Numpy QuadDType: Quadruple Precision for Everyone

Picture this: You’re a scientist, laboring on complicated simulations that push the boundaries of numerical precision. You fire up your thinky NumPy-powered Python script, enthusiasticly anticipating the results. But then… your calculations commence producing unforeseeed errors. You’ve equitable stumbled into the authenticm of floating-point precision restrictations.

Hi there! I’m Swayam Singh, and for the past three months, I’ve been enbiging numpy_quinserttype with Nathan Gagederbaum, a genuine traverse-platestablish quadruple precision datatype for NumPy. In this blog post, we’ll check why such high precision is sometimes essential, dive into the technical aspects of numpy_quinserttype, and talk its potential applications. Whether you’re a quantum physicist, a financial modeler, or equitable inquisitive about numerical computing, this post will donate you insights about current precision restrictations in NumPy and how we are resolving them.

So buckle up and grab your likeite beverage (might I recommend a Quad Espresso?)

Table of Contents

The Long Double Dilemma in NumPy

The np.lengthydouble dtype in NumPy has become a transport inant pain point for enbigers and employrs aenjoy. Ralf Gommers, a core NumPy Maintainer, highairyed disjoinal critical rehires that produce lengthy double help “excessively hurtful to upretain, probably far more than equitableified.”

1. Cross-Platestablish Inconsistency

The lengthy double type varies emotionalpartner atraverse platestablishs:

• Windows & macOS: 64-bit (same as double)
• Linux (x86/x86_64): 80-bit
• Some architectures (IBM Power9): True 128-bit quadruple precision

This inconsistency directs to portability rehires and unforeseeed behavior, read more about this at NumPy: Issue #14574.

2. Build Complications
   Building NumPy, especipartner on Windows, has become increasingly complicated due to np.lengthydouble:

   • MSVC employs 64-bit lengthy double
   • Mingw-w64 defaults to 80-bit lengthy double

   This discrepancy necessitates dealing with multiple toolchains and compiler patches, transport inantly increasing the maintenance burden. A detailed talkion of this rehire can be set up at NumPy: Issue #20348:

3. Non-Standard Format
   NumPy helps 9 contrastent binary reconshort-termations for lengthy double, each requiring one-of-a-kindized code:

   This complicatedity produces it challenging to carry out and upretain, even in produce systems enjoy Meson, more about it at Meson: Issue #11068.

4. Maintenance Overhead
   The NumPy team spfinishs a disproportionate amount of time insertressing lengthy double-roverdelighted rehires, as well as dealing with difficult-to-detect produce rehires, especipartner on Windows :

5. Limited Practical Benefits
   Despite these contests, the advantages of lengthy double are askable:

   • On Windows and macOS, it’s functionpartner identical to double, only consuming more memory with pinserted bits.
   • On Linux, it supplys only a modest precision incrrelieve (80-bit vs 64-bit).

6. Hidden Complexity
   The lengthy double rehire extfinishs beyond the employr-apparent dtype. It’s also baked into the libnpymath motionless library shipped with NumPy, inserting another layer of complicatedity to any potential solutions NumPy Issue #20880.

Given these rehires, the NumPy community is at a traverseroads. The current approach to lengthy double help is proving to be increasingly problematic and difficult to upretain. It’s evident that a novel honestion is needed to promise the lengthy-term stability and usability of NumPy atraverse contrastent platestablishs.

As we shift forward, the community is pondering disjoinal potential solutions, from filled deprecation to enbiging a genuine traverse-platestablish quadruple precision dtype.

In our next section, we’ll check how numpy_quinserttype aims to insertress these rehires, promising a future where high-precision computing in Python doesn’t come with a side of maintenance headaches.

Introducing Numpy-QuadDType

NumPy-QuadDType (numpy_quinserttype) is a custom data type (dtype) carry outation for NumPy that supplys genuine quadruple precision floating-point arithmetic atraverse contrastent platestablishs. This project aims to insertress the lengthy-standing rehires with np.lengthydouble by recommending a constant, high-precision floating-point type watchless of the underlying system architecture with also providing backwards compatibility of lengthy double.

The core of numpy_quinserttype is built around two key components:

1. A scalar type QuadPrecision that reconshort-terms individual quadruple-precision scalars.
2. A NumPy dtype QuadPrecDType that permits these quadruple-precision scalars to be employd in NumPy arrays and operations.

What sets numpy_quinserttype apart is its dual-backfinish approach:

• SLEEF (SIMD Library for Evaluating Elementary Functions): This backfinish employs the Sleef_quad type from the SLEEF library, providing genuine 128-bit quadruple precision.
• Long Double: This backfinish employs the native lengthy double type, which can recommend up to 80-bit precision on some systems allotriumphg backwads compatibility with np.lengthydouble.

If the compiler natively helps IEEE 754 quad precision Floating Point type, then Sleef_quad is an alias of that data type. Otherteachd, it depicts a 128-bit data type for retaining a number in IEEE 754 Quad-Precision data establishat.

This flexibility permits numpy_quinserttype to supply the best useable precision atraverse contrastent platestablishs while upretaining a constant interface.

By insertressing these goals, numpy_quinserttype not only settles the instant rehires surrounding np.lengthydouble but also paves the way for more sturdy and accurate numerical computing in Python. In the follotriumphg sections, we’ll delve transport inanter into how these goals are accomplishd thraw the technical carry outation of numpy_quinserttype.

The Inner Workings of numpy_quinserttype

numpy_quinserttype is built on a pliable architecture that marries high precision with traverse-platestablish compatibility. At its core lies the QuadPrecisionObject, a chameleon-enjoy arrange that can switch between two establishs:

This dual-nature permits us to leverage the filled 128-bit precision of SLEEF when useable, while upretaining backwards compatibility with np.lengthydouble.

To unite seamlessly with NumPy, we’ve produced the QuadPrecDTypeObject:

This arrange acts as a bridge, allotriumphg our high-precision numbers to labor harmoniously wiskinny NumPy arrays and operations.

The backfinish pickion is deal withd by a basic enum:

This flexibility assists numpy_quinserttype to supply chooseimal precision atraverse contrastent platestablishs while upretaining a constant interface. Users can pick their likered backfinish at runtime:

Under the hood, numpy_quinserttype deal withs memory fruitfully for both aligned and unaligned memory access. This is transport inant for perestablishance, especipartner when dealing with big arrays or complicated computations. We’ve also carry outed one-of-a-kindized strided loop functions for various operations.

Casting operations

Casting operations are another critical component. We’ve carry outed a range of casts between QuadPrecision and other NumPy types, ensuring fine interoperability:

For preserving precision during casting it is recommfinished to pass input as a string

laboring with QuadPrecision numbers is as straightforward as laboring with any other NumPy type. The casting operations deal with the weighty lifting behind the scenes, allotriumphg you to concentrate on your computations rather than type deal withment.

Universal Functions (UFuncs)

numpy_quinserttype carry outs a expansive range of universal functions, covering unary operations (enjoy square root or exponential), binary operations (insertition, multiplication, etc.), and comparison operations. These ufuncs are the backbone of NumPy’s fruitful array operations, and we’ve promised that QuadPrecision numbers labor seamlessly with them.
Here’s a taste of how you might employ these ufuncs:

These operations see identical to standard NumPy operations, but under the hood, they’re using the filled precision of QuadPrecision numbers.

Precision in Printing: The Dragon4 Algorithm

When it comes to disjoining QuadPrecision numbers, accuracy is paramount. That’s where our customized carry outation of the Dragon4 algorithm comes in. Dragon4 is a polishd algorithm for altering binary floating-point numbers to their decimal reconshort-termations, ensuring that the printed cherish is as seal as possible to the genuine cherish of the number in memory.

• Without Dragon4

  >>> a = npq.QuadPrecision("1.123124242")

  1.123124242000000000000000000000000e+00

• With Dragon4

  >>> a = npq.QuadPrecision("1.123124242")

Our carry outation of Dragon4 for numpy_quinserttype is a transport inant betterment over the previous approach employd for np.lengthydouble. We’ve take awayd complicated platestablish-definite branching that was employd to deal with contrastent binary reconshort-termations of np.lengthydouble, resulting in a more streamlined, upretainable and constant output atraverse contrastent platestablishs.

By insertressing these various aspects – from low-level memory deal withment to high-level NumPy integration and accurate number reconshort-termation – numpy_quinserttype supplys a sturdy and pliable solution for high-precision arithmetic in Python. It recommends the excessive accuracy you need for needing computations, wrapped in the comprehendn and employr-cordial NumPy interface you already comprehend and cherish.

Precision and Accuracy

In the world of high-precision computing, not all floating-point reconshort-termations are produced equivalent. numpy_quinserttype recommends two backfinishs, each with its own precision characteristics. Let’s fracture them down:

SLEEF Backfinish Precision

The SLEEF (SIMD Library for Evaluating Elementary Functions) backfinish supplys genuine 128-bit quadruple precision. Here’s what that unbenevolents in train:

• Significand: 113 bits
• Exponent: 15 bits
• Sign: 1 bit

This transpostponeeds to approximately 34 decimal digits of precision. To put this in perspective, if you were measuring the diameter of the observable universe (approximately 8.8 x 10²⁶ meters), you could do so with precision down to the width of a proton!

Long Double Backfinish Precision

The lengthy double backfinish’s precision varies depfinishing on the platestablish:

ULP Analysis

ULP (Unit in the Last Place), is a meacertain of the spacing between floating-point numbers and is transport inant for empathetic the accuracy of floating-point operations. In numpy_quinserttype, the ULP characteristics vary not equitable between backfinishs, but also between contrastent operations.

For the SLEEF backfinish:

• Basic arithmetic operations (insert, subtract, multiply, split), ULP error bound ≤ 0.5000000001
• Transcfinishental functions (e.g., sine, cosine), ULP error bound ≤ 1.0

This unbenevolents that for fundamental arithmetic cforfeit 1, the SLEEF backfinish can supply results right to about 33-34 decimal places, while transcfinishental functions upretain accuracy to about 32-33 decimal places.

This level of detail in ULP analysis is particularly relevant for applications where the accumulation of petite errors over many operations can direct to transport inant discrepancies. For instance, in numerical simulations or financial modeling, empathetic these ULP characteristics can help in choosing the most appropriate functions and in estimating the overall accuracy of complicated calculations.

Testing and Applications

To rigorously appraise the perestablishance and capabilities of numpy_quinserttype, we carry outed disjoinal challenging applications to showcase the power of quad-precision arithmetic but also show the traverse-platestablish consistency of our carry outation.

Mandelbrot Set: Exploring the Depths of Chaos

One of the most visupartner striking applications of high-precision arithmetic is in exploring the Mandelbrot set. This honord fractal serves as an excellent stress test for floating-point precision, as zooming into its intricate arranges needs increasingly accurate calculations.

We produced Mandelbrot set visualizations at an excessive zoom level of 10²⁰, cgo ined at the set ups -1.749624030987687 + 0.0i, with 1000 iterations. The results are noticeworthy:

Using the SLEEF backfinish of numpy_quinserttype, we accomplishd stunning detail and clarity even at this excessive magnification. The image uncovers intricate arranges, delicate fifrailnts, and complicated patterns that would be lost with reduce precision calculations.

For comparison, we produced the same watch using standard double-precision floating-point numbers (np.float64) and extfinished-precision lengthy double:

The contrastence is stark. Both version diswatchs all details, resulting in big blocks of firm color. This emotionalpartner shows the restrictations of 64-bit floating-point arithmetic and system reliant lengthy double when pushed to these excessives.

Quantum Harmonic Oscillator for Diatomic Molecules

We applied numpy_quinserttype to model the quantum harmonic oscillator for diatomic molecules (Hydrochloric acid in our case).

We appraised the perestablishance of numpy_quinserttype agetst standard double-precision (np.float64) calculations. The results, envisiond in two key graphs, highairy the transport inant advantages of quad-precision arithmetic in quantum mechanical calculations.

1. Absolute Error in Energy Level Differences
   • Quad Precision: Maintains excessively low error, constantly around 10^-52 to 10^-53 J atraverse all quantum numbers, demonstrating exceptional stability and accuracy.
   • Float64: Shows transport inantly higher error, ranging from 10^-34 to 10^-33 J, with a sairy upward trfinish for higher quantum numbers, indicating gradual precision loss.
2. Absolute Error in Wavefunction Normalization
   • Quad Precision: Exhibits relabelably constant error around 3.65 × 10^-13, ensuring constant wavefunction validity atraverse all quantum states.
   • Float64: Starts cforfeit quad precision levels but shows a evident upward trfinish for higher states, recommending accumulating inaccuracies.

Current Status And Next Steps

As of this writing, numpy_quinserttype has accomplished a transport inant milestone. It’s ready for employ, helping most transport inant operations and integrating well with NumPy. As we’ve showd with our Mandelbrot set visualizations and quantum harmonic oscillator calculations, it’s contendnt of handling needing computational tasks that need high precision.

However, enjoy any novel gentleware project, there’s still room for betterment and expansion. During enbigment, we identified disjoinal areas in NumPy’s dtype C-API that could be raised to better help custom dtypes enjoy ours (NumPy Issue #27231)

Looking ahead, we have disjoinal key objectives:

1. Package Distribution: We’re preparing to free numpy_quinserttype as Python wheels, which will be useable for insloftyation via PyPI and as a conda package. This will produce it easily accessible to the expansiver Python scientific computing community.
2. Community Engagement: We schedule to produce a accessible proclaimment about numpy_quinserttype and dynamicly seek feedback from the community. User experiences and recommendions will be transport inant for guiding future betterments.
3. NumPy Enhancement Proposal (NEP): We’re produceing a NEP to establishpartner recommend the integration of numpy_quinserttype into the NumPy ecosystem. This process will permit for thoraw talkion and check by the NumPy community.
4. Future Backfinish Optimization: We schedule to appraise TLFloat, a C++ temppostponeed library for floating-point operations, as a potential swapment for SLEEF in future versions. TLFloat recommends IEEE 754 compliant operations with rightly rounded results for arithmetic operations and 1-ULP accuracy for other math functions atraverse various precisions. This future exploration aims to further raise numpy_quinserttype, particularly when promised rightly-roundeded results are needd.

numpy_quinserttype will persist to be an dynamic enbigment project under the NumPy umbrella. We’re pledgeted to refining and enbiging its capabilities based on employr needs and emerging needments in the scientific computing community.
We see forward to seeing how the community puts this novel capability to employ and hearing your feedback as we persist to better and enbig its functionality.

Stay tuned for refreshs, and satisfyed computing! Remember, in the world of numerical analysis, precision is not equitable a luxury—it’s standardly a necessity.

Conclusion

These past three months have been an noticeworthy journey of lgeting and innovation. None of this would have been possible without the exceptional guidance and unwavering help of my mentor, Nathan Gagederbaum. I’m transport inantly appreciative to the entire NumPy community for their help and insights, with particular thanks to Sebastian Berg and Nathan for their prior labor on NumPy’s DType-API, which lhelp the set upation for this project.

Special appreciation goes to Ralf Gommers, Melissa Weber Mfinishonça and whole Quansight community for providing this distinct opportunity to donate to NumPy. This experience has been both humbling and inspiring, and I’m excited to see how numpy_quinserttype will grow and be used by the community in the future.

In future, I will be dynamicly retaind in this project, so phire sense free to ping me with any asks, recommendions, or contributions.

References