|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 back-to-back 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 H-P 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
HP-67 6.134E-06 397 3.2127E-05 206
HP-33C 6.134E-06 397 3.2127E-05 206
HP-38C 6.134E-06 397 3.2127E-05 206
HP-41 6.134E-06 397 3.2127E-05 206
HP-11C/12C 6.134E-06 397 3.2127E-05 206
HP-28S/32S 6.803E-08 389 3.1267E-07 204
HP-33s 6.803E-08 389 3.1267E-07 204
TI-55 3.362E-07 3.494E-06
TI-57 7.115E-07 404 7.0811E-05 22
TI-59 6.894E-09 402 3.843E-07 22
TI-66 6.644E-09 401 8.7646E-08 22
TI-95 6.204E-09 396 3.1987E-08 199
TI-80 6.204E-09 396 3.1777E-08 200
TI-81 6.204E-09 396 3.1777E-08 200
TI-82 7.353E-10 386 5.7932E-09 142
TI-83+ 0 500 0 500
TI-85 7.353E-10 386 5.7932E-09 142
TI-86 7.353E-10 386 3.0483E-09 211
TI-89 Auto 0 500 0 500
TI-89 Approx 7.353E-10 386 3.0483E-09 211
CC-40 8.09E-10 424 1.492E-09 415
TI-74 8.09E-10 424 3.33E-10 419
Model 100 1.9564E-09 300 8.2061E-09 84
Durabrand 828 0 500 0 500
6.672E-08 408 8.9125E-07 22
Sharp PC-1201 6.134E-06 3.2127E-05
Sharp PC-1261 0 500 0 500
6.134E-06 397 3.2126E-05 207
fx-7000G 0 500 2.7E-09 498
fx-7700GBus 0 500 2.7E-09 498
For the HP-33C, HP-38C, HP-41 and HP-11C/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 HP-33C absolute differences of 1E-07 from the square root -squared result occur at 39, 62, 65 and 91. For the HP-38C absolute differences occur at 39 and 65. For the HP-41 and HP-11/12 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 HP-67 and HP-41 over the range from 1 to 10,000. There were 66 differences with the hp-67 but only 25 with the hp-41. 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 HP-67. It took only 30 seconds to run the same program on my hp 33s.
It is possible to use the TI-59 (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 EE-INV-EE function after each mathematical operation. If you do so you will find that the emulated ten digit machine yields a sum of 3.2127E-05 (the correct value) with 206 zeroes when doing the square root times the square root, but yields a sum of 3.2227E-05 (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 EE-INV-EE 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 CC-40 and TI-74 (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 non-zero 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 PC-1201 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 PC-1261 a user can use a technique similar to that described above for the Durabrand 828 to yield the non-zero 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 1E-09 due to a non-zero result when i = 10.
For the two Casio machines the first square root results (2.7E-09 for each machine) were obtained using the formula Abs(i - (Sqr(i)x(Sqr(i)). The only non-zero values occur at 207 (1.8E-09) and 327 (9E-10). The second square root results were obtained using the formula Abs(i - (Sqr(i))^2).