# 7.1: Calculating pi

7.1: Calculating pi

## Leibniz formula for π

an alternating series. It is also called the Leibniz-Madhava series as it is a special case of a more general series expansion for the inverse tangent function, first discovered by the Indian mathematician Madhava of Sangamagrama in the 14th century, the specific case first published by Leibniz around 1676. [1] The series for the inverse tangent function, which is also known as Gregory's series, can be given by:

The Leibniz formula for π / 4 can be obtained by putting x = 1 into this series. [2]

It also is the Dirichlet L -series of the non-principal Dirichlet character of modulus 4 evaluated at s = 1 , and therefore the value β(1) of the Dirichlet beta function.

## Calculating Pi: My attempt at breaking the Pi World Record

Before I go into the specifics of my record attempt, it is important for you to have some background information. Over the past few years I have been steadily acquiring server hardware to increase my knowledge of system administration and cyber security. This is not only for personal growth: it has also helped me immensely in my professional career. I have served in various roles including cyber security (Cyber Defense Analyst, Chief Information Security Officer) and system administration (Senior System Administrator). Historically, I have also donated my spare system computing resources to various distributed computing projects through the BOINC 1 platform. In late 2018 I decided to shift the primary focus of my lab to distributed computing. I still maintain a small 1U server for personal use, but most of the other servers run BOINC exclusively.

UPDATE: Partway through the world record attempt (July, 2019), I founded a nonprofit organization, North Alabama Charitable Computing, which repurposes enterprise-grade compute and storage equipment for STEM research. After I break the Pi world record, most of my servers will be donated where they will provide scientists and various research projects with computing power.

Anyways, back to the reason you are here: My primary reason for attempting to break the record set by Emma Haruka Iwao/Google 2 (March, 2019 – 31.4 trillion digits) is to test the limits of my hardware. Once I am finished breaking the Pi world record, the server and hard drives will be repurposed for STEM research (e.g., BOINC, Open Science Grid 3 , university research projects).

One of the first things I did was reach out to Alexander Yee, who wrote and maintains a program called y-cruncher 4 . y-cruncher is a program that computes various mathematical constants based on user input. y-cruncher is able to take advantage of the newest processor features and features many optimizations, which in part explains why it has been used to set several of the past Pi records (and other mathematical constants). After emailing back and forth with Alex, I started to gain an understanding of what would be required for such a feat: 50 trillion digits of Pi. In order to calculate this many digits, y-cruncher determined I would need 256TiB of ephemeral storage, in addition to 38TiB for the final output.

One of the first things that you may notice is that y-cruncher calculates everything in tebibytes (base-2), rather than terabytes (base-10). It can be confusing since these units of measure are sometimes used interchangeably (I’m looking at you, Windows 5 , and Linux 6 df “-h” vs “-H”), but the difference can be significant: for example, 256TiB is equivalent to 281TB. You don’t want to get 3/4 of the way through a record attempt and realize you don’t have enough storage.

Once I had a rough estimate of the amount of equipment and disk required for the Pi computation, I started looking for used hardware, primarily on eBay. From January to March of 2019, I started purchasing equipment that would be required for the attempt: a new server, several hard drives, various PCI-E cards, and a tape libary and drive, to name just a few. Below is a picture of my home lab in it’s entirety (except for an Asus ESC8000 G4 in my office). Everything inside the red box is used for the Pi computation.

## 10,000 digits of Pi formatted for humans

3.
1415926535 8979323846 2643383279 5028841971 6939937510
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3408005355 9849175417 3818839994 4697486762 6551658276
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4043523117 6006651012 4120065975 5851276178 5838292041
9748442360 8007193045 7618932349 2292796501 9875187212
7267507981 2554709589 0455635792 1221033346 6974992356
3025494780 2490114195 2123828153 0911407907 3860251522
7429958180 7247162591 6685451333 1239480494 7079119153
2673430282 4418604142 6363954800 0448002670 4962482017
9289647669 7583183271 3142517029 6923488962 7668440323
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9695946995 5657612186 5619673378 6236256125 2163208628
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9279068212 0738837781 4233562823 6089632080 6822246801
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3084076118 3013052793 2054274628 6540360367 4532865105
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2162484391 4035998953 5394590944 0704691209 1409387001
2645600162 3742880210 9276457931 0657922955 2498872758
4610126483 6999892256 9596881592 0560010165 525637567

## What is Perfusion Index (PI)

Perfusion Index or PI is the ratio of the pulsatile blood flow to the non-pulsatile static blood flow in a patient's peripheral tissue, such as finger tip, toe, or ear lobe. Perfusion index is an indication of the pulse strength at the sensor site. The PI's values range from 0.02% for very weak pulse to 20% for extremely strong pulse. The perfusion index varies depending on patients, physiological conditions, and monitoring sites. Because of this variability, each patient should establish his own "normal" perfusion index for a given location and use this for monitoring purposes.

Perfusion index is normally monitored with pulse oximeters. PI is also a good indicator of the reliability of the pulse oximeter reading. For most pulse oximeters for general use, the reading is unreliable or unavailable if PI is at or below 0.4%. There are oximeters, such as those from Masimo, designed for extreme low PI. Most people that use an oximeter at home would not need a perfusion index indicator because they are considered to be in general good health. A perfusion index adds a lot of sensitivity to the oximeter sensor thus adding to the cost of the oximeter.

The pleth (Plethysmograph), available in many pulse oximeters, is a graphical representation of the perfusion index.

In a hospital, perfusion index, along with many other parameters, is used to monitor critically ill patients. Studies have shown that PI has a high correlation with capillary refill time and central-to-toe temperature difference. In neonatal acute care, a low PI is an objective and accurate measure of acute illness. It is superior to qualitative approach such as foot warmth.

Perfusion index is also used as an early warning of anesthetic failure. Studies have shown that an increase in PI is an early indicator that general or epidural anesthesia has initiated peripheral blood vessel dilation, which typically occurs before the onset of anesthesia. Lacking the spike would indicate the lack of anesthetic effect.

Other uses of the perfusion index can be found throughout various literature. As we learn more about PI, more clinical applications are being discovered.

## Using the Perfusion Index with Pulse Oximeters

The reason having a perfusion index on your pulse oximeter is a good idea is because it can help increase the accuracy of the readings which the pulse oximeter gives you in relation to your blood oxygen levels.

Simply put, the stronger your pulse is at the monitoring site (where the pulse oximeter is attached), the more accurate the readings of it will be.

The perfusion index can help you determine how appropriate the application site is it can help you find the best location to attach the pulse oximeter. Generally speaking, you want to apply the sensor to the area that has the highest percentage. If your earlobe has a perfusion index rating of 3%, and your finger has a rating of 15%, then the finger is where you want to attach the pulse oximeter.

## How do you calculate #Cos^-1 [cos ((7pi)/6) ]#?

You can write #(7pi)/6# as #(pi+pi/6)#
Thus we can clearly see the angle falls in the third quadrant. And the cosine value in third quadrant is always negative.

coming back to the question
#cos^-1[cos((7pi)/6)] = cos^-1[-cos(pi/6)]#
= #pi - cos^-1[cos(pi/6)]#
= #pi- pi/6#
= #(5pi)/6#

# arccos( cos ( <7 pi>/ 6) ) = pm <7pi>/6 + 2pi k,# integer #k#

#### Explanation:

I treat #arccos( cos ( <7 pi>/ 6) ) # as a multivalued expression, all the angles whose cosine equals #cos ( <7 pi>/ 6).#

In general #cos x = cos a # has solution #x=pm a + 2 pi k, # integer #k#

# arccos( cos ( <7 pi>/ 6) ) = pm <7pi>/6 + 2pi k,# integer #k#

The principal value is in the second quadrant, given here by the minus sign and #k=1.#

## Method 2

and writing 1/(1 + t 2 ) = 1 - t 2 + t 4 - t 6 + . and integrating term-by-term, we obtain the infinite series

tan -1 (x) = x - x 3 /3 + x 5 /5 - x 7 /7 + . ,

which converges for -1 <= x <= 1. Substituting x = 1, we obtain Leibniz's series

However, because x = 1 is an endpoint of the interval of convergence, this series converges very slowly and so is impractical for calculation. (For example, the first five terms give the approximations 4., 2.666, 3.466, 2.895, 3.339 to pi.)

On the other hand, we may substitute x = 1/sqrt(3) = tan -1 (pi/6). Then, after some algebraic manipulation, we obtain the series

and the first five terms of this series give the approximations 3.46410, 3.07920, 3.15618, 3.13785, 3.14260 to pi, the last of which differs from pi by litle more than .001.

Indeed, we can do even better. Substituting x = tan(pi/n) for n >= 4 (so that the series converges), we obtain the series

pi = n(tan(pi/n) - tan 3 (pi/n)/3 + tan 5 (pi/n)/5 - tan 7 (pi/n)/7 + tan 9 (pi/n)/9 - . ).

Now tan(x/2) = sqrt((1 - cos(x))/(1 + cos(x))), so we may again compute tan(pi/n) recursively, starting with n = 6. We obtain the following approximations to pi, where in the last two columns of this table, Sk(tan(pi/n)) denotes the approximation obtained by using the first k terms of the above series.

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## Advanced Statistical Computing

There are many introductions and overviews of Markov chain Monte Carlo out there and a quick web search will reveal them. A decent introductory book is Markov Chain Monte Carlo in Practice by Gilks, Richardson, and Spiegelhalter, but there are many others. In this section we will give just the briefest of overviews to get things started.

A Markov chain is a stochastic process that evolves over time by transitioning into different states. The sequence of states is denoted by the collection () and the transition between states is random, following the rule [ mathbb

(X_tmid X_, X_,dots,X_0) = mathbb

(X_tmid X_) ] (The above notation is for discrete distributions we will get to the continuous case later.)

This relationship means that the probability distribution of the process at time (t) , given all of the previous values of the chain, is equal to the probability distribution given just the previous value (this is also known as the Markov property). So in determining the sequence of values that the chain takes, we can determine the distribution of our next value given just our current value.

The collection of states that a Markov chain can visit is called the state space and the quantity that governs the probability that the chain moves from one state to another state is the transition kernel or transition matrix.

### 7.1.1 A Simple Example

A classic example of a Markov chain has a state space and transition probabilities that can be drawn as follows.

Here, we have three states in the state space and the arrows indicate to which state you can travel given your current state. The fractions next to each arrow tell you what is the transition probabilities of going to a given state. Note that in this Markov chain, you can only ever travel to two possible states no matter what your current state happens to be.

We can write the transition probabilities in a matrix, a.k.a. the transition matrix, as [ P = left[ egin 1/2 & 1/2 & 0 1/2 & 0 & 1/2 0 & 1/2 & 1/2 end ight] ] With the matrix notation, the interpretation of the entries is that if we are currently on interation (n) of the chain, then [ mathbb

(X_=jmid X_n=i) = P_ ]

From the matrix, we can see that the probability of going from state (1) to state (3) is (0) , because the ((1, 3)) entry of the matrix is (0) .

Suppose we start this Markov chain in state (3) with probability (1) , so the initial probability distribution over the three states is (pi_0=(0,0,1)) . What is the probability distribution of the states after one iteration? The way the transition matrix works, we can write [ pi_1 = pi_0 P = (0, 1/2, 1/2) ] A quick check of the diagram above confirms that if we start in state (3) , then after one iteration we can only be in either state (3) or in state (2) . So state (1) gets (0) probability.

What is the probability distribution over the states after (n) iterations? We can continue the process above to get [ pi_n = pi_0 overbrace^> = P^ <(n)>] For example, after five iterations, starting in state (3) , we would have (pi_5 = (0.3125, 0.34375, 0.34375)) .

### 7.1.2 Basic Limit Theorem

For a Markov chain with a discrete state space and transition matrix (P) , let (pi_star) be such that (pi_star P = pi_star) . Then (pi_star) is a stationary distribution of the Markov chain and the chain is said to be stationary if it reaches this distribution.

The basic limit theorem for Markov chains says that, under a specific set of assumptions that we will detail below, we have [ |pi_star - pi_n|longrightarrow 0 ] as (n ightarrowinfty) , where (|cdot|) is the total variation distance between the two densities. Therefore, no matter where we start the Markov chain ( (pi_0) ), (pi_n) will eventually approach the stationary distribution. Another way to think of this is that [ lim_ pi_n(i) = pi_star(i) ] for all states (i) in the state space.

There are three assumptions that we must make in order for the basic limit theorem to hold.

The stationary distribution (pi_star) exists.

The chain is irreducible. The basic idea with irreducibility is that there should be a path between any two states and that path should altogether have probability (> 0) . This does not mean that you can transition to any state from any other state in one step. Rather, there is a sequence of steps that can be taken with positive probability that would connect any two states. Mathematically, we can write this as there exists some (n) such that (mathbb

(X_n = jmid X_0 = i)) for all (i) and (j) in the state space.

The chain is aperiodic. A chain is aperiodic if it does not make deterministic visits to a subset of the state space. For example, consider a Markov chain on the space of integers that starts at (X_0=0) , and has a transition rule where [ X_ = X_n + varepsilon_n ] and [ varepsilon_n = left< egin 1 & ext frac<1><2> -1 & ext frac<1> <2>end ight. ] This chain clearly visits the odd numbers when (n) is odd and the even numbers when (n) is even. Furthermore, if we are at any state (i) , we cannot revisit state (i) except in a number of steps that is a multiple of (2) . This chain therefore has a period of (2) . If a chain has a period of (1) it is aperiodic (otherwise it is periodic).

### 7.1.3 Time Reversibility

A Markov chain is time reversible if [ (X_0, X_1,dots,X_n) stackrel <=>(X_n,X_,dots,X_0). ] The sequence of states moving in the “forward” direction (with respect to time) is equal in distribution to the sequence of states moving in the “backward” direction. Further, the definition above implies that ((X_0, X_1)stackrel<=>(X_1,X_0)) , which further implies that (X_0stackrel<=>X_1) . Because (X_1) is equal in distribution to (X_0) , this implies that (pi_1=pi_0) . However, because (pi_1=pi_0 P) , where (P) is the transition matrix, this means that (pi_0) is the stationary distribution, which we will now refer to as (pi) . Therefore, a time reversible Markov chain is stationary.

In addition to stationarity, the time reversibility property tells us that for all (i) , (j) , the following are all equivalent:

[egin (X_0, X_1) & stackrel <=>& (X_1, X_0) mathbb

(X_0=i, X_1=j) & = & mathbb

(X_1=i, X_0 = j) mathbb

(X_0=i)mathbb

(X_1=jmid X_0=i) & = & mathbb

(X_0=j)mathbb

(X_1=imid X_0=j) end]

The last line can also be written as [ pi(i)P(i, j) = pi(j)P(j, i) ] which are called the local balance equations. A key property that we will exploit later is that if the local balance equations hold for a transition matrix (P) and distribution (pi) , then (pi) is the stationary distribution of a chain governed by the transition matrix (P) .

Why is all this important? Time reversibility is relevant to analyses making use of Markov chain Monte Carlo because it allows us to find a way to construct a proper Markov chain from which to simulate. In most MCMC applications, we have no trouble identifying the stationary distribution. We already know the stationary distribution because it is usually a posterior density resulting from a complex Bayesian calculation. Our problem is that we cannot simulate from the distribution. So the problem is constructing a Markov chain that leads to a given stationary distribution.

Time reversibility gives us a way to construct a Markov chain that converges to a given stationary distribution. As long as we can show that a Markov chain with a given transition kernel/matrix satisfies the local balance equations with respect to the stationary distribution, we can know that the chain will converge to the stationary distribution.

### 7.1.4 Summary

To summarize what we have learned in this section, we have that

We want to sample from a complicated density (pi) .

We know that aperiodic and irreducible Markov chains with a stationary distribution (pi) will eventually converge to that stationary distribution.

We know that if a Markov chain with transition matrix (P) is time reversibile with respect to (pi) then (pi) must be the stationary distribution of the Markov chain.

Given a chain governed by transition matrix (P) , we can simulate it for a long time and eventually we will be simulating from (pi) .

That is the basic gist of MCMC techniques and I have given the briefest of introductions here. That said, in the next section we will discuss how to construct a proper Markov chain for simulating from the stationary distribution.

## Y-cruncher - A Multi-Threaded Pi-Program

The first scalable multi-threaded Pi-benchmark for multi-core systems.

How fast can your computer compute Pi?

y-cruncher is a program that can compute Pi and other constants to trillions of digits.

It is the first of its kind that is multi-threaded and scalable to multi-core systems. Ever since its launch in 2009, it has become a common benchmarking and stress-testing application for overclockers and hardware enthusiasts.

y-cruncher has been used to set several world records for the most digits of Pi ever computed.

Windows: Version 0.7.8 Build 9507 (Released: September 14, 2020)

Linux : Version 0.7.8 Build 9507 (Released: September 14, 2020)

Version 0.7.8 Patch with Zen 2 Optimizations: (March 1, 2020) - permalink

A new patch for v0.7.8 has been released. Aside from the usual bug fixes, this release includes a new binary optimized for AMD Zen 2 processors.

I don't usually add new features to patch releases. However, my priorities in recent months have largely shifted away from y-cruncher. As such, very little progress has been made on v0.7.9. With no timeline for when the next feature release will happen, I decided to pull the Zen 2 optimizations into the v0.7.8 branch.

That aside, the new Zen 2 binary seems bring about 10% performance improvement. This is far greater than the 1-2% that I had expected.

This may be a bit of a shocker, but I have no idea why the improvement is so large nor do I have the time to investigate. Most of the tuning is automated by a superoptimizer. So all I did was run the superoptimizer, wait a week for it to finish, then copy-paste the results into the code.

In other words, I have no idea what it did or what it found that is so significant. But it worked, and I'll take it - no questions asked.

The Pi Record Returns to the Personal Computer: (January 29, 2020) - permalink

After a grueling 10 months of computation running from April of last year to today, the world record for Pi has once again fallen. This time to Timothy Mullican who computed 50 trillion digits of Pi on a dated (but still powerful) personal computer.

As the previous record was set by Google using their cloud platform, this computation marks the return of the Pi record to single machine personal computers.

Timothy used a 2012-era computer (Ivy Bridge) along with a very large array of 48 modern hard drives. While the processing power was significantly less than that of Google's computation, it had a better balance of computing power and storage bandwidth - thus leading to a computation of comparable speed to Google's.

Full details of the computation can be found on Tim's blog.

y-cruncher has been used to set a number of world record sized computations.

Blue: Current World Record

Green: Former World Record

Red: Unverified computation. Does not qualify as a world record until verified using an alternate formula.

AMD Threadripper 3995WX @ 2.7 GHz

2 x Intel Xeon E5-2680 v2 @ 2.8 GHz

2 x Intel Xeon E5-2690 v4 @ 2.6 GHz

2 x Intel Xeon E5-2690 v4 @ 2.6 GHz

Not Verified

2 x Intel Xeon Gold 6140 @ 2.3 GHz

4 x Intel Xeon E7-4880 v2 @ 2.5 GHz

Christophe Patris de Broe

Not Verified

2 x Intel Xeon Platinum 8268 @ 2.9 GHz

2 x Intel Xeon E5-2651 v2 @ 1.8 GHz

2 x Intel Xeon Gold 6140 @ 2.3 GHz

2 x Intel Xeon 8260L @ 2.4 GHz

AMD Threadripper 1950X @ 3.4 GHz

Emma Haruka Iwao Pi 31,415,926,535,897 Compute: 121 days

See the complete list including other notably large computations. If you want to set a record yourself, the rules are in that link.

The main computational features of y-cruncher are:

• Able to compute Pi and other constants to trillions of digits.
• Two algorithms are available for most constants. One for computation and one for verification.
• Multi-Threaded - Multi-threading can be used to fully utilize modern multi-core processors without significantly increasing memory usage.
• Vectorized - Able to fully utilize the SIMD capabilities for most processors. (SSE, AVX, AVX512, etc. )
• Swap Space management for large computations that require more memory than there is available.
• Multi-Hard Drive - Multiple hard drives can be used for faster disk swapping.
• Semi-Fault Tolerant - Able to detect and correct for minor errors that may be caused by hardware instability or software bugs.

### Sample Screenshot: 1 trillion digits of Pi

Latest Releases: (September 14, 2020)

 OS Download Link Size Windows y-cruncher v0.7.8.9507.zip 43.4 MB Linux (Static) y-cruncher v0.7.8.9507-static.tar.xz 29.2 MB Linux (Dynamic) y-cruncher v0.7.8.9507-dynamic.tar.xz 22.8 MB

The Linux version comes in both statically and dynamically linked versions. The static version should work on most Linux distributions, but lacks Cilk Plus and NUMA binding. The dynamic version supports all features, but is less portable due to the DLL dependency hell.

The Windows download comes bundled with the HWBOT submitter which allows benchmarks to be submitted to HWBOT.

System Requirements:

• Windows 7 or later.
• The HWBOT submitter requires the Java 8 Runtime.
• 64-bit Linux is required. There is no support for 32-bit.
• The dynamic version has been tested on Ubuntu 18.04.
• An x86 or x64 processor.

Version History:

Comparison Chart: (Last updated: May 31, 2021)

Computations of Pi to various sizes. All times in seconds. All computations done entirely in ram.

The timings include the time needed to convert the digits to decimal representation, but not the time needed to write out the digits to disk.

Blue: Benchmarks are up-to-date with the latest version of y-cruncher.

Green: Benchmarks were done with an old version of y-cruncher that is comparable in performance with the current release.

Red: Benchmarks are significantly out-of-date due to being run with an old version of y-cruncher that is no longer comparable with the current release.

Purple: Benchmarks are from unreleased internal builds that are not speed comparable with the current release.