Accurate floor of natural logarithm for big integers in Python

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Floor of Natural Logarithm on Arbitrarily Large Integers — A Production Postmortem

Summary

Computing the exact floor of the natural logarithm (⌊ln(n)⌋) for arbitrarily large integers is a problem that looks trivial at first glance. The instinctive Python solution is:

import math
result = int(math.log(n))

But this silently produces wrong answers for large inputs. For example, given n = 4311231547115195, this snippet returns 36 when the correct answer is 35. The root issue is floating-point precision loss during the log computation, and it is a class of bug that is extremely difficult to catch without deliberate adversarial or large-input testing.

This post walks through why the bug happens, what real damage it can do, and how to fix it properly.

Root Cause

The root cause is the conversion from arbitrary-precision integer to IEEE 754 double-precision float (float64) that math.log() performs internally.

  • Python’s int type is arbitrary precision — it can represent integers of any size exactly.
  • Python’s float type maps to C’s double, which is a 64-bit IEEE 754 value with only 53 bits of mantissa (roughly 15–17 significant decimal digits).
  • When math.log(n) is called, n is first cast to a float, which rounds the integer if it exceeds the precision of float64.
  • The logarithm is then computed on this already-rounded value, compounding the error.
  • The result can be off by one or more integer units, which is exactly enough to flip the floor value.

Key insight: The bug is not in the log function itself — it is in the silent precision-destroying cast that precedes it. The rounding happens before the logarithm, but the error manifests after it, making it non-obvious during debugging.

Why This Happens in Real Systems

This class of bug appears frequently in real production systems because of several compounding factors:

  • Developers conflate “no error messages” with “correct result.” Python will not raise an exception or warning. The code runs cleanly and returns a plausible — but wrong — value.
  • Small inputs mask the problem. For n < 10^15, float64 precision is sufficient and math.log gives correct results. Bugs only surface at scale.
  • The error boundary is data-dependent. You cannot predict failure from the code alone — it depends on the specific numeric relationship between the input and the boundaries of floating-point rounding.
  • Bit-length tricks inherit the same problem. The observation that ⌊ln(n)⌋ is close to ⌊(n.bit_length() - 1) * ln(2)⌋ is correct, but if you still compute ln(2) as a float, you reintroduce the same precision issue — just shifted by one operation.
  • Convergence-based rational approximations are impractical. The alternating harmonic series for ln(2) converges at O(1/k), meaning you may need millions of terms to get enough precision for large bit_length values, making this approach computationally infeasible.

In production, these bugs surface in:

  • Cryptographic key analysis, where key sizes require exact logarithmic computations
  • Algorithmic complexity estimation, where log-based bucketing determines resource allocation
  • Scientific computing pipelines, where accumulated log errors cascade through subsequent calculations

Real-World Impact

The consequences of an incorrect ⌊ln(n)⌋ are subtle but serious:

  • Off-by-one in digit/bucket calculations: If you use ⌊ln(n)⌋ to determine the number of digits or to bucket values, a wrong floor means misclassification — data goes into the wrong bin.
  • Incorrect loop bounds: Algorithms that iterate ⌊ln(n)⌋ times (e.g., certain number-theoretic algorithms) will execute one too many or too few iterations, causing either wasted computation or silent truncation of results.
  • Cryptanalysis miscalculation: Estimating the bit-strength of a key often involves logarithms. An off-by-one error in the floor can lead to a false sense of security or unnecessary key rotation.
  • Silent propagation: Unlike a crash, a wrong floor value will typically flow silently through the rest of the pipeline, making it hard to detect and hard to trace back to the source.

Example or Code

Here is a direct demonstration of the bug in Python:

import math

n = 4311231547115195
result = int(math.log(n))
print(f"math.log floor result: {result}")   # prints 36 (WRONG)

import decimal
decimal.getcontext().prec = 50
dn = decimal.Decimal(n)
correct = int(dn.ln())
print(f"decimal.Decimal floor result: {correct}")  # prints 35 (CORRECT)

And here is a correct implementation using mpmath for arbitrary-precision arithmetic:

import mpmath

def floor_ln(n: int, prec: int = 100) -> int:
    mpmath.mp.dps = prec  # set decimal places of precision
    return int(mpmath.log(n))

n = 4311231547115195
print(floor_ln(n))  # 35 — correct

For cases where you need ⌊ln(n)ᵏ⌋ (floor of ln(n) raised to a power), the same approach extends:

import mpmath

def floor_ln_pow(n: int, k: int, prec: int = 100) -> int:
    mpmath.mp.dps = prec
    return int(mpmath.log(n) ** k)

n = 4311231547115195
print(floor_ln_pow(n, 2))  # floor(ln(n)^2)
print(floor_ln_pow(n, 3))  # floor(ln(n)^3)
print(floor_ln_pow(n, 4))  # floor(ln(n)^4)

The critical point is that prec=100 decimal places gives a massive safety margin over the ~15 digits that float64 provides.

How Senior Engineers Fix It

A senior engineer approaches this problem with a layered strategy:

  1. Never trust float for exact integer results. The first principle is to recognize that any computation that needs an exact integer output from a transcendental function must avoid lossy intermediate representations.

  2. Use arbitrary-precision libraries. Libraries like mpmath (Python) or boost::multiprecision (C++) provide guaranteed-precision arithmetic. Setting decimal precision to 50–100 places eliminates rounding risk for any realistic input size.

  3. Exploit bit_length() as a safe integer bound. Since ⌊ln(n)⌋ can be bounded using bit_length:

    • We know (b - 1) * ln(2) ≤ ln(n) < b * ln(2) where b = n.bit_length()
    • Compute these bounds with interval arithmetic using a precise enough representation of ln(2)
    • If both bounds have the same floor, you’re done without computing ln(n) at all
    • If they differ, you need a more precise evaluation — but this only happens in a narrow window near integer multiples of 1/ln(2)

  4. Defensive programming with assertions. After computing the result, verify it by checking:

    • e^result ≤ n < e^(result + 1) using arbitrary-precision exponentiation
    • This acts as a runtime invariant check that catches any remaining edge cases

  5. Document the precision contract. A senior engineer would add a comment or docstring explicitly stating: “This function assumes n > 0 and returns exact ⌊ln(n)⌋ for arbitrarily large n. Uses mpmath with 100-digit precision to guarantee correctness.”

Why Juniors Miss It

Junior engineers miss this bug for several overlapping reasons:

  • They test with small numbers. Values like 100, 1000, or even 10^12 work fine with float64, so the test suite passes. The bug only appears at n > ~10^15, which is rarely in ad-hoc test data.
  • They assume standard library functions are exact. There is a widespread but incorrect belief that math.log is “the correct mathematical answer.” It is actually the answer rounded to float64 precision.
  • They don’t think about silent failures. A crash is obvious; a wrong-but-plausible return value is invisible without an explicit correctness check. Juniors rarely write assertions that validate mathematical output against inverse operations.
  • They underestimate float64 limits. Many developers don’t know that float64 has only 53 bits (~15.9 decimal digits) of precision, and therefore can’t judge at what input size math.log starts to become unreliable for integer inputs.
  • They chase alternative algorithms without understanding the precision root cause. As seen in the original question, trying faster-converging series or rational approximations addresses the symptom rather than the disease. The fix is not a better algorithm for ln(2) — it is avoiding float altogether.

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