lib/math/rational.c: fix possible incorrect result from rational fractions helper

In some cases the previous algorithm would not return the closest
approximation.  This would happen when a semi-convergent was the
closest, as the previous algorithm would only consider convergents.

As an example, consider an initial value of 5/4, and trying to find the
closest approximation with a maximum of 4 for numerator and denominator.
The previous algorithm would return 1/1 as the closest approximation,
while this version will return the correct answer of 4/3.

To do this, the main loop performs effectively the same operations as it
did before.  It must now keep track of the last three approximations,
n2/d2 ..  n0/d0, while before it only needed the last two.

If an exact answer is not found, the algorithm will now calculate the
best semi-convergent term, t, which is a single expression with two
divisions:

    min((max_numerator - n0) / n1, (max_denominator - d0) / d1)

This will be used if it is better than previous convergent.  The test
for this is generally a simple comparison, 2*t > a.  But in an edge
case, where the convergent's final term is even and the best allowable
semi-convergent has a final term of exactly half the convergent's final
term, the more complex comparison (d0*dp > d1*d) is used.

I also wrote some comments explaining the code.  While one still needs
to look up the math elsewhere, they should help a lot to follow how the
code relates to that math.

This routine is used in two places in the video4linux code, but in those
cases it is only used to reduce a fraction to lowest terms, which the
existing code will do correctly.  This could be done more efficiently
with a different library routine but it would still be the Euclidean
alogrithm at its heart.  So no change.

The remain users are places where a fractional PLL divider is
programmed.  What would happen is something asked for a clock of X MHz
but instead gets Y MHz, where Y is close to X but not exactly due to the
hardware limitations.  After this change they might, in some cases, get
Y' MHz, where Y' is a little closer to X then Y was.

Users like this are: Three UARTs, in 8250_mid, 8250_lpss, and imx.  One
GPU in vp4_hdmi.  And three clock drivers, clk-cdce706, clk-si5351, and
clk-fractional-divider.  The last is a generic clock driver and so would
have more users referenced via device tree entries.

I think there's a bug in that one, it's limiting an N bit field that is
offset-by-1 to the range 0 ..  (1<<N)-2, when it should be (1<<N)-1 as
the upper limit.

I have an IMX system, one of the UARTs using this, so I can provide a
real example.  If I request a custom baud rate of 1499978, the driver
will program the PLL to produce a baud rate of 1500000.  After this
change, the fractional divider in the UART is programmed to a ratio of
65535/65536, which produces a baud rate of 1499977.0625.  Closer to the
requested value.

Link: http://lkml.kernel.org/r/20190330205855.19396-1-tpiepho@gmail.com
Signed-off-by: Trent Piepho <tpiepho@gmail.com>
Cc: Oskar Schirmer <oskar@scara.com>
Signed-off-by: Andrew Morton <akpm@linux-foundation.org>
Signed-off-by: Linus Torvalds <torvalds@linux-foundation.org>

lib/math/rational.c

@@ -3,6 +3,7 @@
  * rational fractions
  *
  * Copyright (C) 2009 emlix GmbH, Oskar Schirmer <oskar@scara.com>
+ * Copyright (C) 2019 Trent Piepho <tpiepho@gmail.com>
  *
  * helper functions when coping with rational numbers
  */
@@ -10,6 +11,7 @@
 #include <linux/rational.h>
 #include <linux/compiler.h>
 #include <linux/export.h>
+#include <linux/kernel.h>
 
 /*
  * calculate best rational approximation for a given fraction
@@ -33,30 +35,65 @@ void rational_best_approximation(
 	unsigned long max_numerator, unsigned long max_denominator,
 	unsigned long *best_numerator, unsigned long *best_denominator)
 {
-	unsigned long n, d, n0, d0, n1, d1;
+	/* n/d is the starting rational, which is continually
+	 * decreased each iteration using the Euclidean algorithm.
+	 *
+	 * dp is the value of d from the prior iteration.
+	 *
+	 * n2/d2, n1/d1, and n0/d0 are our successively more accurate
+	 * approximations of the rational.  They are, respectively,
+	 * the current, previous, and two prior iterations of it.
+	 *
+	 * a is current term of the continued fraction.
+	 */
+	unsigned long n, d, n0, d0, n1, d1, n2, d2;
 	n = given_numerator;
 	d = given_denominator;
 	n0 = d1 = 0;
 	n1 = d0 = 1;
+
 	for (;;) {
-		unsigned long t, a;
-		if ((n1 > max_numerator) || (d1 > max_denominator)) {
-			n1 = n0;
-			d1 = d0;
-			break;
-		}
+		unsigned long dp, a;
+
 		if (d == 0)
 			break;
-		t = d;
+		/* Find next term in continued fraction, 'a', via
+		 * Euclidean algorithm.
+		 */
+		dp = d;
 		a = n / d;
 		d = n % d;
-		n = t;
-		t = n0 + a * n1;
+		n = dp;
+
+		/* Calculate the current rational approximation (aka
+		 * convergent), n2/d2, using the term just found and
+		 * the two prior approximations.
+		 */
+		n2 = n0 + a * n1;
+		d2 = d0 + a * d1;
+
+		/* If the current convergent exceeds the maxes, then
+		 * return either the previous convergent or the
+		 * largest semi-convergent, the final term of which is
+		 * found below as 't'.
+		 */
+		if ((n2 > max_numerator) || (d2 > max_denominator)) {
+			unsigned long t = min((max_numerator - n0) / n1,
+					      (max_denominator - d0) / d1);
+
+			/* This tests if the semi-convergent is closer
+			 * than the previous convergent.
+			 */
+			if (2u * t > a || (2u * t == a && d0 * dp > d1 * d)) {
+				n1 = n0 + t * n1;
+				d1 = d0 + t * d1;
+			}
+			break;
+		}
 		n0 = n1;
-		n1 = t;
-		t = d0 + a * d1;
+		n1 = n2;
 		d0 = d1;
-		d1 = t;
+		d1 = d2;
 	}
 	*best_numerator = n1;
 	*best_denominator = d1;