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

on totally ordered sets Eric Hughes, Meadhbh Hamrick In brief stdx::interval implements the mathematical sense of an interval on a totally ordered set. The library reasons about intervals as sets, not than raw bounds. Intervals are a basic mathematical concept that appear in all kinds of algorithms, from recursive sorting to adaptive mesh refinement. Interval notation allows a more natural expression of

deeply whether computing made sense, given rounding error6 The Householder convention A standard notation, & a standard way of talking about computations We use math words We mean a computer representation like sort • For hardware experts to optimize • With readable, self-documenting names • From mathematical algorithms • For mathematicians (e.g., experts in rounding error analysis) to implement correctly J. Hanson, “An Extended Set of FORTRAN Basic Linear Algebra Subprograms,” ACM Transactions on Mathematical Software (TOMS), Vol. 14, No. 1, pp. 1 – 17, Mar. 1988. Available online: https://dl.acm.org/doi/10

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 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; context for users Avoid unexpected conversions Leverage mathematical knowledge to optimize Mathematical operations behaviour may depend on mathematical concepts (commutativity for example) The idea Implement

Papers — GraphBLAS standards — SuiteSparse:GraphBLAS releases 0.9 1.0 1.2 1.3 1.0 3. 0 2.2 Mathematical foundations, HPEC Seven good reasons, ICCS Standards for graph algorithm primitives Monoids: Binary Operators with an Identity - Monoids are mathematical objects, consisting of: - A commutative binary operator - A type T - A mathematical identity 113[DISTRIBUTION STATEMENT A] This material Monoids: Binary Operators with an Identity - Monoids are mathematical objects, consisting of: - A commutative binary operator - A type T - A mathematical identity 114[DISTRIBUTION STATEMENT A] This material

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

characterized by the use of mathematical functions and the avoidance of side effects What is Functional Programming?● A programming paradigm characterized by the use of mathematical functions and the avoidance declarative syntax What is Functional Programming?● A programming paradigm characterized by the use of mathematical functions and the avoidance of side effects ● Functional programming is identified by the use and more What is Functional Programming?● A programming paradigm characterized by the use of mathematical functions and the avoidance of side effects ● Functional programming is identified by the use

Such solutions while working in most cases, are not without problems, both in terms of numerical/mathematical soundness (? any hash algorithm), but also in terms of flexibility/composition of the code using Such solutions while working in most cases, are not without problems, both in terms of numerical/mathematical soundness (? any hash algorithm), but also in terms of flexibility/composition of the code using Such solutions while working in most cases, are not without problems, both in terms of numerical/mathematical soundness (? any hash algorithm), but also in terms of flexibility/composition of the code using

term Functor ○ in mathematical terms it is a function which ■ takes one or more functions as its arguments ■ returns a function as the result ○ functor is also defined in mathematical category theory

Program Chair SG20Outline • Main Topic: use C++ to simulate the process of cancer initiation • The mathematical model and simulation study • Generate a single tumor (A single step of evolution) • Generate by changes in genes (DNA). • When cancer is found early, it may be easier to treat or cure • Mathematical models help understand the evolution quantitatively. (e.g. predict the window of opportunity