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Minor.

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Werner Lemberg 2014-08-20 16:19:32 +02:00
parent 8fd87d4e69
commit 70961e5772
1 changed files with 16 additions and 14 deletions
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30
src/base/ftcalc.c
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@ -362,35 +362,37 @@
/* 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' (all positive values). */
/* */
/* The product of two positive numbers never exceeds the square of */
/* their mean. Therefore, we always avoid the overflow by imposing */
/* its mean values. Therefore, we always avoid the overflow by */
/* imposing */
/* */
/* ( a + b ) / 2 <= sqrt( X - c/2 ) */
/* (a + b) / 2 <= sqrt(X - c/2) , */
/* */
/* where X = 2^32 - 1, the maximun unsigned 32-bit value, and using */
/* unsigned arithmetic. Now we replace sqrt with a linear function */
/* that is smaller or equal in the entire range of c from 0 to X/2; */
/* it should be equal to sqrt(X) and sqrt(3X/4) at the termini. */
/* Substituting the linear solution and explicit numbers we get */
/* where X = 2^32 - 1, the maximum unsigned 32-bit value, and using */
/* unsigned arithmetic. Now we replace `sqrt' with a linear function */
/* that is smaller or equal for all values of c in the interval */
/* [0;X/2]; it should be equal to sqrt(X) and sqrt(3X/4) at the */
/* endpoints. Substituting the linear solution and explicit numbers */
/* we get */
/* */
/* a + b <= 131071.99 - c / 122291.84 */
/* a + b <= 131071.99 - c / 122291.84 . */
/* */
/* In practice we should use a faster and even stronger inequality */
/* In practice, we should use a faster and even stronger inequality */
/* */
/* a + b <= 131071 - (c >> 16) */
/* a + b <= 131071 - (c >> 16) */
/* */
/* or, alternatively, */
/* */
/* a + b <= 129895 - (c >> 17) */
/* a + b <= 129895 - (c >> 17) . */
/* */
/* FT_MulFix, on the other hand, is optimized for a small value of */
/* the first argument, when the second argument can be much larger. */
/* This can be achieved by scaling the second argument and the limit */
/* in the above inequalities. For example, */
/* in the above inequalities. For example, */
/* */
/* a + (b >> 8) <= (131071 >> 4) */
/* a + (b >> 8) <= (131071 >> 4) */
/* */
/* should work well to avoid the overflow. */
/* */