Editing Rounding (section)

===Rounding to prepare for shorter precision===
This rounding mode is used to avoid getting a potentially wrong result after [[#Double rounding|multiple roundings]]. This can be achieved if all roundings except the final one are done using rounding to prepare for shorter precision ("RPSP"), and only the final rounding uses the externally requested mode.

With decimal arithmetic, final digits of 0 and 5 are avoided; if there is a choice between numbers with the least significant digit 0 or 1, 4 or 5, 5 or 6, 9 or 0, then the digit different from 0 or 5 shall be selected; otherwise, the choice is arbitrary. IBM defines that, in the latter case, a digit with the smaller magnitude shall be selected.<ref>[https://www.ibm.com/support/pages/zarchitecture-principles-operation IBM z/Architecture Principles of Operation]</ref> RPSP can be applied with the step between two consequent roundings as small as a single digit (for example, rounding to 1/10 can be applied after rounding to 1/100).
For example, when rounding to integer,
* 20.0 is rounded to 20;
* 20.01, 20.1, 20.9, 20.99, 21, 21.01, 21.9, 21.99 are rounded to 21 (avoiding a final 0);
* 22.0, 22.1, 22.9, 22.99 are rounded to 22;
* 24.0, 24.1, 24.9, 24.99 are rounded to 24 (avoiding a final 5);
* 25.0 is rounded to 25;
* 25.01, 25.1 are rounded to 26 (avoiding a final 5).

In the example from "[[#Double rounding|Double rounding]]" section, rounding 9.46 to one decimal gives 9.4, which rounding to integer in turn gives 9.

With binary arithmetic, this rounding is also called "round to odd" (not to be confused with "[[#Rounding half to odd|round half to odd]]"). For example, when rounding to 1/4 (0.01 in binary),
* {{math|1=''x'' = 2.0}} ⇒ result is 2 (10.00 in binary)
* {{math|2.0 < ''x'' < 2.5}} ⇒ result is 2.25 (10.01 in binary)
* {{math|1=''x'' = 2.5}} ⇒ result is 2.5 (10.10 in binary)
* {{math|2.5 < ''x'' < 3.0}} ⇒ result is 2.75 (10.11 in binary)
* {{math|1=''x'' = 3.0}} ⇒ result is 3 (11.00 in binary)

For correct results with binary arithmetic, each rounding step must remove at least 2 binary digits, otherwise, wrong results may appear. For example,
* 3.125 RPSP to 1/4 ⇒ result is 3.25
* 3.25 RPSP to 1/2 ⇒ result is 3.5
* 3.5 round-half-to-even to 1 ⇒ result is 4 (wrong)

If the erroneous middle step is removed, the final rounding to integer rounds 3.25 to the correct value of 3.

RPSP is implemented in hardware in IBM [[zSeries]] and [[pSeries]]. In [[Python (programming language)|Python]] module "Decimal", [[Tcl]] module "math", [[Haskell]] package "decimal-arithmetic", and possibly others, this mode is called ROUND_05UP or round05up.