refactor: improve the implementation of math/base/special/hyp2f1

lib/node_modules/@stdlib/math/base/special/hyp2f1/lib/hyp2f1negcequalbc.js

@@ -0,0 +1,81 @@
+/**
+* @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.
+*
+*
+* ## Notice
+*
+* The original C code and copyright notice are from the [Cephes Mathematical Library]{@link https://www.netlib.org/cephes/}. The implementation has been modified for JavaScript.
+*
+* ```text
+* (C) Copyright Stephen L. Moshier 1984, 1987, 1992, 2000.
+*
+* Use, modification and distribution are subject to the
+* Cephes Mathematical Library License. (See accompanying file
+* LICENSE or copy at https://smath.com/en-US/view/CephesMathLibrary/license)
+* ```
+*/
+
+'use strict';
+
+// MODULES //
+
+var abs = require( '@stdlib/math/base/special/abs' );
+var max = require( '@stdlib/math/base/special/max' );
+
+
+// MAIN //
+
+/**
+* Evaluates 2F1(a, b; b; x) when `b = c` is a negative integer using AMS55 #15.4.2.
+*
+* @private
+* @param {number} a - first parameter
+* @param {number} b - second parameter (equals c, a non-positive integer)
+* @param {number} x - argument
+* @returns {number} function value, or NaN if precision is insufficient
+*/
+function hyp2f1NegCEqualBC( a, b, x ) {
+	var collectorMax;
+	var collector;
+	var sum;
+	var k;
+
+	collectorMax = 1.0;
+	collector = 1.0;
+	sum = 1.0;
+
+	if ( !( abs( b ) < 1.0e5 ) ) {
+		return NaN;
+	}
+
+	for ( k = 1; k <= -b; k++ ) {
+		collector *= ( a + k - 1.0 ) * x / k;
+		collectorMax = max( abs( collector ), collectorMax );
+		sum += collector;
+	}
+
+	if ( 1.0e-16 * ( 1.0 + ( collectorMax / abs( sum ) ) ) > 1.0e-7 ) {
+		return NaN;
+	}
+
+	return sum;
+}
+
+
+// EXPORTS //
+
+module.exports = hyp2f1NegCEqualBC;

lib/node_modules/@stdlib/math/base/special/hyp2f1/lib/hyt2f1.js

@@ -70,7 +70,7 @@
 * @param {number} loss - starting loss of significance
 * @returns {Object} the function value and error
 */
 function hyt2f1( a, b, c, x, loss ) {
 	var negIntA;
 	var negIntB;
 	var qVal;
@@ -122,7 +122,7 @@
 	d = c - a - b;
 	id = round( d );
 
-	if ( x > 0.9 && !negIntA && !negIntB ) {
+	if ( x > 0.85 && !negIntA && !negIntB ) {
 		if ( isInteger( d ) === false ) {
 			// Try the power series first:
 			y = hys2f1( a, b, c, x, err );
@@ -223,7 +223,11 @@
 				y = -y;
 			}
 			q = pow( s, id );
-			y = ( id > 0.0 ) ? y*q : y1*q;
+			if ( id > 0.0 ) {
+				y *= q;
+			} else {
+				y1 *= q;
+			}
 			y += y1;
 		}
 		return {

lib/node_modules/@stdlib/math/base/special/hyp2f1/lib/main.js

@@ -40,6 +40,7 @@
 var pow = require( '@stdlib/math/base/special/pow' );
 var abs = require( '@stdlib/math/base/special/abs' );
 var isNonPositiveInteger = require( './isnonpositiveinteger.js' );
+var hyp2f1NegCEqualBC = require( './hyp2f1negcequalbc.js' );
 var isInteger = require( './isinteger.js' );
 var hys2f1 = require( './hys2f1.js' );
 var hyt2f1 = require( './hyt2f1.js' );
@@ -90,7 +91,7 @@
 * var v = hyp2f1( 1.0, NaN, 2.0, 0.5 );
 * // returns NaN
 */
 function hyp2f1( a, b, c, x ) {
 	var negIntCaOrCb;
 	var negIntC;
 	var negIntB;
@@ -156,6 +157,10 @@
 	if ( ax < 1.0 || x === -1.0 ) {
 		if ( b === c ) {
 			// 2F1(a,b;b;x) = (1-x)**(-a):
+			if ( negIntB ) {
+				// For negative integer b=c use the finite polynomial (AMS55 #15.4.2):
+				return hyp2f1NegCEqualBC( a, b, x );
+			}
 			return pow( s, -a );
 		}
 		if ( a === c ) {

lib/node_modules/@stdlib/math/base/special/hyp2f1/test/fixtures/python/edge_cases5.json

@@ -0,0 +1 @@
+{"a": [1.5, 1.5, 2.5, 0.5, 1.5, 0.5, 2.0, 1.5, 2.5, 1.5], "b": [2.5, 2.5, 1.5, 2.5, 2.5, 1.5, 3.0, 3.5, 2.5, 3.5], "c": [3.0, 3.0, 3.0, 2.0, 2.0, 1.0, 4.0, 4.0, 4.0, 3.0], "x": [0.9, 0.95, 0.9, 0.92, 0.91, 0.9, 0.9, 0.88, 0.93, 0.91], "expected": [14.663397526525134, 30.949523656432504, 14.663397526525134, 6.507774205759818, 106.96827020610523, 7.033214388515233, 21.78942310292968, 13.41253423166626, 36.54504530314331, 88.86455455103395]}

lib/node_modules/@stdlib/math/base/special/hyp2f1/test/fixtures/python/edge_cases6.json

@@ -0,0 +1 @@
+{"a": [2.0, 2.0, 3.0, 0.5, 1.5, 2.0, 4.0, 0.5], "b": [-1.0, -2.0, -2.0, -3.0, -3.0, -4.0, -2.0, -1.0], "c": [-1.0, -2.0, -2.0, -3.0, -3.0, -4.0, -2.0, -1.0], "x": [0.5, 0.5, 0.3, 0.7, -0.5, 0.6, 0.8, -0.9], "expected": [2.0, 2.75, 2.44, 1.6409375, 0.4453125, 4.792, 10.600000000000001, 0.55]}

lib/node_modules/@stdlib/math/base/special/hyp2f1/test/fixtures/python/outliers.json

@@ -1 +1 @@
-{"a": [-1.8, 1.8, 1.8, 1.8, 1.8, 5.0, 10.0, 10.0, 2.0, 2.0, -1.8, 1.8, 10.0, 10.0, 0.5, -8.0, -0.1156544430516313, -27.911722328444608, -10.184355055477743, -5.0083982772911, -0.7429095709749414, -14.55029516886238, -5.428179853908536, -1.8190858888165087, -0.9534205693212994, -31.70680212982947, -0.18350917503602782, -200.19574300099214, -8.837706832399117, -3.2052905962805456, -13.37614633859299], "b": [7.4, -2.5, 1.0, 7.4, 7.4, 7.4, 7.4, 7.4, 3.0, 2.5, -2.5, 7.4, 1.0, 7.4, -269.5, 18.016500331508873, 12.722140489351698, 14.018736008860946, 6.5450429328481174, 171.09919289684245, 11.725525989335528, 0.46553367087834685, 26.782963689614604, 21.025202609254364, 10.024920055648606, 0.11445767340979862, 14.696533286274349, 1.9444493115386567, 0.19096384651567044, 0.3048596186043564, 1.337786902832312], "c": [20.4, 20.4, 20.4, -1.8, 20.4, 20.4, 20.4, 20.4, 5.0, -3.25, 20.4, -1.8, -1.8, -1.8, 1.5, 10.805295997850628, -0.6030985338257255, -1.0168630024136034, -0.44051751620383994, -1.4923865283013589, -7.299203351011158, -0.5349901418098844, -4.162402828198655, -0.6960141045675334, -0.23845367675425955, -0.5570625216964069, -6.338990250643928, -0.6933677849479691, -2.3808990396202425, -0.7039276707364683, -0.05015281580946018], "x": [-10.0, -10.0, -10.0, -10.0, -10.0, -10.0, -10.0, 0.95, 0.99, 0.999, -10.0, -1.01, -0.99, -0.99, 0.998001, 0.90875647507, -0.0005448214561780684, 0.6228483452847298, 0.6555896492302491, 0.7073764056574465, 0.11534401417791407, 0.6086682776938632, 0.10047548647104909, 0.2570211738060222, 0.4419851803780206, 0.020574313007314604, -0.9003519736816863, 0.24492811068225362, -0.27624718180517105, 0.8496565195400692, -0.0022109253176725296], "expected": [16.334204105729413, 5.883122144128727, 0.5820646473844754, 0.010147094307759376, 0.07291374137469146, 0.0011628419587190388, 8.042653908304492e-06, 566.0092271657466, 27.699347904322664, 2.183739328012162e+26, -0.5786871007671348, 0.10921062752340924, -0.8752587026378827, 0.4234752106080307, 0.053963052503373715, -3.566216341436626e-09, 0.998681836639102, 5.33299803869569, 1.1177709865124041, 3.491332086988064e+82, 1.132062691110089, -0.0006927158077474787, 98.41698380414908, -221.57018712798285, 43.53554136829992, 1.0540592696289446, 0.5371016318543966, 0.0028040224553278668, -2.4067466563131665, 0.028751519134363426, 0.18419839456905535]}
+{"a": [-1.8, 1.8, 1.8, 1.8, 1.8, 5.0, 10.0, 10.0, 2.0, 2.0, -1.8, 1.8, 10.0, 10.0, 0.5, -8.0, -0.1156544430516313, -27.911722328444608, -10.184355055477743, -5.0083982772911, -0.7429095709749414, -14.55029516886238, -5.428179853908536, -1.8190858888165087, -0.9534205693212994, -31.70680212982947, -0.18350917503602782, -200.19574300099214, -8.837706832399117, -3.2052905962805456, -13.37614633859299], "b": [7.4, -2.5, 1.0, 7.4, 7.4, 7.4, 7.4, 7.4, 3.0, 2.5, -2.5, 7.4, 1.0, 7.4, -269.5, 18.016500331508873, 12.722140489351698, 14.018736008860946, 6.5450429328481174, 171.09919289684245, 11.725525989335528, 0.46553367087834685, 26.782963689614604, 21.025202609254364, 10.024920055648606, 0.11445767340979862, 14.696533286274349, 1.9444493115386567, 0.19096384651567044, 0.3048596186043564, 1.337786902832312], "c": [20.4, 20.4, 20.4, -1.8, 20.4, 20.4, 20.4, 20.4, 5.0, -3.25, 20.4, -1.8, -1.8, -1.8, 1.5, 10.805295997850628, -0.6030985338257255, -1.0168630024136034, -0.44051751620383994, -1.4923865283013589, -7.299203351011158, -0.5349901418098844, -4.162402828198655, -0.6960141045675334, -0.23845367675425955, -0.5570625216964069, -6.338990250643928, -0.6933677849479691, -2.3808990396202425, -0.7039276707364683, -0.05015281580946018], "x": [-10.0, -10.0, -10.0, -10.0, -10.0, -10.0, -10.0, 0.95, 0.99, 0.999, -10.0, -1.01, -0.99, -0.99, 0.998001, 0.90875647507, -0.0005448214561780684, 0.6228483452847298, 0.6555896492302491, 0.7073764056574465, 0.11534401417791407, 0.6086682776938632, 0.10047548647104909, 0.2570211738060222, 0.4419851803780206, 0.020574313007314604, -0.9003519736816863, 0.24492811068225362, -0.27624718180517105, 0.8496565195400692, -0.0022109253176725296], "expected": [16.334204105729413, 5.883122144128727, 0.5820646473844754, 0.010147094307759376, 0.07291374137469146, 0.0011628419587190388, 8.042653908304492e-06, 566.0092271658314, 27.699347904322664, 2.1837393280121743e+26, -0.5786871007671348, 0.10921062752340924, -0.8752587026378827, 0.4234752106080307, 0.053963052503373715, -3.566216341436626e-09, 0.998681836639102, 5.33299803869569, 1.1177709865124041, 3.491332086988064e+82, 1.132062691110089, -0.0006927158077474787, 98.41698380414908, -221.57018712798285, 43.53554136829992, 1.0540592696289446, 0.5371016318543966, 0.0028040224553278668, -2.4067466563131665, 0.028751519134363426, 0.18419839456905535]}

lib/node_modules/@stdlib/math/base/special/hyp2f1/test/fixtures/python/runner.py

@@ -256,6 +256,46 @@ def main():
     # Outliers:
     gen_outliers("outliers.json")
 
+    # Edge Cases #5
+    # d = c - a - b is a negative integer, x > 0.85, a and b are positive integers.
+    pts = [
+        (1.5, 2.5, 3.0, 0.90),
+        (1.5, 2.5, 3.0, 0.95),
+        (2.5, 1.5, 3.0, 0.90),
+        (0.5, 2.5, 2.0, 0.92),
+        (1.5, 2.5, 2.0, 0.91),
+        (0.5, 1.5, 1.0, 0.90),
+        (2.0, 3.0, 4.0, 0.90),
+        (1.5, 3.5, 4.0, 0.88),
+        (2.5, 2.5, 4.0, 0.93),
+        (1.5, 3.5, 3.0, 0.91),
+    ]
+    a, b, c, x = zip(*pts)
+    a = np.array(a)
+    b = np.array(b)
+    c = np.array(c)
+    x = np.array(x)
+    gen(a, b, c, x, "edge_cases5.json")
+
+    # Edge Cases #6
+    # b = c is a negative integer: 2F1(a, b; b; x) -> finite polynomial sum.
+    pts = [
+        (2.0, -1.0, -1.0, 0.5),
+        (2.0, -2.0, -2.0, 0.5),
+        (3.0, -2.0, -2.0, 0.3),
+        (0.5, -3.0, -3.0, 0.7),
+        (1.5, -3.0, -3.0, -0.5),
+        (2.0, -4.0, -4.0, 0.6),
+        (4.0, -2.0, -2.0, 0.8),
+        (0.5, -1.0, -1.0, -0.9),
+    ]
+    a, b, c, x = zip(*pts)
+    a = np.array(a)
+    b = np.array(b)
+    c = np.array(c)
+    x = np.array(x)
+    gen(a, b, c, x, "edge_cases6.json")
+
 
 if __name__ == "__main__":
     main()

lib/node_modules/@stdlib/math/base/special/hyp2f1/test/test.js

@@ -34,6 +34,8 @@ var edgeCases1 = require( './fixtures/python/edge_cases1.json' );
 var edgeCases2 = require( './fixtures/python/edge_cases2.json' );
 var edgeCases3 = require( './fixtures/python/edge_cases3.json' );
 var edgeCases4 = require( './fixtures/python/edge_cases4.json' );
+var edgeCases5 = require( './fixtures/python/edge_cases5.json' );
+var edgeCases6 = require( './fixtures/python/edge_cases6.json' );
 var outliers = require( './fixtures/python/outliers.json' );
 
 
@@ -238,6 +240,50 @@ tape( 'the function correctly evaluates the hypergeometric function', function t
 	t.end();
 });
 
+tape( 'the function correctly evaluates the hypergeometric function', function test( t ) {
+	var expected;
+	var a;
+	var b;
+	var c;
+	var x;
+	var v;
+	var i;
+
+	a = edgeCases5.a;
+	b = edgeCases5.b;
+	c = edgeCases5.c;
+	x = edgeCases5.x;
+	expected = edgeCases5.expected;
+
+	for ( i = 0; i < x.length; i++ ) {
+		v = hyp2f1( a[ i ], b[ i ], c[ i ], x[ i ] );
+		t.strictEqual( isAlmostSameValue( v, expected[ i ], 12 ), true, 'returns expected value.' );
+	}
+	t.end();
+});
+
+tape( 'the function correctly evaluates the hypergeometric function', function test( t ) {
+	var expected;
+	var a;
+	var b;
+	var c;
+	var x;
+	var v;
+	var i;
+
+	a = edgeCases6.a;
+	b = edgeCases6.b;
+	c = edgeCases6.c;
+	x = edgeCases6.x;
+	expected = edgeCases6.expected;
+
+	for ( i = 0; i < x.length; i++ ) {
+		v = hyp2f1( a[ i ], b[ i ], c[ i ], x[ i ] );
+		t.strictEqual( isAlmostSameValue( v, expected[ i ], 1 ), true, 'returns expected value.' );
+	}
+	t.end();
+});
+
 tape( 'the function correctly evaluates the hypergeometric function', function test( t ) {
 	var expected;
 	var a;