(12C) Decimal to Fraction - Printable Version

(12C) Decimal to Fraction - Printable Version

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RE: (12C) Decimal to Fraction - Thomas Klemm - 08-10-2018 05:02 PM

(08-10-2018 12:28 PM)Dieter Wrote: n_max is stored in the n-register

That's a clever idea for the HP-12C.

(08-09-2018 10:15 PM)Albert Chan Wrote: I had revised my Python code after your HP-11C translation. Now:
1. hi = lo + 1 instead of 1/0 (infinity), thus saved 1 iteration
2. Both abs() calls are removed

These are the relevant lines:

Code:

    n1, d1 = int(x), 1

    n2, d2 = n1+1, 1        # hi = lo + 1

    pick_hi = (n2/d2 - x) < (x - n1/d1)

    return (n2, d2) if pick_hi else (n1, d1)

In both cases we use that \(\frac{n1}{d1} < x < \frac{n2}{d2}\).

This allows to remove squaring the differences in:

Code:

47 RCL 0

48 RCL 3

49 RCL 4

50 /

51 -

52 ENTER

53 x

54 RCL 0

55 RCL 1

56 RCL 2

57 /

58 -

59 ENTER

60 x

61 x≤y?

Instead we can just use:

Code:

47 RCL 3

48 RCL 4

49 /

50 RCL 0

51 -

52 RCL 0

53 RCL 1

54 RCL 2

55 /

56 -

57 x≤y?

Kind regards
Thomas

RE: (12C) Decimal to Fraction - Dieter - 08-10-2018 07:06 PM

(08-10-2018 05:02 PM)Thomas Klemm Wrote: In both cases we use that \(\frac{n1}{d1} < x < \frac{n2}{d2}\).

Fine. This saves four lines to that GTO 66 becomes GTO 62.
Here is a revised version:

Code:

01 STO 0

02 INTG

03 STO 1

04 1

05 STO 2

06 STO 3

07 RCL 0

08 FRAC

09 x=0?

10 GTO 62

11 Clx

12 STO 4

13 RCL 1

14 RCL 3

15 +

16 STO 5

17 RCL n

18 RCL 2

19 RCL 4

20 +

21 STO 6

22 x≤y?

23 GTO 25

24 GTO 47

25 RCL 5

26 RCL 0

27 RCL 6

28 x

29 x≤y?

30 GTO 36

31 RCL 5

32 STO 1

33 RCL 6

34 STO 2

35 GTO 13

36 -

37 x=0?

38 GTO 44

39 RCL 5

40 STO 3

41 RCL 6

42 STO 4

43 GTO 13

44 RCL 6

45 RCL 5

46 GTO 00

47 RCL 3

48 RCL 4

49 ÷

50 RCL 0

51 -

52 RCL 0

53 RCL 1

54 RCL 2

55 ÷

56 -

57 x≤y?

58 GTO 62

59 RCL 4

60 RCL 3

61 GTO 00

62 RCL 2

63 RCL 1

64 GTO 00

Dieter

RE: (12C) Decimal to Fraction - Albert Chan - 08-11-2018 12:58 AM

Code:

    pick_hi = (n2/d2 - x) < (x - n1/d1)

    return (n2, d2) if pick_hi else (n1, d1)

Discovered a pick_hi bug:
If x = midpoint between the Farey pair, it always pick n1/d1, but n2/d2 might be better.

Example: x = midpoint of a Farey Pair

farey(111/308, 20) => 5/14 --> 4/11 is better
farey( 51/130, 15 ) => 5/13 --> 2/5 is better

It is not technically wrong, but we prefer smaller denominator.
Patch below:

Code:

    diff = 2*d1*d2*x - (n1*d2 + n2*d1)  # x vs Farey Pair midpoint

    if diff == 0: diff = d1 - d2        # x is midpoint, pick smaller d

    return (n2, d2) if diff > 0 else (n1, d1)

RE: (12C) Decimal to Fraction - Albert Chan - 08-11-2018 02:13 AM

(08-09-2018 11:13 AM)Thomas Klemm Wrote: As we know \(\frac{113}{355}\) is closer for a long time. We have to wait until \(\frac{33215}{104348}\) to get closer

Hi, Thomas,

If you meant closer from below (i.e fraction < 1/Pi), above is correct.

However, if closer in absolute term, you don't have to wait that long.
For semiconvergent better than 113/355, solve for k

1/Pi - 113/355 = 2.703e-8 = (106 + 113k)/(333 + 355k) - 1/Pi
k ~ 145.8

If k >= 146, the fraction is closer.
For k = 146, we get 16604/52163, absolute error = 2.697e-8

RE: (12C) Decimal to Fraction - Thomas Klemm - 08-11-2018 09:38 AM

(08-11-2018 02:13 AM)Albert Chan Wrote: If you meant closer from below (i.e fraction < 1/Pi), above is correct.

Exactly. I just checked when would it switch from \(\frac{113}{355}\) to the next value:

Code:

(…)

113 355 32876 103283

113 355 32989 103638

113 355 33102 103993

33215 104348 33102 103993

Sorry for my poor wording.

Thanks for clarifying
Thomas

RE: (12C) Decimal to Fraction - Thomas Klemm - 08-11-2018 10:05 AM

(08-09-2018 10:15 PM)Albert Chan Wrote: 1. hi = lo + 1 instead of 1/0 (infinity), thus saved 1 iteration

Code:

n2, d2 = n1+1, 1        # hi = lo + 1

(08-10-2018 07:06 PM)Dieter Wrote: Here is a revised version:

Code:

01 STO 0 02 INTG 03 STO 1 04 1 05 STO 2 06 STO 3 07 RCL 0 08 FRAC 09 x=0? 10 GTO 62 11 Clx 12 STO 4 (…)

Thus we can still save an iteration using:

Code:

01 STO 0

02 INTG

03 STO 1

04 1

05 STO 2

06 STO 4

07 +

08 STO 3

09 RCL 0

10 FRAC

11 x=0?

12 GTO 62

(…)

Best regards
Thomas

RE: (12C) Decimal to Fraction - Gamo - 05-07-2019 01:44 AM

Quote:999 [n]
3,141592654 [R/S] => 355 [X↔Y] 113

Now I wonder how long all this takes on an original hardware 12C... ;-)

Just acquired an Original HP-12C (made in Singapore) ran this and took
about 40 seconds.

Gamo