7 ms·

Calculator Forensics (2002)

91 points by ColinWright 1y ago

kazinator 1y ago
Forensics? Have these calculators committed a crime (against numerical analysis)?
- gs17 1y ago
  If they have, this site has their fingerprints.
- bobmcnamara 1y ago
  Failure to identify an identity! Straight to calculator jail!
- maxbond 1y ago
  I think it's forensic in the sense that, should you perform this operation on an unknown calculator, this chart could be used to identify it.
  - maxbond 1y ago
    On second thought what you'd probably do is identify the chipset of a calculator.
    - kstrauser 1y ago
      I think you'd identify the "OS" of a calculator. For example, HP famously had lots of long, complex, highly accurate algorithms in its firmware. They could reuse large portions of that between different models, regardless of the hardware executing it.
      - maxbond 1y ago
        For sure. I was imagining that the software would be pretty thin and the FP behavior would be primarily a function of chipset, but I stand corrected.
- gblargg 1y ago
  This is a good way to find out what fugitive chip an unknown calculator is harboring.
_trampeltier 1y ago
Didn't expect to see this site here. From time to time I show this site to young guys.
quirkot 1y ago
I always wonder... if there was an AGI and it's chipset gave the wrong answer, how would it ever know?
- GuB-42 1y ago
  The neural networks we use today have really terrible accuracy, and we tend to make them worse, not better, as having more neurons is better than having more precision. Human brains are also a mess, but somehow, they work, and we are usually able to correct our own mistakes. Since by AGI, we usually mean human-like, that system should be able to self correct the same way we do.
- gs17 1y ago
  I'd presume it could reason around the wrong answer, at least to realize something was off. Current LLMs will sometimes hallucinate that this has happened when they're "thinking".
- nenenejej 1y ago
  How do humans know? usually someone corrects someone else. we have repeatability in physics, or we wait 30 years and quash convictions etc. etc.
mysteria 1y ago
So what's the correct answer?
- gs17 1y ago
  9 degrees. arcsin(arccos(arctan(tan(cos(sin(9)))))) basically makes a set of sin-cos-tan layers that arctan-arccos-arcsin unwrap one-by-one, which should result in nothing having changed, unless the functions used weren't accurate.
  - measurablefunc 1y ago
    That's incorrect, you have to choose the proper inverse branch if you want the answer to be 9.
    - madars 1y ago
      There is no choice here - each inverse is uniquely determined. That's similar to how 3 and -3 are both square roots of 9 (i.e., solutions to x^2=9), but sqrt(9)=3 as it denotes the principal square root, which by convention is always the non-negative value. Of course, in a different context we might design functions to have multi-valued properties, like atan2(x,y) != atan(y/x) in general (atan2 takes quadrant in account and returns full range [-pi, pi], atan only returns principal values in [-pi/2, pi/2]) as practical applications benefit from preserving quadrant beyond just the principal inverse (or not failing when x=0!)
      - measurablefunc 1y ago
        The inverse branches are not unique, you might think there is no choice being made but picking the standard branch is a choice b/c I can always shift the result by 2π by picking a different branch of the inverse. The answer is not unique & the assumption is that the calculators are using the standard branch.
        madars 1y ago
tomhow 1y ago
Previously: Calculator Forensics (2002) - https://news.ycombinator.com/item?id=42757455 https://news.ycombinator.com/item?id=42757455 - Jan 2025 (1 comment) Calculator Forensics (2002) - https://news.ycombinator.com/item?id=28561298 https://news.ycombinator.com/item?id=28561298 - Sept 2021 (19 comments) Calculator Forensics (2002) - https://news.ycombinator.com/item?id=7682045 https://news.ycombinator.com/item?id=7682045 - May 2014 (2 comments)
OskarS 1y ago
Couldn't resist checking what math.h does for floats and doubles on godbolt, it returns slightly above 9, but curiously not exactly equal to any answer in that list [1]. Maybe not that surprising, these transcendental functions are always iffy in the last digits. [1]: https://godbolt.org/z/dK85Eq8r6 https://godbolt.org/z/dK85Eq8r6
- masfuerte 1y ago
  CORE-MATH provides correctly rounded transcendental functions. (Though that doesn't mean that this calculation will return exactly 9. And I haven't tried it.) https://core-math.gitlabpages.inria.fr/ https://core-math.gitlabpages.inria.fr/
- chongli 1y ago
  I would be really surprised if any physical calculator used binary floating point numbers instead of binary-coded decimal. The appearance of repeating bits even for relatively simple decimal numbers like 0.1 is likely to confuse and annoy a calculator user expecting an exact answer.
  - OskarS 1y ago
    I would assume most basic calculators use regular floating point circuits, and then just round aggressively for the display.
    - inejge 1y ago
      Cheap calculators use microcontroller-like CPUs which simply have no room for floating-point hardware. BCD or fixed-point integers in software are much cheaper and simpler.
    - kragen 1y ago
      Microcontrollers with floating point hardware are very expensive (60¢ and up) compared to microcontrollers without (3¢), and since the 01980s, software floating point has been plenty fast enough to use in pocket calculators, even on the slowest microcontrollers available. To be somewhat concrete, on amd64 with valgrind --tool=callgrind my non-BCD decimal fixed-point RPN calculator program https://gitlab.com/kragen/bubbleos/tree/master/yeso/rpncalc.c https://gitlab.com/kragen/bubbleos/tree/master/yeso/rpncalc.... compiled for the console (rpncalc_linux.c) takes 228,141 instructions to process the input "2 3q" from the keyboard, and 229,824 instructions to process the input "2 3/q", an additional 1,683 instructions for the division, mostly to display the result. Division is probably the worst case for regular arithmetic. An additional division operation takes another 1,601 instructions. A subtraction instead takes 1,563. On an 8-bit processor like a PIC or AVR, you need to run more instructions to do the same work, but it's typically maybe a factor of 8 for data processing and less for things like looping. So we're talking about maybe 10,000 instructions per keystroke, probably less. On a 16MHz 1T microcontroller, that's less than a millisecond. You might think that if the microcontroller doesn't have space for floating-point circuitry it also doesn't have space for enough memory for software floating point, but that would be wrong. My decimal floating point library is 1204 bytes (≈602 instructions) compiled for AVR, and the calculator UI logic is 2134 bytes (≈1067 instructions). This fits in all but the smallest microcontrollers. Like, it doesn't quite fit in an ATTiny25.
tiahura 1y ago
Disappointed in the HP-48.
ani17 1y ago
thanks for sharing!
qiqitori 1y ago
Nice. Checked on a Commodore SR-37, produces the same result as the SR-36 per the site. (9.08210803 Commodore SR-36)
deleted 1y ago
[deleted]
petters 1y ago
Numerically difficult since intermediate values end up close to a local maximum of cos
LeoPanthera 1y ago
It's better to start from the parent page otherwise you're missing all context: https://www.rskey.org/~mwsebastian/miscprj/forensics.htm https://www.rskey.org/~mwsebastian/miscprj/forensics.htm This is where this post should probably point, IMHO.
harshreality 1y ago
It would be nice to see an update. HP Prime? TI Nspire series? Swissmicros? Numworks? (the numworks app gives 9, but are the algorithms hardware-independent?)
avipars 1y ago
Wolfram alpha result - https://www.wolframalpha.com/input?i=arcsin+%28arccos+%28arctan+%28tan+%28cos+%28sin+%289%29+%29+%29+%29+%29+%29 https://www.wolframalpha.com/input?i=arcsin+%28arccos+%28arc... 3pi - 9 = 0.42477796076937971538793014983850865259150819812531746292483377692344921885
- nneonneo 1y ago
  This is the result if you do it in radians, but the test actually requires that you set the calculator to degrees mode first. Note that all of the inverse trig functions are multivalued because the trig functions are periodic. Here, Wolfram Alpha is giving you one of the possible answers. The entire family of answers should be +/-9 + n*pi for any integer n; the sign on the 9 is due to cos being an even function.
  - dominicrose 1y ago
    irb(main):012> Math.asin(Math.acos(Math.atan(Math.tan(Math.cos(Math.sin(9*Math::PI/180))))))*180/Math::PI => 9.00000000000001

91 points by ColinWright 1y ago

kazinator 1y ago

Forensics? Have these calculators committed a crime (against numerical analysis)?

gs17 1y ago
If they have, this site has their fingerprints.
bobmcnamara 1y ago
Failure to identify an identity! Straight to calculator jail!
maxbond 1y ago
I think it's forensic in the sense that, should you perform this operation on an unknown calculator, this chart could be used to identify it.
- maxbond 1y ago
  On second thought what you'd probably do is identify the chipset of a calculator.
  - kstrauser 1y ago
    I think you'd identify the "OS" of a calculator. For example, HP famously had lots of long, complex, highly accurate algorithms in its firmware. They could reuse large portions of that between different models, regardless of the hardware executing it.
    - maxbond 1y ago
      For sure. I was imagining that the software would be pretty thin and the FP behavior would be primarily a function of chipset, but I stand corrected.
gblargg 1y ago
This is a good way to find out what fugitive chip an unknown calculator is harboring.

_trampeltier 1y ago

Didn't expect to see this site here. From time to time I show this site to young guys.

quirkot 1y ago

I always wonder... if there was an AGI and it's chipset gave the wrong answer, how would it ever know?

GuB-42 1y ago
The neural networks we use today have really terrible accuracy, and we tend to make them worse, not better, as having more neurons is better than having more precision. Human brains are also a mess, but somehow, they work, and we are usually able to correct our own mistakes. Since by AGI, we usually mean human-like, that system should be able to self correct the same way we do.
gs17 1y ago
I'd presume it could reason around the wrong answer, at least to realize something was off. Current LLMs will sometimes hallucinate that this has happened when they're "thinking".
nenenejej 1y ago
How do humans know? usually someone corrects someone else. we have repeatability in physics, or we wait 30 years and quash convictions etc. etc.

mysteria 1y ago

So what's the correct answer?

gs17 1y ago
9 degrees. arcsin(arccos(arctan(tan(cos(sin(9)))))) basically makes a set of sin-cos-tan layers that arctan-arccos-arcsin unwrap one-by-one, which should result in nothing having changed, unless the functions used weren't accurate.
- measurablefunc 1y ago
  That's incorrect, you have to choose the proper inverse branch if you want the answer to be 9.
  - madars 1y ago
    There is no choice here - each inverse is uniquely determined. That's similar to how 3 and -3 are both square roots of 9 (i.e., solutions to x^2=9), but sqrt(9)=3 as it denotes the principal square root, which by convention is always the non-negative value. Of course, in a different context we might design functions to have multi-valued properties, like atan2(x,y) != atan(y/x) in general (atan2 takes quadrant in account and returns full range [-pi, pi], atan only returns principal values in [-pi/2, pi/2]) as practical applications benefit from preserving quadrant beyond just the principal inverse (or not failing when x=0!)
    - measurablefunc 1y ago
      The inverse branches are not unique, you might think there is no choice being made but picking the standard branch is a choice b/c I can always shift the result by 2π by picking a different branch of the inverse. The answer is not unique & the assumption is that the calculators are using the standard branch.
      - madars 1y ago

tomhow 1y ago

Previously: Calculator Forensics (2002) - https://news.ycombinator.com/item?id=42757455 https://news.ycombinator.com/item?id=42757455 - Jan 2025 (1 comment) Calculator Forensics (2002) - https://news.ycombinator.com/item?id=28561298 https://news.ycombinator.com/item?id=28561298 - Sept 2021 (19 comments) Calculator Forensics (2002) - https://news.ycombinator.com/item?id=7682045 https://news.ycombinator.com/item?id=7682045 - May 2014 (2 comments)

OskarS 1y ago

Couldn't resist checking what math.h does for floats and doubles on godbolt, it returns slightly above 9, but curiously not exactly equal to any answer in that list [1]. Maybe not that surprising, these transcendental functions are always iffy in the last digits. [1]: https://godbolt.org/z/dK85Eq8r6 https://godbolt.org/z/dK85Eq8r6

masfuerte 1y ago
CORE-MATH provides correctly rounded transcendental functions. (Though that doesn't mean that this calculation will return exactly 9. And I haven't tried it.) https://core-math.gitlabpages.inria.fr/ https://core-math.gitlabpages.inria.fr/
chongli 1y ago
I would be really surprised if any physical calculator used binary floating point numbers instead of binary-coded decimal. The appearance of repeating bits even for relatively simple decimal numbers like 0.1 is likely to confuse and annoy a calculator user expecting an exact answer.
- OskarS 1y ago
  I would assume most basic calculators use regular floating point circuits, and then just round aggressively for the display.
  - inejge 1y ago
    Cheap calculators use microcontroller-like CPUs which simply have no room for floating-point hardware. BCD or fixed-point integers in software are much cheaper and simpler.
  - kragen 1y ago
    Microcontrollers with floating point hardware are very expensive (60¢ and up) compared to microcontrollers without (3¢), and since the 01980s, software floating point has been plenty fast enough to use in pocket calculators, even on the slowest microcontrollers available. To be somewhat concrete, on amd64 with valgrind --tool=callgrind my non-BCD decimal fixed-point RPN calculator program https://gitlab.com/kragen/bubbleos/tree/master/yeso/rpncalc.c https://gitlab.com/kragen/bubbleos/tree/master/yeso/rpncalc.... compiled for the console (rpncalc_linux.c) takes 228,141 instructions to process the input "2 3q" from the keyboard, and 229,824 instructions to process the input "2 3/q", an additional 1,683 instructions for the division, mostly to display the result. Division is probably the worst case for regular arithmetic. An additional division operation takes another 1,601 instructions. A subtraction instead takes 1,563. On an 8-bit processor like a PIC or AVR, you need to run more instructions to do the same work, but it's typically maybe a factor of 8 for data processing and less for things like looping. So we're talking about maybe 10,000 instructions per keystroke, probably less. On a 16MHz 1T microcontroller, that's less than a millisecond. You might think that if the microcontroller doesn't have space for floating-point circuitry it also doesn't have space for enough memory for software floating point, but that would be wrong. My decimal floating point library is 1204 bytes (≈602 instructions) compiled for AVR, and the calculator UI logic is 2134 bytes (≈1067 instructions). This fits in all but the smallest microcontrollers. Like, it doesn't quite fit in an ATTiny25.

tiahura 1y ago

Disappointed in the HP-48.

ani17 1y ago

thanks for sharing!

qiqitori 1y ago

Nice. Checked on a Commodore SR-37, produces the same result as the SR-36 per the site. (9.08210803 Commodore SR-36)

deleted 1y ago

[deleted]

petters 1y ago

Numerically difficult since intermediate values end up close to a local maximum of cos

LeoPanthera 1y ago

It's better to start from the parent page otherwise you're missing all context: https://www.rskey.org/~mwsebastian/miscprj/forensics.htm https://www.rskey.org/~mwsebastian/miscprj/forensics.htm This is where this post should probably point, IMHO.

harshreality 1y ago

It would be nice to see an update. HP Prime? TI Nspire series? Swissmicros? Numworks? (the numworks app gives 9, but are the algorithms hardware-independent?)

avipars 1y ago

Wolfram alpha result - https://www.wolframalpha.com/input?i=arcsin+%28arccos+%28arctan+%28tan+%28cos+%28sin+%289%29+%29+%29+%29+%29+%29 https://www.wolframalpha.com/input?i=arcsin+%28arccos+%28arc... 3pi - 9 = 0.42477796076937971538793014983850865259150819812531746292483377692344921885

nneonneo 1y ago
This is the result if you do it in radians, but the test actually requires that you set the calculator to degrees mode first. Note that all of the inverse trig functions are multivalued because the trig functions are periodic. Here, Wolfram Alpha is giving you one of the possible answers. The entire family of answers should be +/-9 + n*pi for any integer n; the sign on the 9 is due to cos being an even function.
- dominicrose 1y ago
  irb(main):012> Math.asin(Math.acos(Math.atan(Math.tan(Math.cos(Math.sin(9*Math::PI/180))))))*180/Math::PI => 9.00000000000001