Welcome to dogduck.com — a powerful tool designed for professionals who need to approximate decimal numbers using mathematical expressions. Whether you're exploring mathematical constants, verifying formulas, or simply curious about elegant expressions, this tool will find a simple mathematical expression equal to your target value.

How to Use

1. Enter Your Target Number

Input the decimal number you want to approximate. This could be any number you're curious about. Enter at least 3 decimal numbers after the comma, to ensure the program finds an accurate expression. The program automatically determines the search margin based on the amount of decimals you give it. For example, if your number is 1.618, the margin will automatically be set to 0.001.

2. Choose Thinking Time

Select how long the program should search for expressions:

• Very Short: Quick results
• Short: Balanced speed and quality
• Medium: More thorough search
• Long: Extensive search for elegant expressions
• Very Long: Maximum effort for the simplest possible result

Note: The program always returns a value within your specified margin, regardless of thinking time.

3. Advanced Options (Optional)

Click "More options" to access advanced settings:

• Margin Mode: Choose between ± margin (symmetric) or + margin (one-sided precision). Use + margin if you want all your inputed decimals to be matched.
• Allowed Operations: Select which mathematical operations and symbols the program may use. For example, if you want to find an expression π, you might want to unselect π itself and the trigonometric functions.
• Depth & Loops: Fine-tune computational parameters
  • Computational time grows approximately as loops^depth
• Batch Size: Control processing speed (automatic if set to 0)
• Show Search Time: Display how long the search took

4. Find Expression

Click "Find Expression" to start the search. The tool will process your request and display the best expression within you margin. For example if your number is 1.618, and ± margin is selected, the program will search for an expression between the intevals 1.617 and 1.619.

You can stop the search at any time by clicking "Stop" — the best result found so far will be displayed.

Understanding Results

The program displays the least complex expression found, along with its calculated value.

Complexity Definition

An expression's complexity is determined by an algorithm that considers:

• Number of operations used
• Size of numbers in the expression
• Types of operations involved (simpler operations = lower complexity)

The goal is to find the simplest and most elegant expression that matches your target within the specified margin.

Examples of How to Use

Example 1: The Golden Ratio

Input: 1.618033
Thinking Time: Very short
Result: (1+√5)/2

This is one of the most famous mathematical constants - the golden ratio φ (phi).

Example 2: Pi Approximation

Input: 3.14159
Thinking Time: Medium
Advanced Options: Click "More options" and unmark π, sin, cos, tan, asin, acos, and atan
Result: 16/113+3

By blocking trigonometric functions and π itself, the program finds alternative ways to approximate this famous constant.

Example 3: Euler's Number

Input: 2.71828
Thinking Time: Long
Advanced Options: Unmark e, e^x and ln and choose + margin to get the las decimal correct
Result: 3-√(5/63)

Finding Euler's number without using e directly shows how the program can discover creative expressions.

Example 4: Square Root of 2

Input: 1.41421356
Thinking Time: Very Short
Result: √2

Simple constants are recognized quickly even with minimal thinking time.

Example 5: Custom Number

Input: 0.57721566
Thinking Time: Very long
Result: 3-sin(1)^(4*(e-4))

Some constants don't have known simple closed-form expressions - the program will find a simple available approximation. In this, case the resulting expression match with the input digits, but is not an exact expression for the Euler-Mascheroni constant γ.

Example 6: Custom Advanced Search

Input: 7.41221376523
Advanced Options: Set Depth to 2 and Loops to 500
Result: ln(45) + √13

By manually adjusting depth and loops, you can control the search more precisely. Higher loop values explore more constants, potentially finding more elegant expressions.

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Tip: Start with shorter thinking times and gradually increase if you want simpler expressions. Use "More options" to fine-tune the search for your specific needs.

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Note: The calculations and the print-outs in this program is made using f64 precision.

About the Creator

Hello, my name is Tom and am from Sweden. I have always been interested in math and programming, especially when you come up with something new. I like working on programming projects that are both fun, interesting and also can be useful. Which hopefully this program is.

Why I Built This Tool

There have been multiple times when I have done some calculation or something, and ended up with an unknown decimal number, and often I wonder if it can be written in an mathematical expression. I also built this website to see if already known constants can be approximated in simple expressions.

This is actually the second version of the website. The previous version was much worse at finding good expressions and was a little lower quality. The previous version just generated random expression and showed the expression that evaluated to a number as close as possible to the input number. Compared to the previous version, this new version is many orders of magnitudes more accurate, and allows for more operations.

The Technology

This tool is built using Rust compiled to WebAssembly (WASM) for maximum performance in the browser.

In short, the program have a look-up table of about 300 000 stored expression and their value, if it cant find the input number in there, it makes a fraction of it (which can be made using an algorithm), then it tries making transformations of your number, such as adding something to it and see if the new number can be approximated. It does this in a recrusive loop.

I have used lots of AI while building this program. The prototype was built in python by myself, but AI have helped me translated it to rust to make this website look proffesional. The only thing on the website not written fully or partly by AI is this about me page, since here I want the viewer to read my own words and not an AI's words.