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Adding to Number.MAX_VALUE

The answer to this question may be painfully obvious but I can't find it in the Mozilla docs nor on Google from a cursory search.

If you have some code like this

Number.MAX_VALUE + 1; // Infinity, right?
Number.MIN_VALUE - 1; // -Infinity, right?

Then I would expect adding anything to Number.MAX_VALUE would push it over to Infinity. The result is just Number.MAX_VALUE spat right back at me.

However, when playing around in the Chrome JS console, I noticed that it didn't actually become Infinity until I added/subtracted enough:

Number.MAX_VALUE + Math.pow(100,1000); // now we hit Infinity
Number.MIN_VALUE - Math.pow(100,1000); // -Infinity at last

What is the explanation for this "buffer" between Number.MAX_VALUE and Infinity?

Solution

  • Standardwise...

    In ECMAScript, addition of two nonzero finite numbers is implemented as (ECMA-262 §11.6.3 "Applying the Additive Operators to Numbers"):

    the sum is computed and rounded to the nearest representable value using IEEE 754 round-to-nearest mode. If the magnitude is too large to represent, the operation overflows and the result is then an infinity of appropriate sign.

    IEEE-754's round-to-nearest mode specifies that (IEEE-754 2008 §4.3.1 "Rounding-direction attributes to nearest")

    In the following two rounding-direction attributes, an infinitely precise result with magnitude at least bemax ( b − ½ b1-p ) shall round to ∞ with no change in sign; here emax and p are determined by the destination format (see 3.3). With:

    • roundTiesToEven, the floating-point number nearest to the infinitely precise result shall be delivered; if the two nearest floating-point numbers bracketing an unrepresentable infinitely precise result are equally near, the one with an even least significant digit shall be delivered
    • roundTiesToAway, the floating-point number nearest to the infinitely precise result shall be delivered; if the two nearest floating-point numbers bracketing an unrepresentable infinitely precise result are equally near, the one with larger magnitude shall be delivered.

    ECMAScript does not specify which of the round-to-nearest, but it doesn't matter here because both gives the same result. The number in ECMAScript is "double", in which

    • b = 2
    • emax = 1023
    • p = 53,

    so the result must be at least 21024 - 2970 ~ 1.7976931348623158 × 10308 in order to round to infinity. Otherwise it will just round to MAX_VALUE, because that is the closer than Infinity.

    Notice that MAX_VALUE = 21024 - 2971, so you need to add at least 2971 - 2970 = 2970 ~ 9.979202 × 10291 in order to get infinity. We could check:

    >>> Number.MAX_VALUE + 9.979201e291
1.7976931348623157e+308
>>> Number.MAX_VALUE + 9.979202e291
Infinity

    Meanwhile, your Math.pow(100,1000) ~ 26643.9 is well beyond 21024 - 2970. It is already infinity.