Algorithm to convert an IEEE 754 double to a string?
See Ryan Juckett's Printing Floating-Point Numbers (2014), which describes history and implementations of floating-point to string conversions. In this four-part post, Ryan also provides a C++ implementation of Dragon4 based on Steele and White (1990), which is an efficient algorithm to convert a binary number in floating point format to a decimal number in string format.
You can also see a C implementation of Ryan's Dragon4 for Numpy here, and use it within Python/Numpy 1.14 format_float_positional and format_float_scientific functions.
In 2018, an algorithm/library Ryu was published, with bindings in many modern programming languages (C, Java, C++, C#, Scala, Rust, Julia, Go, ...)
I believe you are looking for Printing Floating-Point Numbers Quickly and Accurately
I found that link on another post: here.
The code used by various software environments to convert floating-point numbers to string representations is typically based on the following publications (the work by Steele and White is particularly frequently cited):
Jerome T. Coonen: "An Implementation Guide to a Proposed Standard for Floating-Point Arithmetic." Computer, Vol. 13, No. 1, January 1980, pp. 68-79
Guy. L. Steele Jr. and J. L. White: "How to print floating-point numbers accurately". In proceedings of ACM SIGPLAN '90 Conference on Programming Language Design and Implementation, White Plains, New York, June 1990, pp. 112-126
David M. Gay: "Correctly rounded binary-decimal and decimal-binary conversions." Technical Report 90-10, AT&T Bell Laboratories, November 1990.
Some relevant followup work:
Robert G. Burger and R. Kent Dybvig: "Printing floating-point numbers quickly and accurately." In proceedings of ACM SIGPLAN 1996 conference on Programming Language Design and Implementation, Philadelphia, PA, USA, May 1996, pp. 108-116
Guy L. Steele Jr. and Jon L. White: "Retrospective: How to print floating-point numbers accurately." ACM SIGPLAN Notices, Vol. 39, No. 4, April 2004, pp. 372–389
Florian Loitsch: "Printing floating-point numbers quickly and accurately with integers." In proceedings of 2010 ACM SIGPLAN Conference on Programming Language Design and Implementation, Toronto, ON, Canada, June 2010, pp. 233-243
Marc Andrysco, Ranjit Jhala, and Sorin Lerner: "Printing floating-point numbers: a faster, always correct method." ACM SIGPLAN Notices, Vol. 51, No. 1, January 2016, pp. 555-567
Ulf Adams: "Ryū: fast float-to-string conversion." ACM SIGPLAN Notices, Vol. 53, No. 4, April 2018, pp. 270-282