Commit 2f52df4a0dd1bfeb645c12b28f27970549acd809
2014-07-05T22:29:26
[base] Improve comment.
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diff --git a/src/base/ftcalc.c b/src/base/ftcalc.c
index b2eba9b..2216880 100644
--- a/src/base/ftcalc.c
+++ b/src/base/ftcalc.c
@@ -358,20 +358,24 @@
/* documentation is in freetype.h */
/* The FT_MulDiv function has been optimized thanks to ideas from */
- /* Graham Asher. The trick is to optimize computation when everything */
- /* fits within 32-bits (a rather common case). */
+ /* Graham Asher and Alexei Podtelezhnikov. The trick is to optimize */
+ /* a rather common case when everything fits within 32-bits. */
/* */
- /* we compute 'a*b+c/2', then divide it by 'c'. (positive values) */
+ /* We compute 'a*b+c/2', then divide it by 'c'. (positive values) */
/* */
- /* A sufficient condition to avoid overflow is as follows. */
+ /* The product of two positive numbers never exceeds the square of */
+ /* their mean. Therefore, we always avoid the overflow by imposing */
/* */
- /* a + b <= 2 * sqrt( X - c/2 ) */
+ /* ( a + b ) / 2 <= sqrt( X - c/2 ) */
/* */
- /* where X = 2^31 - 1. After Taylor expansion, we make it stronger */
+ /* where X = 2^31 - 1. Now we replace sqrt with a linear function */
+ /* that is smaller or equal in the entire range of c from 0 to X; */
+ /* it should be equal to sqrt(X) and sqrt(X/2) at the range termini. */
+ /* Substituting the linear solution and explicit numbers we get */
/* */
- /* a + b <= 92681.9 - c / 92681.9 */
+ /* a + b <= 92681.9 - c / 79108.95 */
/* */
- /* with explicit 2*sqrt(X) = 92681.9. What we actually use is this */
+ /* In practice we use a faster and even stronger inequality */
/* */
/* a + b <= 92681 - (c >> 16) */
/* */