## Overview ¶

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.

Indexing uses [] notation: x[1], x[1][2], and so on. Indexing by a vector selects multiple elements: x[1 2] creates a new item from x[1] and x[2].

Only a subset of APL's functionality is implemented, but the intention is to have most numerical operations supported eventually.

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.

Unary operators

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 Floor ⌊B floor Greatest integer less than or equal to B Shape ⍴B rho Number of components in each dimension of 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 Exponential ⋆B ** e to the B power Negation −B - Changes sign of B Identity +B + No change to B Signum ×B sgn ¯1 if B<0; 0 if B=0; 1 if B>0 Reciprocal ÷B / 1 divided by B Ravel ,B , Reshapes B into a vector Matrix inverse ⌹B Inverse of matrix B 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

Binary operators

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. asin arcsin(B); ivy uses traditional name. acos arccos(B); ivy uses traditional name. atan arctan(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. 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 ×A Drop A↓B drop Remove the first (or last) A elements of B according to ×A Decode A⊥B decode Value of a polynomial whose coefficients are B at A Encode A⊤B encode Base-A representation of the value of 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 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 Solution to system of linear equations Ax = 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 ('e', 'f', 'g'). General transpose A⍉B 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)

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)

Type-converting operations

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

### 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, but since there are no conditional or looping constructs (yet), such operators are problematic when executed.

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).

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

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, identifiers are local to the invocation unless they are undefined in the operator but defined globally, in which case they refer to the global variable. A mechanism to declare locals may come later.

### 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 Print this list of special commands. ) 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. If the base is greater than 10, any identifier formed from valid numerals in the base system, such as abe for base 16, is taken to be a number. TODO: 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. Requires a Go installation. ) 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. ) 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. ) 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.