The HP-55 is a unique calculator. A cheaper alternative to the HP-65, it had no magnetic cards; what it had instead was a built-in quartz timer, making it probably the first pocket-size calculator ever with such a capability. In timer mode, the R/S key was used to start or stop the timer, and while the timer was running, the number keys could be used to store the current value in a numbered register.

The story of my HP-55, received in non-working condition, began with a crushed transistor. Two display driver chips replaced, with inhuman effort, by identical chips taken from a thoroughly dead HP-65. Still, no working calculator. Then came the oscilloscope, the voltmeter, some headscratching, more measurements, circuit tracing, even more headscratching, mumbling, swearing, then a spark of inspiration: where, exactly, is this chip getting power from? And why isn't it getting it? I wonder... could this diode be faulty? Five minutes later: a working calculator in my hands. Now I can start exploring this beast that I just brought back to life...

The HP-55 had a fair number (20) of registers but a very small program memory. Only 49 program steps, and these were completely unmerged, with the exception of the GTO instruction that required only one step for the instruction itself and the line number. (Could this be the only HP calculator in which even register references are unmerged?) I would like to create a Gamma function program on this calculator for demonstration purposes, but in 49 program steps, it just doesn't seem possible.

Or does it? After all, I have not one, but two alternative solutions! I can sacrifice accuracy and satisfy myself with a program for Stirling's formula. Or, I can sacrifice speed and write a program that calculates the incomplete Gamma function (and, by extension, approximates the Gamma function itself) iteratively. Hmmm... why don't I do both?

Here's what Stirling's formula looks like on the HP-55:

01. 41    ENTER
02. 41    ENTER
03. 22    x-y
04. 02    2
05. 71    ×
06. 31    f
07. 83    π
08. 71    ×
09. 31    f
10. 42    √
11. 22    x-y
12. 41    ENTER
13. 12    yx
14. 71    ×
15. 22    x-y
16. 32    g
17. 22    ex
18. 81    ÷
19. 22    x-y
20. 13    1/x
21. 01    1
22. 02    2
23. 81    ÷
24. 01    1
25. 61    +
26. 71    ×

Fast, simple, and inaccurate. To use the program, enter the argument and press BST (to reset the program counter) followed by R/S.

The incomplete Gamma function program is listed below. To use, enter the function argument, hit ENTER, then enter the integration limit, hit BST, and hit R/S. Execution time may be several minutes, depending on the integration limit used. Note that this version works only for positive arguments.

01.  33   STO
02.  01   1
03.  22   x-y
04.  33   STO
05.  02   2
06.  12   yx
07.  34   RCL
08.  02   2
09.  81   ÷
10.  33   STO
11.  03   3
12.  34   RCL
13.  01   1
14.  34   RCL
15.  02   2
16.  01   1
17.  61   +
18.  33   STO
19.  02   2
20.  81   ÷
21.  34   RCL
22.  03   3
23.  71   ×
24.  33   STO
25.  03   3
26.  61   +
27.  32   g
28. -30   x=y 30
29. -12   GTO 12
30.  34   RCL
31.  01   1
32.  32   g
33.  22   ex
34.  81   ÷