Why is Math.pow(0, 0) === 1?

Question

We all know that 0⁰ is indeterminate.

But, javascript says that:

Math.pow(0, 0) === 1 // true 

and C++ says the same thing:

pow(0, 0) == 1 // true 

WHY?

I know that:

>Math.pow(0.001, 0.001) 0.9931160484209338 

But why does Math.pow(0, 0) throw no errors? Or maybe a NaN would be better than 1.

Answer 1

In C++ The result of pow(0, 0) the result is basically implementation defined behavior since mathematically we have a contradictory situation where N^0 should always be 1 but 0^N should always be 0 for N > 0, so you should have no expectations mathematically as to the result of this either. This Wolfram Alpha forum posts goes into a bit more details.

Although having pow(0,0) result in 1 is useful for many applications as the Rationale for International Standard—Programming Languages—C states in the section covering IEC 60559 floating-point arithmetic support:

Generally, C99 eschews a NaN result where a numerical value is useful. [...] The results of pow(∞,0) and pow(0,0) are both 1, because there are applications that can exploit this definition. For example, if x(p) and y(p) are any analytic functions that become zero at p = a, then pow(x,y), which equals exp(y*log(x)), approaches 1 as p approaches a.

Update C++

As leemes correctly pointed out I originally linked to the reference for the complex version of pow while the non-complex version claims it is domain error the draft C++ standard falls back to the draft C standard and both C99 and C11 in section 7.12.7.4 The pow functions paragraph 2 says (emphasis mine):

[...]A domain error may occur if x is zero and y is zero.[...]

which as far as I can tell means this behavior is unspecified behavior Winding back a bit section 7.12.1 Treatment of error conditions says:

[...]a domain error occurs if an input argument is outside the domain over which the mathematical function is defined.[...] On a domain error, the function returns an implementation-defined value; if the integer expression math_errhandling & MATH_ERRNO is nonzero, the integer expression errno acquires the value EDOM; [...]

So if there was a domain error then this would be implementation defined behavior but in both the latest versions of gcc and clang the value of errno is 0 so it is not a domain error for those compilers.

Update Javascript

For Javascript the ECMAScript® Language Specification in section 15.8 The Math Object under 15.8.2.13 pow (x, y) says amongst other conditions that:

If y is +0, the result is 1, even if x is NaN.

Answer 2

In JavaScript Math.pow is defined as follows:

• If y is NaN, the result is NaN.
• If y is +0, the result is 1, even if x is NaN.
• If y is −0, the result is 1, even if x is NaN.
• If x is NaN and y is nonzero, the result is NaN.
• If abs(x)>1 and y is +∞, the result is +∞.
• If abs(x)>1 and y is −∞, the result is +0.
• If abs(x)==1 and y is +∞, the result is NaN.
• If abs(x)==1 and y is −∞, the result is NaN.
• If abs(x)<1 and y is +∞, the result is +0.
• If abs(x)<1 and y is −∞, the result is +∞.
• If x is +∞ and y>0, the result is +∞.
• If x is +∞ and y<0, the result is +0.
• If x is −∞ and y>0 and y is an odd integer, the result is −∞.
• If x is −∞ and y>0 and y is not an odd integer, the result is +∞.
• If x is −∞ and y<0 and y is an odd integer, the result is −0.
• If x is −∞ and y<0 and y is not an odd integer, the result is +0.
• If x is +0 and y>0, the result is +0.
• If x is +0 and y<0, the result is +∞.
• If x is −0 and y>0 and y is an odd integer, the result is −0.
• If x is −0 and y>0 and y is not an odd integer, the result is +0.
• If x is −0 and y<0 and y is an odd integer, the result is −∞.
• If x is −0 and y<0 and y is not an odd integer, the result is +∞.
• If x<0 and x is finite and y is finite and y is not an integer, the result is NaN.

_{emphasis mine}

as a general rule, native functions to any language should work as described in the language specification. Sometimes this includes explicitly "undefined behavior" where it's up to the implementer to determine what the result should be, however this is not a case of undefined behavior.