# 1.5 is the midpoint between 0 and infinity in Ruby

What’s the midpoint between 0 and infinity? Well the answer differs depending on whether you are asking a mathematician, philosopher, or a Ruby developer. I’m not a mathematician or a philosopher, but I am a Ruby developer, so I can tell you than 1.5 is the midpoint between 0 and infinity.

# `Range#bsearch`

`Range#bsearch`

performs binary search within a range. For example, lets use it to find the first integer that’s larger than 42 (which is 43) and see the values it inspects to find it.

```
values = []
found_value = (0..).bsearch do |i|
values << i
i > 42
end
puts "The integer larger than 42 is: #{found_value}"
puts "The following values were inspected:"
puts values
```

And when we run it, we get the following output:

```
The integer larger than 42 is: 43
The following values were inspected:
1
2
4
8
16
32
64
32
48
40
44
42
43
```

We see typical binary search behavior.

A co-worker recently asked me about some odd behavior when we change the starting value to a float. For example, consider the following code, which is the same as the one above but the range is between 0 and `Float::INFINITY`

:

```
values = []
found_value = (0..Float::INFINITY).bsearch do |i|
values << i
i > 42
end
puts "The float larger than 42 is: #{found_value}"
puts "The following values were inspected:"
puts values
```

We then get the following output when we run it (note that the output has been truncated because it’s too long):

```
The float larger than 42 is: 42.00000000000001
The following values were inspected:
1.5
1.6759759912428246e+154
1.5921412270130977e+77
4.8915590244884904e+38
2.7093655358260904e+19
6375342080.0
97792.0
383.0
23.96875
95.8125
47.921875
31.96484375
39.92578125
43.923828125
41.9248046875
42.92431640625
42.424560546875
42.1746826171875
...
```

The first few values we inspect in the binary search are rather odd: `1.5`

, `1.6759759912428246e+154`

, `1.5921412270130977e+77`

, etc. Where do these numbers come from? To explain these values, we first have to understand how IEEE 754 floating-point numbers work.

# Ruby float implementation

Ruby floats are double-precision IEEE 754 floating-point numbers. If you don’t know the basics about how floating-point numbers are represented in memory, there are plenty of resources on the internet, here’s one.

Of special interest to us is how infinity is represented in floating-point. Infinity is a special value where the exponent bits are all 1’s and the significand bits (also known as fraction or mantissa) are all 0’s.

# How `Range#bsearch`

works for floats

Ruby’s `Range#bsearch`

is implemented in a C function called `range_bsearch`

. There are several cases it deals with, and the one of interest is the case when either endpoint is a float. In this case, it performs a clever trick. It reads the C double type of the Ruby float endpoints as 64-bit integers (`int64_t`

)^{1}. Note that this is **not** the same as casting the double into integer type, this is directly reading the double as a 64-bit integer. Are you confused? I sure was when I first read this code so if you are too, it’ll make more sense later on. Trust me.

Let’s revisit how infinity is represented in floating-point. Here’s it visualized (through the help of the amazing website float.exposed). We can see that the sign is 0 (this is a positive value), all exponent bits are 1, and all significand bits are 0.

And of course, `0`

is represented in floating-point with all the bits set to 0.

So what if we read these bits of the endpoint as if they were integers? Then our range would be between `0`

and `9218868437227405312`

. What’s the midpoint of this range? It’s `4609434218613702656`

. Now let’s plug this value back into float.exposed.

Oh look, it’s 1.5!

Using this same technique, you can now find out why the binary search examines `1.6759759912428246e+154`

and `1.5921412270130977e+77`

after this. This is left as an exercise for the reader.

# Why does this even work?

The reason why this works is simple once you wrap your head around it. It’s because the more significant bits are always in a more significant position (than the bits to the right of it) in floating-point numbers. This is obviously true for the significand. But this is also true for the exponent because the next power of 2 can’t be reached by just increasing the significand, the exponent has to be increased. So thus a larger exponent will always mean that the floating-point number has a larger magnitude.

Once this is true, we can see why binary search works using this technique. It works because if `x`

and `y`

are doubles and `x > y`

, then we have shown that `double_as_int64(x) > double_as_int64(y)`

is also true. This is the requirement for binary search because the values remain strictly increasing (*technically* binary search only requires monotonically increasing, but strictly increasing is a tighter guarantee than monotonically increasing).

# Conclusion

Ruby’s binary search in a range uses a clever technique to perform binary search when the endpoints are doubles while maintaining a worst-case runtime of `O(n log n)`

. In fact, this technique isn’t specific to Ruby and can be used in any language that uses IEEE 754 floating-point numbers.