosm1( x[i] );
-		expected = cos( x[i] ) - 1.0;
-		t.strictEqual( isAlmostSameValue( y, expected, 1 ), true, 'returns expected value' );
+		t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' );
 	}
+	t.end();
+});
+
+tape( 'the function computes the cosine minus one close to multiples of 2π', function test( t ) {
+	var expected;
+	var x;
+	var y;
+	var i;
 
+	x = nearTwoPi.x;
+	expected = nearTwoPi.expected;
+	for ( i = 0; i < x.length; i++ ) {
+		y = cosm1( x[i] );
+		t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' );
+	}
 	t.end();
 });
 

lib/node_modules/@stdlib/math/base/special/cosm1/test/test.native.js

@@ -22,20 +22,15 @@
 
 var resolve = require( 'path' ).resolve;
 var tape = require( 'tape' );
-var linspace = require( '@stdlib/array/base/linspace' );
 var isnan = require( '@stdlib/math/base/assert/is-nan' );
-var cos = require( '@stdlib/math/base/special/cos' );
 var PINF = require( '@stdlib/constants/float64/pinf' );
 var NINF = require( '@stdlib/constants/float64/ninf' );
-var PI = require( '@stdlib/constants/float64/pi' );
 var isAlmostSameValue = require( '@stdlib/assert/is-almost-same-value' );
 var tryRequire = require( '@stdlib/utils/try-require' );
 
 
 // VARIABLES //
 
-var NPIO4 = -7.85398163397448309616e-1; // -π/4
-var PIO4 = 7.85398163397448309616e-1; // π/4
 var cosm1 = tryRequire( resolve( __dirname, './../lib/native.js' ) );
 var opts = {
 	'skip': ( cosm1 instanceof Error )
@@ -45,6 +40,9 @@ var opts = {
 // FIXTURES //
 
 var data = require( './fixtures/python/data.json' );
+var positive = require( './fixtures/julia/positive.json' );
+var negative = require( './fixtures/julia/negative.json' );
+var nearTwoPi = require( './fixtures/julia/near_two_pi.json' );
 
 
 // TESTS //
@@ -72,28 +70,48 @@ tape( 'the function computes the cosine minus one more accurately inside the int
 	t.end();
 });
 
-tape( 'the function computes `cos(x) - 1.0` outside the interval [-π/4,π/4]', opts, function test( t ) {
+tape( 'the function computes the cosine minus one outside the interval [-π/4,π/4] (positive values)', opts, function test( t ) {
 	var expected;
 	var x;
 	var y;
 	var i;
 
-	x = linspace( PIO4, 4.0*PI, 100 );
-
+	x = positive.x;
+	expected = positive.expected;
 	for ( i = 0; i < x.length; i++ ) {
 		y = cosm1( x[i] );
-		expected = cos( x[i] ) - 1;
-		t.strictEqual( isAlmostSameValue( y, expected, 1 ), true, 'returns expected value' );
+		t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' );
 	}
+	t.end();
+});
 
-	x = linspace( -4.0*PI, NPIO4, 100 );
+tape( 'the function computes the cosine minus one outside the interval [-π/4,π/4] (negative values)', opts, function test( t ) {
+	var expected;
+	var x;
+	var y;
+	var i;
 
+	x = negative.x;
+	expected = negative.expected;
 	for ( i = 0; i < x.length; i++ ) {
 		y = cosm1( x[i] );
-		expected = cos( x[i] ) - 1.0;
-		t.strictEqual( isAlmostSameValue( y, expected, 1 ), true, 'returns expected value' );
+		t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' );
 	}
+	t.end();
+});
 
+tape( 'the function computes the cosine minus one close to multiples of 2π', opts, function test( t ) {
+	var expected;
+	var x;
+	var y;
+	var i;
+
+	x = nearTwoPi.x;
+	expected = nearTwoPi.expected;
+	for ( i = 0; i < x.length; i++ ) {
+		y = cosm1( x[i] );
+		t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'returns expected value' );
+	}
 	t.end();
 });