Spaceship operator in Python

Some programming languages, such as Perl, have an infix operator <=> that returns a three-state comparison. The expression

    a <=> b

evaluates to -1, 0, or 1 depending on whether a < b, a = b, or a > b. You could think of <=> as a concatenation of <, =, and >.

The <=> operator is often called the “spaceship operator” because it looks like Darth Vader’s ship in Star Wars.

Python doesn’t have a spaceship operator, but you can get the same effect with numpy.sign(a-b). For example, suppose you wanted to write a program to compare two integers.

You could write

    from numpy import sign
    def compare(x, y):
        cmp = ["equal to", "greater than", "less than"][sign(x-y)]
        print(f"{x} is {cmp} {y}.")

Here we take advantage of the fact that an index of -1 points to the last element of a list.

The sign function will return an integer if its argument is an integer or a float if its argument is a float. The code above will break if you pass in floating point numbers because sign will return -1.0, 0.0, or 1.0. But if you replace sign(x-y) with int(sign(x-y)) it will work for floating point arguments.

Related post: Symbol pronunciation

Fraction comparison trick

If you want to determine whether

a/b > c/d,

it is often enough to test whether

a+d > b+c.

Said another way a/b is usually greater than c/d when a+d is greater than b+c.

This sounds imprecise if not crazy. But it is easy to make precise and [1] shows that it is true.


Consider 4/11 vs 3/7. In this case 4 + 7 = 11 < 11 + 3 = 14, which suggests 4/11 < 3/7, which is correct.

But the rule of thumb can lead you astray. For example, suppose you want to compare 3/7 and 2/5. We have 3 + 5 = 8 < 7 + 2 = 9, but 3/7 > 2/5.

The claim isn’t that the rule always works; clearly it doesn’t. The claim is that it usually works, in a sense that will be made precise in the next section.

Rigorous statement

Let N be a large integer, and pick integers a, b, c, and d at random uniformly between 1 and N. Let

x = a/bc/d


y = (a+d) – (b+c).

Then the probability x and y are both positive, or both negative, or both zero, approaches 11/12 as N goes to infinity.


I won’t repeat the proof of the theorem above; see [1] for that. But I’ll give a simulation that illustrates the theorem.

    import numpy as np
    N = 1_000_000
    numreps = 10_000
    count = 0
    for _ in range(numreps):
        (a, b, c, d) = np.random.randint(1, N+1, 4, dtype=np.int64)
        x = a*d - b*c
        y = (a+d) - (b+c)
        if np.sign(x) == np.sign(y):
            count += 1

This prints 0.9176, and 11/12 = 0.91666….

The random number generator randint defaults to 32-bit integers, but this could lead to overflow since 106 × 106 > 232. So I used 64-bit integers.

Instead of computing a/bc/d, I multiply by bd and compute adbc instead because this avoids any possible floating point issues.

In case you’re not familiar with the sign function, it returns 1 for positive numbers, 0 for 0, and -1 for negative numbers.

The code suggests a different statement of the theorem: if you generate two pairs of integers, their sums and their products are probably ordered the same way.


This post has assumed that the numbers a, b, c, and d are all chosen uniformly at random. But the components of the fractions for which you have any doubt whether a/b is greater or less than c/d are not uniformly distributed. For example, consider 17/81 versus 64/38. Clearly the former is less than 1 and the latter is greater than 1.

It would be interesting to try to assess how often the rule of thumb presented here is correct in practice. You might try to come up with a model for the kinds of fractions people can’t compare instantly, such as proper fractions that have similar size.

Related posts

[1] Kenzi Odani. A Rough-and-Ready Rule for Fractions. The Mathematical Gazette, Vol. 82, No. 493 (Mar., 1998), pp. 107-109

Python triple quote strings and regular expressions

There are several ways to quote strings in Python. Triple quotes let strings span multiple lines. Line breaks in your source file become line break characters in your string. A triple-quoted string in Python acts something like “here doc” in other languages.

However, Python’s indentation rules complicate matters because the indentation becomes part of the quoted string. For example, suppose you have the following code outside of a function.

x = """\

Then you move this into a function foo and change its name to y.

def foo():
    y = """\

Now x and y are different strings! The former begins with a and the latter begins with four spaces. (The backslash after the opening triple quote prevents the following newline from being part of the quoted string. Otherwise x and y would begin with a newline.) The string y also has four spaces in front of def and four spaces in front of ghi. You can’t push the string contents to the left margin because that would violate Python’s formatting rules.

We now give three solutions to this problem.

Solution 1: textwrap.dedent

There is a function in the Python standard library that will strip the unwanted space out of the string y.

import textwrap 

def foo():
    y = """\
    y = textwrap.dedent(y)

This works, but in my opinion a better approach is to use regular expressions [1].

Solution 2: Regular expression with a flag

We want to remove white space, and the regular expression for a white space character is \s. We want to remove one or more white spaces so we add a + on the end. But in general we don’t want to remove all white space, just white space at the beginning of a line, so we stick ^ on the front to say we want to match white space at the beginning of a line.

import re 

def foo():
    y = """\
    y = re.sub("^\s+", "", y)

Unfortunately this doesn’t work. By default ^ only matches the beginning of a string, not the beginning of a line. So it will only remove the white space in front of the first line; there will still be white space in front of the following lines.

One solution is to add the flag re.MULTILINE to the substitution function. This will signal that we want ^ to match the beginning of every line in our multi-line string.

    y = re.sub("^\s+", "", y, re.MULTILINE)

Unfortunately that doesn’t quite work either! The fourth positional argument to re.sub is a count of how many substitutions to make. It defaults to 0, which actually means infinity, i.e. replace all occurrences. You could set count to 1 to replace only the first occurrence, for example. If we’re not going to specify count we have to set flags by name rather than by position, i.e. the line above should be

    y = re.sub("^\s+", "", y, flags=re.MULTILINE)

That works.

You could also abbreviate re.MULTILINE to re.M. The former is more explicit and the latter is more compact. To each his own. There’s more than one way to do it. [2]

Solution 3: Regular expression with a modifier

In my opinion, it is better to modify the regular expression itself than to pass in a flag. The modifier (?m) specifies that in the rest of the regular the ^ character should match the beginning of each line.

    y = re.sub("(?m)^\s+", "", y)

One reason I believe this is better is that moves information from a language-specific implementation of regular expressions into a regular expression syntax that is supported in many programming languages.

For example, the regular expression


would have the same meaning in Perl and Python. The two languages have the same way of expressing modifiers [3], but different ways of expressing flags. In Perl you paste an m on the end of a match operator to accomplish what Python does with setting flasgs=re.MULTILINE.

One of the most commonly used modifiers is (?i) to indicate that a regular expression should match in a case-insensitive manner. Perl and Python (and other languages) accept (?i) in a regular expression, but each language has its own way of adding modifiers. Perl adds an i after the match operator, and Python uses




as a function argument.

More on regular expressions

[1] Yes, I’ve heard the quip about two problems. It’s funny, but it’s not a universal law.

[2] “There’s more than one way to do it” is a mantra of Perl and contradicts The Zen of Python. I use the line here as a good-natured jab at Python. Despite its stated ideals, Python has more in common with Perl than it would like to admit and continues to adopt ideas from Perl.

[3] Python’s re module doesn’t support every regular expression modifier that Perl supports. I don’t know about Python’s regex module.

Minimizing boolean expressions

This post will look at how to take an expression for a Boolean function and look for a simpler expression that corresponds to the same function. We’ll show how to use a Python implementation of the Quine-McCluskey algorithm.


We will write AND like multiplication, OR like addition, and use primes for negation. For example,

wx + z


(w AND x) OR (NOT z).

Minimizing expressions

You may notice that the expression

wxz + wxz

can be simplified to wz, for example, but it’s not feasible to simplify complicated expressions without a systematic approach.

One such approach is the Quine-McCluskey algorithm. Its run time increases exponentially with the problem size, but for a small number of terms it’s quick enough [1]. We’ll show how to use the Python module qm which implements the algorithm.

Specifying functions

How are you going to pass a Boolean expression to a Python function? You could pass it an expression as a string and expect the function to parse the string, but then you’d have to specify the grammar of the little language you’ve created. Or you could pass in an actual Python function, which is more work than necessary, especially if you’re going to be passing in a lot of expressions.

A simpler way is pass in the set of places where the function evaluates to 1, encoded as numbers.

For example, suppose your function is

wxyz + wxyz

This function evaluates to 1 when either the first term evaluates to 1 or the second term evaluates to 1. That is, when either

(w, x, y, z) = (1, 1, 0, 1)


(w, x, y, z) = (0, 1, 1, 0).

Interpreting the left sides as binary numbers, you could specify the expression with the set {13, 6} which describes where the function is 1.

If you prefer, you could express your numbers in binary to make the correspondence to terms more explicit, i.e. {0b1101,0b110}.

Using qm

One more thing before we use qm: your Boolean expression might not be fully specified. Maybe you want it to be 1 on some values, 0 on others, and you don’t care what it equals on the rest.

The qm module lets you specify these with arguments ones, zeroes, and dc. If you specify two out of these three sets, qm will infer the third one.

For example, in the code below

    from qm import qm
    print(qm(ones={0b111, 0b110, 0b1101}, dc={}))

we’re asking qm to minimize the expression

xyz + xyz‘ + wxyz.

Since the don’t-care set is empty, we’re saying our function equals 0 everywhere we haven’t said that it equals 1. The function prints

    ['1101', '011X']

which corresponds to

wxyz + wxy,

the X meaning that the fourth variable, z, is not part of the second term.

Note that the minimized expression is not unique: we could tell by inspection that

xyz + xyz‘ + wxyz.

could be reduced to

xy + wxyz.

Also, our code defines a minimum expression to be one with the fewest sums. Both simplifications in this example have two sums. But xy + wxyz is simpler than wxyz + wxy in the sense of having one less term, so there’s room for improvement, or at least discussion, as to how to quantify the complexity of an expression.

In the next post I use qm to explore how much minimization reduces the size of Boolean expressions.


[1] The Boolean expression minimization problem is in NP, and so no known algorithm that always produces an exact answer will scale well. But there are heuristic algorithms like Espresso and its variations that usually provide optimal or near-optimal results.

Solving Van der Pol equation with ivp_solve

Van der Pol’s differential equation is

{d^2x \over dt^2}-\mu(1-x^2){dx \over dt}+x= 0

The equation describes a system with nonlinear damping, the degree of nonlinearity given by μ. If μ = 0 the system is linear and undamped, but as μ increases the strength of the nonlinearity increases. We will plot the phase portrait for the solution to Van der Pol’s equation in Python using SciPy’s new ODE solver ivp_solve.

The function ivp_solve does not solve second-order systems of equations directly. It solves systems of first-order equations, but a second-order differential equation can be recast as a pair of first-order equations by introducing the first derivative as a new variable.

\begin{align*} {dx \over dt} &= y \\ {dy \over dt}&= \mu(1-x^2)y -x \\ \end{align*}

Since y is the derivative of x, the phase portrait is just the plot of (x, y).

Phase portait of Van der Pol oscillator

If μ = 0, we have a simple harmonic oscillator and the phase portrait is simply a circle. For larger values of μ the solutions enter limiting cycles, but the cycles are more complicated than just circles. These limiting cycles are periodic attractors: every non-trivial solution converges to the limit cycle.

Here’s the Python code that made the plot.

from scipy import linspace
from scipy.integrate import solve_ivp
import matplotlib.pyplot as plt

def vdp(t, z):
    x, y = z
    return [y, mu*(1 - x**2)*y - x]

a, b = 0, 10

mus = [0, 1, 2]
styles = ["-", "--", ":"]
t = linspace(a, b, 500)

for mu, style in zip(mus, styles):
    sol = solve_ivp(vdp, [a, b], [1, 0], t_eval=t)
    plt.plot(sol.y[0], sol.y[1], style)
# make a little extra horizontal room for legend
plt.legend([f"$\mu={m}$" for m in mus])

To plot the solutions as a function of time, rather than plotting phase portraits, change the line

    plt.plot(sol.y[0], sol.y[1], style)


    plt.plot(sol.t, sol.y[0], style)

and comment out the line setting xlim This gives the following plot.

Van der Pol oscillator solutions as a function of time

More dynamical system posts

Higher order Newton-like methods for root finding

There are variations on Newton’s root finding method that use higher derivatives and converge faster. Alston Householder developed a sequence of such methods of the form

H_n(x) = x + (n-1) \frac{\left( \frac{1}{f(x)}\right)^{(n-2)}}{\left( \frac{1}{f(x)}\right)^{(n-1)}}

where the superscripts in parentheses indicate derivatives. When n = 2, Householder’s method reduces to Newton’s method.

Once Newton’s method is close enough to a root, the error is squared at each iteration. Said another way, the number of correct decimal places roughly doubles. With the Householder method of order n, the number of correct decimal places roughly increases by a factor of n.

The Householder method Hn can be written as a rational function of f and its derivatives up to order n-1, though the formulas are complicated.

Maybe it’s time for Householder’s methods to be more widely used: with automatic differentiation [1], the complexity of computing higher order derivatives is not the drawback it was when these computations were done solely by hand.


We’ll illustrate the convergence of the Householder methods by finding roots of the quintic polynomial

f(x) = x5 + ax + b.

I used Mathematica to calculate expressions for the Householder methods and export the results as code. Mathematica does not have a function to export Python code, but it does have a function CForm to export expressions as C code. I started to use this, converting exponents from C to Python form with search and replace. Then I tried using FortranForm instead, and that was much easier because Fortran and Python both represent powers with a ** operator.

def f(x, a, b):
    return x**5 + a*x + b

def H2(x, a, b):
    return x - (b + a*x + x**5)/(a + 5*x**4)
def H3(x, a, b):
    num = (a + 5*x**4)*(b + a*x + x**5)
    den = a**2 - 10*b*x**3 + 15*x**8
    return x - num/den

def H4(x, a, b):
    num = (b + a*x + x**5)*(-a**2 + 10*b*x**3 - 15*x**8)
    den = a**3 + 10*b**2*x**2 + 5*a**2*x**4 - 80*b*x**7 - 25*a*x**8 + 35*x**12
    return x + num/den

To set values of a and b, I arbitrarily chose a = 2 and solved for b so that π is a root. That way we have a known value of the root for comparison. I used x = 4 as my starting guess at the root.

I first used -log10|x – π| to measure the number of correct decimals. That didn’t quite work because the method would converge exactly (to the limits of machine precision) which led to taking the log of zero. So I modified my accuracy function to the following.

def decimals(x):
    d = abs(x - pi)
    if d != 0:
        return( round(-log10(d),2) )
        return "inf"

Then I tested each of the Householder methods with the following.

epsilon = 1e-14
for H in [H2, H3, H4]:
    x = 4
    for i in range(1, 10):
        if abs(f(x,a,b)) > epsilon:
            x = H(x,a,b)
            print(i, x, decimals(x))

Here are the results in tabular form. I replaced all the decimals after the first incorrect decimal with underscores to make it easier to see the convergence.

| i |                 x | digits |
| 1 | 3.4______________ |   0.53 |
| 2 | 3.18_____________ |   1.33 |
| 3 | 3.142____________ |   2.87 |
| 4 | 3.141593_________ |   5.93 |
| 5 | 3.141592653590___ |  12.07 |
| 6 | 3.141592653589793 |    inf |
| 1 | 3.2______________ |   1.11 |
| 2 | 3.1416___________ |   4.03 |
| 3 | 3.1415926535899__ |  12.79 |
| 4 | 3.141592653589793 |    inf |
| 1 | 3.15_____________ |   1.84 |
| 2 | 3.141592654______ |   8.85 |
| 3 | 3.141592653589793 |    inf |

Note that, as promised, the number of correct decimal places for Hn increases by roughly a factor of n with each iteration.

Update: See this post where the symbolic calculations are done in SymPy rather than Mathematica. Everything stays in Python, and there’s no need to edit the code for the Householder methods.

Related posts

[1] Note that in this post I used Mathematica to do symbolic differentiation, not automatic differentiation. Automatic differentiation differs from both symbolic and numerical differentiation. See an explanation here.

Drawing with Unicode block characters

My previous post contained the image below.

The point of the post was that the numbers that came up in yet another post made the fractal image above when you wrote them in binary. Here’s the trick I used to make that image.

To make the pattern of 0’s and 1’s easier to see, I wanted to make a graphic with black and white squares instead of the characters 0 and 1. I thought about writing a program to create an image using strings of 0’s and 1’s as input, but then I thought of something more direct.

The Unicode character U+2588 is just a solid rectangle. I replaced the 1’s with U+2588 and replaced the 0’s with spaces. So I made a text file that looked like the image above, and took a screen shot of it. The only problem was that the image was rectangular. I thought the pattern would be easier to see if it were a square, so I changed the aspect ratio on the image.

Here’s another example using Unicode block elements, this time to make a little bar graph, sorta like Edward Tufte’s idea of a sparkline. The characters U+2581 through U+2588 produce bars of increasing height.

bar graph of English letter frequencies

To create images like this, your font needs to include glyphs for Unicode block elements. The font needs to be monospaced as well so the letters will line up under the blocks. The example above was created using the APL385. It also looks good in Hack and Input Mono. In some fonts the block characters don’t align consistently on the baseline.

Here’s the Python code that produced the above graph of English letter frequencies.

# Letter frequencies via
freq = [ 
  8.04, 1.48, 3.34, 3.82, 12.49,
  2.40, 1.87, 5.05, 7.57,  0.16,
  0.54, 4.07, 2.51, 7.23,  7.64,
  2.14, 0.12, 6.28, 6.51,  9.28,
  2.73, 1.05, 1.68, 0.23,  1.66,
m = max(freq)

for i in range(26):
    u = int(8*freq[i]/m)
    ch = chr(0x2580+u) if u > 0 else " "
    print(ch, end="")

for i in range(26):
    print(chr(0x41+i), end="")

Predicted distribution of Mersenne primes

Mersenne primes are prime numbers of the form 2p – 1. It turns out that if 2p – 1 is a prime, so is p; the requirement that p is prime is a theorem, not part of the definition.

So far 51 Mersenne primes have discovered [1]. Maybe that’s all there are, but it is conjectured that there are an infinite number Mersenne primes. In fact, it has been conjectured that as x increases, the number of primes px is asymptotically

eγ log x / log 2

where γ is the Euler-Mascheroni constant. For a heuristic derivation of this conjecture, see Conjecture 3.20 in Not Always Buried Deep.

How does the actual number of Mersenne primes compared to the number predicted by the conjecture? We’ll construct a plot below using Python. Note that the conjecture is asymptotic, and so it could make poor predictions for now and still be true for much larger numbers. But it appears to make fairly good predictions over the range where we have discovered Mersenne primes.

Predicted vs actual Mersenne primes

import numpy as np
import matplotlib.pyplot as plt

# p's for which 2^p - 1 is prime.
# See
ps = [2, 3, 5, ... , 82589933]

# x has 200 points from 10^1 to 10^8 
# spaced evenly on a logarithmic scale
x = np.logspace(1, 8, 200)

# number of p's less than x such that 2^p - 1 is prime
actual = [np.searchsorted(ps, t) for t in x]

exp_gamma = np.exp(0.5772156649)
predicted = [exp_gamma*np.log2(t) for t in x]

plt.plot(x, actual)
plt.plot(x, predicted, "--")

plt.ylabel(r"Mersenne primes $\leq 2^p-1$")
plt.legend(["actual", "predicted"])

Related posts

[1] Fifty one Mersenne primes have been verified. But these may not be the smallest Mersenne primes. It has not yet been verified that there are no Mersenne primes yet to be discovered between the 47th and 51st known ones. The plot in this post assumes the known Mersenne primes are consecutive, and so it is speculative toward the right end.

String interpolation in Python and R

One of the things I liked about Perl was string interpolation. If you use a variable name in a string, the variable will expand to its value. For example, if you a variable $x which equals 42, then the string

    "The answer is $x."

will expand to “The answer is 42.” Perl requires variables to start with sigils, like the $ in front of scalar variables. Sigils are widely considered to be ugly, but they have their benefits. Here, for example, $x is clearly a variable name, whereas x would not be.

You can do something similar to Perl’s string interpolation in Python with so-called f-strings. If you put an f in front of an opening quotation mark, an expression in braces will be replaced with its value.

    >>> x = 42
    >>> f"The answer is {x}."
    'The answer is 42.'

You could also say

    >>> f"The answer is {6*7}."

for example. The f-string is just a string; it’s only printed because we’re working from the Python REPL.

The glue package for R lets you do something very similar to Python’s f-strings.

    > library(glue)
    > x <- 42
    > glue("The answer is {x}.")
    The answer is 42.
    > glue("The answer is {6*7}.")
    The answer is 42.

As with f-strings, glue returns a string. It doesn’t print the string, though the string is displayed because we’re working from the REPL, the R REPL in this case.

Regular expressions and special characters

Special characters make text processing more complicated because you have to pay close attention to context. If you’re looking at Python code containing a regular expression, you have to think about what you see, what Python sees, and what the regular expression engine sees. A character may be special to Python but not to regular expressions, or vice versa.

This post goes through an example in detail that shows how to manage special characters in several different contexts.

Escaping special TeX characters

I recently needed to write a regular expression [1] to escape TeX special characters. I’m reading in text like ICD9_CODE and need to make that ICD9\_CODE so that TeX will understand the underscore to be a literal underscore, and a subscript instruction.

Underscore isn’t the only special character in TeX. It has ten special characters:

    \ { } $ & # ^ _ % ~

The two that people most commonly stumble over are probably $ and % because these are fairly common in ordinary prose. Since % begins a comment in TeX, importing a percent sign without escaping it will fail silently. The result is syntactically valid. It just effectively cuts off the remainder of the line.

So whenever my script sees a TeX special character that isn’t already escaped, I’d like it to escape it.

Raw strings

First I need to tell Python what the special characters are for TeX:

    special = r"\\{}$&#^_%~"

There’s something interesting going on here. Most of the characters that are special to TeX are not special to Python. But backslash is special to both. Backslash is also special to regular expressions. The r prefix in front of the quotes tells Python this is a “raw” string and that it should not interpret backslashes as special. It’s saying “I literally want a string that begins with two backslashes.”

Why two backslashes? Wouldn’t one do? We’re about to use this string inside a regular expression, and backslashes are special there too. More on that shortly.


Here’s my regular expression:

    re.sub(r"(?<!\\)([" + special + "])", r"\\\1", line)

I want special characters that have not already been escaped, so I’m using a negative lookbehind pattern. Negative lookbehind expressions begin with (?<! and end with ). So if, for example, I wanted to look for the string “ball” but only if it’s not preceded by “charity” I could use the regular expression

    (?<!charity )ball

This expression would match “foot ball” or “foosball” but not “charity ball”.

Our lookbehind expression is complicated by the fact that the thing we’re looking back for is a special character. We’re looking for a backslash, which is a special character for regular expressions [2].

After looking behind for a backslash and making sure there isn’t one, we look for our special characters. The reason we used two backslashes in defining the variable special is so the regular expression engine would see two backslashes and interpret that as one literal backslash.


The second argument to re.sub tells it what to replace its match with. We put parentheses around the character class listing TeX special characters because we want to capture it to refer to later. Captures are referred to by position, so the first capture is \1, the second is \2, etc.

We want to tell re.sub to put a backslash in front of the first capture. Since backslashes are special to the regular expression engine, we send it \\ to represent a literal backslash. When we follow this with \1 for the first capture, the result is \\\1 as above.


We can test our code above on with the following.

    line = r"a_b $200 {x} %5 x\y"

and get

    a\_b \$200 \{x\} \%5 x\\y

which would cause TeX to produce output that looks like

a_b $200 {x} %5 x\y.

Note that we used a raw string for our test case. That was only necessary for the backslash near the end of the string. Without that we could have dropped the r in front of the opening quote.

P.S. on raw strings

Note that you don’t have to use raw strings. You could just escape your special characters with backslashes. But we’ve already got a lot of backslashes here. Without raw strings we’d need even more. Without raw strings we’d have to say

    special = "\\\\{}$&#^_%~"

starting with four backslashes to send Python two to send the regular expression engine one.

Related posts

[1] Whenever I write about using regular expressions someone will complain that my solution isn’t completely general and that they can create input that will break my code. I understand that, but it works for me in my circumstances. I’m just writing scripts to get my work done, not claiming to have written hardened production software for anyone else to use.

[2] Keep context in mind. We have three languages in play: TeX, Python, and regular expressions. One of the keys to understanding regular expressions is to see them as a small language embedded inside other languages like Python. So whenever you hear a character is special, ask yourself “Special to whom?”. It’s especially confusing here because backslash is special to all three languages.