2026-06-03 23:13:24
Suppose you run across the power series for the exponential function and decide to code it up. Good idea: you’ll probably learn something, though maybe not what you expect.
Maybe you decide a tolerance of 10−12 is good enough, and so you sum the terms until the next term to add is below the tolerance.
from math import factorial, exp
def naive_exp(x):
tolerance = 1e-12
s = 0
n = 0
while True:
delta = x**n / factorial(n)
s += delta
if abs(delta) < tolerance:
return s
n += 1
You want to try your program out, so you compute e by calling the function at 1. If you compare this to calling exp(1) you find that you got all the digits correct.
Now you try computing exp(-20). Calling naive_exp(-20) gives
5.47893091802112e-10
but calling exp(-20) gives
2.061153622438558e-09
Don’t brush things like this as flukes or compiler bugs [1]. This is your golden opportunity to learn something.
Maybe you add a print statement to see the intermediate values of the sum stored in the variable s. If you do, you’ll see that the partial sums oscillate wildly before settling down.
Maybe that seems wrong, but then you look more carefully at the series. The nth term is xn/n!. Since x is negative, the terms alternate in sign. And the absolute values of the term get bigger before they get smaller. When x = −20, each numerator is 20 times larger than the previous, and each denominator is n times larger than the previous. So the terms will get bigger until n > 20. So the wild oscillations are real, not a bug.
The largest partial sum is 21822593.77927747 in absolute value. You know that exp(−20) is a very small number, so there’s going to have to be a lot of cancellation before the partial sums settle down to a small number. Maybe you’ve heard that cancellation is where numerical calculations lose precision. If not, now you know!
Look again at the largest partial sum. There are eight figures to the right of the decimal point. The code is printing out results to as much precision as it has, so the error at this point is on the order of 10−8. We’re trying to compute a number on the order of 10−9, and if any digits in our result are correct, it would be a coincidence.
If you go back and try your code on x = −22, the result is even worse, giving a negative result for a quantity that for theoretical reasons cannot be negative. But you can see why: you’re asking the code to compute a number that is closer to zero than the accuracy of the code.
Computers don’t represent numbers in base 10 internally, but the argument above is sufficient in this case. If you want to dig deeper, look into the anatomy of a floating point number.
There is a simple way around the problem above, but discovering it sooner would short-circuit the learning process. You could calculate exp(−20) as 1/exp(20) and avoid all the cancellation because the series for exp(20) does not alternate.
[1] Compilers do have bugs occasionally, but it’s orders of magnitude more likely that something is wrong with your code.
The post Naively summing an alternating series first appeared on John D. Cook.2026-06-01 20:42:12
There is still active discussion on X about the approximation
exp(−x²) ≈ (1 + cos(sin(x) + x))/2
and some are saying this can just be explained by Taylor series: the series for the two sides differ for the first time at the x6 term, so that’s why you get a good approximation. As I wrote yesterday, that’s only part of it.
If it were just about Taylor series you could use
exp(−x²) ≈ 1 − x² + x4/2
which also has error O(x6). But this approximation is only good for fairly small x, say in [−0.5, 0.5], whereas the approximation at the top of the post is good over [−4, 4]. When x = 4, the error in the cosine approximation is 0.002579 but the error in the Taylor approximation is 113, five orders of magnitude larger.
If the accuracy of the cosine approximation were due to Taylor series, then we’d expect the function
exp(−x²) − (1 + cos(sin(x) + x))/2
to be small not just over the interval [−4, 4] but over a disk of radius 4 in the complex plane. But it’s not. When x = 4i the function is on the order of 1013.
Both the cosine approximation and the Taylor approximation are good for small disks, say of radius 0.5. They’re both bad for much larger disks, and in fact the cosine approximation is worse.
The post It’s not just Taylor series first appeared on John D. Cook.
2026-06-01 19:01:26
Readers have subscribed to this blog via email almost from its beginning in 2008, but how they have subscribed has changed several times. I’ve used several services to provide email subscription that have come and gone.
For the past two years I’ve been using Substack to send out emails announcing new blog posts. That has worked out well. Substack delivers email reliably, and that’s all I wanted. I’m not active on Substack other than using it to an email. I give a brief introduction to the latest two or three blog posts in each email, and sometimes I include additional ideas that occurred to me later.
If you’d like to get blog post announcements and a little extra commentary via email, sign up for my free Substack newsletter here.
If you’d like to learn about new posts sooner, you can subscribe via RSS or follow me on X or Mastodon.
The post Subscribe by email first appeared on John D. Cook.2026-06-01 00:15:49
The function
(1 + cos(x))/2
gives a fair approximation to the Gaussian density
exp(−x²)
You can make the approximation much better by raising it to a power. The function
((1 + cos(x))/2)4
gives a good lower bound and
((1 + cos(x))/2)3.5597
gives a good upper bound. More on that here.
There are other ways of improving the cosine approximation to the Gaussian. Yesterday I came across one I hadn’t seen before, adding a sin(x) term to x.
(1 + cos(sin(x) + x))/2
This function matches the first few terms of the power series for exp(−x²) and has an error on the order of x6/240. You can’t see the difference between the two functions in a plot for −4 ≤ x ≤ 4.
***
There’s a tension between the previous two statements. If the error in on the order of x6/240 then we’d expect the error to be huge at x = 4. We have
46/240 = 17.07
and yet
exp(−4²) − ((1 + cos(4 + sin(4)))/2) = −0.002579,
i.e. the error is between 3 and 4 orders of magnitude smaller than we might expect.
We have an alternating series, so the truncation error should be roughly equal to the first term after the truncation, right? No, the alternating series theorem doesn’t apply because the absolute values of the terms in the series are not decreasing yet for x = 4. The terms have to decrease eventually because the series has infinite radius of convergence, but they’re not decreasing at the 6th term; the terms will get much larger in absolute value before they get smaller.
The basic alternating series theorem gives only an upper bound on truncation error, but there are extensions that also give a lower bound. I wrote about these extensions a few weeks ago. But they don’t apply here because the terms have not started decreasing in absolute value.
Update: See further discussion in the post It’s not just Taylor series.
The post Another Gaussian approximation first appeared on John D. Cook.2026-05-31 05:06:49
If a polynomial identity holds at a few random points, it’s very like true. We’ll make this statement more precise, but first let’s look at some applications.
You may want to test an identity that naturally presents itself as a statement that two polynomials are equal. Or you might use something like the binomial coefficient trick to reframe a problem that isn’t obviously an identity about polynomials. And with algebraic circuits, you can reformulate a wide range of computations as polynomial identities; this is widely used in zero-knowledge proofs.
The theorem alluded to at the top of the post is the Schwartz-Zippel lemma. It is formulated in terms of the probability of a non-zero polynomial P evaluating to zero. To prove that two polynomials Q1 and Q2 are equal, you can show that
P = Q1(x) − Q2(x) = 0.
Let F be a (typically large) finite field and let P be a non-zero polynomial in n variables
P(x1, x2, x3, …, xn)
of total degree d. If we choose the x‘s randomly from F then the probability that P evaluates to zero is no more than d/|F|. [1]
If the total degree d is small relative to the size of the field, then the probability of P evaluating to zero is small. As long as d is less than |F|, you can evaluate the polynomial k times to make
(d / |F|)k
as small as you’d like. If d isn’t too large, and F is large, like the integers mod p = 2255 − 19 used in cryptography, one polynomial evaluation might be enough to give convincing evidence that the polynomial is zero.
[1] The Schwartz-Zippel lemma in its full generality applies to polynomials over an integral domain R with variables drawn from S, a finite subset of R. Here we’re setting R = S = F.
The post Spot checking polynomial identities first appeared on John D. Cook.2026-05-29 20:24:30
The sample variance of a set of numbers is defined in terms of the sum of the squared distances from each point to the mean.
So it would seem that you first need to calculate the mean, then go back and compute the squared differences from the mean. And yet sample variance can be computed in one pass through the data.
You’ll find two equivalent equations in statistics books: the one described above and another based on the sum of the data points and the sum of the data points squared.
While this equation is theoretically correct, it is numerically unstable. Code that directly implements this equation can return a negative value for a quantity that is theoretically positive. I’ve seen this happen with real data, causing a program to crash when taking the square root of the variance to get the standard deviation.
However, there is an algorithm that computes mean and variance in one pass that is accurate and numerically stable. This algorithm was developed by B. P. Welford in 1962. I discuss Welford’s algorithm and give code for implementing it here.
Welford’s algorithm is known in computer science as an “online” algorithm. This term was coined well before the Internet. For example, see the paper [1] from 1965.
But of course now “online” means something else, and so the technical and colloquial uses of “online algorithm” have split. Technical literature uses the phrase to describe the kinds of algorithms in this post. Most people would take “online algorithm” to mean code that runs on a remote server. You may see “streaming algorithm” as a contemporary technical term, but I’d still search on “online algorithm” to find papers.
Welford’s algorithm computes the first two moments, mean and variance, of a data set online. It is also possible to compute skewness and kurtosis online, as well as higher moments.
Simple linear regression is closely related to calculating mean and variance, and it is possible to compute simple regression coefficients online. I have some old notes on this here.
This post was motivated by an email asking me about multiple regression. It is also possible to compute multiple regression coefficients online, but I haven’t done this. I found a couple references, [2] and [3], but I have not read them. There is a simple procedure for two predictor variables but I believe things get a little more complicated with three or more predictors, requiring a recursive least squares algorithm.
The notion of online algorithms is closely related to the notion of a fold in functional programming. Here are several posts on computing things with folds.
[1] One-Tape, Off-Line Turing Machine Computations by F. C. Hennie. Information and Control. 8, 553-578 (1965). Available here. In this paper Hennie writes “In an on-line computation the input data are supplied to the machine, one symbol at a time, at a special input terminal. … In an off-line computation all of the input symbols are written on one of the machine’s tapes prior to the start of the computation.
[2] Arthur Albert and Robert W. Sittler, “A Method for Computing Least Squares Estimators that Keep Up with the Data,” Journal of the Society for Industrial and Applied Mathematics, Series A: Control, 3(3), 384–417, 1965. DOI: 10.1137/0303026.
[3] Petre Stoica and Per Ashgren. Exact initialization of the recursive least-squares algorithm. Int. J. Adapt. Control Signal Process. 2002; 16:219&ndashh;230.
The post Online (one-pass) algorithms first appeared on John D. Cook.
