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Metrology

Preventing Integer Overflow in Physical Computations

Integers overflow. That is not a controversial statement. What is surprising is how easily overflow can hide behind the abstraction of a units library.

Most developers immediately think of explicit or implicit scaling operations — calling .in(unit) to convert a quantity, constructing a quantity from a different unit, or assigning between quantities with different units. These are indeed places where overflow can occur, and the library cannot prevent it at compile time when the values are only known at runtime. But at least these operations are visible in your code: you wrote the conversion, you asked for the scaling, and you can reason about whether the multiplication or division might overflow your integer type.

The far more insidious problem is what happens when you don't ask for a conversion.

When you write 1 * m + 1 * ft, the library must automatically convert both operands to a common unit before performing the addition. That conversion — which you never explicitly requested — involves multiplication or division by scaling factors. With integer representations, those scaling operations can overflow silently, producing garbage results that propagate through your calculations undetected.

No compile-time programming can prevent this. The values are only known at runtime. But very few libraries provide proper tools to detect it.

This article explains why that limitation is real, how other libraries have tried to work around it, and what mp-units provides to close the gap as tightly as the language allows.

Range-Validated Quantity Points

Physical units libraries have always been very good at preventing dimensional errors and unit mismatches. But there is a category of correctness that they have universally ignored: domain constraints on quantity point values.

A latitude is not just a length divided by a radius. It is a value that lives in \([-90°, +90°]\); anything outside that range is physically meaningless. An angle used in bearing navigation wraps cyclically around a circle; treating it as an unbounded real number ignores a fundamental property of the domain. A clinical body-temperature sensor should reject a reading of \(44\ \mathrm{°C}\) at the API boundary, not silently pass it downstream.

Type-level constraint enforcement for quantity points with this level of flexibility is a relatively unexplored area in mainstream physical units libraries. The approach we present here is novel and experimental — we are certain there are edge cases and design considerations we haven't yet discovered.

This article describes the motivation in depth, the design we arrived at, and the open questions we would love the community's help to answer.

Understanding Safety Levels in Physical Units Libraries

Physical quantities and units libraries exist primarily to prevent errors at compile time. However, not all libraries provide the same level of safety. Some focus only on dimensional analysis and unit conversions, while others go further to prevent representation errors, semantic misuse of same-dimension quantities, and even errors in the mathematical structure of equations.

This article explores six distinct safety levels that a comprehensive quantities and units library can provide. We'll examine each level in detail with practical examples, then compare how leading C++ libraries and units libraries from other languages perform across these safety dimensions. Finally, we'll analyze the performance and memory costs associated with different approaches, helping you understand the trade-offs between safety guarantees and runtime efficiency.

We'll pay particular attention to the upper safety levels—especially quantity kind safety (distinguishing dimensionally equivalent concepts such as work vs. torque, or Hz vs. Bq) and quantity safety (enforcing correct quantity hierarchies and scalar/vector/tensor mathematical rules)—which are well-established concepts in metrology and physics, yet remain widely overlooked in the C++ ecosystem. Most units library authors and users simply do not realize these guarantees are achievable, or how much they matter in practice. These levels go well beyond dimensional analysis, preventing subtle semantic errors that unit conversions alone cannot catch, and are essential for realizing truly strongly-typed numerics in C++.

Introducing Absolute Quantities

Until now, mp-units forced users to choose between points (no arithmetic) and deltas (no physical semantics) — missing the most common case: a non-negative absolute amount.

An absolute quantity represents an absolute amount of a physical property — measured from a true, physically meaningful zero. Examples include mass in kilograms, temperature in Kelvin, or length in meters (as a size, not a position). Such quantities live on a ratio scale and are anchored at a physically meaningful zero; negative values are typically meaningless.

Absolute quantities stand in contrast to:

  • Affine points (e.g., \(20\ \mathrm{°C}\), \(100\ \mathrm{m}\ \mathrm{AMSL}\)) — values measured relative to an arbitrary or conventional origin.
  • Deltas (e.g., \(10\ \mathrm{K}\), \(–5\ \mathrm{kg}\)) — differences between two values.

Arithmetic on absolute quantities behaves like ordinary algebra: addition, subtraction, and scaling are well-defined and map naturally to physical reasoning. This article proposes making absolute quantities the default abstraction in mp-units V3, reflecting how scientists express equations in practice.

Note: Revised on March 23, 2026 for clarity, accuracy, and completeness.

Bringing Quantity-Safety To The Next Level

All quantities and units libraries need to be unit-safe. Most of the libraries on the market do this correctly. Some of them are also dimension-safe, which adds another level of protection for their users.

mp-units is probably the only library on the market that additionally is quantity-safe. This gives a new quality and possibilities. I've described the major idea behind it, implementation details, and benefits to the users in the series of posts about the International System of Quantities.

However, this is only the beginning. We've always planned more and worked on the extensions in our free time. In this post, I will describe:

  • What a quantity character is?
  • The importance of using proper representation types for the quantities.
  • The power of providing character-specific operations for the quantities.
  • Discuss implementation challenges and possible solutions.

International System of Quantities (ISQ): Part 5 - Benefits

In the previous articles, we introduced the International System of Quantities, described how we can model and implement it in a programming language, and presented the issues of software that does not use such abstraction to implement a units library.

Some of the issues raised in Part 2 of our series were addressed in Part 3 already. This article will present how our ISQ model elegantly addresses the remaining problems.

International System of Quantities (ISQ): Part 3 - Modeling ISQ

The physical units libraries on the market typically only focus on modeling one or more systems of units. However, as we have learned, this is not the only system kind to model. Another, and maybe even more important, is a system of quantities. The most important example here is the International System of Quantities (ISQ) defined by ISO/IEC 80000.

This article continues our series about the International System of Quantities. This time, we will learn about the main ideas behind the ISQ and describe how it can be modelled in a programming language.