Precision issues

[ChinhQT]

Hi,
I don't know why this one is wrong. The Arity's result is tanh^-1(2%^9) = 5.5511151E-16
But the correct result is tanh^-1(2%^9) = 5.12E-16.

[woheller69]

are you sure? 2%^9 equals 5.12E-16
Should atanh of this value be the same?

[ChinhQT]

tanh and atanh (2%^9) = 5.12E-16. They have the same result. I've just tested on other app. It still has the same result. You can check them again.

[woheller69]

I checked the formula and it is correct.
So I guess these are precision issues of Java's Math library.

[Fontan030]

Yes, looks like Java's library has some general issues with trigonometry. For example, sin(11π) gives 4.899E-15, sin(2000π) is evaluated as -6.428E-13 (while both expressions obviously should be equal to 0)

[woheller69]

Java double seems to be good for about 15 digits...

[ChinhQT]

Are you sure ? I don't think it's an error of Java's library. The ClevCal on Android, it works well. The result is correct. I don't why.

[woheller69]

Maybe ClevCal uses some other arithmetic library.

The formula used by ArityEngine is: 0.5 * Math.log(1. + (x + x) / (1. - x));
which equals 0.5 * Math.log(1. + x) / (1. - x));

https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions

[woheller69]

btw, Java has already problems subtracting 56.3 - 55.7. It gives 0.59999999
I recently fixed that by using BigDecimal for subtractions

[ChinhQT]

With the formula 0.5 * Math.log(1. + x) / (1. - x))
x = 2%^9. The result is wrong.
When you use x = 2%^8. The Arity works well. The result is correct. I don't know why.

[woheller69]

well I guess it is the 1 + 2%^9 or 1 - 2%^9. One or both of them probably exceeds the possibilty of "double"

[ChinhQT]

Do you have any suggestion to resolve this issue, please ?

[woheller69]

No. For high-precision stuff use another calculator.

[woheller69]

next version will have a better check for pi multiples, so sin(11PI) and sin(2000PI) should be 0
But that is all I can do.

[supsm]

well I guess it is the 1 + 2%^9 or 1 - 2%^9. One or both of them probably exceeds the possibilty of "double"

Yep, I looked into this and it's actually a known issue.
libc (llvm) gets around this by using a taylor series approximation for small values, the code boils down to 1/11 x^11 + 1/9 x^9 + 1/7 x^7 + 1/5 x^5 + 1/3 x^3 + x. glibc (gnu) uses log1p, which calculates ln(1 + x) with greater precision around x=0.
Neither clang (llvm), gcc (gnu), nor msvc (microsoft) optimized 0.5 * log(1 + (x + x) / (1 - x)) to 0.5 * log1p((x + x) / (1 - x)), so I doubt java would as well.

If java has log1p, it should be an easy replacement for small input values, and if not, the taylor approximation also works.

These methods worked with the given value, even on 32-bit float.

[woheller69]

I am using my own fork https://github.com/woheller69/ArityEngine
Will apply the fix there :-)
Maybe we should add a test case also

[woheller69]

if 0.5 * log1p((x + x) / (1 - x)) should be used:

Why are you proposing

 0.5 * Math.log1p(twox + twox * x / (1 - x));

Shouldn't it be simply

 0.5 * Math.log1p(twox / (1 - x));

[supsm]

They're mathematically the same thing. I'm not exactly sure why the gcc wizards chose this, but it must have been purposeful somehow as they had the other option for another case. Besides, they're some of the world's smartest people working on critical infrastructure, so there's plenty of reason for it to be as optimized as possible.

I think it might be because for exceedingly small values of x, twox + twox * x / (1 - x) guarantees at least twox (x + x is just an increase in floating point exponent with no effect on precision, and operator precedence dictates that the addition is done last; twox * x / (1 - x) can never be negative), which is an alright approximation at those values. twox / (1 - x), on the other hand, has no such guarantee and could easily be rounded down. This is just my personal opinion, perhaps the authors had a different rationale.
Either way, their approach should translate to java pretty effectively as the code is already "generic" and not tailored to a specific architecture or platform.

[woheller69]

Are there similar tweaks for things like

sin(11Pi) or sin(2000Pi) ? (See above)

For exact multiples of PI I fixed it already in my fork. But maybe there are better solutions to improve precision

Modulo causes issues with complex numbers...

[supsm]

I don't believe so. Neither gcc nor clang fixes this issue (they give the same result as java), and in fact the input value is simply not representable very well using double-precision floating point. The closest representable value is 6283.18530718, and its sin value is pretty close to what java and C output (2.6666141e-13). This value is slightly different from reported above, so I'm not sure if there was some input rounding somewhere. I used an input value of 6283.1853071795864 as 2000pi in my testing.

[woheller69]

Maybe this library is an option?

https://github.com/eobermuhlner/big-math

[woheller69]

But probably that does not help either, as the input value does not come as BigDecimal. The whole Arity Engine would have to be changed...

[woheller69]

@supsm I added your modifications and some test cases to my fork of Arity engine:
https://github.com/woheller69/ArityEngine

Here is an updated version of Arity. tanh and atanh (2%^9) now yield 5.12E-16

-uninstall
-remove .zip and install

app-release.apk.zip

[woheller69]

This should fix some more issues with sin and cos calculations for multiples of PI or PI/2
woheller69/ArityEngine@fab02d6

app-release.apk.zip

[Murmele]

I found a similar problem:

4÷3×π×(((6371+0.1)×1000)^3−(6371×1000)^3×2÷1000. Dividing this formula by 1000:

4÷3×π×(((6371+0.1)×1000)^3−(6371×1000)^3×2÷1000÷1000 does not result in a lower exponent, only the mantissa changes

[woheller69]

If you want to divide the whole formula by 1000 you should put it in brackets

(4÷3×π×(((6371+0.1)×1000)^3−(6371×1000)^3×2÷1000)÷1000

[Murmele]

Sorry I completely overseen that one bracket is not closed