Copyright (C) 2001, 2002 Mike Sebastian

### Hints and Tricks for Getting the Hidden Guard Digits

Most calculators, including many of the classic calculators, compute intermediate results to greater precision than what is displayed. These undisplayed digits are frequently refered to as "guard digits." An important part of evaluating a calculator with the forensics algorithm is obtaining these guard digits so that a full precision forensics result can be recorded.

#### First!

First, before doing any of the steps below, make sure the display format of the calculator is set to display the maximum number of digits possible. Remember how the classic LED Hewlett-Packard calculators defaulted to two digits after the decimal point? Many modern calculators also default to less than all possible digits being displayed.

#### Calculators that do not round displayed results.

The TI-36X, like many of the classic LED calculators, is a calculator for which it is easy to obtain the guard digits. Simply execute the algorithm, which displays 9.000002295. Then subtract 9, and multiply that result by 1,000,000. The least significant digits, including the two least significant guard digits are displayed as 2.29461. With these two results, it is easy to "stitch" the two intermediate results together to get the full precision forensics result of 9.00000229461.

A Slightly more difficult situation is presented by those calculators that produce a forensics result slightly less than 9.0. The generic 56-function scientific calculator illustrates this twist to the problem of obtaining the guard digits. Simply execute the algorithm, which displays 8.999998637. Then subtract 8.99999, and multiply the result by 1,000,000. The least significant digits, including the two least significant guard digits are displayed as 8.63704. With these two results, it is again easy to "stitch" the two intermediate results together to get the full precision forensics result of 8.99999863704.

#### Calculators that do round displayed results.

When the calculator rounds displayed results, more ingenuity is required to get at the guard digits. The trick for these calculators is to do as much of the calculation as possible before any result is displayed.

Some calculators round displayed results, but retain the full precision result in the "ANS" register/variable. If this is the case, obtaining the least significant digits will be almost as easy as it is for the calculators that don't round results (discussed above).

The Radio Shack PC-5 illustrates the problem of rounding displayed results. Initially executing the forensics formula results in 9.000007168 being displayed. Simply subtracting 9 does not display any extra guard digits. But, the least significant digits, including the guard digits are displayed if the forensics formula calculation minus nine is computed, as shown in this PC-5 "formula": ASN ACS ATN TAN COS SIN9-9
The least significant digits are displayed: 7.16758E-06. With these two intermediate results, it is easy to "stitch" the full precision forensics result together: 9.00000716758.

The TI-60X rounds displayed results and produces a forensics result slightly less than 9. Initially executing the forensics formula results in 8.999999008 being displayed. Next, subtract 8.999999 from the forensics formula - the not very helpful result of 0.000000008 is displayed. So, to get the least significant digits, subtract 8.999999 from the forensics formula, then multiply that quantity by 1,000,000. The TI-60X formula that does all this is: ( SIN-1 COS-1 TAN-1 TAN COS SIN9-8.999999)x1000000
This formula displays the least significant digits: 0.007884. With these intermediate results, the full forensics result of 8.999999007884 can be "stitched" together.

"Classic" Texas Instruments scientific calculators such as the SR-51 or TI-58 internally compute their results to thirteen digits, usually producing a forensic result of 9.000004661314. However, some of these calculators, such as the SR-52, SR-56 and SR-60, will only produce a twelve digit forensics result (i.e. 9.00000466131, the 4 is missing) when the methods described above are utilized. It is possible to extract the missing thirteenth digit for the SR-56 and SR-60 as follows. First, of course, obtain the most significant digits by computing the forensics formula; 9.000004661 is displayed. Next, utilize the following "formula" to display all of the guard digits (for convenience, the forensics formula is referred to as "f(9)"): -90000 + 10000 x f(9)
This formula produces 0.04661314 on the SR-56 and SR-60. With these two results, it is easy to produce the full forensics result of 9.000004661314. (The SR-52 steadfastly continues to refuse to reveal that thirteenth digit!)

#### Another Hint.

If you ever get a negative number when trying to extract the least significant digits of a forensics result, you may have done something wrong. Make sure you are not subtracting nine from a forensics result that is slightly less than nine. Or, if your are subtracting 8.999... from a forensics result, make sure there are not too many nines in the number.

#### But I still cannot see the guard digits.

The above discussion and examples address ninety-five percent of the problems associated with obtaining the full precision forensics result. Sometimes, additional ingenuity is required. Other times, it just isn't possible due to the design of the firmware inside the calculator.

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Last updated March 30, 2002