为什么舍入浮点数1.4999999999999999产生2？

Question

I've been reading a book Write Great code - Understanding the Machine . In the section about rounding it says:

Numbers should be rounded to the smallest bigger number if the decimal bit value is more than or equal half the total decimal value that can be represented.

which means:

round(1.5) // equals 2
round(1.49) // equals 1

but when I tried this with Python:

x1 = 1.4999  # rounds to 1

x2 = 1.4999999999999999  # rounds to 2

print(round(x1))

print(round(x2))

the output was:

1

2

I tried the same thing with C# and Swift and it gave the same output. So I assume it's a language-agnostic topic.

But why does this happen?

My assumption is that the floating-point unit rounds the extra bits which convert the "1.4999999999999999999" to "1.5" before applying the programmer's rounding.

Answer 1

In x2 = 1.4999999999999999 and print(round(x2)) , there are two operations that affect the value. The round function cannot operate directly on the number 1.4999999999999999 or the numeral “1.4999999999999999”. Its operand must be in the floating-point format that the Python implementation uses.

So, first, 1.4999999999999999 is converted to the floating-point format. Python is not strict about which floating-point format a Python implementation uses, but the IEEE-754 basic 64-bit binary format is common. In this format, the closest representable values to 1.4999999999999999 are 1.5 and 1.4999999999999997779553950749686919152736663818359375. The former is closer to 1.4999999999999999 than the latter is, so the former is used.

Thus, converting 1.4999999999999999 to the floating-point format produces 1.5. Then round(1.5) produces 2.