Interval arithmetic with paceval’s TINC (Trusted interval computing):
• Allows users to determine the maximum error of a calculation based on the available platform
• Provides a range of values in which the true calculation value is located
• Ensures calculations are precise in complex mathematical models and long functions with multiple variables
• Helps users decide whether the result, system, or product is operating within an acceptable tolerance range

Benefit of exporting neural networks for paceval:
• Reduces energy consumption: Exported neural networks can be used for inference on any hardware, including low-power CPUs and FPGAs, which are more energy-efficient than GPUs.
• Enables faster inference: Exported neural networks can be executed more quickly on low-power hardware, reducing the time it takes to make predictions.
• Supports sustainable hardware development: Exported neural networks can be implemented on low-power hardware, which is more environmentally friendly than using GPUs for inference.

paceval. offers a method to export neural networks into mathematical functions, which is not only helpful to better understand the functioning of a neural network, but also to compare, evaluate, optimize and certify neural networks according to regulatory requirements.

paceval. is used in different areas, including:
• Applied mathematics: paceval. is used to solve problems in fields such as physics, engineering, statistics and finance.
• Software development: paceval. is used in software development projects to implement mathematical calculations.
• AI research and innovation: paceval. has invented a method to export neural networks into mathematical functions. This is helpful to better understand the functioning of a neural network and to compare, evaluate, optimize and certify neural networks according to regulatory requirements.
• Sustainable hardware development: paceval. has developed a new type of mathematical coprocessor for the Federal Agency for Disruptive Innovation SPRIND, which performs mathematical calculations more efficiently on any hardware (e.g. APPLE Silicon, INTEL/AMD, ARM, RISC-V, FPGA or ASIC).
• Mathematical research and education: paceval. serves as an effective tool for solving mathematical problems and creating mathematical models. It is often used in math education to teach students how to use mathematical expressions.

paceval.’s exceptional computational speed is powered by its innovative and groundbreaking mathematical technology.
• Multi-threading: paceval. utilizes a unique multi-threading approach that distributes calculations across all available processors, maximizing speed and energy efficiency.
• Parallel Processing: The library can process large mathematical expressions in parallel, effectively reducing computational time.
• Caching and Optimization: paceval. employs advanced caching and optimization techniques to reuse previous calculations and minimize redundant operations, leading to faster execution.
• Interval Arithmetic: paceval. supports interval arithmetic, providing accurate results with error bounds, ensuring precision and reducing the need for multiple calculations.
• Customizable Precision: paceval. allows users to select from single, double, and extended precision levels, enabling them to balance accuracy and computational time.

paceval. offers several advantages over traditional acceleration technologies for complex calculations:
• paceval. is system independent. Traditional acceleration technologies are often tied to specific hardware or software platforms, which can limit their flexibility and portability. paceval., on the other hand, can be used on any system with a compatible operating system.
• paceval. supports expressions of any length and with any number of variables. Traditional acceleration technologies often have limitations on the length and complexity of the expressions they can handle. paceval. does not have these limitations, making it suitable for a wide range of complex calculations.
• paceval. can evaluate expressions very quickly. The paceval. library is highly optimized for speed and can perform complex calculations in milliseconds. This makes it ideal for applications that require real-time or near-real-time results.

paceval.’s implementation of “Trusted interval computing” (TINC) allows for the determination of result intervals within all mathematical sub-functions, making it possible to achieve a level of accuracy unparalleled by other solutions in the field of complex mathematical calculations.
• TINC is implemented in all mathematical sub-functions in paceval.
• This allows for the determination of result intervals, or upper and lower boundaries, in which the true calculation value will lie.
• This level of accuracy is critical for complex mathematical calculations, as it provides a way to assess the potential for error and ensure the reliability of results.
• As a result, paceval. is able to deliver unmatched accuracy in the realm of complex mathematical calculations.

paceval. differs from common accelerators in its architectural design by focusing on MIMD (Multiple Instruction Multiple Data) parallelism rather than the more popular SIMD (Single Instruction Multiple Data) parallelism used in most existing accelerators like vector units or graphics processors. paceval.‘s MIMD approach allows for a higher degree of parallelism in complex calculations.

paceval. enables the parallel execution of mathematical calculations by distributing them across all available processors, thereby achieving maximum speed and energy efficiency. This is achieved through a process called parallelization, where individual calculations are divided into smaller tasks that can be executed simultaneously. paceval. also has the ability to perform calculations with selectable precision (single, double, and extended), ensuring accuracy while reducing computational overhead.

paceval. simplifies a developer’s life by offering a user-friendly interface and straightforward integration into various programming languages. Its well-documented API and extensive examples enable seamless integration, while its community support and active development ensure ongoing improvements. By offloading complex mathematical calculations to paceval., developers can focus on higher-level tasks, resulting in increased productivity and more efficient development processes.

paceval. is a groundbreaking technology that redefines the way complex mathematical calculations are performed and used. Here are some of the key differences between paceval. and other mathematical calculation tools:

• paceval. can calculate almost any complex mathematical expression, while other tools may have limitations on the types of calculations they can perform.
• paceval. is a software library, while other tools may be standalone applications. This makes paceval. more versatile and easier to integrate with other software.
• paceval. is system independent, while other tools may only run on certain operating systems. This makes paceval. more portable and easier to use in a variety of environments.
• paceval. supports interval arithmetic, which allows users to compute the error bounds of their calculations. This can be useful for ensuring the accuracy of results.
• paceval. is highly optimized for speed and energy efficiency. This makes it ideal for use in applications where performance is critical, such as real-time systems and mobile devices.

Overall, paceval. is a powerful and versatile mathematical calculation tool that offers a number of advantages over other tools. It is ideal for users who need to perform complex mathematical calculations quickly and accurately, and who want to be able to use their calculations in a variety of applications.

paceval.‘s interface enables easy integration with other programming languages through its use of a Foreign Function Interface (FFI). This FFI allows programmers to call functions or use services written in one programming language from within another. paceval. supports FFI for a wide range of languages, including .NET, ABAP, C, C#, COBOL, Delphi, Erlang, Fortran, Golang, Java, Maple, Julia, Kotlin, Lua, Mathematica, MATLAB, Mendix, Node.js, Octave, Pascal, Perl, PHP, Python, R, Ruby, Rust, Scratch, Swift, TypeScript, or Visual Basic. This makes it easy to integrate paceval.‘s mathematical engine into a variety of software applications and development environments, regardless of the programming language being used.

Hyperscalers can scale paceval. with Kubernetes through various methods. First, they can use Docker containers to package and distribute the paceval. service. This allows the service to be easily deployed and scaled on a Kubernetes cluster. Second, they can use Helm charts to manage the deployment and configuration of paceval. on Kubernetes. Third, they can use Kubernetes to orchestrate the deployment of paceval. across multiple nodes in a cluster. Finally, they can use Kubernetes to monitor and manage the performance of paceval. on a cluster.

Here are some additional details about each of these methods:
• Docker containers are a lightweight virtualization technology that allows applications to be packaged and distributed in a portable way. This makes it easy to deploy and scale applications on a Kubernetes cluster.
• Helm charts are a collection of YAML files that define the deployment and configuration of a Kubernetes application. This makes it easy to manage the deployment and configuration of paceval. on a Kubernetes cluster.
• Kubernetes is a container orchestration system that automates the deployment, management, and scaling of containerized applications. This makes it easy to deploy and scale paceval. on a Kubernetes cluster.
• Kubernetes monitoring and management tools can be used to monitor and manage the performance of paceval. on a Kubernetes cluster. This allows hyperscalers to ensure that paceval. is running smoothly and efficiently.

By using these methods, hyperscalers can easily scale paceval. to meet the demands of their customers. This allows them to provide a reliable and scalable mathematical engine as a service.

paceval. can help with software development projects by:
• Reducing development time by providing a mathematical engine that can be used to perform complex calculations quickly and easily.
• Improving code quality by providing a way to test and verify mathematical calculations.
• Lowering development costs by eliminating the need to develop and maintain complex mathematical algorithms.
• Enabling innovation by providing a platform for developing new and innovative mathematical applications.

paceval. is an excellent choice for developers who need to implement complex mathematical calculations into their software projects. It is a simple and flexible solution that allows users to bypass the need for non-standard software code or database solutions. paceval. is also very efficient, so it can help to reduce development time and costs.

One of the biggest challenges that developers face when integrating complex mathematical models into software projects is the need to ensure that the models are accurate and reliable. paceval. addresses this challenge by providing a mathematical engine that is based on sound mathematical principles. This engine can be used to verify the accuracy of mathematical models, and it can also be used to generate test cases that can be used to test the models.

Another challenge that developers face when integrating complex mathematical models into software projects is the need to optimize the models for performance. paceval. addresses this challenge by providing a number of features that can help to improve the performance of mathematical models. These features include:
• A parallel processing engine that can be used to distribute calculations across multiple processors.
• A caching mechanism that can be used to store frequently used calculations.
• A number of optimization techniques that can be used to improve the efficiency of mathematical models.

By using paceval., developers can overcome the challenges of integrating complex mathematical models into software projects. paceval. can help to ensure that models are accurate and reliable, and it can also help to optimize models for performance.

paceval. can be integrated with existing software projects using text files and standard mathematical notations.

paceval. handles references to calculation objects by providing a unique reference ID for each object. This allows users to access and manipulate calculation objects without having to pass the entire function string and variable values each time. The benefits of this include:
• Reduced bandwidth: Only the reference ID needs to be transmitted over the network, which can save bandwidth, especially for complex calculations.
• Improved performance: By caching calculation objects, paceval. can avoid having to re-parse the function string and perform the calculations multiple times. This can significantly improve performance, especially for repeated calculations.
• Easier debugging: The reference ID can be used to track down errors in calculations, making it easier to debug and maintain mathematical models.

For Developers
• Easy to use
• The simplicity of reading the source code
• Functionality for error handling

For Mathematicians
• Solving mathematical problems
• Creating mathematical models
• Used in math education

For Scientists
• Serves as an effective tool for solving mathematical problems
• Creating mathematical models

paceval. is a software library that can solve mathematical problems with selectable precision and distributed across all available processors to achieve maximum speed and energy efficiency.

Advantages of using paceval. in scientific research and education:
• It is a groundbreaking technology that redefines the way complex mathematical calculations are performed and used.
• paceval. can calculate almost any complex mathematical expressions.
• Expressions can contain any number of placeholders (variables).
• Calculations are performed with selectable precision (single, double and extended).
• The library can also output an interval that specifies the error limits due to the limited precision of floating point number formats.
• paceval. serves as an effective tool for solving mathematical problems and creating mathematical models.
• It is often used in math education to teach students how to use mathematical expressions.

paceval. enhances the readability and error handling of mathematical models compared to standard source code in the following ways:
• Readability: paceval. uses standard mathematical notation, making it easy to understand and modify mathematical models.
• Error handling: paceval. automatically detects and localizes errors in mathematical models, such as division by zero. This helps to prevent errors from causing crashes or incorrect results.

The implementation of long functions in paceval. is more efficient than in standard source code because paceval. uses caching methods to speed up repeated calculations. In addition, paceval. supports parallel processing, which can further improve performance for long functions.
In standard source code, long functions are typically implemented as a single, monolithic block of code. This can make them difficult to read and maintain, and it can also lead to performance problems if the function is called multiple times with different arguments.
In contrast, paceval. allows you to break long functions into smaller, more manageable pieces. These pieces can be cached and reused, which can significantly improve performance. In addition, paceval. supports parallel processing, which can further improve performance for long functions.

Using paceval. over Traditional non-standard proprietary software code for programming mathematical calculations provides numerous advantages, including:
• Easy-to-use interface: paceval. is designed with a user-friendly interface, allowing users to create and manage mathematical models with ease.
• High performance: paceval.is optimized for speed and efficiency, ensuring fast and reliable calculations.
• Extensive library of functions: paceval. comes with a comprehensive library of mathematical functions, eliminating the need for users to develop their own functions.
• Parallelization capabilities: paceval. supports parallelization, enabling complex calculations to be distributed across multiple processors for improved performance.
• High precision: paceval. offers high-precision calculations, ensuring accurate and reliable results.
• Error handling and debugging: paceval. provides robust error handling and debugging capabilities, making it easy to identify and resolve any issues that may arise during calculations.
• Manageability: paceval. is easy to manage and maintain, allowing users to focus on their mathematical models rather than the underlying software.
• Portability: paceval. is highly portable, making it suitable for use on a wide range of platforms and operating systems.

paceval.‘s caching mechanism contributes to its efficiency in mathematical calculations by storing previously calculated results, which can then be reused for future calculations with the same input values. This eliminates the need to recalculate the same results multiple times, saving time and computational resources. Additionally, the caching mechanism is designed to prioritize the storage of frequently used results, further improving performance for common calculations.

paceval. is a unique mathematical tool that offers several advantages over other software libraries:
• Handles complex mathematical expressions: paceval. can calculate almost any complex mathematical expression, including those with multiple placeholders (variables), transcendental functions (trigonometry, exponential function, etc.), and other common operations.
• Supports various precision levels: Calculations can be performed with selectable precision (single, double, and extended) to meet specific accuracy requirements.
• Distributes calculations across processors: paceval. leverages all available processors to achieve maximum speed and energy efficiency during calculations.
• Provides error limits: paceval. can output an interval that specifies the error limits due to the limited precision of floating-point number formats, helping ensure calculation accuracy.
• Small binary footprint and fast compilation: paceval. has a small binary size and compiles quickly, making it suitable for integration into various software applications.
• Wide range of programming language compatibility: paceval. can be used with multiple programming languages through its Foreign function interface (FFI), allowing for easy integration into existing software projects.
• Supports neural network inference on any hardware: paceval. can export neural networks into mathematical functions, enabling neural network inference on any hardware, including non-GPU devices, for improved energy efficiency.

TINC is particularly useful in applications where precision and reliability are important, such as:
• Financial forecasting: TINC can be used to improve the accuracy of financial forecasts by accounting for uncertainty in the input data.
• Drug discovery: TINC can be used to improve the accuracy of drug discovery simulations by accounting for uncertainty in the molecular interactions.
• Scientific research: TINC can be used to improve the accuracy of scientific research results by accounting for uncertainty in the experimental data.
• Quality control: TINC can be used to improve the accuracy of quality control measurements by accounting for uncertainty in the measurement process.
• Security: TINC can be used to improve the accuracy of security systems by accounting for uncertainty in the data collected by the system.

Here is a bulleted list of the main mathematical functions and operations that can be implemented in hardware as part of the paceval. engine:
• Elementary operations: +/-/*/:/exp
• Parallel execution of (partial) functions
• Minimum and maximum: min, max
• Modulo: %, mod
• Trigonometric functions and associated inverse: sin(), cos(), tan(), cot(), asin(), acos(), atan(), acot()
• Hyperbolic functions and associated inverse: sinh(), cosh(), tanh(), coth(), arsinh(), arcosh(), artanh(), arcoth()
• Numerical manipulation: sgn(), abs(), round(), ceil(), floor()
• Additional paceval.-specific numerical manipulation: ispos(), isposq(), isneg(), isnegq(), isnull()

A software company can use paceval. to optimize its existing software, increase its hardware development, and improve its research and innovation.
• paceval.‘s ability to accelerate complex mathematical calculations would enable the software company to develop more sophisticated and accurate analytics solutions.
• With these solutions, the software company could help its customers make better decisions, improve their operations, and gain a competitive edge.
• Additionally, paceval. could open up new opportunities for the software company to develop innovative products and services that leverage its enhanced analytics capabilities.

The data center could gain improved energy efficiency and cost savings, increased performance and scalability, and enhanced security and compliance by leveraging paceval’s hardware-accelerated mathematical engine.
• Improved Energy Efficiency and Cost Savings: paceval.‘s hardware-accelerated mathematical engine is designed to be highly energy-efficient, consuming significantly less power than traditional software-based solutions. By utilizing paceval., the data center could reduce its energy consumption and operating costs, making its cloud services more sustainable and cost-effective.
• Increased Performance and Scalability: paceval.’s hardware acceleration capabilities can significantly enhance the performance and scalability of cloud services. This would enable the data center to handle complex mathematical workloads more efficiently, improving the overall user experience and allowing it to scale its services to meet growing customer demands.
• Enhanced Security and Compliance: paceval.’s hardware-based approach provides enhanced security and compliance benefits. The dedicated hardware ensures that mathematical calculations are performed in a secure and isolated environment, mitigating risks associated with software vulnerabilities and unauthorized access. This would enable the data center to meet industry regulations and standards, building trust with its customers and partners.

The hyperscalar can gain a revolutionary energy-efficient mathematical coprocessor from paceval. that can be integrated into its cloud services and hardware products.
• Cost efficiency: paceval.‘s low running costs and energy efficiency can reduce expenses for data centers and cloud services of the hyperscalar.
• Performance enhancement: paceval.‘s ability to perform complex mathematical operations quickly and efficiently can enhance the performance of AI and machine learning applications.
• Product innovation: paceval.’s unique capabilities can enable the hyperscalar to develop innovative products and services in areas such as AI, IoT, and cloud computing.
• Competitive advantage: By partnering with paceval., the hyperscalar can gain a competitive advantage in the rapidly growing market for AI and cloud-based services.
• Market expansion: paceval.’s compatibility with various hardware platforms allows the hyperscalar to expand its market reach to a wider range of customers.

A semiconductor can derive the following benefits by collaborating with paceval.:
• Enhanced performance in AI and machine learning applications: paceval.’s ability to accelerate complex mathematical computations can significantly boost the performance of AI and machine learning algorithms, leading to more accurate and efficient results.
• Reduced power consumption: paceval.’s energy-efficient design can help the semiconductor develop low-power processors and hardware solutions, reducing the overall energy consumption of data centers and edge devices.
• Increased cost-effectiveness: The low operational costs associated with paceval. can translate into reduced expenses for the customers of the semiconductor, making their AI and machine learning solutions more cost-effective.
• Reduced cost: paceval.’s cost-effective technology can help reduce the production costs of its semiconductors, making them more competitive in the market.
• Competitive edge: By incorporating paceval.’s technology into its offerings, the semiconductor can differentiate its products and gain a competitive advantage in the rapidly growing AI and cloud computing market.
• Broader market reach: paceval.’s compatibility with a variety of hardware platforms allows the semiconductor to expand its market reach and cater to a wider range of customers.
• Access to new markets: paceval.‘s technology can help the semiconductor expand into new markets, such as the IoT and automotive industries, where energy efficiency and performance are critical.
• Stronger customer relationships: By partnering with paceval., the semiconductor can strengthen its relationships with customers who are looking for energy-efficient and high-performance semiconductor solutions.

A hardware component manufacturer can derive the following benefits by partnering with paceval.:
• Enhanced performance in IoT and IIoT applications: paceval.‘s ability to accelerate mathematical computations will improve the real-time performance of IoT and IIoT devices. This can be crucial for applications such as smart manufacturing, autonomous vehicles, and healthcare devices that rely on timely and accurate data processing.
• Reduced cost: paceval.’s cost-effective technology can help the hardware component manufacturer reduce the production costs of its devices, making them more competitive in the market.
• Reduced latency: paceval.‘s low-latency processing can help devices of the hardware component manufacturer respond more quickly to external stimuli, making them ideal for applications that require real-time decision-making.
• Energy efficiency: paceval.‘s energy-efficient design can extend the battery life of devices of the hardware component manufacturer, making them more suitable for long-term deployment in remote or power-constrained environments.
• Increased reliability: paceval.‘s robust and reliable technology can help devices of the hardware component manufacturer operate more consistently and minimize downtime, which is critical for mission-critical applications.
• Competitive edge: By incorporating paceval.‘s technology into its products, the hardware component manufacturer can differentiate its offerings and gain a competitive advantage in the growing IoT and IIoT market.
• Stronger customer relationships: By partnering with paceval, the hardware component manufacturer can strengthen its relationships with customers who are looking for energy-efficient and high-performance devices.