Composing Ancient Mathematical Knowledge Into Powerful Bit-fiddling techniques Jamie Pond -- CppCon 2024TLDW; New insights from Ancient Egyptian Multiplication gives us the freedom of synthesising

lambda trick Comparison Binding Constraints Architecture Substitution Construction Conclusion Mathematical expressions Normal distribution PDF f = 1 √ 2πσ2 e− (x−µ)2 2σ2 2 + + 2 Normal distribution symbols Expression templates are stateful Stateless Expression Templates Formulas are stateless Data is injected in formulas Both live their lives in different world What changed since then? C++20 Conclusion Symbols with constraints It would be nice to be able to provide information on the mathematical entity it represents 1 symbol a; 2 symbol w; 3 symbol t; 4 symbol phi; 1 symbol a;

Theoretical foundations Outline of an implementation Conclusion Mathematical and software architecture Question How to abstract the problem? Mathematical abstractions Well-posed and abstracted in mathematics mathematics ⇒ ‘‘easy” to abstract in software Mathematical architecture ⇒ Software architecture Counter-example: units Intuitively understood in physics But very fuzzy foundations in mathematics ⇒ Incredibly DedicationIntroduction Firsts steps Context Theoretical foundations Outline of an implementation Conclusion The mathematical architecture of algorithms John C. Butcher Numerical Methods for Ordinary Differential Equations

Literal zero and one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5 Mathematical Operations and Elementary Functions 26 5.1 Arithmetic Operators . . . . . . . . . . . . . . . . . . . . . 717 43 Mathematics 719 43.1 Mathematical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 43.2 Mathematical Functions . . . . . . . . . . . . . . . . . . Python, but also supports general programming. To achieve this, Julia builds upon the lineage of mathematical programming languages, but also borrows much from popular dynamic languages, including Lisp,

Literal zero and one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5 Mathematical Operations and Elementary Functions 26 5.1 Arithmetic Operators . . . . . . . . . . . . . . . . . . . . . 717 43 Mathematics 719 43.1 Mathematical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 43.2 Mathematical Functions . . . . . . . . . . . . . . . . . . Python, but also supports general programming. To achieve this, Julia builds upon the lineage of mathematical programming languages, but also borrows much from popular dynamic languages, including Lisp,

Literal zero and one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5 Mathematical Operations and Elementary Functions 26 5.1 Arithmetic Operators . . . . . . . . . . . . . . . . . . . . . 718 43 Mathematics 720 43.1 Mathematical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720 43.2 Mathematical Functions . . . . . . . . . . . . . . . . . . Python, but also supports general programming. To achieve this, Julia builds upon the lineage of mathematical programming languages, but also borrows much from popular dynamic languages, including Lisp,

Literal zero and one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6 Mathematical Operations and Elementary Functions 30 6.1 Arithmetic Operators . . . . . . . . . . . . . . . . . . . . . 806 44 Mathematics 808 44.1 Mathematical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808 44.2 Mathematical Functions . . . . . . . . . . . . . . . . . . Python, but also supports general programming. To achieve this, Julia builds upon the lineage of mathematical programming languages, but also borrows much from popular dynamic languages, including Lisp,

Literal zero and one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6 Mathematical Operations and Elementary Functions 30 6.1 Arithmetic Operators . . . . . . . . . . . . . . . . . . . . . 806 44 Mathematics 808 44.1 Mathematical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808 44.2 Mathematical Functions . . . . . . . . . . . . . . . . . . Python, but also supports general programming. To achieve this, Julia builds upon the lineage of mathematical programming languages, but also borrows much from popular dynamic languages, including Lisp,

Literal zero and one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6 Mathematical Operations and Elementary Functions 30 6.1 Arithmetic Operators . . . . . . . . . . . . . . . . . . . . . 809 44 Mathematics 811 44.1 Mathematical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811 44.2 Mathematical Functions . . . . . . . . . . . . . . . . . . Python, but also supports general programming. To achieve this, Julia builds upon the lineage of mathematical programming languages, but also borrows much from popular dynamic languages, including Lisp,

Literal zero and one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6 Mathematical Operations and Elementary Functions 30 6.1 Arithmetic Operators . . . . . . . . . . . . . . . . . . . . . 808 44 Mathematics 810 44.1 Mathematical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810 44.2 Mathematical Functions . . . . . . . . . . . . . . . . . . Python, but also supports general programming. To achieve this, Julia builds upon the lineage of mathematical programming languages, but also borrows much from popular dynamic languages, including Lisp,