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

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.

Accuracy

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.

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

Factorial Sign	!
Factorial Function (paceval. specific)	fac()

Constants

π (~3.1415384626433832795)	pi
Euler’s number (~2.718281828459045235)	e

Brackets

Circle Bracket Open Character	(
Circle Bracket Close Character	)

Variables

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).

Exponentiation

Power Sign	^
Square Function	sqr()
Square Root Function	sqrt()
Exponential Function	exp()

Logarithm

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()

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.

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 16-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.

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]