When I published Grisu (Google double-conversion), it was multiple times faster than the existing algorithms. I knew that there was still room for improvement, but I was at most expecting a factor 2 or so. Six times faster is really impressive.
It seems that most research effort goes into better dtoa, and not enough in a better atod. There are probably a dozen dtoa algorithms now, and (I think?) two for atod. Anyone know why?
Good question. I am not familiar with string-to-double algorithms but maybe it's an easier problem? double-to-string is relatively complex, people even doing PhD in this area. There is also some inherent asymmetry: formatting is more common than parsing.
In implementing Rust's serde_json library, I have dealt with both string-to-double and double-to-string. Of the two, I found string-to-double was more complex.
Unlike formatting, correct parsing involves high precision arithmetic.
Example: the IEEE 754 double closest to the exact value "0.1" is 7205759403792794*2^-56, which has an exact value of A (see below). The next higher IEEE 754 double has an exact value of C (see below). Exactly halfway between these values is B=(A+C)/2.
So for correctness the algorithm needs the ability to distinguish the following extremely close values, because the first is closer to A (must parse to A) whereas the second is closer to C:
The problem of "string-to-double for the special case of strings produced by a good double-to-string algorithm" might be relatively easy compared to double-to-string, but correct string-to-double for arbitrarily big inputs is harder.
Is it, though? It's genuinely hard for me to tell.
There's both serialization and deserialization of data sets with, e.g., JSON including floating point numbers, implying formatting and parsing, respectively.
Source code (including unit tests etc.) with hard-coded floating point values is compiled, linted, automatically formatted again and again, implying lots of parsing.
Code I usually work with ingests a lot of floating point numbers, but whatever is calculated is seldom displayed as formatted strings and more often gets plotted on graphs.
And there’s one detail I found confusing. Suppose I go through the steps to find the rounding interval and determine that k=-3, so there is at most one integer multiple of 10^-3 in the interval (and at least one multiple of 10^-4). For the sake of argument, let’s say that -3 worked: m·10^-3 is in the interval.
Then, if m is not a multiple of 10, I believe that m·10^-3 is the right answer. But what if m is a multiple of 10? Then the result will be exactly equal, numerically, to the correct answer, but it will have trailing zeros. So maybe I get 7.460 instead of 7.46 (I made up this number and I have no idea whether any double exists gives this output.) Even though that 6 is definitely necessary (there is no numerically different value with decimal exponent greater than -3 that rounds correctly), I still want my formatter library to give me the shortest decimal representation of the result.
Is this impossible for some reason? Is there logic hiding in the write function to simplify the answer? Am I missing something?
This is possible and the trailing zeros are indeed removed (with the exponent adjusted accordingly) in the write function. The post mentions removing trailing zeros without going into details but it's a pretty interesting topic and was recently changed to use lzcnt/bsr instead of a lookup table.
I wonder how Teju Jaguá compares. I don't see it in the C++ benchmark repo you linked and whose graph you included.
I have contributed an implementation in Rust :) https://crates.io/crates/teju it includes benchmarks which compare it vs Ryu and vs Rust's stdlib, and the readme shows a graph with some test cases. It's quite easy to run if you're interested!
I am not sure how it compares but I did use one idea from Cassio's talk on Teju:
> A more interesting improvement comes from a talk by Cassio Neri Fast Conversion From Floating Point Numbers. In Schubfach, we look at four candidate numbers. The first two, of which at most one is in the rounding interval, correspond to a larger decimal exponent. The other two, of which at least one is in the rounding interval, correspond to the smaller exponent. Cassio’s insight is that we can directly construct a single candidate from the upper bound in the first case.
Indeed! I saw that you linked to Neri's work, so you were aware of Teju Jaguá. I might make a pull request to add it to the benchmark repo when I have some time :)
Another nice thing about your post is mentioning the "shell" of the algorithm, that is, actually translating the decimal significand and exponent into a string (as opposed to the "core", turning f * 2^e into f' * 10^e'). A decent chunk of the overall time is spent there, so it's worth optimising it as well.
This blows my mind TBH. I used to say a few years back that Ryu is my favorite modern algorithm but it felt so complicated. Your C implentation almost feels... simple!
Congratulations, can't wait to have some time to study this further
Thank you! The simplicity is mostly thanks to Schubfach although I did simplify it a bit more. Unfortunately the paper makes it appear somewhat complex because of all the talk about generic bases and Java workarounds.
I've just started a Julia port and I think it will be even cleaner than the C version (mostly because Julia gives you a first class (U)Int128 and count leading zeros (and also better compile time programming that lets you skip on writing the first table out explicitly).
Nice, but it's too late I needed a different API for future use in my custom sprintf so I made mulle-dtostr (https://github.com/mulle-core/mulle-dtostr). On my machine (AMD) that benchmarked in a quick try quite a bit faster even, but I was just checking that it didn't regress too badly and didn't look at it closer.
It depends on the input distribution, specifically exponents. It is also possible to compress the table at the cost of additional computation using the method from Dragonbox.
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When I published Grisu (Google double-conversion), it was multiple times faster than the existing algorithms. I knew that there was still room for improvement, but I was at most expecting a factor 2 or so. Six times faster is really impressive.
Unlike formatting, correct parsing involves high precision arithmetic.
Example: the IEEE 754 double closest to the exact value "0.1" is 7205759403792794*2^-56, which has an exact value of A (see below). The next higher IEEE 754 double has an exact value of C (see below). Exactly halfway between these values is B=(A+C)/2.
So for correctness the algorithm needs the ability to distinguish the following extremely close values, because the first is closer to A (must parse to A) whereas the second is closer to C: The problem of "string-to-double for the special case of strings produced by a good double-to-string algorithm" might be relatively easy compared to double-to-string, but correct string-to-double for arbitrarily big inputs is harder.Is it, though? It's genuinely hard for me to tell.
There's both serialization and deserialization of data sets with, e.g., JSON including floating point numbers, implying formatting and parsing, respectively.
Source code (including unit tests etc.) with hard-coded floating point values is compiled, linted, automatically formatted again and again, implying lots of parsing.
Code I usually work with ingests a lot of floating point numbers, but whatever is calculated is seldom displayed as formatted strings and more often gets plotted on graphs.
https://vitaut.net/posts/2025/smallest-dtoa/
And there’s one detail I found confusing. Suppose I go through the steps to find the rounding interval and determine that k=-3, so there is at most one integer multiple of 10^-3 in the interval (and at least one multiple of 10^-4). For the sake of argument, let’s say that -3 worked: m·10^-3 is in the interval.
Then, if m is not a multiple of 10, I believe that m·10^-3 is the right answer. But what if m is a multiple of 10? Then the result will be exactly equal, numerically, to the correct answer, but it will have trailing zeros. So maybe I get 7.460 instead of 7.46 (I made up this number and I have no idea whether any double exists gives this output.) Even though that 6 is definitely necessary (there is no numerically different value with decimal exponent greater than -3 that rounds correctly), I still want my formatter library to give me the shortest decimal representation of the result.
Is this impossible for some reason? Is there logic hiding in the write function to simplify the answer? Am I missing something?
I wonder how Teju Jaguá compares. I don't see it in the C++ benchmark repo you linked and whose graph you included.
I have contributed an implementation in Rust :) https://crates.io/crates/teju it includes benchmarks which compare it vs Ryu and vs Rust's stdlib, and the readme shows a graph with some test cases. It's quite easy to run if you're interested!
> A more interesting improvement comes from a talk by Cassio Neri Fast Conversion From Floating Point Numbers. In Schubfach, we look at four candidate numbers. The first two, of which at most one is in the rounding interval, correspond to a larger decimal exponent. The other two, of which at least one is in the rounding interval, correspond to the smaller exponent. Cassio’s insight is that we can directly construct a single candidate from the upper bound in the first case.
Another nice thing about your post is mentioning the "shell" of the algorithm, that is, actually translating the decimal significand and exponent into a string (as opposed to the "core", turning f * 2^e into f' * 10^e'). A decent chunk of the overall time is spent there, so it's worth optimising it as well.
Congratulations, can't wait to have some time to study this further
I have been doing some tests. Is it correct to assume that it converts 1.0 to "0.000000000000001e+15".
Is there a test suite it is passing?