Math

65 words. Sorted alphabetically.

-

( a:number b:number -- diff:number )
( a:complex b:complex -- result:complex )
( a:complex b:number -- result:complex )
( a:number b:complex -- result:complex )
( a:complex b:integer -- result:complex )
( a:integer b:complex -- result:complex )
( a:vec3 b:vec3 -- result:vec3 )
( v:vec3 n:number -- result:vec3 )
( v:vec3 n:integer -- result:vec3 )
( n:number v:vec3 -- result:vec3 )
( n:integer v:vec3 -- result:vec3 )
( a:vector b:vector -- diff:vector )
( a:matrix b:matrix -- diff:matrix )
( a:mx3 b:mx3 -- diff:mx3 )
( a:quantity b:quantity -- diff:quantity )
( q:quantity n:number -- diff:quantity )
( n:number q:quantity -- diff:quantity )
( int frac )
( double frac )
( frac int )
( frac double )
( frac frac )
( hms hms -- hms )
( dms dms -- dms )
( hms qtime -- quantity )
( qtime hms -- quantity )
( dms qangle -- quantity )
( qangle dms -- quantity )
( symbolic symbolic )

Quantity subtraction. Q-Q auto-converts; Q-number and number-Q promote the bare to Q's display unit. Offset units refuse the promotion form.

->COMPLEX

( double double )
( integer double )
( double integer )
( integer integer )

Create a complex number from real (NOS) and imaginary (TOS) parts.

->EXPR

( string )

->POLAR

( m:number a:number -- z:complex )

Construct a Complex from magnitude (NOS) and angle (TOS). Angle interpreted in current angle mode.

!

( n:number -- result:number )

Factorial (generalized). n! = Gamma(n+1). Works for real n.

*

( a:number b:number -- product:number )
( a:complex b:complex -- result:complex )
( a:complex b:number -- result:complex )
( a:number b:complex -- result:complex )
( a:complex b:integer -- result:complex )
( a:integer b:complex -- result:complex )
( v:vector s:number -- scaled:vector )
( v:vector s:integer -- scaled:vector )
( s:number v:vector -- scaled:vector )
( s:integer v:vector -- scaled:vector )
( a:matrix b:matrix -- product:matrix )
( m:matrix v:vector -- result:vector )
( m:matrix s:number -- scaled:matrix )
( m:matrix s:integer -- scaled:matrix )
( s:number m:matrix -- scaled:matrix )
( s:integer m:matrix -- scaled:matrix )
( v:vec3 s:number -- scaled:vec3 )
( v:vec3 s:integer -- scaled:vec3 )
( s:number v:vec3 -- scaled:vec3 )
( s:integer v:vec3 -- scaled:vec3 )
( m:mx3 v:vec3 -- result:vec3 )
( a:mx3 b:mx3 -- product:mx3 )
( m:mx3 s:number -- scaled:mx3 )
( m:mx3 s:integer -- scaled:mx3 )
( s:number m:mx3 -- scaled:mx3 )
( s:integer m:mx3 -- scaled:mx3 )
( a:quantity b:quantity -- product:quantity )
( q:quantity n:number -- scaled:quantity )
( n:number q:quantity -- scaled:quantity )
( int frac )
( double frac )
( frac int )
( frac double )
( frac frac )
( hms double -- hms )
( double hms -- hms )
( hms integer -- hms )
( integer hms -- hms )
( dms double -- dms )
( double dms -- dms )
( dms integer -- dms )
( integer dms -- dms )
( symbolic symbolic )

Quantity multiplication. Compound dimensions/symbols; refuses offset units.

/

( a:number b:number -- quotient:number )
( a:complex b:complex -- result:complex )
( a:complex b:number -- result:complex )
( a:number b:complex -- result:complex )
( a:complex b:integer -- result:complex )
( a:integer b:complex -- result:complex )
( m:mx3 s:number -- result:mx3 )
( m:mx3 s:integer -- result:mx3 )
( s:number m:mx3 -- result:mx3 )
( s:integer m:mx3 -- result:mx3 )
( a:quantity b:quantity -- quotient:quantity )
( q:quantity n:number -- scaled:quantity )
( n:number q:quantity -- inverse:quantity )
( int frac )
( double frac )
( frac int )
( frac double )
( frac frac )
( hms double -- hms )
( hms integer -- hms )
( hms hms -- double )
( dms double -- dms )
( dms integer -- dms )
( dms dms -- double )
( symbolic symbolic )

Quantity division. Compound dimensions/symbols; refuses offset units and divide-by-zero.

^

( base:number exponent:number -- result:number )
( base:complex exponent:complex -- result:complex )
( base:complex exponent:number -- result:complex )
( base:number exponent:complex -- result:complex )
( base:complex exponent:integer -- result:complex )
( base:integer exponent:complex -- result:complex )
( q:quantity n:integer -- q':quantity )
( int frac )
( double frac )
( frac int )
( frac double )
( frac frac )
( symbolic symbolic )

Quantity exponentiation (integer exponent only). Refuses offset units and non-integer exponents.

+

( a:number b:number -- sum:number )
( a:complex b:complex -- result:complex )
( a:complex b:number -- result:complex )
( a:number b:complex -- result:complex )
( a:complex b:integer -- result:complex )
( a:integer b:complex -- result:complex )
( object:object key:string value:any -- object:object )
( value:any key:string object:object -- object:object )
( array:array item:any -- array:array )
( item:any array:array -- array:array )
( a:vec3 b:vec3 -- result:vec3 )
( v:vec3 n:number -- result:vec3 )
( v:vec3 n:integer -- result:vec3 )
( n:number v:vec3 -- result:vec3 )
( n:integer v:vec3 -- result:vec3 )
( a:vector b:vector -- sum:vector )
( a:matrix b:matrix -- sum:matrix )
( a:mx3 b:mx3 -- sum:mx3 )
( a:quantity b:quantity -- sum:quantity )
( q:quantity n:number -- sum:quantity )
( n:number q:quantity -- sum:quantity )
( int frac )
( double frac )
( frac int )
( frac double )
( frac frac )
( hms hms -- hms )
( dms dms -- dms )
( hms qtime -- quantity )
( qtime hms -- quantity )
( dms qangle -- quantity )
( qangle dms -- quantity )
( symbolic symbolic )

Quantity addition. Q+Q auto-converts right to left's units; Q+number and number+Q promote the bare to Q's display unit. Offset units refuse the promotion form.

ABS

( z:complex -- magnitude:number )
( a:number -- result:number )
( hms -- hms )
( dms -- dms )
( symbolic )

Absolute value.

ACOS

( z:complex -- angle:complex )
( x:number -- angle:number )
( symbolic )

Arccosine. Result in current angle mode.

ACOSH

( z:complex -- result:complex )
( x:number -- result:number )

Inverse hyperbolic cosine. Domain: x >= 1.

ARG

( z:complex -- angle:number )

Argument (angle) of a complex in current angle mode.

ASIN

( z:complex -- angle:complex )
( x:number -- angle:number )
( symbolic )

Arcsine. Result in current angle mode.

ASINH

( z:complex -- result:complex )
( x:number -- result:number )

Inverse hyperbolic sine.

ATAN

( z:complex -- angle:complex )
( x:number -- angle:number )
( symbolic )

Arctangent. Result in current angle mode.

ATAN2

( x:number y:number -- angle:number )
( symbolic symbolic )

Four-quadrant arctangent of (x, y). TOS=y, NOS=x. Result in current angle mode.

ATANH

( z:complex -- result:complex )
( x:number -- result:number )

Inverse hyperbolic arctangent. Domain: |x| < 1.

CEIL

( a:number -- result:number )
( symbolic )

Round up to nearest integer value (result is double).

CHS

( a:number -- result:number )

Change sign: negate TOS.

CONJ

( z:complex -- result:complex )

Complex conjugate: (a+bi) → (a-bi).

COS

( angle:complex -- result:complex )
( angle:number -- result:number )
( angle:quantity -- result:number )
( symbolic )

Cosine. Angle interpreted in current mode (DEG / RAD / GRAD).

COSH

( z:complex -- result:complex )
( x:number -- result:number )

Hyperbolic cosine.

CUBIC

( number number number number -- array )

( a b c d -- roots-array ) Solve ax^3 + bx^2 + cx + d = 0 via Cardano's formula. Returns 3 roots (Real ascending then Complex).

D->R

( degrees:number -- radians:number )

Convert degrees to radians.

DEGREE

( mathexpr name -- integer )

( mathexpr target-name -- degree ) Polynomial degree of the equation in target-name IF polynomial-normal; Integer(-1) otherwise. No CAS distribution.

EXP

( z:complex -- result:complex )
( a:number -- result:number )
( symbolic )

e raised to the power of TOS.

FACT

( n:number -- result:number )

Factorial. Alias for !.

FLOOR

( a:number -- result:number )
( symbolic )

Round down to nearest integer value (result is double).

GAMMA

( a:number -- result:number )

Gamma function Γ(x). Generalizes factorial: n! = GAMMA(n+1).

GCD

( a:number b:number -- result:number )

Greatest common divisor (Euclidean algorithm).

HYPOT

( a:number b:number -- c:number )

Euclidean distance: sqrt(a² + b²).

IM

( z:complex -- im:number )

Extract the imaginary part of a complex.

INV

( z:complex -- result:complex )
( a:number -- result:number )
( m:matrix -- inverse:matrix )
( m:mx3 -- inverse:mx3 )
( f:fraction -- inv:fraction )
( symbolic )

Reciprocal or inverse — overloaded by type (1/x for numbers, fraction reciprocal, matrix / Mx3 inverse).

LCM

( a:number b:number -- result:number )

Least common multiple.

LET

( symbolic symbolic symbolic )

LGAMMA

( a:number -- result:number )

Natural logarithm of the gamma function ln Γ(x). Useful for large arguments.

LHS

( mathexpr -- mathexpr )

( mathexpr -- lhs-mathexpr ) Left-hand side of an Equal-rooted MathExpr. Non-Equal roots return the whole expression (implicit = 0 model).

LN

( z:complex -- result:complex )
( a:number -- result:number )
( symbolic )

Natural logarithm (base e).

LOG10

( z:complex -- result:complex )
( a:number -- result:number )
( symbolic )

Logarithm base 10.

LOG2

( a:number -- result:number )
( symbolic )

Logarithm base 2.

MAG

( z:complex -- magnitude:number )

Magnitude |z| of a complex.

MAX

( a:number b:number -- max:number )
( symbolic symbolic )

Return the larger of two values.

MIN

( a:number b:number -- min:number )
( symbolic symbolic )

Return the smaller of two values.

MOD

( a:number b:number -- remainder:number )

Modulo: NOS mod TOS. Integer or double.

nCr

( n:number r:number -- result:number )

Combinations: n choose r = n! / (r! * (n-r)!).

nPr

( n:number r:number -- result:number )

Permutations: n P r = n! / (n-r)!.

QUAD

( double double double )
( integer double double )
( double integer double )
( integer integer double )
( double double integer )
( integer double integer )
( double integer integer )
( integer integer integer )
( symbolic symbolic symbolic )

Solve quadratic a·x²+b·x+c=0. Pops a, b, c; pushes two roots (real or complex).

QUARTIC

( number number number number number -- array )

( a b c d e -- roots-array ) Solve ax^4 + bx^3 + cx^2 + dx + e = 0 via Ferrari's method. Returns 4 roots (Real ascending then Complex).

R->D

( radians:number -- degrees:number )

Convert radians to degrees.

RE

( z:complex -- re:number )

Extract the real part of a complex.

RHS

( mathexpr -- mathexpr )

( mathexpr -- rhs-mathexpr ) Right-hand side of an Equal-rooted MathExpr. Non-Equal roots return Value(Integer(0)) — the implicit RHS in the = 0 model.

ROUND

( a:number -- result:number )
( symbolic )

Round to nearest integer value (result is double).

SIGN

( a:number -- sign:number )

SIN

( angle:complex -- result:complex )
( angle:number -- result:number )
( angle:quantity -- result:number )
( symbolic )

Sine. Angle interpreted in current mode (DEG / RAD / GRAD).

SINH

( z:complex -- result:complex )
( x:number -- result:number )

Hyperbolic sine.

SOLUTION

( mathexpr -- any )
( mathexpr name -- any )
( mathexpr object -- any )
( mathexpr name object -- any )

Same as SOLVE but returns Object { ...bindings, target: value } — the complete picture. Auto-degrade case adds __satisfied__ field.

SOLVE

( mathexpr -- any )
( mathexpr name -- any )
( mathexpr object -- any )
( mathexpr name object -- any )

Solve an equation MathExpr for its sole unknown; return the bare value (or Array for multi-root). Four arities: ( eq -- ), ( eq name -- ), ( eq obj -- ), ( eq name obj -- ). Arity-2-obj auto-degrades to Boolean when all bound. See SOLUTION for the Object shape and SOLVE-EQ for the equation shape.

SOLVE-EQ

( mathexpr -- any )
( mathexpr name -- any )
( mathexpr object -- any )
( mathexpr name object -- any )

Same as SOLVE but returns MathExpr(target = value) — the equation form. Chains into further MathExpr work (LHS, RHS, DEPARSE, SUBST). Auto-degrade case returns MathExpr(target = <same-bindings-object>) — not typically useful; use SOLVE for that case.

SOLVE-RESID

( mathexpr object -- number )
( mathexpr -- number )

( eq [bindings] -- residual ) Evaluate lhs - rhs (or the whole expression if root is non-Equal) at the given bindings. Not a solver — no iteration, no target inference. All free variables must be resolvable.

SQ

( z:complex -- result:complex )
( n:number -- result:number )
( f:fraction -- sq:fraction )

Square a fraction.

SQRT

( z:complex -- result:complex )
( a:number -- result:number )
( f:fraction -- root:fraction )
( symbolic )

Square root of a fraction (returns a fraction approximation).

SYMBOLS

( mathexpr -- array )

( mathexpr -- names-array ) Free-Name set of the MathExpr with Constants filtered out. <expr> SYMBOLS SIZE 1_ == is the arity-1 SOLVE precondition idiom (Integer literal — SIZE returns Integer).

TAN

( angle:complex -- result:complex )
( angle:number -- result:number )
( angle:quantity -- result:number )
( symbolic )

Tangent. Angle interpreted in current mode (DEG / RAD / GRAD).

TANH

( z:complex -- result:complex )
( x:number -- result:number )

Hyperbolic tangent.