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PossibleZeroQ

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

PossibleZeroQ[expr] gives True if symbolic and numerical methods suggest that expr has value zero, and False otherwise.
The general problem of deciding whether an expression is zero is undecidable; PossibleZeroQ is a quick but not always accurate test.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= PossibleZeroQ[E^(I Pi/4) - (-1)^(1/4)]
Out[1]= True

In[2]:= PossibleZeroQ[(x + 1)(x - 1) - x^2 + 1]
Out[2]= True

In[3]:= PossibleZeroQ[(E + Pi)^2 - E^2 - Pi^2 - 2 E Pi]
Out[3]= True

In[4]:= PossibleZeroQ[E^Pi - Pi^E]
Out[4]= False

In[5]:= PossibleZeroQ[2^(2 I) - 2^(-2 I) - 2 I Sin[Log[4]]]
Out[5]= True

In[6]:= PossibleZeroQ[Sqrt[x^2] - x]
Out[6]= False

In[7]:= PossibleZeroQ[Sin[x]^2 + Cos[x]^2 - 1]
Out[7]= True

Implementation notes

Algorithm. builtin_possible_zero_q (src/zero_test.c) calls zero_test_decide, a staged hybrid symbolic-numeric pipeline that early-exits on the first definite verdict:

  • Stage 0 — structural: O(1) shortcuts for literal Integer/Real/BigInt/MPFR zero, Complex[0,0], lists of zeros, and unbound symbols (decide_structural).
  • Stage 1 — rational normalisation: Together ∘ Cancel plus Expand, then a polynomial zero test, deciding every identity in Q(x_1,...,x_n) (decide_rational). A True here is trusted; a False is not trusted alone.
  • Stage 2 — numeric: for symbol-free inputs, numericalize at machine precision and compare |z| against an IEEE catastrophic-cancellation threshold, climbing a precision ladder (53 -> 200 -> 500 -> 1000 bits) while ambiguous; a true zero shrinks geometrically across precisions (decide_numeric).
  • Stage 3 — Schwartz–Zippel: for inputs with free symbols, substitute each free symbol with a random rational drawn from Q[i] (to probe branch cuts) and require several independent Stage-2 confirmations (decide_schwartz_zippel).

Result mapping. A definite False returns False; both True and UNKNOWN return True, matching Mathematica's documented "assume zero when uncertain" behaviour (the accompanying PossibleZeroQ::ztest1 message is not emitted). The symbol is Listable.

Attributes: Listable, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= PossibleZeroQ[(x - 1) (x + 1) - (x^2 - 1)]
Out[1]= True

In[1]:= PossibleZeroQ[x^2 + 1]
Out[1]= False

In[1]:= PossibleZeroQ[Sin[x]^2 + Cos[x]^2 - 1]
Out[1]= True

In[1]:= PossibleZeroQ[Sqrt[2] + Sqrt[3] - Sqrt[5 + 2 Sqrt[6]]]
Out[1]= True

In[1]:= PossibleZeroQ[Log[2] + Log[3] - Log[6]]
Out[1]= True

Notes

PossibleZeroQ[expr] uses combined symbolic and numerical heuristics to decide whether expr is identically zero. It sees through polynomial cancellation ((x-1)(x+1) - (x^2-1) = 0), the Pythagorean identity Sin[x]^2 + Cos[x]^2 - 1, the nested-radical denesting Sqrt[2] + Sqrt[3] = Sqrt[5 + 2 Sqrt[6]], and the logarithm law Log[2] + Log[3] = Log[6]. A nonzero expression like x^2 + 1 returns False. Because deciding whether a closed-form expression is exactly zero is undecidable in general, PossibleZeroQ is a fast but not infallible test: True strongly suggests a zero and False rules one out for the cases it can analyse.