Highest and lowest possible result of this javascript expression?

Question

What is the highest number this javascript expression can evaluate to? What is the lowest number? Why?

+(''+Math.random()).substring(2)

Extra credit: How many different values can the expression evaluate to? Can it be every value from the minimum to the maximum, or are some intermediate values not obtainable due to rounding issues?

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Response to Daniel's answer (deleted, was 10000000000000000 max, 0 min):

I was playing around in Chrome's console and got this:

    Math.random();

>> 0.00012365682050585747

    '12365682050585747'.length

>> 17

    12365682050585747 > 10000000000000000

>> true

... so 10000000000000000 can't be the max!

Answer 1

It depends on how the random number is generated, and how the number will be converted to string. The ECMAScript spec doesn't specify both of these.

In practice, the number will have at most 17 significant figures, so the maximum should be at most 10¹⁷.

The spec does specify that a number will be displayed in decimal form (instead of scientific form) when the exponent is between -6 and 20 (10^-6 ≤ x < 10²¹), so we just need to restrict our attention on numbers in [10^-6, 1) when trying to seek the maximum exhaustively.

However, in this range a number must be representable as s × 2^e, where 1 ≤ s ≤ 2 − 2^-52 with a precision of Δs = 2^-52 and -20 ≤ e ≤ -1. The spec recommends that ToNumber(ToString(x)) == x, so the number should be precise down to 2^-52+e for a given e. Thus the "17-digit" number with (2 − n × 2^-52) × 2^e with the smallest n will be the biggest number representable with a given e, after chopping the initial 0..

             v                 
(-20) 0.0000019073486328124998
(-19) 0.0000038146972656249996
(-18) 0.0000076293945312499975 (n=3)
(-17)  0.000015258789062499998
(-16)  0.000030517578124999997
(-15)  0.000061035156249999986 (n=2)
(-14)   0.00012207031249999999
(-13)   0.00024414062499999997
(-12)   0.00048828124999999995
(-11)    0.0009765624999999999 (always 16-digit?)
(-10)    0.0019531249999999998
(-9)     0.0039062499999999996
(-8)     0.0078124999999999965 (n=4)
(-7)      0.015624999999999998
(-6)      0.031249999999999997
(-5)      0.062499999999999986 (n=2)
(-4)       0.12499999999999999
(-3)       0.24999999999999997
(-2)       0.49999999999999994
(-1)        0.9999999999999999 (always 16-digit?)

From here we know that the absolute maximum is 78,124,999,999,999,965.

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Math.random() can return any nonnegative numbers in the interval [0, 1), so the safe minimum is -324 from 5e-324 (the smallest subnormal number in double precision is 4.94 × 10^-324).