Log and sqrt require floating point?

Does log or sqrt need to be computed using floating-point representation? This excludes the larger part of bignums.

1. log requires bignum?

This is a test:

(define x (expt 10 2480))
(log x)

If the implementation wants to answer (log x) with the correct number, the logarithm must be calculated without the floating-point C library log function, since the number is far too large to fit in current days 64-bit doubles.

2. How is the log of bignum rational computed?

Now we can try another thing:

(define n (/ (expt 2 3000)
             (+ (expt 2 3000) 1)))
(log n)

An implementation may first obtain an inexact then compute the log, or compute separate logs then subtract. The results may be different, depending on how the implementation deals with logs of bignums.

3. What about a bignum that does fit a double?

Another interesting test is this -- what if we have a bignum that does fit a double?

(define x (expt 2 200))
(log x)

4. Does sqrt work on bignums?

A third test:

(define S (+ (expt 2 2030) 111111111111111111111111117))
(sqrt S)

• Biwa and Tinyscheme already compute x as the inexact infinity +inf.0.
• Bigloo computes (expt 2 2030) as zero

Another experiment:

(log 1/6319748715279270675921934218987893281199411530039296)

Should return -119.27551918982401. (This was posted to the ECL mailing list on 2022-Jul-03 -- ECL does not compute the value, and complains about the user requesting log of zero).

• Cyclone needs the number to be given as explicit division (doesn't support rationals), and returns -inf.0
• Loko returns -inf.0
• Tinyscheme needs the number to be given as explicit division (doesn't support rationals), and returns -43.66827238

All others return the correct result; Bigloo and Scheme9 need the rational to be described as division, not as rational type.

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