Ivy is an interpreter for an APL-like language. It is a plaything and a work in progress.
Ivy has a custom domain. Do not install using github directly. Instead, run:
go install robpike.io/ivy@latest
Documentation at https://pkg.go.dev/robpike.io/ivy.
Try the demo: within ivy, type
Prototype apps for iOS and Android are available in the App store and Google Play store. (At the moment, the iOS one is unavailable but will return.) These are very old but should be updated before too long. To find them, search for "ivy bignum calculator".
Slides for a talk at: https://talks.godoc.org/github.com/robpike/ivy/talks/ivy.slide Video for the talk at: https://www.youtube.com/watch?v=PXoG0WX0r_E The talk predates a lot of the features, including floating point, text, and user-defined operators.
To be built, ivy requires Go 1.21.
Ivy is an interpreter for an APL-like language. It is a plaything and a work in progress.
Unlike APL, the input is ASCII and the results are exact (but see the next paragraph). It uses exact rational arithmetic so it can handle arbitrary precision. Values to be input may be integers (3, -1), rationals (1/3, -45/67) or floating point values (1e3, -1.5 (representing 1000 and -3/2)).
Some functions such as sqrt are irrational. When ivy evaluates an irrational function, the result is stored in a high-precision floating-point number (default 256 bits of mantissa). Thus when using irrational functions, the values have high precision but are not exact.
Unlike in most other languages, operators always have the same precedence and expressions are evaluated in right-associative order. That is, unary operators apply to everything to the right, and binary operators apply to the operand immediately to the left and to everything to the right. Thus, 3*4+5 is 27 (it groups as 3*(4+5)) and iota 3+2 is 1 2 3 4 5 while 3+iota 2 is 4 5. A vector is a single operand, so 1 2 3 + 3 + 3 4 5 is (1 2 3) + 3 + (3 4 5), or 7 9 11.
As a special but important case, note that 1/3, with no intervening spaces, is a single rational number, not the expression 1 divided by 3. This can affect precedence: 3/6*4 is 2 while 3 / 6*4 is 1/8 since the spacing turns the / into a division operator. Use parentheses or spaces to disambiguate: 3/(6*4) or 3 /6*4.
Ivy has complex numbers, which are constructed using the unary or binary j operator. As with rationals, the token 1j2 (the representation of 1+2i) is a single token. The individual parts can be rational, so 1/2j-3/2 is the complex number 0.5-1.5i and scans as a single value.
Indexing uses  notation: x, x[1; 2], and so on. Indexing by a vector selects multiple elements: x[1 2] creates a new item from x and x. An empty index slot is a shorthand for all the elements along that dimension, so x is equivalent to x, and x[;3] gives the third column of two-dimensional array x.
Only a subset of APL's functionality is implemented, but all numerical operations are supported.
Semicolons separate multiple statements on a line. Variables are alphanumeric and are assigned with the = operator. Assignment is an expression.
After each successful expression evaluation, the result is stored in the variable called _ (underscore) so it can be used in the next expression.
The APL operators, adapted from https://en.wikipedia.org/wiki/APL_syntax_and_symbols, and their correspondence are listed here. The correspondence is incomplete and inexact.
Name APL Ivy Meaning Roll ?B ? One integer selected randomly from the first B integers Ceiling ⌈B ceil Least integer greater than or equal to B If B is complex, the complex ceiling, as defined by McDonnell Floor ⌊B floor Greatest integer less than or equal to B If B is complex, the complex floor, as defined by McDonnell Shape ⍴B rho Vector of number of components in each dimension of B Count ≢B count Scalar number of elements at top level of B Flatten ∊B flatten Vector of all the scalar elements within B Not ∼B not Logical: not 1 is 0, not 0 is 1 Absolute value ∣B abs Magnitude of B Index generator ⍳B iota Vector of the first B integers If B is a vector, matrix of coordinates Unique ∪B unique Remove all duplicate elements from B Enclose ⊂B box Wrap B in one level of nesting Disclose ⊃B first First element of B in ravel order Split ↓B split Create vector of nested elements from matrix B; inverse of mix Mix ↑B mix Create matrix from elements of vector B; inverse of split Exponential ⋆B ** e to the B power Negation −B - Change sign of B Identity +B + No change to B Signum ×B sgn -1 if B<0; 0 if B=0; 1 if B>0. More generally: B/abs B if B!=0 Reciprocal ÷B / 1 divided by B Ravel ,B , Reshapes B into a vector Matrix inverse ⌹B inv Inverse of B; for vector (conj v)/v+.*conj v Pi times ○B Multiply by π Logarithm ⍟B log Natural logarithm of B Reversal ⌽B rot Reverse elements of B along last axis Reversal ⊖B flip Reverse elements of B along first axis Grade up ⍋B up Indices of B which will arrange B in ascending order Grade down ⍒B down Indices of B which will arrange B in descending order Execute ⍎B ivy Execute an APL (ivy) expression Monadic format ⍕B text A character representation of B Monadic transpose ⍉B transp Reverse the axes of B Factorial !B ! Product of integers 1 to B Bitwise not ^ Bitwise complement of B (integer only) Square root B⋆.5 sqrt Square root of B. Sine sin sin(A); APL uses binary ○ (see below) Cosine cos cos(A); ditto Tangent tan tan(A); ditto Arcsine asin arcsin(B) Arccosine acos arccos(B) Arctangent atan arctan(B) Hyperbolic sine sinh sinh(B) Hyperbolic cosine cosh cosh(B) Hyperbolic tangent tanh tanh(B) Hyperbolic arcsine asinh arcsinh(B) Hyperbolic arccosine acosh arccosh(B) Hyperbolic arctangent atanh arctanh(B) Rotation by 90° j Multiplication by sqrt(-1) Real part real Real component of the value Imaginary part imag Imaginary component of the value Phase phase Phase of the value in the complex plane (-π to π) Conjugate +B conj Complex conjugate of the value System functions ⎕ sys Argument is a string; run "sys 'help'" for details
Name APL Ivy Meaning Add A+B + Sum of A and B Subtract A−B - A minus B Multiply A×B * A multiplied by B Divide A÷B / A divided by B (exact rational division) div A divided by B (Euclidean) idiv A divided by B (Go) Exponentiation A⋆B ** A raised to the B power Circle A○B Trigonometric functions of B selected by A A=1: sin(B) A=2: cos(B) A=3: tan(B); ¯A for inverse sin sin(B); ivy uses traditional name. cos cos(B); ivy uses traditional name. tan tan(B); ivy uses traditional name. Deal A?B ? A distinct integers selected randomly from the first B integers Membership A∈B in 1 for elements of A present in B; 0 where not. Intersection A∩B intersect A with all elements that are also in B removed Union A∪B union A followed by all members of B not already in A Maximum A⌈B max The greater value of A or B Minimum A⌊B min The smaller value of A or B Reshape A⍴B rho Array of shape A with data B Take A↑B take Select the first (or last) A elements of B according to sgn A Drop A↓B drop Remove the first (or last) A elements of B according to sgn A Decode A⊥B decode Value of a polynomial whose coefficients are B at A 'T' decode B creates a seconds value from the time vector B Encode A⊤B encode Base-A representation of the value of B 'T' encode B creates a time vector from the seconds value B Residue A∣B B modulo A mod A modulo B (Euclidean) imod A modulo B (Go) Catenation A,B , Elements of B appended to the elements of A along last axis Catenation A,B ,% Elements of B appended to the elements of A along first axis Expansion A\B fill Insert zeros (or blanks) in B corresponding to zeros in A In ivy: abs(A) gives count, A <= 0 inserts zero (or blank) Compression A/B sel Select elements in B corresponding to ones in A In ivy: abs(A) gives count, A <= 0 inserts zero Index of A⍳B iota The location (index) of B in A; 1+⌈/⍳⍴A if not found In ivy: origin-1 if not found (i.e. 0 if one-indexed) Matrix divide A⌹B mdiv Solution to system of linear equations Ax = B For real vectors, the magnitude of A projected on B Rotation A⌽B rot The elements of B are rotated A positions left Rotation A⊖B flip The elements of B are rotated A positions along the first axis Logarithm A⍟B log Logarithm of B to base A Dyadic format A⍕B text Format B into a character matrix according to A A is the textual format (see format special command); otherwise result depends on length of A: 1 gives decimal count, 2 gives width and decimal count, 3 gives width, decimal count, and style ('d', 'e', 'f', etc.). 'T' text B formats seconds value B as a Unix date General transpose A⍉B transp The axes of B are ordered by A Combinations A!B ! Number of combinations of B taken A at a time Less than A<B < Comparison: 1 if true, 0 if false Less than or equal A≤B <= Comparison: 1 if true, 0 if false Equal A=B == Comparison: 1 if true, 0 if false Greater than or equal A≥B >= Comparison: 1 if true, 0 if false Greater than A>B > Comparison: 1 if true, 0 if false Not equal A≠B != Comparison: 1 if true, 0 if false Or A∨B or Logic: 0 if A and B are 0; 1 otherwise And A∧B and Logic: 1 if A and B are 1; 0 otherwise Nor A⍱B nor Logic: 1 if both A and B are 0; otherwise 0 Nand A⍲B nand Logic: 0 if both A and B are 1; otherwise 1 Xor xor Logic: 1 if A != B; otherwise 0 Bitwise and & Bitwise A and B (integer only) Bitwise or | Bitwise A or B (integer only) Bitwise xor ^ Bitwise A exclusive or B (integer only) Left shift << A shifted left B bits (integer only) Right Shift >> A shifted right B bits (integer only) Complex construction j The complex number A+Bi
Operators and axis indicator
Name APL Ivy APL Example Ivy Example Meaning (of example) Reduce (last axis) / / +/B +/B Sum across B Reduce (first axis) ⌿ /% +⌿B Sum down B Scan (last axis) \ \ +\B +\B Running sum across B Scan (first axis) ⍀ \% +⍀B Running sum down B Inner product . . A+.×B A +.* B Matrix product of A and B Outer product ∘. o. A∘.×B A o.* B Outer product of A and B (lower case o; may need preceding space)
Name APL Ivy Meaning Code code B The integer Unicode value of char B Char char B The character with integer Unicode value B Float float B The floating-point representation of B; for complex numbers, the result is (float A)j(float B)
Pre-defined constants ¶
The constants e (base of natural logarithms) and pi (π) are pre-defined to high precision, about 3000 decimal digits truncated according to the floating point precision setting.
Character data ¶
Strings are vectors of "chars", which are Unicode code points (not bytes). Syntactically, string literals are very similar to those in Go, with back-quoted raw strings and double-quoted interpreted strings. Unlike Go, single-quoted strings are equivalent to double-quoted, a nod to APL syntax. A string with a single char is just a singleton char value; all others are vectors. Thus “, "", and ” are empty vectors, `a`, "a", and 'a' are equivalent representations of a single char, and `ab`, `a` `b`, "ab", "a" "b", 'ab', and 'a' 'b' are equivalent representations of a two-char vector.
Unlike in Go, a string in ivy comprises code points, not bytes; as such it can contain only valid Unicode values. Thus in ivy "\x80" is illegal, although it is a legal one-byte string in Go.
Strings can be printed. If a vector contains only chars, it is printed without spaces between them.
Chars have restricted operations. Printing, comparison, indexing and so on are legal but arithmetic is not, and chars cannot be converted automatically into other singleton values (ints, floats, and so on). The unary operators char and code enable transcoding between integer and char values.
User-defined operators ¶
Users can define unary and binary operators, which then behave just like built-in operators. Both a unary and a binary operator may be defined for the same name.
The syntax of a definition is the 'op' keyword, the operator and formal arguments, an equals sign, and then the body. The names of the operator and its arguments must be identifiers. For unary operators, write "op name arg"; for binary write "op leftarg name rightarg". The final expression in the body is the return value. Operators may have recursive definitions; see the paragraph about conditional execution for an example.
The body may be a single line (possibly containing semicolons) on the same line as the 'op', or it can be multiple lines. For a multiline entry, there is a newline after the '=' and the definition ends at the first blank line (ignoring spaces).
Conditional execution is done with the ":" binary conditional return operator, which is valid only within the code for a user-defined operator. The left operand must be a scalar. If it is non-zero, the right operand is returned as the value of the function. Otherwise, execution continues normally. The ":" operator has a lower precedence than any other operator; in effect it breaks the line into two separate expressions.
Example: average of a vector (unary):
op avg x = (+/x)/rho x avg iota 11 result: 6
Example: n largest entries in a vector (binary):
op n largest x = n take x[down x] 3 largest 7 1 3 24 1 5 12 5 51 result: 51 24 12
Example: multiline operator definition (binary):
op a sum b = a = a+b a iota 3 sum 4 result: 1 2 3 4 5 6 7
Example: primes less than N (unary):
op primes N = (not T in T o.* T) sel T = 1 drop iota N primes 50 result: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Example: greatest common divisor (binary):
op a gcd b = a == b: a a > b: b gcd a-b a gcd b-a 1562 gcd !11 result: 22
On mobile platforms only, due to I/O restrictions, user-defined operators must be presented on a single line. Use semicolons to separate expressions:
op a gcd b = a == b: a; a > b: b gcd a-b; a gcd b-a
To declare an operator but not define it, omit the equals sign and what follows.
op foo x op bar x = foo x op foo x = -x bar 3 result: -3 op foo x = /x bar 3 result: 1/3
Within a user-defined operator body, identifiers are local to the invocation if they are assigned before being read, and global if read before being written. To write to a global without reading it first, insert an unused read.
total = 0 last = 0 op save x = total = total + x # total is global because total is read before written last; last = x # unused read makes last global save 9; save 3 total last result: 12 3
To remove the definition of a unary or binary user-defined operator,
opdelete foo x opdelete a gcd b
Special commands ¶
Ivy accepts a number of special commands, introduced by a right paren at the beginning of the line. Most report the current value if a new value is not specified. For these commands, numbers are always read and printed base 10 and must be non-negative on input.
) help Describe the special commands. Run )help <topic> to learn more about a topic, )help <op> to learn more about an operator. ) base 0 Set the number base for input and output. The commands ibase and obase control setting of the base for input and output alone, respectively. Base 0 allows C-style input: decimal, with 037 being octal and 0x10 being hexadecimal. Bases above 16 are disallowed. To output large integers and rationals, base must be one of 0 2 8 10 16. Floats are always printed base 10. ) cpu Print the duration of the last interactive calculation. ) debug name 0|1 Toggle or set the named debugging flag. With no argument, lists the settings. ) demo Run a line-by-line interactive demo. On mobile platforms, use the Demo menu option instead. ) format "" Set the format for printing values. If empty, the output is printed using the output base. If non-empty, the format determines the base used in printing. The format is in the style of golang.org/pkg/fmt. For floating-point formats, flags and width are ignored. ) get "save.ivy" Read input from the named file; return to interactive execution afterwards. If no file is specified, read from "save.ivy". (Unimplemented on mobile.) ) maxbits 1e6 To avoid consuming too much memory, if an integer result would require more than this many bits to store, abort the calculation. If maxbits is 0, there is no limit; the default is 1e6. ) maxdigits 1e4 To avoid overwhelming amounts of output, if an integer has more than this many digits, print it using the defined floating-point format. If maxdigits is 0, integers are always printed as integers. ) maxstack 1e5 To avoid using too much stack, the number of nested active calls to user-defined operators is limited to maxstack. ) op X If X is absent, list all user-defined operators. Otherwise, show the definition of the user-defined operator X. Inside the definition, numbers are always shown base 10, ignoring the ibase and obase. ) origin 1 Set the origin for indexing a vector or matrix. Must be non-negative. ) prec 256 Set the precision (mantissa length) for floating-point values. The value is in bits. The exponent always has 32 bits. ) prompt "" Set the interactive prompt. ) save "save.ivy" Write definitions of user-defined operators and variables to the named file, as ivy textual source. If no file is specified, save to "save.ivy". (Unimplemented on mobile.) ) seed 0 Set the seed for the ? operator. ) timezone "Local" Set the time zone to be used for display. If the argument is missing, print the name and zone offset in seconds east. ) var X If X is absent, list all defined variables. Otherwise, show the definition of the variable X in a form that can be evaluated to recreate the value.
Package demo implements the I/O for running the )demo special command.
|Package demo implements the I/O for running the )demo special command.|
The mobile package provides a very narrow interface to ivy, suitable for wrapping in a UI for mobile applications.
|The mobile package provides a very narrow interface to ivy, suitable for wrapping in a UI for mobile applications.|
Package run provides the execution control for ivy.
|Package run provides the execution control for ivy.|