1  /* SPDX-License-Identifier: GPL-2.0-only */
 2  /*
 3   * Helper functions for rational numbers.
 4   *
 5   * Copyright (C) 2009 emlix GmbH, Oskar Schirmer <oskar@scara.com>
 6   * Copyright (C) 2019 Trent Piepho <tpiepho@gmail.com>
 7   */
 8  
 9  #include <commonlib/helpers.h>
10  #include <commonlib/rational.h>
11  #include <limits.h>
12  
13  /*
14   * For theoretical background, see:
15   * https://en.wikipedia.org/wiki/Continued_fraction
16   */
17  void rational_best_approximation(
18  	unsigned long numerator, unsigned long denominator,
19  	unsigned long max_numerator, unsigned long max_denominator,
20  	unsigned long *best_numerator, unsigned long *best_denominator)
21  {
22  	/*
23  	 * n/d is the starting rational, where both n and d will
24  	 * decrease in each iteration using the Euclidean algorithm.
25  	 *
26  	 * dp is the value of d from the prior iteration.
27  	 *
28  	 * n2/d2, n1/d1, and n0/d0 are our successively more accurate
29  	 * approximations of the rational.  They are, respectively,
30  	 * the current, previous, and two prior iterations of it.
31  	 *
32  	 * a is current term of the continued fraction.
33  	 */
34  	unsigned long n, d, n0, d0, n1, d1, n2, d2;
35  	n = numerator;
36  	d = denominator;
37  	n0 = d1 = 0;
38  	n1 = d0 = 1;
39  
40  	for (;;) {
41  		unsigned long dp, a;
42  
43  		if (d == 0)
44  			break;
45  		/*
46  		 * Find next term in continued fraction, 'a', via
47  		 * Euclidean algorithm.
48  		 */
49  		dp = d;
50  		a = n / d;
51  		d = n % d;
52  		n = dp;
53  
54  		/*
55  		 * Calculate the current rational approximation (aka
56  		 * convergent), n2/d2, using the term just found and
57  		 * the two prior approximations.
58  		 */
59  		n2 = n0 + a * n1;
60  		d2 = d0 + a * d1;
61  
62  		/*
63  		 * If the current convergent exceeds the maximum, then
64  		 * return either the previous convergent or the
65  		 * largest semi-convergent, the final term of which is
66  		 * found below as 't'.
67  		 */
68  		if ((n2 > max_numerator) || (d2 > max_denominator)) {
69  			unsigned long t = ULONG_MAX;
70  
71  			if (d1)
72  				t = (max_denominator - d0) / d1;
73  			if (n1)
74  				t = MIN(t, (max_numerator - n0) / n1);
75  
76  			/*
77  			 * This tests if the semi-convergent is closer than the previous
78  			 * convergent.  If d1 is zero there is no previous convergent as
79  			 * this is the 1st iteration, so always choose the semi-convergent.
80  			 */
81  			if (!d1 || 2u * t > a || (2u * t == a && d0 * dp > d1 * d)) {
82  				n1 = n0 + t * n1;
83  				d1 = d0 + t * d1;
84  			}
85  			break;
86  		}
87  		n0 = n1;
88  		n1 = n2;
89  		d0 = d1;
90  		d1 = d2;
91  	}
92  
93  	*best_numerator = n1;
94  	*best_denominator = d1;
95  }