How does JavaScript determine the number of digits to produce when formatting floating-point values?

Question

In JavaScript, everyone knows the famous calculation: 0.1 + 0.2 = 0.30000000000000004. But why does JavaScript print this value instead of printing the more accurate and precise 0.300000000000000044408920985006?

Answer 1

The default rule for JavaScript when converting a Number value to a decimal numeral is to use just enough digits to distinguish the Number value. (You can request more or fewer digits by using the toPrecision method.)

JavaScript uses IEEE-754 basic 64-bit binary floating-point for its Number type. Using IEEE-754, the result of .1 + .2 is exactly 0.3000000000000000444089209850062616169452667236328125. This results from:

• Converting “.1” to the nearest value representable in the Number type.
• Converting “.2” to the nearest value representable in the Number type.
• Adding the above two values and rounding the result to the nearest value representable in the Number type.

When formatting this Number value for display, “0.30000000000000004” has just enough significant digits to uniquely distinguish the value. To see this, observe that the neighboring values are:

• 0.299999999999999988897769753748434595763683319091796875,
• 0.3000000000000000444089209850062616169452667236328125, and
• 0.300000000000000099920072216264088638126850128173828125.

If the conversion to a decimal numeral produced only “0.3000000000000000”, it would be nearer to 0.299999999999999988897769753748434595763683319091796875 than to 0.3000000000000000444089209850062616169452667236328125. Therefore, another digit is needed. When we have that digit, “0.30000000000000004”, then the result is closer to 0.3000000000000000444089209850062616169452667236328125 than to either of its neighbors. Therefore, “0.30000000000000004” is the shortest decimal numeral (neglecting the leading “0” which is there for aesthetic purposes) that uniquely distinguishes which possible Number value the original value was.

This rules comes from step 5 in clause 7.1.12.1 of the ECMAScript 2017 Language Specification, which is one of the steps in converting a Number value m to a decimal numeral for the ToString operation:

Otherwise, let n, k, and s be integers such that k ≥ 1, 10^k‐1 ≤ s < 10^k, the Number value for s × 10^n‐k is m, and k is as small as possible.

The phrasing here is a bit imprecise. It took me a while to figure out that by “the Number value for s × 10^n‐k”, the standard means the Number value that is the result of converting the mathematical value s × 10^n‐k to the Number type (with the usual rounding). In this description, k is the number of significant digits that will be used, and this step is telling us to minimize k, so it says to use the smallest number of digits such that the numeral we produce will, when converted back to the Number type, produce the original number m.