#1 - Update README.md

README.md

@@ -5,7 +5,9 @@ The 3n+1, collatz or hailstone numbers problem - https://en.wikipedia.org/wiki/C
 ### Optimization 1: Combine odd/even steps
 The following optimization will speed up a run by up to 21%
 
-When we have an odd number, the next step is usually 3n + 1 applied to the current value.  However, the number resulting from 3n + 1 will always be positive - which will require at least one divide by 2.
+When we have an odd number, the next step is usually 3n + 1 applied to the current value.  However, the number resulting from 3n + 1 will always be positive - which will require at least one divide by 2.  If we combine the double step optimization with the fact that a shift right (or divide by 2) is always floor truncated (where the 1/2 is removed on an odd number).  If we combine the floor with an implicit round up (ceil) by adding 1 (where for example 27 /2  = 13.5 = 13 rounded, with + 1 = 14) - we have the following math...
+
+(3n + 1) / 2 = 3/2 * n + 1/2, where we drop the 1/2 due to rounding on a shift right.  We then have 3/2 * n which is also n + n/2.  We add 1 to this to get a round up (it will hold as we only perform this round up for odd numbers) - of - 1 + n + n/2.
 
 The operation...
 ```
@@ -14,9 +16,9 @@ with a subsequent
 n / 2 = n >> 1
 ```
 
-can be expressed as a single
+can be expressed as a single shift right with a add + 1 or effectively a divide by 
 ```
-n / 2 + n + 1 = n > 1 + n + 1
+n / 2 + n + 1 = n >> 1 + n + 1
 ```
 ### Optimization 2: Roll up all divide by 2 sequences
 When whe have for example a power of 2 like 256 - this will represent a straight path to 1 via 8 shift right operations.