More Results with Rodger's Test Message #1 Posted by Palmer O. Hanson, Jr. on 10 Aug 2007, 10:39 p.m.
I have been intrigued with the possibilities of Rodger's calculator tests using backtoback reciprocals and the square of the square root. I have now completed the tests on all of the programmables in my possession. The following table combines my new results with those in the table I previously published. The table also includes a corrected entry for the square root test with my Model 100 where, inexplicably, I somehow managed to forget to include the ABS part of the equation in my previously published result. Finally, the table includes two entries in the square root squared column for the HP machines and for some others where the results for the square root multiplied by the square root are not the same as the square root raised to the second power using exponential techniques.
Reciprocal Square of the Square Root
Sum Zeroes Sum Zeroes
HP67 6.134E06 397 3.2127E05 206
3.2167E05 202
HP33C 6.134E06 397 3.2127E05 206
3.2167E05 202
HP38C 6.134E06 397 3.2127E05 206
3.2147E05 204
HP41 6.134E06 397 3.2127E05 206
3.2137E05 205
HP11C/12C 6.134E06 397 3.2127E05 206
3.2137E05 205
HP28S/32S 6.803E08 389 3.1267E07 204
3.1367E07 204
HP33s 6.803E08 389 3.1267E07 204
TI55 3.362E07 3.494E06
TI57 7.115E07 404 7.0811E05 22
1.234234E04 9
TI59 6.894E09 402 3.843E07 22
TI66 6.644E09 401 8.7646E08 22
TI95 6.204E09 396 3.1987E08 199
TI80 6.204E09 396 3.1777E08 200
TI81 6.204E09 396 3.1777E08 200
TI82 7.353E10 386 5.7932E09 142
TI83+ 0 500 0 500
TI85 7.353E10 386 5.7932E09 142
TI86 7.353E10 386 3.0483E09 211
TI89 Auto 0 500 0 500
TI89 Approx 7.353E10 386 3.0483E09 211
CC40 8.09E10 424 1.492E09 415
TI74 8.09E10 424 3.33E10 419
Model 100 1.9564E09 300 8.2061E09 84
Durabrand 828 0 500 0 500
6.672E08 408 8.9125E07 22
Sharp PC1201 6.134E06 3.2127E05
Sharp PC1261 0 500 0 500
6.134E06 397 3.2126E05 207
fx7000G 0 500 2.7E09 498
6.878104E08 22
fx7700GBus 0 500 2.7E09 498
8.9829E08 24
Some Comments:
For the HP33C, HP38C, HP41 and HP11C/12C the results for the reciprocal test and the square root test are the same and are equal to Rodger's expected values. The square root test results are the same whether one uses sqr(i)xsqr(i) or (sqr(i))^2 . The results when using the sequence sqr(i) 2 y^x are not quite equal to the expected value and are different for the different machines. For the HP33C absolute differences of 1E07 from the square root squared result occur at 39, 62, 65 and 91. For the HP38C absolute differences occur at 39 and 65. For the HP41 and HP11/12 the only absolute difference occurs at 39. I suspect that this implies some evolution in the exponential function as the ten digit hp product line matured. I wondered if the higher density of differences was maintained over a larger range so I tested my HP67 and HP41 over the range from 1 to 10,000. There were 66 differences with the hp67 but only 25 with the hp41. There were only six numbers for which the difference occurred on both machines: 39, 3446, 6221, 6430, 7421 and 7560. One of the things that impressed me during the exercise was just how slow those old machines were. Doing the sum of the errors and counting the errors for the exponential test over the range from 1 to 500 took eighteen minutes on my HP67. It took only 30 seconds to run the same program on my hp 33s.
It is possible to use the TI59 (which uses thirteen digits without rounding and has the famous multiplication anomaly) to emulate the ten digit HP machines for the square root case. All one has to do is perform the EEINVEE function after each mathematical operation. If you do so you will find that the emulated ten digit machine yields a sum of 3.2127E05 (the correct value) with 206 zeroes when doing the square root times the square root, but yields a sum of 3.2227E05 (an incorrect value) and 205 zeroes when doing the square root raised to the second power using the y^x function. The difference comes from different results when the input integer is 423 where the square root of 423 times the square root of 423 yields 423 but the square root of 423 raised to the second power yields 422.9999999. Doing the calculations without the final EEINVEE sequence will yield 422.9999999505 which rounds to 423 with the square root times the square root calculation but 422.9999999498 which rounds to 422.9999999 for the square root raised to the second power calculation.
The CC40 and TI74 (base 100 machines) yield the same result for the reciprocal test but different results for the square root test. I suspected that the difference would occur at only four input values in a manner similar to that seen with the second square root tests with the HP machines. It turns out that is not the case. There is another interesting phenomena in these machines. For both machines the sum when running the square root test from 1 to 99 is the same as when running from 1 to 500, or when running from 1 to 999.
A user can get the nonzero sums in the second line for the Durabrand 828 by transferring each intermediate result to a memory location before using it again. For example, for the reciprocal case use the sequence 1/i > M : 1/M > M : S + Abs (i  M) > S . This process essentially changes the result to that which would be obtained with a twelve digit machine which truncates rather than rounds. One caution: It turns out that there are two versions of the Durabrand 828 out there. One version, the version that I have, provides sixteen digit arithmetic. The other version, which I don't have, does NOT provide sixteen digit arithmetic. Since I don't have that version I can't say what the results would be with it.
The Sharp PC1201 yields the correct results for ten digit machines. But when I changed the sequence for the square root test from square root squared to square root y^x 2 the sum is zero.
With the Sharp PC1261 a user can use a technique similar to that described above for the Durabrand 828 to yield the nonzero sums on the second line. This essentially changes the machine to a ten digit machine with rounding at each calculation. But, the sum for the square root test is in error by 1E09 due to a nonzero result when i = 10.
For the two Casio machines the first square root results (2.7E09 for each machine) were obtained using the formula Abs(i  (Sqr(i)x(Sqr(i)). The only nonzero values occur at 207 (1.8E09) and 327 (9E10). The second square root results were obtained using the formula Abs(i  (Sqr(i))^2).
