fix: add the missing sqrt(-ln(x)) >= 18 branch to math/base/special/erfcinv

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

@@ -43,6 +43,7 @@ var rationalFcnR2 = require( './rational_p2q2.js' );
 var rationalFcnR3 = require( './rational_p3q3.js' );
 var rationalFcnR4 = require( './rational_p4q4.js' );
 var rationalFcnR5 = require( './rational_p5q5.js' );
+var rationalFcnR6 = require( './rational_p6q6.js' );
 
 
 // VARIABLES //
@@ -52,6 +53,7 @@ var Y2 = 2.249481201171875;
 var Y3 = 8.07220458984375e-1;
 var Y4 = 9.3995571136474609375e-1;
 var Y5 = 9.8362827301025390625e-1;
+var Y6 = 9.9714565277099609375e-1;
 
 
 // MAIN //
@@ -214,9 +216,15 @@ function erfcinv( x ) {
 		return sign * ( (Y4*q) + (r*q) );
 	}
 	// q < 18
-	qs = q - 6.0;
-	r = rationalFcnR5( qs );
-	return sign * ( (Y5*q) + (r*q) );
+	if ( q < 18.0 ) {
+		qs = q - 6.0;
+		r = rationalFcnR5( qs );
+		return sign * ( (Y5*q) + (r*q) );
+	}
+	// q < 44 (the smallest double gives `q = 27.28...`)
+	qs = q - 18.0;
+	r = rationalFcnR6( qs );
+	return sign * ( (Y6*q) + (r*q) );
 }
 
 

lib/node_modules/@stdlib/math/base/special/erfcinv/lib/rational_p1q1.js

@@ -1,7 +1,7 @@
 /**
 * @license Apache-2.0
 *
-* Copyright (c) 2022 The Stdlib Authors.
+* 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.

lib/node_modules/@stdlib/math/base/special/erfcinv/lib/rational_p2q2.js

@@ -1,7 +1,7 @@
 /**
 * @license Apache-2.0
 *
-* Copyright (c) 2022 The Stdlib Authors.
+* 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.

lib/node_modules/@stdlib/math/base/special/erfcinv/lib/rational_p3q3.js

@@ -1,7 +1,7 @@
 /**
 * @license Apache-2.0
 *
-* Copyright (c) 2022 The Stdlib Authors.
+* 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.

lib/node_modules/@stdlib/math/base/special/erfcinv/lib/rational_p4q4.js

@@ -1,7 +1,7 @@
 /**
 * @license Apache-2.0
 *
-* Copyright (c) 2022 The Stdlib Authors.
+* 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.

lib/node_modules/@stdlib/math/base/special/erfcinv/lib/rational_p5q5.js

@@ -1,7 +1,7 @@
 /**
 * @license Apache-2.0
 *
-* Copyright (c) 2022 The Stdlib Authors.
+* 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.

lib/node_modules/@stdlib/math/base/special/erfcinv/lib/rational_p6q6.js

@@ -0,0 +1,64 @@
+/**
+* @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.
+*/
+
+/* This is a generated file. Do not edit directly. */
+'use strict';
+
+// MAIN //
+
+/**
+* Evaluates a rational function (i.e., the ratio of two polynomials described by the coefficients stored in \\(P\\) and \\(Q\\)).
+*
+* ## Notes
+*
+* -   Coefficients should be sorted in ascending degree.
+* -   The implementation uses [Horner's rule][horners-method] for efficient computation.
+*
+* [horners-method]: https://en.wikipedia.org/wiki/Horner%27s_method
+*
+* @private
+* @param {number} x - value at which to evaluate the rational function
+* @returns {number} evaluated rational function
+*/
+function evalrational( x ) {
+	var ax;
+	var s1;
+	var s2;
+	if ( x === 0.0 ) {
+		return -0.002497821279189813;
+	}
+	if ( x < 0.0 ) {
+		ax = -x;
+	} else {
+		ax = x;
+	}
+	if ( ax <= 1.0 ) {
+		s1 = -0.002497821279189813 + (x * (-0.00000779190719229054 + (x * (0.000025472303741302746 + (x * (0.0000016239777734251093 + (x * (3.963410113048012e-8 + (x * (4.116328311909442e-10 + (x * (1.4559628671867504e-12 + (x * -1.1676501239718427e-18))))))))))))); // eslint-disable-line max-len
+		s2 = 1.0 + (x * (0.2071231122144225 + (x * (0.01694108381209759 + (x * (0.0006905382656226846 + (x * (0.000014500735981823264 + (x * (1.4443775662814415e-7 + (x * (5.097612765997785e-10 + (x * 0.0))))))))))))); // eslint-disable-line max-len
+	} else {
+		x = 1.0 / x;
+		s1 = -1.1676501239718427e-18 + (x * (1.4559628671867504e-12 + (x * (4.116328311909442e-10 + (x * (3.963410113048012e-8 + (x * (0.0000016239777734251093 + (x * (0.000025472303741302746 + (x * (-0.00000779190719229054 + (x * -0.002497821279189813))))))))))))); // eslint-disable-line max-len
+		s2 = 0.0 + (x * (5.097612765997785e-10 + (x * (1.4443775662814415e-7 + (x * (0.000014500735981823264 + (x * (0.0006905382656226846 + (x * (0.01694108381209759 + (x * (0.2071231122144225 + (x * 1.0))))))))))))); // eslint-disable-line max-len
+	}
+	return s1 / s2;
+}
+
+
+// EXPORTS //
+
+module.exports = evalrational;

lib/node_modules/@stdlib/math/base/special/erfcinv/scripts/evalrational.js

@@ -39,7 +39,7 @@
 
 // Polynomial coefficients ordered in ascending degree...
 
 // Coefficients for erfinv on [0, 0.5]:
 var P1 = [
 	-5.08781949658280665617e-4,
 	-8.36874819741736770379e-3,
@@ -65,7 +65,7 @@
 	8.86216390456424707504e-4
 ];
 
 // Coefficients for erfinv for 0.5 > 1-x >= 0:
 var P2 = [
 	-2.02433508355938759655e-1,
 	1.05264680699391713268e-1,
@@ -89,7 +89,7 @@
 	1.72114765761200282724
 ];
 
 // Coefficients for erfinv for sqrt( -log(1-x) ):
 var P3 = [
 	-1.31102781679951906451e-1,
 	-1.63794047193317060787e-1,
@@ -163,6 +163,27 @@
 	0.0
 ];
 
+var P6 = [
+	-2.4978212791898131227e-3,
+	-7.79190719229053954292e-6,
+	2.54723037413027451751e-5,
+	1.62397777342510920873e-6,
+	3.96341011304801168516e-8,
+	4.11632831190944208473e-10,
+	1.45596286718675035587e-12,
+	-1.16765012397184275695e-18
+];
+var Q6 = [
+	1.0,
+	2.07123112214422517181e-1,
+	1.69410838120975906478e-2,
+	6.90538265622684595676e-4,
+	1.45007359818232637924e-5,
+	1.44437756628144157666e-7,
+	5.09761276599778486139e-10,
+	0.0
+];
+
 // Header to add to output files:
 var header = licenseHeader( 'Apache-2.0', 'js', {
 	'year': currentYear(),
@@ -218,31 +239,35 @@
 
 	fpath = resolve( __dirname, '..', 'lib', 'rational_p1q1.js' );
 	str = header + compile( P1, Q1 );
 	writeFileSync( fpath, str, opts );
 
 	fpath = resolve( __dirname, '..', 'lib', 'rational_p2q2.js' );
 	str = header + compile( P2, Q2 );
 	writeFileSync( fpath, str, opts );
 
 	fpath = resolve( __dirname, '..', 'lib', 'rational_p3q3.js' );
 	str = header + compile( P3, Q3 );
 	writeFileSync( fpath, str, opts );
 
 	fpath = resolve( __dirname, '..', 'lib', 'rational_p4q4.js' );
 	str = header + compile( P4, Q4 );
 	writeFileSync( fpath, str, opts );
 
 	fpath = resolve( __dirname, '..', 'lib', 'rational_p5q5.js' );
 	str = header + compile( P5, Q5 );
 	writeFileSync( fpath, str, opts );
 
+	fpath = resolve( __dirname, '..', 'lib', 'rational_p6q6.js' );
+	str = header + compile( P6, Q6 );
+	writeFileSync( fpath, str, opts );
+
 	copts = {
 		'dtype': 'double',
 		'name': ''
 	};
 
 	fpath = resolve( __dirname, '..', 'src', 'main.c' );
 	file = readFileSync( fpath, opts );
 
 	copts.name = 'rational_p1q1';
 	str = compileC( P1, Q1, copts );
@@ -264,6 +289,10 @@
 	str = compileC( P5, Q5, copts );
 	file = insert( file, copts.name, str );
 
+	copts.name = 'rational_p6q6';
+	str = compileC( P6, Q6, copts );
+	file = insert( file, copts.name, str );
+
 	writeFileSync( fpath, file, opts );
 }
 

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

@@ -43,6 +43,7 @@ static const double Y2 = 2.249481201171875;
 static const double Y3 = 8.07220458984375e-1;
 static const double Y4 = 9.3995571136474609375e-1;
 static const double Y5 = 9.8362827301025390625e-1;
+static const double Y6 = 9.9714565277099609375e-1;
 
 /* Begin auto-generated functions. The following functions are auto-generated. Do not edit directly. */
 
@@ -251,6 +252,47 @@ static double rational_p5q5( const double x ) {
 
 // END: rational_p5q5
 
+// BEGIN: rational_p6q6
+
+/**
+* Evaluates a rational function (i.e., the ratio of two polynomials described by the coefficients stored in \\(P\\) and \\(Q\\)).
+*
+* ## Notes
+*
+* -   Coefficients should be sorted in ascending degree.
+* -   The implementation uses [Horner's rule][horners-method] for efficient computation.
+*
+* [horners-method]: https://en.wikipedia.org/wiki/Horner%27s_method
+*
+* @param x    value at which to evaluate the rational function
+* @return     evaluated rational function
+*/
+static double rational_p6q6( const double x ) {
+	double ax;
+	double ix;
+	double s1;
+	double s2;
+	if ( x == 0.0 ) {
+		return -0.002497821279189813;
+	}
+	if ( x < 0.0 ) {
+		ax = -x;
+	} else {
+		ax = x;
+	}
+	if ( ax <= 1.0 ) {
+		s1 = -0.002497821279189813 + (x * (-0.00000779190719229054 + (x * (0.000025472303741302746 + (x * (0.0000016239777734251093 + (x * (3.963410113048012e-8 + (x * (4.116328311909442e-10 + (x * (1.4559628671867504e-12 + (x * -1.1676501239718427e-18)))))))))))));
+		s2 = 1.0 + (x * (0.2071231122144225 + (x * (0.01694108381209759 + (x * (0.0006905382656226846 + (x * (0.000014500735981823264 + (x * (1.4443775662814415e-7 + (x * (5.097612765997785e-10 + (x * 0.0)))))))))))));
+	} else {
+		ix = 1.0 / x;
+		s1 = -1.1676501239718427e-18 + (ix * (1.4559628671867504e-12 + (ix * (4.116328311909442e-10 + (ix * (3.963410113048012e-8 + (ix * (0.0000016239777734251093 + (ix * (0.000025472303741302746 + (ix * (-0.00000779190719229054 + (ix * -0.002497821279189813)))))))))))));
+		s2 = 0.0 + (ix * (5.097612765997785e-10 + (ix * (1.4443775662814415e-7 + (ix * (0.000014500735981823264 + (ix * (0.0006905382656226846 + (ix * (0.01694108381209759 + (ix * (0.2071231122144225 + (ix * 1.0)))))))))))));
+	}
+	return s1 / s2;
+}
+
+// END: rational_p6q6
+
 /* End auto-generated functions. */
 
 /**
@@ -396,7 +438,13 @@ double stdlib_base_erfcinv( const double x ) {
 		return sign * ( ( Y4 * q ) + ( r * q ) );
 	}
 	// q < 18
-	qs = q - 6.0;
-	r = rational_p5q5( qs );
-	return sign * ( ( Y5 * q ) + ( r * q ) );
+	if ( q < 18.0 ) {
+		qs = q - 6.0;
+		r = rational_p5q5( qs );
+		return sign * ( ( Y5 * q ) + ( r * q ) );
+	}
+	// q < 44 (the smallest double gives `q = 27.28...`)
+	qs = q - 18.0;
+	r = rational_p6q6( qs );
+	return sign * ( ( Y6 * q ) + ( r * q ) );
 }

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

@@ -20,7 +20,7 @@ import JSON
 using SpecialFunctions
 
 """
-    gen( x, filepath )
+	gen( x, filepath )
 
 Generate fixture data and write to file.
 
@@ -53,6 +53,35 @@ function gen( x, filepath )
 	close( outfile );
 end
 
+"""
+	gen_tiny( x, filepath )
+
+Generate fixture data for tiny `x` in extended precision and write to file.
+
+# Arguments
+
+* `x`: domain
+* `filepath::AbstractString`: filepath of the output file
+"""
+function gen_tiny( x, filepath )
+	y = Array{Float64}( undef, length(x) );
+	setprecision( BigFloat, 2048 ) do
+		for i in eachindex(x)
+			y[i] = Float64( erfcinv( BigFloat( x[i] ) ) );
+		end
+	end
+
+	data = Dict([
+		("x", x),
+		("expected", y)
+	]);
+
+	outfile = open( filepath, "w" );
+	write( outfile, JSON.json(data) );
+	write( outfile, "\n" );
+	close( outfile );
+end
+
 # Get the filename:
 file = @__FILE__;
 
@@ -93,3 +122,8 @@ gen( x, out );
 x = range( 1.9999999999999998, stop = 2, length = 500 );
 out = joinpath( dir, "x_1.9999..8_2.json" );
 gen( x, out );
+
+# Tiny values, `sqrt(-ln(x)) >= 6` (log-spaced down to the smallest subnormal):
+x = vcat( exp10.( range( -15.0, stop = -323.0, length = 1000 ) ), [ 5.0e-324 ] );
+out = joinpath( dir, "x_tiny.json" );
+gen_tiny( x, out );

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

@@ -38,6 +38,7 @@ var x4 = require( './fixtures/julia/x_1.75_1.9998.json' );
 var x5 = require( './fixtures/julia/x_0.0002_0.25.json' );
 var x6 = require( './fixtures/julia/x_1.9998_1.9999..8.json' );
 var x7 = require( './fixtures/julia/x_1.9999..8_2.json' );
+var x8 = require( './fixtures/julia/x_tiny.json' );
 
 
 // TESTS //
@@ -200,3 +201,18 @@ tape( 'the function evaluates the inverse complementary error function for `x` o
 	}
 	t.end();
 });
+
+tape( 'the function evaluates the inverse complementary error function for tiny `x`', function test( t ) {
+	var expected;
+	var x;
+	var y;
+	var i;
+
+	expected = x8.expected;
+	x = x8.x;
+	for ( i = 0; i < x.length; i++ ) {
+		y = erfcinv( x[i] );
+		t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'x: '+x[i]+'. y: '+y+'. E: '+expected[i] );
+	}
+	t.end();
+});

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

@@ -39,6 +39,7 @@ var x4 = require( './fixtures/julia/x_1.75_1.9998.json' );
 var x5 = require( './fixtures/julia/x_0.0002_0.25.json' );
 var x6 = require( './fixtures/julia/x_1.9998_1.9999..8.json' );
 var x7 = require( './fixtures/julia/x_1.9999..8_2.json' );
+var x8 = require( './fixtures/julia/x_tiny.json' );
 
 
 // VARIABLES //
@@ -209,3 +210,18 @@ tape( 'the function evaluates the inverse complementary error function for `x` o
 	}
 	t.end();
 });
+
+tape( 'the function evaluates the inverse complementary error function for tiny `x`', opts, function test( t ) {
+	var expected;
+	var x;
+	var y;
+	var i;
+
+	expected = x8.expected;
+	x = x8.x;
+	for ( i = 0; i < x.length; i++ ) {
+		y = erfcinv( x[i] );
+		t.strictEqual( isAlmostSameValue( y, expected[ i ], 2 ), true, 'x: '+x[i]+'. y: '+y+'. E: '+expected[i] );
+	}
+	t.end();
+});