Log is non-monotonic in PHP and Lua

Posted by ibobev 1 day ago

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Comment by lioeters 1 day ago

Recently I was reading an article on the EML operator (exp-min-log) that uses exponentiation and logarithm to build elementary math functions including arithmetic operations. There was a table of results from testing across languages.

  Language         Result of 2 x 3    Error
  ---
  Node.js v25.3.0  6.000000000000000  0
  Python 3.9.6     6.000000000000001  8.88e-16
  PHP 8.5.1        6.000000000000001  8.88e-16
  Go 1.26.2        6.000000000000000  0
  Rust             6.000000000000001  8.88e-16
It was speculated that this miniscule margin of error, 1 ULP (unit in the last place), is likely due to the difference in how log() is implemented by the language. Supposedly Python, PHP, and Rust use LLVM's libm (C math library) but maybe Go and Node.js internally compile to CPU instructions directly? There was no evidence presented, so I was skeptical of this explanation.

Comment by purplesyringa 1 day ago

Python, PHP, and Rust do indeed use libm. Not LLVM's libm specifically, though, but rather just about anything they can find in runtime, and those libraries can have suboptimal accuracy.

Node uses V8, which notably implements a ton of math manually, because it needs the results to be deterministic across devices. It seemingly uses LLVM's libm, which IIRC promises 0.5 ulp for most operations. (https://github.com/v8/v8/blob/f24c62fbc342d616032734b714116e...)

Go avoids dynamic linking, so they also have their own implementation. (https://github.com/golang/go/blob/543ead71a8e7acc2bd6f326327...) They only promise 1 ulp, but I guess in this specific case it works out better than approximations used by the default libm on their system by pure chance?

Comment by lioeters 1 day ago

Nice, thanks for tracking down those links, fascinating. Apparently being off by 1 ULP is within IEEE 754 floating-point spec, and "effectively negligible".

Comment by AlotOfReading 1 day ago

Correctly rounding transcendental functions is very difficult because of something called the table maker's dilemma, so the standard didn't want to impose a potentially extreme performance cost if they didn't have to.

Correctly rounded single precision functions at a reasonable cost are much easier these days. LLVM 18+ implements them, but the gnu libraries have been slower to improve.

Comment by cwt137 1 day ago

I find it interesting the blog author tied PHP and Lua together. PHP uses a JIT from Lua. Is this related to the log issue?

Comment by purplesyringa 1 day ago

> PHP uses a JIT from Lua.

Wow, TIL! I was certain this is a mistake, but apparently PHP uses DynAsm (https://wiki.php.net/rfc/jit), which was developed for LuaJIT (https://luajit.org/dynasm.html). Cool stuff!

To answer your question, probably not. I dated the PHP change that added this "optimization" back to 2014 (https://github.com/php/php-src/commit/b547e1358d3846fad4cd0c...), while DynAsm only started being used around 2019 (https://wiki.php.net/rfc/jit). I think this is just convergent evolution.

(I'm putting "optimization" in quotes because it changes semantics.)

Comment by kristianp 1 day ago

Wouldn't a more applicable title be that the log base is non-monotonic?

Comment by purplesyringa 19 hours ago

The log base is a parameter, not a function, so that doesn't typecheck. Multivariate monotonicity isn't really a popular term, as far as I'm aware, so I'm not aware of good terminology for this. Maybe "log is non-monotonic with respect to the base"?

Comment by dmitrygr 1 day ago

  > you can compute any logarithm from two base-e ones.
They need not be base e. Any base will do as long as it is the same for both the numerator and the denominator

Comment by purplesyringa 1 day ago

Yeah, I simplified it a little. Though I must say I'm surprised pretty much every library I looked at uses the natural logarithm specifically, and not log2, which would seemingly be easier to compute with floats. Does anyone here know why, by any chance?

Comment by AlotOfReading 1 day ago

You can change base with a single multiplication by special constants if you accept 1ULP error, so working in one or the other doesn't really gain you much. But exp/logs are often implemented as a combination of a table lookup and polys, so you can just bake it into the constants even more easily.

Comment by stackghost 1 day ago

Not sure why they wouldn't use log2 internally, but the mathematical reason is that e^x is the function whose derivative is itself, so it and the natural logarithm are fundamental to lots of other theories/operations like Euler's formula.