From 5ba85da29e51a6c0eb2fc5f08a8c897ece83aecf Mon Sep 17 00:00:00 2001 From: Abhist17 Date: Tue, 13 Oct 2026 19:05:02 +0530 Subject: [PATCH] =?UTF-8?q?fix:=20avoid=20cancellation=20near=20multiples?= =?UTF-8?q?=20of=202=CF=80=20in=20`math/base/special/cosm1`?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Outside `[-π/4,π/4]`, `cosm1` returned `cos(x) - 1`, which cancels where `cos(x)` is close to one; near `x = 2πk` the result could be off by about 2^30 ULPs. Use `-2 sin^2(x/2)` there instead, the same half-angle identity as #15609 uses for `versin`. The test for that range compared against `cos(x) - 1` itself; it now uses fixtures evaluated in BigFloat, including values close to multiples of 2π. Signed-off-by: Abhist17 --- .../math/base/special/cosm1/lib/main.js | 7 +- .../math/base/special/cosm1/manifest.json | 6 +- .../math/base/special/cosm1/src/main.c | 7 +- .../special/cosm1/test/fixtures/julia/REQUIRE | 2 + .../test/fixtures/julia/near_two_pi.json | 1 + .../cosm1/test/fixtures/julia/negative.json | 1 + .../cosm1/test/fixtures/julia/positive.json | 1 + .../cosm1/test/fixtures/julia/runner.jl | 82 +++++++++++++++++++ .../math/base/special/cosm1/test/test.js | 48 +++++++---- .../base/special/cosm1/test/test.native.js | 44 +++++++--- 10 files changed, 162 insertions(+), 37 deletions(-) create mode 100644 lib/node_modules/@stdlib/math/base/special/cosm1/test/fixtures/julia/REQUIRE create mode 100644 lib/node_modules/@stdlib/math/base/special/cosm1/test/fixtures/julia/near_two_pi.json create mode 100644 lib/node_modules/@stdlib/math/base/special/cosm1/test/fixtures/julia/negative.json create mode 100644 lib/node_modules/@stdlib/math/base/special/cosm1/test/fixtures/julia/positive.json create mode 100644 lib/node_modules/@stdlib/math/base/special/cosm1/test/fixtures/julia/runner.jl diff --git a/lib/node_modules/@stdlib/math/base/special/cosm1/lib/main.js b/lib/node_modules/@stdlib/math/base/special/cosm1/lib/main.js index f2d2e1a3426f..e020dd76742a 100644 --- a/lib/node_modules/@stdlib/math/base/special/cosm1/lib/main.js +++ b/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 ) ); diff --git a/lib/node_modules/@stdlib/math/base/special/cosm1/manifest.json b/lib/node_modules/@stdlib/math/base/special/cosm1/manifest.json index fec32d6cb6ec..083d8fc81535 100644 --- a/lib/node_modules/@stdlib/math/base/special/cosm1/manifest.json +++ b/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" ] } ] diff --git a/lib/node_modules/@stdlib/math/base/special/cosm1/src/main.c b/lib/node_modules/@stdlib/math/base/special/cosm1/src/main.c index ec4290169883..760d180cf86f 100644 --- a/lib/node_modules/@stdlib/math/base/special/cosm1/src/main.c +++ b/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 ) ); diff --git a/lib/node_modules/@stdlib/math/base/special/cosm1/test/fixtures/julia/REQUIRE b/lib/node_modules/@stdlib/math/base/special/cosm1/test/fixtures/julia/REQUIRE new file mode 100644 index 000000000000..308c3be89c85 --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/cosm1/test/fixtures/julia/REQUIRE @@ -0,0 +1,2 @@ +julia 1.5 +JSON 0.21 diff --git a/lib/node_modules/@stdlib/math/base/special/cosm1/test/fixtures/julia/near_two_pi.json b/lib/node_modules/@stdlib/math/base/special/cosm1/test/fixtures/julia/near_two_pi.json new file mode 100644 index 000000000000..d6a36d2202d4 --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/cosm1/test/fixtures/julia/near_two_pi.json @@ -0,0 +1 @@ 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diff --git a/lib/node_modules/@stdlib/math/base/special/cosm1/test/fixtures/julia/negative.json b/lib/node_modules/@stdlib/math/base/special/cosm1/test/fixtures/julia/negative.json new file mode 100644 index 000000000000..c77f5e55f62b --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/cosm1/test/fixtures/julia/negative.json @@ -0,0 +1 @@ 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diff --git a/lib/node_modules/@stdlib/math/base/special/cosm1/test/fixtures/julia/positive.json b/lib/node_modules/@stdlib/math/base/special/cosm1/test/fixtures/julia/positive.json new file mode 100644 index 000000000000..7faf6d921835 --- /dev/null +++ b/lib/node_modules/@stdlib/math/base/special/cosm1/test/fixtures/julia/positive.json @@ -0,0 +1 @@ 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diff --git a/lib/node_modules/@stdlib/math/base/special/cosm1/test/fixtures/julia/runner.jl b/lib/node_modules/@stdlib/math/base/special/cosm1/test/fixtures/julia/runner.jl new file mode 100644 index 000000000000..c9c37a5c3666 --- /dev/null +++ b/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" ); diff --git a/lib/node_modules/@stdlib/math/base/special/cosm1/test/test.js b/lib/node_modules/@stdlib/math/base/special/cosm1/test/test.js index 1cd0004b12ff..7a17eb0d8809 100644 --- a/lib/node_modules/@stdlib/math/base/special/cosm1/test/test.js +++ b/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(); }); diff --git a/lib/node_modules/@stdlib/math/base/special/cosm1/test/test.native.js b/lib/node_modules/@stdlib/math/base/special/cosm1/test/test.native.js index b20865ee047c..b0dc67ddfb6a 100644 --- a/lib/node_modules/@stdlib/math/base/special/cosm1/test/test.native.js +++ b/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(); });