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Re: Significant digits -- well, yes and no...



#5

Quote:
HP-42S: sin (3.14159265358 rad) = 9.79323846264 x 10-12 -- correct result to 12 significant digits

HP-41: sin (3.141592653 rad) = 5.9 x 10-10 -- correct result to 2 significant digits


Sorry Karl, shouldn't that be:
HP-41: sin (3.141592654 rad) = -4.1 x 10-10 vs -4.10206761537 x 10-10 ?

Greetings,
Massimo


#6

Hi, Massimo --

Quote:
Sorry Karl, shouldn't that be:
HP-41: sin (3.141592654 rad) = -4.1 x 10-10 vs -4.10206761537 x 10-10 ?

That also is a correct calculation, but my point was to reveal the ensuing digits of pi by calculating a truncated (not rounded) value of pi in radians mode. I've gone through the exercise several times in the Forum, but didn't save a bookmark to those posts:

sin(pi - x) = sin(pi)*cos(x) - cos(pi)*sin(x)

= 0 * cos(x) - (-1)*sin(x)

= sin(x)

x represents the truncated digits. For very small x, sin(x) ~= x, so the result produces a limited string of those digits.

The excellent mathematical routines developed for the Saturn microprocessor (debuting with the HP-71B) were ported to the Pioneer-series calculators. No other calculator I own matches the quality of the Saturn mathematics, although the TI-89 might. It also seems likely that Valentin's vintage Sharp pocket computers could meet or exceed the accuracy.

-- KS


#7

Hi, Karl:

Karl posted:

    "It also seems likely that Valentin's vintage Sharp pocket computers could meet or exceed the accuracy."

      Likely. These are your results as computed by my SHARP PC-1475,
      rounded to 12 digits:

             sin (3.14159265358 rad) = 9.79323846265E-12

      sin (3.141592653 rad) = 5.89793238463E-10

      cos (89.9999999 deg) = 1.74532925199E-09

      Matter of fact, this last result actually comes out in full precision
      as:

             cos (89.9999999 deg) = 1.7453292519943295760D-09   

      which has all its 20 significant digits absolutely correct.

Best regards from V.


#8

Hi, Valntin --

Quote:
These are your results as computed by my SHARP PC-1475, rounded to 12 digits:

sin (3.14159265358 rad) = 9.79323846265E-12


But, V! The last mantissa digit should be a "4", whether it's the actual digit or rounded. Who was responsible for the rounding? :-)

Seriously though, 20 significant digits was quite impressive for the era. I safely considered the 12+ digit performance "likely" from your photos I had previously viewed, at least one of which depicted a Sharp "PC" with many digits showing in its display.

Best regards,

-- KS


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