8/19/2021

Operator's Soundtrack

Julia provides a complete collection of basic arithmetic and bitwise operators across all of its numeric primitive types, as well as providing portable, efficient implementations of a comprehensive collection of standard mathematical functions.

About At one point during the bizarre music video for Operator Music Band’s 2016 single, “Bebop Radiohaus,” singer/guitarist Jared Hiller looks straight into the camera and asks, blankly, “is it unseen? Is it something real?”. Find Operators song information on AllMusic. Synth pop project led by Dan Boeckner, with his Divine Furs bandmate Sam Brown and Devojka. Operator is an American post-grunge band from Los Angeles, California, United States.The name Operator was used for a solo project created by Johnny Strong, an actor and musician, who has appeared in movies such as Black Hawk Down (2001), The Fast and the Furious (2001), Get Carter (2000) and The Glimmer Man (1996). ‎Listen to songs and albums by Operator Music Band, including 'Mondo,' 'Slim Spin,' 'Nul,' and many more. Songs by Operator Music Band start at $0.99. Dan Boeckner (vocals, guitar), Sam Brown (drums), Devojka (keyboards, effects), Dustin Hawthorne (bass, 2015-present).

The following arithmetic operators are supported on all primitive numeric types:

ExpressionNameDescription
+xunary plusthe identity operation
-xunary minusmaps values to their additive inverses
x + ybinary plusperforms addition
x - ybinary minusperforms subtraction
x * ytimesperforms multiplication
x / ydivideperforms division
x ÷ yinteger dividex / y, truncated to an integer
x yinverse divideequivalent to y / x
x ^ ypowerraises x to the yth power
x % yremainderequivalent to rem(x,y)

as well as the negation on Bool types:

ExpressionNameDescription
!xnegationchanges true to false and vice versa

A numeric literal placed directly before an identifier or parentheses, e.g. 2x or 2(x+y), is treated as a multiplication, except with higher precedence than other binary operations. See Numeric Literal Coefficients for details.

Julia's promotion system makes arithmetic operations on mixtures of argument types 'just work' naturally and automatically. See Conversion and Promotion for details of the promotion system.

Here are some simple examples using arithmetic operators:

(By convention, we tend to space operators more tightly if they get applied before other nearby operators. For instance, we would generally write -x + 2 to reflect that first x gets negated, and then 2 is added to that result.)

When used in multiplication, false acts as a strong zero:

This is useful for preventing the propagation of NaN values in quantities that are known to be zero. See Knuth (1992) for motivation.

The following bitwise operators are supported on all primitive integer types:

Operators Band Members

ExpressionName
~xbitwise not
x & ybitwise and
x ybitwise or
x ⊻ ybitwise xor (exclusive or)
x >>> ylogical shift right
x >> yarithmetic shift right
x << ylogical/arithmetic shift left

Here are some examples with bitwise operators:

Every binary arithmetic and bitwise operator also has an updating version that assigns the result of the operation back into its left operand. The updating version of the binary operator is formed by placing a = immediately after the operator. For example, writing x += 3 is equivalent to writing x = x + 3:

Operator

The updating versions of all the binary arithmetic and bitwise operators are:

An updating operator rebinds the variable on the left-hand side. As a result, the type of the variable may change.

For every binary operation like ^, there is a corresponding 'dot' operation .^ that is automatically defined to perform ^ element-by-element on arrays. For example, [1,2,3] ^ 3 is not defined, since there is no standard mathematical meaning to 'cubing' a (non-square) array, but [1,2,3] .^ 3 is defined as computing the elementwise (or 'vectorized') result [1^3, 2^3, 3^3]. Similarly for unary operators like ! or , there is a corresponding .√ that applies the operator elementwise.

More specifically, a .^ b is parsed as the 'dot' call(^).(a,b), which performs a broadcast operation: it can combine arrays and scalars, arrays of the same size (performing the operation elementwise), and even arrays of different shapes (e.g. combining row and column vectors to produce a matrix). Moreover, like all vectorized 'dot calls,' these 'dot operators' are fusing. For example, if you compute 2 .* A.^2 .+ sin.(A) (or equivalently @. 2A^2 + sin(A), using the @. macro) for an array A, it performs a single loop over A, computing 2a^2 + sin(a) for each element of A. In particular, nested dot calls like f.(g.(x)) are fused, and 'adjacent' binary operators like x .+ 3 .* x.^2 are equivalent to nested dot calls (+).(x, (*).(3, (^).(x, 2))).

Furthermore, 'dotted' updating operators like a .+= b (or @. a += b) are parsed as a .= a .+ b, where .= is a fused in-place assignment operation (see the dot syntax documentation).

Note the dot syntax is also applicable to user-defined operators. For example, if you define ⊗(A,B) = kron(A,B) to give a convenient infix syntax A ⊗ B for Kronecker products (kron), then [A,B] .⊗ [C,D] will compute [A⊗C, B⊗D] with no additional coding.

Combining dot operators with numeric literals can be ambiguous. For example, it is not clear whether 1.+x means 1. + x or 1 .+ x. Therefore this syntax is disallowed, and spaces must be used around the operator in such cases.

Standard comparison operations are defined for all the primitive numeric types:

OperatorName
equality
!=, inequality
<less than
<=, less than or equal to
'>>greater than
='>>=, ='>greater than or equal to

Here are some simple examples:

Integers are compared in the standard manner – by comparison of bits. Floating-point numbers are compared according to the IEEE 754 standard:

  • Finite numbers are ordered in the usual manner.
  • Positive zero is equal but not greater than negative zero.
  • Inf is equal to itself and greater than everything else except NaN.
  • -Inf is equal to itself and less than everything else except NaN.
  • NaN is not equal to, not less than, and not greater than anything, including itself.

The last point is potentially surprising and thus worth noting:

Song Smooth Operator Original

and can cause headaches when working with arrays:

Julia provides additional functions to test numbers for special values, which can be useful in situations like hash key comparisons:

FunctionTests if
isequal(x, y)x and y are identical
isfinite(x)x is a finite number
isinf(x)x is infinite
isnan(x)x is not a number

isequal considers NaNs equal to each other:

isequal can also be used to distinguish signed zeros:

Mixed-type comparisons between signed integers, unsigned integers, and floats can be tricky. A great deal of care has been taken to ensure that Julia does them correctly.

For other types, isequal defaults to calling , so if you want to define equality for your own types then you only need to add a method. If you define your own equality function, you should probably define a corresponding hash method to ensure that isequal(x,y) implies hash(x) hash(y).

Unlike most languages, with the notable exception of Python, comparisons can be arbitrarily chained:

Chaining comparisons is often quite convenient in numerical code. Chained comparisons use the && operator for scalar comparisons, and the & operator for elementwise comparisons, which allows them to work on arrays. For example, 0 .< A .< 1 gives a boolean array whose entries are true where the corresponding elements of A are between 0 and 1.

Note the evaluation behavior of chained comparisons:

The middle expression is only evaluated once, rather than twice as it would be if the expression were written as v(1) < v(2) && v(2) <= v(3). However, the order of evaluations in a chained comparison is undefined. It is strongly recommended not to use expressions with side effects (such as printing) in chained comparisons. If side effects are required, the short-circuit && operator should be used explicitly (see Short-Circuit Evaluation).

Operator's Soundtrack Torrent

Julia provides a comprehensive collection of mathematical functions and operators. These mathematical operations are defined over as broad a class of numerical values as permit sensible definitions, including integers, floating-point numbers, rationals, and complex numbers, wherever such definitions make sense.

Moreover, these functions (like any Julia function) can be applied in 'vectorized' fashion to arrays and other collections with the dot syntaxf.(A), e.g. sin.(A) will compute the sine of each element of an array A.

Run or fight soundtrack. Julia applies the following order and associativity of operations, from highest precedence to lowest:

CategoryOperatorsAssociativity
Syntax. followed by ::Left
Exponentiation^Right
Unary+ - √Right[1]
Bitshifts<< >> >>>Left
Fractions//Left
Multiplication* / % & ÷Left[2]
Addition+ - ⊻Left[2]
Syntax: .Left
Syntax >Left
Syntax< Right
Comparisons> < >= <= != ! <:Non-associative
Control flow&& followed by followed by ?Right
Pair=>Right
Assignments= += -= *= /= //= = ^= ÷= %= = &= ⊻= <<= >>= >>>=Right

For a complete list of every Julia operator's precedence, see the top of this file: src/julia-parser.scm

Numeric literal coefficients, e.g. 2x, are treated as multiplications with higher precedence than any other binary operation, and also have higher precedence than ^.

You can also find the numerical precedence for any given operator via the built-in function Base.operator_precedence, where higher numbers take precedence:

A symbol representing the operator associativity can also be found by calling the built-in function Base.operator_associativity:

Note that symbols such as :sin return precedence 0. This value represents invalid operators and not operators of lowest precedence. Similarly, such operators are assigned associativity :none.

Julia supports three forms of numerical conversion, which differ in their handling of inexact conversions.

  • The notation T(x) or convert(T,x) converts x to a value of type T.

    • If T is a floating-point type, the result is the nearest representable value, which could be positive or negative infinity.
    • If T is an integer type, an InexactError is raised if x is not representable by T.
  • x % T converts an integer x to a value of integer type T congruent to x modulo 2^n, where n is the number of bits in T. In other words, the binary representation is truncated to fit.

  • The Rounding functions take a type T as an optional argument. For example, round(Int,x) is a shorthand for Int(round(x)).

The following examples show the different forms.

See Conversion and Promotion for how to define your own conversions and promotions.

FunctionDescriptionReturn type
round(x)round x to the nearest integertypeof(x)
round(T, x)round x to the nearest integerT
floor(x)round x towards -Inftypeof(x)
floor(T, x)round x towards -InfT
ceil(x)round x towards +Inftypeof(x)
ceil(T, x)round x towards +InfT
trunc(x)round x towards zerotypeof(x)
trunc(T, x)round x towards zeroT
FunctionDescription
div(x,y), x÷ytruncated division; quotient rounded towards zero
fld(x,y)floored division; quotient rounded towards -Inf
cld(x,y)ceiling division; quotient rounded towards +Inf
rem(x,y)remainder; satisfies x div(x,y)*y + rem(x,y); sign matches x
mod(x,y)modulus; satisfies x fld(x,y)*y + mod(x,y); sign matches y
mod1(x,y)mod with offset 1; returns r∈(0,y] for y>0 or r∈[y,0) for y<0, where mod(r, y) mod(x, y)
mod2pi(x)modulus with respect to 2pi; 0 <= mod2pi(x) < 2pi
divrem(x,y)returns (div(x,y),rem(x,y))
fldmod(x,y)returns (fld(x,y),mod(x,y))
gcd(x,y..)greatest positive common divisor of x, y,..
lcm(x,y..)least positive common multiple of x, y,..
FunctionDescription
abs(x)a positive value with the magnitude of x
abs2(x)the squared magnitude of x
sign(x)indicates the sign of x, returning -1, 0, or +1
signbit(x)indicates whether the sign bit is on (true) or off (false)
copysign(x,y)a value with the magnitude of x and the sign of y
flipsign(x,y)a value with the magnitude of x and the sign of x*y
FunctionDescription
sqrt(x), √xsquare root of x
cbrt(x), ∛xcube root of x
hypot(x,y)hypotenuse of right-angled triangle with other sides of length x and y
exp(x)natural exponential function at x
expm1(x)accurate exp(x)-1 for x near zero
ldexp(x,n)x*2^n computed efficiently for integer values of n
log(x)natural logarithm of x
log(b,x)base b logarithm of x
log2(x)base 2 logarithm of x
log10(x)base 10 logarithm of x
log1p(x)accurate log(1+x) for x near zero
exponent(x)binary exponent of x
significand(x)binary significand (a.k.a. mantissa) of a floating-point number x

For an overview of why functions like hypot, expm1, and log1p are necessary and useful, see John D. Cook's excellent pair of blog posts on the subject: expm1, log1p, erfc, and hypot.

All the standard trigonometric and hyperbolic functions are also defined:

These are all single-argument functions, with atan also accepting two arguments corresponding to a traditional atan2 function.

Additionally, sinpi(x) and cospi(x) are provided for more accurate computations of sin(pi*x) and cos(pi*x) respectively.

In order to compute trigonometric functions with degrees instead of radians, suffix the function with d. For example, sind(x) computes the sine of x where x is specified in degrees. The complete list of trigonometric functions with degree variants is:

Many other special mathematical functions are provided by the package SpecialFunctions.jl.

  • 1The unary operators + and - require explicit parentheses around their argument to disambiguate them from the operator ++, etc. Other compositions of unary operators are parsed with right-associativity, e. g., √√-a as √(√(-a)).
  • 2The operators +, ++ and * are non-associative. a + b + c is parsed as +(a, b, c) not +(+(a, b), c). However, the fallback methods for +(a, b, c, d..) and *(a, b, c, d..) both default to left-associative evaluation.
Songs

Operators were just a name when they opened for Future Islands at the Gothic Theatre in 2014, but as soon as Boeckner opened his mouth during the first track from their EP1, I immediately knew who he was. That distinct voice could belong to no other. The set showed promise as a guitar-driven, synth-heavy electro mix between Handsome Furs and Wolf Parade, but it was too short to do anything but drive expectations as to what would come next. The Lost Lake Lounge performance met and exceeded those expectations. Performing in front of 150 people on the day they released their debut album, Blue Wave, Operators were surprisingly tight for a band who hadn’t been performing the songs for any length of time. Boeckner was also surprisingly humble for a man getting ready to play Coachella after selling out a Bowery residency in mere minutes. And the audience, who couldn’t have been familiar with many of the tracks being performed, were surprisingly hyped. Backed by (Divine Fits drummer) Sam Brown, (solo synth artist) Devojka, and a bassist, Boeckner sported a black blazer and grey beard as he introduced the album to the packed house. “This is fucking crazy! Thank you all for coming out tonight!”

The set consisted of a perfect mixture of punked-out rockers and electronic dance anthems. Inspired by ‘80s electro-pop, as well as the more aggressive post-punk/hardcore from that era, the performance brought me back to the days of watching MTV’s 120 Minutes after returning home from a Fugazi show. When he wasn’t abusing his guitar, Boeckner was like an animal trapped in a small cage, sticking his head out as far as he could to get a better look at his captors, all while wrapping his mic around himself like a comfort blanket. Honestly, the place was so packed I could only catch glimpses of him between the old fireplace I was standing next to and the tall man in front of me, but as his voice sonically skipped over each chord (just as it did in Wolf Parade), I could see his movements being mimicked by those around me. “Does anyone have one of those creepy-ass Dennis Hopper Blue Velvet things? Just in case? I’m asking for my health, not because I’m fishing for nitrous.” Joking about the altitude that affects so many musicians who play the Mile High City, Devojka followed up Boeckner’s comment with a quick rendition of Giogorio Moroder’s “Take My Breath Away”. It was a funny little interlude, but if the band was experiencing any real effects from the elevation, they did nothing to show it.

Operator's Soundtrack Download

- See more at: http://ilistensoyoudonthaveto.com/2016/04/03/operators-lost-lake-04-01-16/#sthash.ACCJrefM.dpuf