Adding to Number.MAX_VALUE

Question

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?

Answer 1

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 b^emax ( b − ½ b^1-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

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Meanwhile, your Math.pow(100,1000) ~ 2^6643.9 is well beyond 2¹⁰²⁴ - 2⁹⁷⁰. It is already infinity.

Answer 2

If you look at Number.MAX_VALUE.toString(2), you'll see that the binary representation of MAX_VALUE is 53 ones followed by 971 zeros. This because IEEE 754 floating points are made of a mantissa coefficient multiplied by a power of 2 (so the other half of the floating point number is the exponent). With MAX_VALUE, both the mantissa and the exponent are maxed out, so you see a bunch of ones bit-shifted up a lot.

In short, you need to increase MAX_VALUE enough to actually affect the mantissa, otherwise your additional value gets lost and rounded out.

Math.pow(2, 969) is the lowest power of 2 that will not tip MAX_VALUE into Infinity.