A mathematical model must be transparent traceable and demonstrably continuously improvable, without having to be entirely re-qualified.
[EU legal requirement]

A mathematical model must be easy to integrate on any hardware and software, and consistently deliver mathematically correct results.
[simplicity and quality requirement]

Date: September 2023

PRODUCT BRIEF – paceval.
mathematical engine, Version 4.24

Computing Excellence

Calculation Speed Calculates “real time” in milliseconds. Even faster for repeated calculations because we use caching methods. The exact speed depends on the hardware capacity and the functional complexity.
Data Volume

No limits through paceval. The limit values are determined by the hardware system used or the development environment used.


Depending on the compiler:

long double 80 bits value area
1.1E +/- 4932
19 digits
double 64 bits value area
1.7E +/- 308
16 digits
float 32 bits value area
3.4E +/- 38
7 digits
Accuracy optimization

Trusted Interval Computation, TINC™ (paceval. specific Interval arithmetic), limits rounding errors and measurement errors of the calculation system to achieve reliable results.

Formula length and numbers of variables No limits through paceval. The limit values are determined by the hardware system used or the development environment used.


Computing Excellence
Supported terms in closed-form expressions and interval arithmetic

       + Allows the definition of up to 1000 custom user functions.

Elementary arithmetic
Addition Sign +
Subtraction Sign
Multiplication Sign *
Division Sign /
Logical operators returning Boolean values either 0 (for false) or 1 (for true)
Negation NOT()
AND Operator Conjunction AND
OR Operator Disjunction OR
XOR Operator Exclusive Disjunction XOR
NAND Operator NAND
NOR Operator NOR
XNOR Operator XNOR
Annotation: For the operands, 0 is false and all other values ​​are true.
Relational operators
less than. greater than, not equal to <, >, <>
equal to, greater than or equal to, less than or equal to =, >=, <=
Annotation: The relation operators return 0 for false and 1 for true, e.g. 4 > 5 = 0.

Other operators

Least element min
Greatest element max
Annotation: min/max are implemented as operators, e.g. 4 min 5 = 4
Modulo as a symmetrical variant (like C++ or Java), e.g. (-1) % 10 = -1 %
Modulo as a mathematical variant (like Python), e.g. (-1) mod 10 = 9 mod
Annotation: The following applies to the mathematical variant
a mod b = ((a % b) + b) % b
Factorial Sign !
Factorial Function (paceval. specific) fac()
π (~3.1415384626433832795) pi
Euler’s number (~2.718281828459045235) e
Circle Bracket Open Character (
Circle Bracket Close Character )
Variables as symbolic names for values A paceval. identifier must begin with a letter (A-Z) or (a-z); Subsequent characters can also be digits (0-9).
Power Sign ^
Square Function sqr()
Square Root Function sqrt()
Exponential Function exp()
Logarithm Function lg()
Natural Logarithm Function ln()
Standard Logistic Sigmoid Function sig()
Trigonometric functions
Sine Function sin()
Cosine Function cos()
Tangent Function tan()
Cotangent Function cot()
Inverse trigonometric functions
Arc Sine Function asin()
Arc Cosine Function acos()
Arc Tangent Function atan()
Arc Cotangent Function acot()
Hyperbolic functions
Hyperbolic Sine Function sinh()
Hyperbolic Cosine Function cosh()
Hyperbolic Tangent Function tanh()
Hyperbolic Cotangent Function coth()
Inverse hyperbolic functions
Area Hyperbolic Sine Function arsinh()
Area Hyperbolic Cosine Function arcosh()
Area Hyperbolic Tangent Function artanh()
Area Hyperbolic Cotangent Function arcoth()
Numerical manipulations
Sign Function sgn()
Absolute Value Function abs()
Rounding Value Function round()
Ceiling Value Function ceil()
Floor Value Function floor()
Other Numerical manipulations (paceval. specific)
Greater Than Zero Function ispos()
Greater Than Zero Or Zero Function isposq()
Less Than Zero Function isneg()
Less Than Zero Or Zero Function isnegq()
Is Zero Function isnull()


small and safe
Use cases, security and size

Watchdog and System monitoring applications

Large hardware systems with many small subsystems and processors or sensors (e.g. cars, automation, etc.) can be monitored by a mathematical model implemented by a finite-state machine processed by paceval.

Black-box testing, Boundary and Stress testing

In order to detect anomalies in system and software/hardware applications and quickly fix them, paceval. can be used to define and create black box test cases based on a mathematical model. Usually this mathematical model is derived from the specification or is already part of the specification for the application.

Internet of Things applications

Calculations on computers and servers are increasingly becoming self-calculating (intelligent) things. Fast and reliable calculations with paceval. help you to improve your development cycles and overall schedules, resulting in lower costs and broad support of different (hardware) systems.

Pattern matching algorithms You can easily create your own pattern matching algorithms that identify patterns and regularities in data processed by paceval.
Safety Concept Using a special technique, we can quickly monitor whether data manipulation, i.e. hacking of your mathematical models, has taken place.
Object code footprint Only a few hundred kilobytes in size; The exact size depends on your compiler, development environment, and operating system.
Memory usage footprint A few hundred kilobytes to several gigabytes in size; The exact size depends on your compiler, development environment, and mathematical model.

easy handling fast implementation
Handling and integration

Programming language The source code of paceval is based on the ISO Standard (ISO/IEC 14882:1998) and therefore runs on any compiler since the 1990s.
Supported operating system Any; Works best with 32-bit and higher, e.g. B. Linux, Windows, iOS, macOS, Android, as well proprietary or specialized operating systems
Supported development environment Any; All you need is a Foreign function interface (FFI) for integration into your target development environment, e.g. Golang, C++, Python, Object Pascal, Fortran, Visual Basic, Java, C#, Perl, Ruby or PHP, see https://github.com/paceval. Enables a low code/no code approach.
Integration concept Integrate a paceval. library into your software easily with text files.
Commands Just use the standard mathematical notations with paceval.



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