Functions
number	log1p (numeric x)
	$f(x) = \ln{\left(1+x\right)}$

number	expm1 (numeric x)
	$f(x) = e^x - 1$

number	xlogx (numeric x)
	$f(x) = x \ln{\left(x\right)}$

number	xlogy (numeric x, numeric y)
	$f(x, y) = x \ln{\left(y\right)}$

number	xlog1py (numeric x, numeric y)
	$f(x, y) = x \ln{\left(1 + y\right)}$

number	xexpx (numeric x)
	$f(x) = x e^x$

number	xexpy (numeric x, numeric y)
	$f(x, y) = x e^y$

number	logistic (numeric x)
	Standard logistic function or expit function $\mathcal{L}$.

number	logit (numeric x)
	Logit function or log-odds function. Inverse of the standard logistic function $\sigma = \mathcal{L}^{-1}$.

number	logcosh (numeric x)
	$f(x) = \ln{\left(\cosh{(x)}\right)}$

number	log1psq (numeric x)
	$f(x) = \ln{\left(1 + x^2\right)}$

number	log1pexp (numeric x)
	$f(x) = \ln{\left(1 + e^x\right)}$

number	log1pmx (numeric x)
	$f(x) = \ln{\left(1 + x\right)} - x$

number	logmxp1 (numeric x)
	$f(x) = \ln{\left(x\right)} -x + 1$

number	log1mexp (numeric x)
	$f(x) = \ln{\left(1 - e^x\right)}$

number	log2mexp (numeric x)
	$f(x) = \ln{\left(2 - e^x\right)}$

number	cloglog (numeric x)
	Complementary log-log function.

number	cexpexp (numeric x)
	Complementary double exponential function.

number	eq (any x, any y)
	Equality operator `==`. Mathematical equality relation for numeric values $=$.

number	ne (any x, any y)
	Inequality operator `!=`. Mathematical inequality relation for numeric values $\ne$.

number	lt (any x, any y)
	Less than operator `<`. Mathematical less-than relation for numeric values $\lt$.

number	le (any x, any y)
	Less than or equal operator `<=`. Mathematical less-than or equal relation for numeric values $\leq$.

number	gt (any x, any y)
	Greater than operator `>`. Mathematical greater-than relation for numeric values $\gt$.

number	ge (any x, any y)
	Greater than or equal operator `>=`. Mathematical greater-than or equal relation for numeric values $\geq$.

number	rematch (any x, any y)
	Regular expression matching operator `~`

number	renomatch (any x, any y)
	Negated regular expression matching operator `!~`

number	is (any x, any y)
	Identity operator: Equality in value and type.

number	ni (any x, any y)
	Non-identity operator: Inequality in value or type.

bool	eqBool (any x, any y)
	Equality operator `==` that returns a `number\|bool`

bool	neBool (any x, any y)
	Equality operator `!=` that returns a `number\|bool`

bool	ltBool (any x, any y)
	Equality operator `<` that returns a `number\|bool`

bool	leBool (any x, any y)
	Equality operator `<=` that returns a `number\|bool`

bool	gtBool (any x, any y)
	Equality operator `>` that returns a `number\|bool`

bool	geBool (any x, any y)
	Equality operator `>=` that returns a `number\|bool`

bool	rematchBool (any x, any y)
	Equality operator `~` that returns a `number\|bool`

bool	renomatchBool (any x, any y)
	Equality operator `!~` that returns a `number\|bool`

auto	eqCollate (x, y)
	Check if strings collate equally.

number	cmp (any x, any y)
	Comparator or "spaceship operator".

number	add (numeric x, numeric y)
	Addition operator `+`. Arithmetic addition for numeric values $+$.

number	subt (numeric x, numeric y)
	Subtraction operator `-`. Arithmetic subtraction for numeric values $-$.

number	mul (numeric x, numeric y)
	Multiplication operator `*`. Arithmetic multiplication for numeric values $\times$.

number	div (numeric x, numeric y)
	Division operator `/`. Arithmetic division for numeric values $\div$.

number	rdiv (numeric x, numeric y)
	Right division operator. Arithmetic division for numeric values $\div$ with operands in reversed order.

number	mod (numeric x, numeric y)
	Modulo or remainder operator `%`. Binary operation for numeric values $mod$.

number	pow (numeric x, numeric y)
	Exponentiation operator `^`. Arithmetic exponentiation for numeric values $x^y$.

number	floorDiv (numeric x, numeric y)
	Floor division.

number	ln (numeric x)
	Natural logarithm (base $e$) function $\ln$ or $\log_e$.

number	logb (numeric x, numeric b)
	Logarithm function $\log_b$.

number	log10 (numeric x)
	Common logarithm (base 10) function $\text{lg}$ or $\log_{10}$.

number	log2 (numeric x)
	Binary logarithm (base 2) function $\text{lb}$ or $\log_2$.

number	cbrt (numeric x)
	Cube root function $\sqrt[3]{\phantom{x}}$.

number	root (numeric x, numeric n)
	Nth root function $\sqrt[n]{\phantom{x}}$.

number	abs (numeric x)
	Absolute value function $\|\phantom{x}\|$.

number	abs2 (numeric x)
	Absolute value squared function $\|\phantom{x}\|^2$.

number	floor (numeric x)
	Floor function $\lfloor \phantom{x} \rfloor$.

number	ceil (numeric x)
	Ceil function $\lceil \phantom{x} \rceil$.

number	frac (numeric x)
	Fractional or decimal part function $\{~~\}$.

unassigned	modf (array A, numeric x)
	Modulus fraction function.

number	round (numeric x)
	Round function.

number	roundUp (numeric x, numeric y)
	Round to next multiple of.

number	roundDown (numeric x, numeric y)
	Round to previous multiple of.

number	max (any x, any y)
	Maximum function $\max$.

number	min (any x, any y)
	Minimum function $\min$.

unassigned	minmax (array A, any x, any y)
	Min-max function $\text{minmax}$.

number	muladd (numeric x, numeric y, numeric z)
	Multiply then add operation (MAD)

number	isBetween (numeric a, numeric x, numeric b, bool eq)
	Check if `x` is bounded between `a` and `b`

number	approx (numeric v1, numeric v2, numeric rt=0.01, numeric at=0.00)
	Approximately function $\approx$.

number	sgn (numeric x)
	Sign function (or signum) $sgn$.

number	signbit (numeric x)
	Signbit function.

number	flipsign (numeric x, numeric y)
	Flip the sign of `x` if `y` is negative.

number	copysign (numeric x, numeric y)
	Copy the sign of `y` into `x`

number	triangular (numeric x)
	Triangular function. $x?$.

number	factorial (numeric x)
	Factorial function. $x!$.

number	sumSeq (numeric x, numeric y)
	Sum of sequence function. $\sum$.

number	prodSeq (numeric x, numeric y)
	Product of sequence function. $\prod$.

number	hypot (numeric x, numeric y, numeric z)
	Hypotenuse function (or euclidean distance function, or polar length, or pythagorean addition) $\oplus$.

number	gcd (numeric x, numeric y)
	Greatest common denominator function $\gcd$.

number	lcm (numeric x, numeric y)
	Least common multiple function $\text{lcm}$.

number	sine (numeric x)
	Sine function $\sin$.

number	cosine (numeric x)
	Cosine function $\cos$.

number	tan (numeric x)
	Tangent function $\tan$.

number	cot (numeric x)
	Cotangent function $\cot$.

number	sec (numeric x)
	Secant function $\sec$.

number	csc (numeric x)
	Cosecant function $\csc$.

number	asin (numeric x)
	Arc sine function. Inverse of the sine function. $\arcsin = \sin^{-1}$.

number	acos (numeric x)
	Arc cosine function. Inverse of the cosine function. $\arccos = \cos^{-1}$.

number	atan (numeric x)
	Arc tangent function. Inverse of the tangent function. $\arctan = \tan^{-1}$.

number	acot (numeric x)
	Arc cotangent function. Inverse of the cotangent function. $\text{arccot} = \cot^{-1}$.

number	asec (numeric x)
	Arc secant function. Inverse of the secant function. $\text{arcsec} = \sec^{-1}$.

number	acsc (numeric x)
	Arc cosecant function. Inverse of the cosecant function. $\text{arccsc} = \csc^{-1}$.

number	sinh (numeric x)
	Hyperbolic sine function $\sinh$.

number	cosh (numeric x)
	Hyperbolic cosine function $\cosh$.

number	tanh (numeric x)
	Hyperbolic tangent function $\tanh$.

number	coth (numeric x)
	Hyperbolic cotangent function $\coth$.

number	sech (numeric x)
	Hyperbolic secant function $\text{sech}$.

number	csch (numeric x)
	Hyperbolic cosecant function $\text{csch}$.

number	asinh (numeric x)
	Arc hyperbolic sine function. Inverse of the hyperbolic sine function. $\text{arcsinh} = \sinh^{-1}$.

number	acosh (numeric x)
	Arc hyperbolic cosine function. Inverse of the hyperbolic cosine function. $\text{arccosh} = \cosh^{-1}$.

number	atanh (numeric x)
	Arc hyperbolic tangent function. Inverse of the hyperbolic tangent function. $\text{arctanh} = \tanh^{-1}$.

number	acoth (numeric x)
	Arc hyperbolic cotangent function. Inverse of the hyperbolic cotangent function. $\text{arccoth} = \coth^{-1}$.

number	asech (numeric x)
	Arc hyperbolic secant function. Inverse of the hyperbolic sine function. $\text{arcsech} = \text{sech}^{-1}$.

number	acsch (numeric x)
	Arc hyperbolic cosecant function. Inverse of the hyperbolic sine function. $\text{arccsch} = \text{csch}^{-1}$.

Variables
number	e = 2.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713821785251664274
	$e$ number. The natural rate of exponential growth $f(x) = f'(x) = e^x$. $$e = \lim\limits_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$$

number	pi = 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679
	$\pi$ number. The ratio of a circle's circumference to its diameter. $\pi$ radians represent a half rotation ( $180°$). $$\pi = \frac{C}{d} = 2 \prod _{n=1}^{\infty}{ \left(1 + \frac{1}{n}\right)^{\left(-1\right)^{n+1}} }$$

number	tau = 6.283185307179586476925286766559005768394338798750211641949889184615632812572417997256069650684234136
	$\tau$ number. The ratio of a circle's circumference to its radius. $\tau$ radians represent a full rotation ( $360°$). $$\tau = \frac{C}{r} = 2\pi$$

number	sqrt2 = 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
	$\sqrt{2}$. Square root of 2

number	sqrt3 = 1.7320508075688772935274463415058723669428052538103806280558069794519330169088000370811461867572485756
	$\sqrt{3}$. Square root of 3

number	sqrt5 = 2.2360679774997896964091736687312762354406183596115257242708972454105209256378048994144144083787822749
	$\sqrt{5}$. Square root of 5

number	sqrt6 = 2.4494897427831780981972840747058913919659474806566701284326925672509603774573150265398594331046402348
	$\sqrt{6}$. Square root of 6

number	sqrt7 = 2.6457513110645905905016157536392604257102591830824501803683344592010688232302836277603928864745436106
	$\sqrt{7}$. Square root of 7

number	sqrt8 = 2.8284271247461900976033774484193961571393437507538961463533594759814649569242140777007750686552831454
	$\sqrt{8}$. Square root of 8

number	sqrt10 = 3.1622776601683793319988935444327185337195551393252168268575048527925944386392382213442481083793002951
	$\sqrt{10}$. Square root of 10

number	sqrt11 = 3.3166247903553998491149327366706866839270885455893535970586821461164846426090438467088433991282906509
	$\sqrt{11}$. Square root of 11

number	sqrt12 = 3.4641016151377545870548926830117447338856105076207612561116139589038660338176000741622923735144971513
	$\sqrt{12}$. Square root of 12

number	sqrt13 = 3.6055512754639892931192212674704959462512965738452462127104530562271669482930104452046190820184907176
	$\sqrt{13}$. Square root of 13

number	cbrt2 = 1.2599210498948731647672106072782283505702514647015079800819751121552996765139594837293965624362550941
	$\sqrt[3]{2}$. Cube root of 2

number	cbrt3 = 1.4422495703074083823216383107801095883918692534993505775464161945416875968299973398547554797056452566
	$\sqrt[3]{3}$. Cube root of 3

number	cbrt4 = 1.5874010519681994747517056392723082603914933278998530098082857618252165056242191732735442132622209570
	$\sqrt[3]{4}$. Cube root of 4

number	cbrt5 = 1.7099759466766969893531088725438601098680551105430549243828617074442959205041732162571870100201890022
	$\sqrt[3]{5}$. Cube root of 5

number	cbrt6 = 1.8171205928321396588912117563272605024282104631412196714813342979313097394593018656471417041264170721
	$\sqrt[3]{6}$. Cube root of 6

number	cbrt7 = 1.9129311827723891011991168395487602828624390503458757662106476404472342761792307560075254414772857099
	$\sqrt[3]{7}$. Cube root of 7

number	cbrt9 = 2.0800838230519041145300568243578853863378053403732621096975910802001063113972687736060566367907574867
	$\sqrt[3]{8}$. Cube root of 9

number	cbrt10 = 2.1544346900318837217592935665193504952593449421921085824892355063464111066483408001854415035432432761
	$\sqrt[3]{10}$. Cube root of 10

number	lnhalf = -0.6931471805599453094172321214581765680755001343602552541206800094933936219696947156058633269964186875
	$\ln\left(\frac{1}{2}\right)$. Natural logarithm of $\frac{1}{2}$

number	ln2 = 0.6931471805599453094172321214581765680755001343602552541206800094933936219696947156058633269964186875
	$\ln(2)$. Natural logarithm of 2

number	ln3 = 1.0986122886681096913952452369225257046474905578227494517346943336374942932186089668736157548137320887
	$\ln(3)$. Natural logarithm of 3

number	ln4 = 1.3862943611198906188344642429163531361510002687205105082413600189867872439393894312117266539928373750
	$\ln(4)$. Natural logarithm of 4

number	ln5 = 1.6094379124341003746007593332261876395256013542685177219126478914741789877076577646301338780931796107
	$\ln(5)$. Natural logarithm of 5

number	ln10 = 2.302585092994045684017991454684364207601101488628772976033327900967572609677352480235997205089598298
	$\ln(10)$. Natural logarithm of 10

number	phi = 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374
	$\varphi = \frac{1 + \sqrt{5}}{2}$

number	psi = -0.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374
	$\psi = \frac{1 - \sqrt{5}}{2} = - \varphi^{-1}$

number	invphi = -psi
	$\varphi^{-1} = \frac{-1 + \sqrt{5}}{2} = -\psi$

number	invpsi = -phi
	$\psi^{-1} = \frac{-1 - \sqrt{5}}{2} = -\varphi$

number	golden = phi
	Golden ratio $\varphi$. 1° metallic mean. $\frac{1 + \sqrt{5}}{2}$. Continued fraction $[1;\overline{1}]$ and positive solution to the equation $x^2 = x + 1$.

number	silver = 2.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
	Silver ratio. 2° metallic mean. $1 + \sqrt{2}$. Continued fraction $[2;\overline{2}]$ and positive solution to the equation $x^2 = 2x + 1$.

number	bronze = 3.302775637731994646559610633735247973125648286922623106355226528113583474146505222602309541009245359
	Bronze ratio. 3° metallic mean. $\frac{3 + \sqrt{13}}{2}$. Continued fraction $[3;\overline{3}]$ and positive solution to the equation $x^2 = 3x + 1$.

number	copper = 4.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
	Copper ratio. 4° metallic mean. $2 + \sqrt{5}$. Continued fraction $[4;\overline{4}]$ and positive solution to the equation $x^2 = 4x + 1$.

number	nickel = 5.192582403567252015625355245770164778147560080822394418840194335008322981413829346438316890839917742
	Nickel ratio. 5° metallic mean. $\frac{5 + \sqrt{29}}{2}$. Continued fraction $[5;\overline{5}]$ and positive solution to the equation $x^2 = 5x + 1$.

number	super_golden = 1.4655712318767680266567312252199391080255775684722857016431831112492629966850178404781258011949092701
	Supergolden ratio. Real solution to the equation $x^3 = x^2 + 1$.

number	plastic = 1.324717957244746025960908854478097340734404056901733364534015050302827851245547594054699347981787280
	Plastic ratio. Real solution to the equation $x^3 = x + 1$.

number	tribonacci = 1.839286755214161132551852564653286600424178746097592246778758639404203222081966425738435419428307014
	Tribonacci constant. Real solution to the equation $x^3 = x^2 + x + 1$.

number	lemniscate = 2.622057554292119810464839589891119413682754951431623162816821703800790587070414250230295532961429093
	$\varpi$ Lemniscate constant. The ratio of a Bernoulli's lemniscate's perimeter to its diameter. $$\varpi = \frac{p}{d} = 2 \prod _{n=1}^{\infty}{ \left(1 + \frac{1}{2n}\right)^{\left(-1\right)^{n+1}} }$$

number	parabolic = 2.295587149392638074034298049189490387597832203638583483929975346644109662684133126684094426237897616
	$P$ Universal parabolic constant. The ratio of a parabola's latus rectum arc to its semi latus rectum. $$P = \sqrt{2} + \ln{\left(1 + \sqrt{2}\right)}$$

number	gauss = 0.8346268416740731862814297327990468089939930134903470024498273701036819927095264118696911603512753241
	$G$ Gauss constant. $$G = \frac{\varpi}{\pi}$$

number	golden_angle = 2.399963229728653322231555506633613853124999011058115042935112750731307338239438790779962060660583964
	Golden angle in radians $g$. The smaller angle that sections a circles's circumference according to the golden ratio $\varphi$.

number	euler_gamma = 0.57721566490153286060651209
	Euler–Mascheroni constant.

number	catalan = 0.9159655941772190150546035149323841107741934
	Catalan constant.

number	twopi = tau
	$2\pi = \tau$

number	fourpi = 12.56637061435917295385057353311801153678867759750042328389977836923126562514483599451213930136846827
	$4\pi$

number	halfpi = 1.570796326794896619231321691639751442098584699687552910487472296153908203143104499314017412671058534
	$\frac{\pi}{2}$

number	quartpi = 0.7853981633974483096156608458198757210492923498437764552437361480769541015715522496570087063355292670
	$\frac{\pi}{4}$

number	invpi = 0.3183098861837906715377675267450287240689192914809128974953346881177935952684530701802276055325061719
	$\frac{1}{\pi}$

number	twoinvpi = 0.6366197723675813430755350534900574481378385829618257949906693762355871905369061403604552110650123438
	$\frac{2}{\pi}$

number	fourinvpi = 1.273239544735162686151070106980114896275677165923651589981338752471174381073812280720910422130024688
	$\frac{4}{\pi}$

number	inv2pi = 0.1591549430918953357688837633725143620344596457404564487476673440588967976342265350901138027662530860
	$\frac{1}{2\pi}$

number	inv4pi = 0.07957747154594766788444188168625718101722982287022822437383367202944839881711326754505690138312654298
	$\frac{1}{4\pi}$

number	sqrtpi = 1.772453850905516027298167483341145182797549456122387128213807789852911284591032181374950656738544665
	$\sqrt{\pi}$

number	sqrt2pi = 2.506628274631000502415765284811045253006986740609938316629923576342293654607841974946595838378057266
	$\sqrt{2\pi}$

number	sqrt4pi = 3.544907701811032054596334966682290365595098912244774256427615579705822569182064362749901313477089331
	$\sqrt{4\pi}$

number	sqrthalfpi = 1.253314137315500251207882642405522626503493370304969158314961788171146827303920987473297919189028633
	$\sqrt{\frac{\pi}{2}}$

number	invsqrt2 = 0.7071067811865475244008443621048490392848359376884740365883398689953662392310535194251937671638207864
	$\frac{1}{\sqrt{2}}$

number	invsqrtpi = 0.5641895835477562869480794515607725858440506293289988568440857217106424684414934144867436602021073634
	$\frac{1}{\sqrt{\pi}}$

number	invsqrt2pi = 0.3989422804014326779399460599343818684758586311649346576659258296706579258993018385012523339073069364
	$\frac{1}{\sqrt{2\pi}}$

number	lnpi = 1.144729885849400174143427351353058711647294812915311571513623071472137769884826079783623270275489708
	$\ln{(\pi)}$. Natural logarithm of $\pi$

number	ln2pi = 1.837877066409345483560659472811235279722794947275566825634303080965531391854520795389486597271908395
	$\ln{(2\pi)}$. Natural logarithm of $2\pi$

number	ln4pi = 2.531024246969290792977891594269411847798295081635822079754983090458925013824215510995349924268327083
	$\ln{(4\pi)}$. Natural logarithm of $4\pi$

Function Documentation

◆ abs()

number math::abs ( numeric x )

Absolute value function $|\phantom{x}|$.

Parameters

x Number

Returns: Absolute value $|x|$

◆ abs2()

number math::abs2 ( numeric x )

Absolute value squared function $|\phantom{x}|^2$.

Parameters

x Number

Returns: Squared of the absolute value $|x|^2$

◆ acos()

number math::acos ( numeric x )

Arc cosine function. Inverse of the cosine function. $\arccos = \cos^{-1}$.

Parameters

x	Cosine value. Domain: $\left[-1, 1\right]$

Returns: Angle in radians in range $\left[0, π\right]$ whose cosine is x

◆ acosh()

number math::acosh ( numeric x )

Arc hyperbolic cosine function. Inverse of the hyperbolic cosine function. $\text{arccosh} = \cosh^{-1}$.

Parameters

x	Hyperbolic cosine value. Domain: $\left[1, \infty\right)$

Returns: Hyperbolic angle in range $\left[0, \infty\right)$ whose hyperbolic cosine is x

◆ acot()

number math::acot ( numeric x )

Arc cotangent function. Inverse of the cotangent function. $\text{arccot} = \cot^{-1}$.

Parameters

x	Cotangent value

Returns: Angle in radians in range $\left[0, π\right]$ whose cotangent is x

◆ acoth()

number math::acoth ( numeric x )

Arc hyperbolic cotangent function. Inverse of the hyperbolic cotangent function. $\text{arccoth} = \coth^{-1}$.

Parameters

x	Hyperbolic cotangent value. Domain: $\left(-\infty, -1\right) \cup \left(1, \infty\right)$

Returns: Hyperbolic angle in range $\left(0, \infty\right)$ whose hyperbolic cotangent is x

◆ acsc()

number math::acsc ( numeric x )

Arc cosecant function. Inverse of the cosecant function. $\text{arccsc} = \csc^{-1}$.

Parameters

x	Cosecant value: Domain: $\left(-\infty, -1\right] \cup \left[1, \infty\right)$

Returns: Angle in radians in range $\left[-\frac{π}{2}, 0\right) \cup \left(0, \frac{π}{2}\right]$ whose cosecant is x

◆ acsch()

number math::acsch ( numeric x )

Arc hyperbolic cosecant function. Inverse of the hyperbolic sine function. $\text{arccsch} = \text{csch}^{-1}$.

Parameters

x	Hyperbolic cosecant value. Domain: $\left(-\infty, 0\right) \cup \left(0, \infty\right)$

Returns: Hyperbolic angle in range $\left(-\infty, -1\right) \cup \left(1, \infty\right)$ whose hyperbolic cosecant is x

◆ add()

number math::add	(	numeric	x,
		numeric	y )

Addition operator +. Arithmetic addition for numeric values $+$.

Parameters

x	Summand
y	Summand

Returns: Sum of the summands. $x + y$

◆ approx()

number math::approx	(	numeric	v1,
		numeric	v2,
		numeric	rt = 0.01,
		numeric	at = 0.00 )

Approximately function $\approx$.

Parameters

v1	Number
v2	Number
rt	Relative tolerance
at	Absolute tolerance

Returns: 1 if numbers are approximately equal given the tolerances. 0 otherwise

◆ asec()

number math::asec ( numeric x )

Arc secant function. Inverse of the secant function. $\text{arcsec} = \sec^{-1}$.

Parameters

x	Secant value. Domain: $\left(-\infty, -1\right] \cup \left[1, \infty\right)$

Returns: Angle in radians in range $\left[0, \frac{π}{2}\right) \cup \left(\frac{π}{2}, π\right]$ whose secant is x

◆ asech()

number math::asech ( numeric x )

Arc hyperbolic secant function. Inverse of the hyperbolic sine function. $\text{arcsech} = \text{sech}^{-1}$.

Parameters

x	Hyperbolic secant value. Domain: $\left(0, 1\right]$

Returns: Hyperbolic angle in range $\left[0, \infty\right)$ whose hyperbolic secant is x

◆ asin()

number math::asin ( numeric x )

Arc sine function. Inverse of the sine function. $\arcsin = \sin^{-1}$.

Parameters

x	Sine value. Domain: $\left[-1, 1\right]$

Returns: Angle in radians in range $\left[-\frac{π}{2}, \frac{π}{2}\right]$ whose sine is x

◆ asinh()

number math::asinh ( numeric x )

Arc hyperbolic sine function. Inverse of the hyperbolic sine function. $\text{arcsinh} = \sinh^{-1}$.

Parameters

x	Hyperbolic sine value

Returns: Hyperbolic angle whose hyperbolic sine is x

◆ atan()

number math::atan ( numeric x )

Arc tangent function. Inverse of the tangent function. $\arctan = \tan^{-1}$.

Parameters

x	Tangent value

Returns: Angle in radians in range $\left[-\frac{π}{2}, \frac{π}{2}\right]$ whose tangent is x

◆ atanh()

number math::atanh ( numeric x )

Arc hyperbolic tangent function. Inverse of the hyperbolic tangent function. $\text{arctanh} = \tanh^{-1}$.

Parameters

x	Hyperbolic tangent value. Domain: $\left(-1, 1\right)$

Returns: Hyperbolic angle whose hyperbolic tangent is x

◆ cbrt()

number math::cbrt ( numeric x )

Cube root function $\sqrt[3]{\phantom{x}}$.

Parameters

x Radicand

Returns: Cube root of radicand. $\sqrt[3]{x}$

◆ ceil()

number math::ceil ( numeric x )

Ceil function $\lceil \phantom{x} \rceil$.

Parameters

x Number

Returns: Nearest integer up. $\lceil x \rceil$

◆ cexpexp()

number math::cexpexp ( numeric x )

Complementary double exponential function.

Parameters

x Number

Returns: $f(x) = 1 - e^{-e^x}$

◆ cloglog()

number math::cloglog ( numeric x )

Complementary log-log function.

Parameters

x Number

Returns: $f(x) = \ln{\left(- \ln{\left(1-x\right)} \right)}$

◆ cmp()

number math::cmp	(	any	x,
		any	y )

Comparator or "spaceship operator".

Parameters

x	Left operand
y	Right operand

Returns: 1 if x is greater than y, 0 if they are equal, -1 otherwise

◆ copysign()

number math::copysign	(	numeric	x,
		numeric	y )

Copy the sign of y into x

Parameters

x	Number
y	Number

Returns: x with sign copied from y

◆ cosh()

number math::cosh ( numeric x )

Hyperbolic cosine function $\cosh$.

Parameters

x	Hyperbolic angle

Returns: x-coordinate of the point on the unit hyperbola

◆ cosine()

number math::cosine ( numeric x )

Cosine function $\cos$.

Parameters

x	Angle in radians

Returns: Ratio of the adjacent side to the hypotenuse in a right triangle. x-coordinate of the point on the unit circle. Range: $\left[-1, 1\right]$

◆ cot()

number math::cot ( numeric x )

Cotangent function $\cot$.

Parameters

x	Angle in radians

Returns: Ratio of the adjacent side to the opposite side in a right triangle. Ratio of the x-coordinate to the y-coordinate of the point on the unit circle

◆ coth()

number math::coth ( numeric x )

Hyperbolic cotangent function $\coth$.

Parameters

x	Hyperbolic angle

Returns: Ratio of the x-coordinate to the y-coordinate of the point on the unit hyperbola

◆ csc()

number math::csc ( numeric x )

Cosecant function $\csc$.

Parameters

x	Angle in radians

Returns: Ratio of the hypotenuse to the opposite side in a right triangle. Reciprocal of the y-coordinate of the point on the unit circle. Range: $\left(-\infty, -1\right] \cup \left[1, \infty\right)$

◆ csch()

number math::csch ( numeric x )

Hyperbolic cosecant function $\text{csch}$.

Parameters

x	Hyperbolic angle

Returns: Reciprocal of the y-coordinate of the point on the unit hyperbola

◆ div()

number math::div	(	numeric	x,
		numeric	y )

Division operator /. Arithmetic division for numeric values $\div$.

Parameters

x	Dividend
y	Divisor

Returns: Quotient of dividend and divisor. $x \div y$

◆ eq()

number math::eq	(	any	x,
		any	y )

Equality operator ==. Mathematical equality relation for numeric values $=$.

Parameters

x	Left operand
y	Right operand

Returns: 1 if the left operand is equal to the right operand. 0 otherwise

◆ eqBool()

bool math::eqBool	(	any	x,
		any	y )

Equality operator == that returns a number|bool

Parameters

x	Left operand
y	Right operand

Returns: True if the left operand is equal to the right operand. False otherwise

◆ eqCollate()

auto math::eqCollate	(	x	,
		y	)

Check if strings collate equally.

◆ expm1()

number math::expm1 ( numeric x )

$f(x) = e^x - 1$

Parameters

x Number

Returns

◆ factorial()

number math::factorial ( numeric x )

Factorial function. $x!$.

Parameters

x	Non-negative integer

Returns: x factorial: The product of all positive integers not greater than x. $$x! = \prod\limits_{i=1}^{x} i = x \cdot (x - 1)!$$

Todo: implement gamma function and possibly extend this factorial

◆ flipsign()

number math::flipsign	(	numeric	x,
		numeric	y )

Flip the sign of x if y is negative.

Parameters

x	Number
y	Number

Returns: x with possibly sign flipped

◆ floor()

number math::floor ( numeric x )

Floor function $\lfloor \phantom{x} \rfloor$.

Parameters

x Number

Returns: Nearest integer down. $\lfloor x \rfloor$

◆ floorDiv()

number math::floorDiv	(	numeric	x,
		numeric	y )

Floor division.

Parameters

x	Dividend
y	Divisor

Returns: Quotient of dividend and divisor rounded down to nearest integer

◆ frac()

number math::frac ( numeric x )

Fractional or decimal part function $\{~~\}$.

Parameters

x Number

Returns: Fractional part

◆ gcd()

number math::gcd	(	numeric	x,
		numeric	y )

Greatest common denominator function $\gcd$.

Parameters

x	Integer
y	Integer

Returns: Greatest common denominator of the integers $\gcd{\left(x, y\right)}$

Todo: handle decimals

◆ ge()

number math::ge	(	any	x,
		any	y )

Greater than or equal operator >=. Mathematical greater-than or equal relation for numeric values $\geq$.

Parameters

x	Left operand
y	Right operand

Returns: 1 if the left operand is greater than or equal to the right operand. 0 otherwise

◆ geBool()

bool math::geBool	(	any	x,
		any	y )

Equality operator >= that returns a number|bool

Parameters

x	Left operand
y	Right operand

Returns: True if the left operand is greater than or equal to the right operand. False otherwise

◆ gt()

number math::gt	(	any	x,
		any	y )

Greater than operator >. Mathematical greater-than relation for numeric values $\gt$.

Parameters

x	Left operand
y	Right operand

Returns: 1 if the left operand is greater than the right operand. 0 otherwise

◆ gtBool()

bool math::gtBool	(	any	x,
		any	y )

Equality operator > that returns a number|bool

Parameters

x	Left operand
y	Right operand

Returns: True if the left operand is greater than the right operand. False otherwise

◆ hypot()

number math::hypot	(	numeric	x,
		numeric	y,
		numeric	z )

Hypotenuse function (or euclidean distance function, or polar length, or pythagorean addition) $\oplus$.

Parameters

x	Number
y	Number. Optional
z	Number. Optional

Returns: Squared root of the sum of squares of the arguments

◆ is()

number math::is	(	any	x,
		any	y )

Identity operator: Equality in value and type.

Parameters

x	Left operand
y	Right operand

Returns: 1 if the left operand is equal to the right operand in value and type. 0 otherwise

◆ isBetween()

number math::isBetween	(	numeric	a,
		numeric	x,
		numeric	b,
		bool	eq )

Check if x is bounded between a and b

Parameters

a	Lower bound
x	Number
b	Upper bound
eq	Whether to allow equality

Returns: 1 if x is bounded between a and b. 0 otherwise

◆ lcm()

number math::lcm	(	numeric	x,
		numeric	y )

Least common multiple function $\text{lcm}$.

Parameters

x	Integer
y	Integer

Returns: Least common multiple of the integers $\text{lcm}\left(x, y\right)$

Todo: handle decimals

◆ le()

number math::le	(	any	x,
		any	y )

Less than or equal operator <=. Mathematical less-than or equal relation for numeric values $\leq$.

Parameters

x	Left operand
y	Right operand

Returns: 1 if the left operand is less than or equal to the right operand. 0 otherwise

◆ leBool()

bool math::leBool	(	any	x,
		any	y )

Equality operator <= that returns a number|bool

Parameters

x	Left operand
y	Right operand

Returns: True if the left operand is less than or equal to the right operand. False otherwise

◆ ln()

number math::ln ( numeric x )

Natural logarithm (base $e$) function $\ln$ or $\log_e$.

Parameters

x	Antilogarithm

Returns: Natural logarithm (of base $e$) of the antilogarithm. $\ln(x)$ or $\log_e(x)$

◆ log10()

number math::log10 ( numeric x )

Common logarithm (base 10) function $\text{lg}$ or $\log_{10}$.

Parameters

x	Antilogarithm

Returns: Logarithm of antilogarithm to the base 10. $\log_{10}(x)$

◆ log1mexp()

number math::log1mexp ( numeric x )

$f(x) = \ln{\left(1 - e^x\right)}$

Parameters

x Number

Returns

◆ log1p()

number math::log1p ( numeric x )

$f(x) = \ln{\left(1+x\right)}$

Parameters

x Number

Returns

◆ log1pexp()

number math::log1pexp ( numeric x )

$f(x) = \ln{\left(1 + e^x\right)}$

Parameters

x Number

Returns

◆ log1pmx()

number math::log1pmx ( numeric x )

$f(x) = \ln{\left(1 + x\right)} - x$

Parameters

x Number

Returns

◆ log1psq()

number math::log1psq ( numeric x )

$f(x) = \ln{\left(1 + x^2\right)}$

Parameters

x Number

Returns

◆ log2()

number math::log2 ( numeric x )

Binary logarithm (base 2) function $\text{lb}$ or $\log_2$.

Parameters

x	Antilogarithm

Returns: Logarithm of antilogarithm to the base 2. $\log_2(x)$

◆ log2mexp()

number math::log2mexp ( numeric x )

$f(x) = \ln{\left(2 - e^x\right)}$

Parameters

x Number

Returns

◆ logb()

number math::logb	(	numeric	x,
		numeric	b )

Logarithm function $\log_b$.

Parameters

x	Antilogarithm
b	Base

Returns: Logarithm of antilogarithm to the base b. $\log_b(x)$

◆ logcosh()

number math::logcosh ( numeric x )

$f(x) = \ln{\left(\cosh{(x)}\right)}$

Parameters

x Number

Returns

◆ logistic()

number math::logistic ( numeric x )

Standard logistic function or expit function $\mathcal{L}$.

Parameters

x Number

Returns: $$\mathcal{L}(x) = \frac{1}{1 + e^{-x}}$$

◆ logit()

number math::logit ( numeric x )

Logit function or log-odds function. Inverse of the standard logistic function $\sigma = \mathcal{L}^{-1}$.

Parameters

x	Number. Domain: $\left(0, 1\right)$

Returns: $$\sigma(x) = \ln{\left(\frac{x}{1-x}\right)} = \mathcal{L}^{-1}(x)$$

◆ logmxp1()

number math::logmxp1 ( numeric x )

$f(x) = \ln{\left(x\right)} -x + 1$

Parameters

x Number

Returns

◆ lt()

number math::lt	(	any	x,
		any	y )

Less than operator <. Mathematical less-than relation for numeric values $\lt$.

Parameters

x	Left operand
y	Right operand

Returns: 1 if the left operand is less than the right operand. 0 otherwise

◆ ltBool()

bool math::ltBool	(	any	x,
		any	y )

Equality operator < that returns a number|bool

Parameters

x	Left operand
y	Right operand

Returns: True if the left operand is less than the right operand. False otherwise

◆ max()

number math::max	(	any	x,
		any	y )

Maximum function $\max$.

Parameters

x	Any value comparable by `>`
y	Any value comparable by `>`

Returns: Maximum (or greatest) between the arguments

◆ min()

number math::min	(	any	x,
		any	y )

Minimum function $\min$.

Parameters

x	Any value comparable by `>`
y	Any value comparable by `>`

Returns: Minimum (or smallest) between the arguments

◆ minmax()

unassigned math::minmax	(	array	A,
		any	x,
		any	y )

Min-max function $\text{minmax}$.

Parameters

[out]	A	Array to fill with: Minimum Maximum
	x	Any value comparable by `>`
	y	Any value comparable by `>`

Returns

◆ mod()

number math::mod	(	numeric	x,
		numeric	y )

Modulo or remainder operator %. Binary operation for numeric values $mod$.

Parameters

x	Dividend
y	Divisor or modulus

Returns: Remainder of dividend and divisor. $x \: \text{mod} \: y$

◆ modf()

unassigned math::modf	(	array	A,
		numeric	x )

Modulus fraction function.

Parameters

[out]	A	Array to fill with: Fractional part Integer part
	x	Number

Returns

◆ mul()

number math::mul	(	numeric	x,
		numeric	y )

Multiplication operator *. Arithmetic multiplication for numeric values $\times$.

Parameters

x	Factor
y	Factor

Returns: Product of the factors. $x \times y$

◆ muladd()

number math::muladd	(	numeric	x,
		numeric	y,
		numeric	z )

Multiply then add operation (MAD)

Parameters

x	Factor
y	Factor
z	Summand

Returns: Sum of the product of factors and the summand $x * y + z$

◆ ne()

number math::ne	(	any	x,
		any	y )

Inequality operator !=. Mathematical inequality relation for numeric values $\ne$.

Parameters

x	Left operand
y	Right operand

Returns: 1 if the left operand is not equal to the right operand. 0 otherwise

◆ neBool()

bool math::neBool	(	any	x,
		any	y )

Equality operator != that returns a number|bool

Parameters

x	Left operand
y	Right operand

Returns: True if the left operand is not equal to the right operand. False otherwise

◆ ni()

number math::ni	(	any	x,
		any	y )

Non-identity operator: Inequality in value or type.

Parameters

x	Left operand
y	Right operand

Returns: 1 if the left operand is not equal to the right operand in value or in type, 0 otherwise

◆ pow()

number math::pow	(	numeric	x,
		numeric	y )

Exponentiation operator ^. Arithmetic exponentiation for numeric values $x^y$.

Parameters

x	Base
y	Exponent

Returns: Power of base and exponent $x^y$

◆ prodSeq()

number math::prodSeq	(	numeric	x,
		numeric	y )

Product of sequence function. $\prod$.

Parameters

x	Lower bound integer
y	Upper bound integer

Returns: Product of all integers from x to y. $$\prod\limits_{i=x}^{y} i$$

◆ rdiv()

number math::rdiv	(	numeric	x,
		numeric	y )

Right division operator. Arithmetic division for numeric values $\div$ with operands in reversed order.

Parameters

x	Divisor
y	Dividend

Returns: Quotient of dividend and divisor. $y \div x$

◆ rematch()

number math::rematch	(	any	x,
		any	y )

Regular expression matching operator ~

Parameters

x	String or expression coerced to a string to be matched
y	Regular expression or expression coerced to a string to match against

Returns: 1 if the string matches against the regular expression. 0 otherwise

◆ rematchBool()

bool math::rematchBool	(	any	x,
		any	y )

Equality operator ~ that returns a number|bool

Parameters

x
y

Returns

◆ renomatch()

number math::renomatch	(	any	x,
		any	y )

Negated regular expression matching operator !~

Parameters

x	String to be matched or expression coerced to a string to be matched
y	Regular expression or expression coerced to a string to match against

Returns: 1 if the string does not match against the regular expression. 0 otherwise

◆ renomatchBool()

bool math::renomatchBool	(	any	x,
		any	y )

Equality operator !~ that returns a number|bool

Parameters

x
y

Returns

◆ root()

number math::root	(	numeric	x,
		numeric	n )

Nth root function $\sqrt[n]{\phantom{x}}$.

Parameters

x	Radicand
n	Degree

Returns: Nth root of radicand. $\sqrt[n]{x}$

◆ round()

number math::round ( numeric x )

Round function.

Parameters

x Number

Returns: Nearest integer

Todo: to nearest of multiple

◆ roundDown()

number math::roundDown	(	numeric	x,
		numeric	y )

Round to previous multiple of.

Parameters

x	Number
y	Number

Returns

◆ roundUp()

number math::roundUp	(	numeric	x,
		numeric	y )

Round to next multiple of.

Parameters

x	Number
y	Number

Returns

◆ sec()

number math::sec ( numeric x )

Secant function $\sec$.

Parameters

x	Angle in radians

Returns: Ratio of the hypotenuse to the adjacent side in a right triangle. Reciprocal of the x-coordinate of the point on the unit circle. Range: $\left(-\infty, -1\right] \cup \left[1, \infty\right)$

◆ sech()

number math::sech ( numeric x )

Hyperbolic secant function $\text{sech}$.

Parameters

x	Hyperbolic angle

Returns: Reciprocal of the x-coordinate of the point on the unit hyperbola

◆ sgn()

number math::sgn ( numeric x )

Sign function (or signum) $sgn$.

Parameters

x Number

Returns: 1 if positive, -1 if negative, 0 if 0

◆ signbit()

number math::signbit ( numeric x )

Signbit function.

Parameters

x Number

Returns: 1 if negative, 0 if non-negative

◆ sine()

number math::sine ( numeric x )

Sine function $\sin$.

Parameters

x	Angle in radians

Returns: Ratio of the opposite side to the hypotenuse in a right triangle. y-coordinate of the point on the unit circle. Range: $\left[-1, 1\right]$

◆ sinh()

number math::sinh ( numeric x )

Hyperbolic sine function $\sinh$.

Parameters

x	Hyperbolic angle

Returns: y-coordinate of the point on the unit hyperbola

◆ subt()

number math::subt	(	numeric	x,
		numeric	y )

Subtraction operator -. Arithmetic subtraction for numeric values $-$.

Parameters

x	Minuend
y	Subtrahend

Returns: Difference of minued and subtrahend. $x - y$

◆ sumSeq()

number math::sumSeq	(	numeric	x,
		numeric	y )

Sum of sequence function. $\sum$.

Parameters

x	Lower bound integer
y	Upper bound integer

Returns: Sum of all integers from x to y. $$\sum\limits_{i=x}^{y} i$$

Todo: step parameter

◆ tan()

number math::tan ( numeric x )

Tangent function $\tan$.

Parameters

x	Angle in radians

Returns: Ratio of the opposite side to the adjacent side in a right triangle. Ratio of the y-coordinate to the x-coordinate of the point on the unit circle

◆ tanh()

number math::tanh ( numeric x )

Hyperbolic tangent function $\tanh$.

Parameters

x	Hyperbolic angle

Returns: Ratio of the y-coordinate to the x-coordinate of the point on the unit hyperbola

◆ triangular()

number math::triangular ( numeric x )

Triangular function. $x?$.

Parameters

x	Non-negative integer

Returns: x termial: The sum of all positive integers not greater than x $$x? = \sum\limits_{i=1}^{x} i = x + (x - 1)?$$

◆ xexpx()

number math::xexpx ( numeric x )

$f(x) = x e^x$

Parameters

x Number

Returns

◆ xexpy()

number math::xexpy	(	numeric	x,
		numeric	y )

$f(x, y) = x e^y$

Parameters

x	Number
y	Number

Returns

◆ xlog1py()

number math::xlog1py	(	numeric	x,
		numeric	y )

$f(x, y) = x \ln{\left(1 + y\right)}$

Parameters

x	Number
y	Number. Domain: $\left[-1, \infty\right)$

Returns

◆ xlogx()

number math::xlogx ( numeric x )

$f(x) = x \ln{\left(x\right)}$

Parameters

x	Number. Domain: $\left[0, \infty\right)$

Returns

◆ xlogy()

number math::xlogy	(	numeric	x,
		numeric	y )

$f(x, y) = x \ln{\left(y\right)}$

Parameters

x	Number
y	Number. Domain: $\left(0, \infty\right)$

Returns

Variable Documentation

◆ bronze

number math::bronze = 3.302775637731994646559610633735247973125648286922623106355226528113583474146505222602309541009245359

Bronze ratio. 3° metallic mean. $\frac{3 + \sqrt{13}}{2}$. Continued fraction $[3;\overline{3}]$ and positive solution to the equation $x^2 = 3x + 1$.

◆ catalan

number math::catalan = 0.9159655941772190150546035149323841107741934

Catalan constant.

◆ cbrt10

number math::cbrt10 = 2.1544346900318837217592935665193504952593449421921085824892355063464111066483408001854415035432432761

$\sqrt[3]{10}$. Cube root of 10

◆ cbrt2

number math::cbrt2 = 1.2599210498948731647672106072782283505702514647015079800819751121552996765139594837293965624362550941

$\sqrt[3]{2}$. Cube root of 2

◆ cbrt3

number math::cbrt3 = 1.4422495703074083823216383107801095883918692534993505775464161945416875968299973398547554797056452566

$\sqrt[3]{3}$. Cube root of 3

◆ cbrt4

number math::cbrt4 = 1.5874010519681994747517056392723082603914933278998530098082857618252165056242191732735442132622209570

$\sqrt[3]{4}$. Cube root of 4

◆ cbrt5

number math::cbrt5 = 1.7099759466766969893531088725438601098680551105430549243828617074442959205041732162571870100201890022

$\sqrt[3]{5}$. Cube root of 5

◆ cbrt6

number math::cbrt6 = 1.8171205928321396588912117563272605024282104631412196714813342979313097394593018656471417041264170721

$\sqrt[3]{6}$. Cube root of 6

◆ cbrt7

number math::cbrt7 = 1.9129311827723891011991168395487602828624390503458757662106476404472342761792307560075254414772857099

$\sqrt[3]{7}$. Cube root of 7

◆ cbrt9

number math::cbrt9 = 2.0800838230519041145300568243578853863378053403732621096975910802001063113972687736060566367907574867

$\sqrt[3]{8}$. Cube root of 9

◆ copper

number math::copper = 4.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275

Copper ratio. 4° metallic mean. $2 + \sqrt{5}$. Continued fraction $[4;\overline{4}]$ and positive solution to the equation $x^2 = 4x + 1$.

◆ e

number math::e = 2.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713821785251664274

$e$ number. The natural rate of exponential growth $f(x) = f'(x) = e^x$. $$e = \lim\limits_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$$

◆ euler_gamma

number math::euler_gamma = 0.57721566490153286060651209

Euler–Mascheroni constant.

◆ fourinvpi

number math::fourinvpi = 1.273239544735162686151070106980114896275677165923651589981338752471174381073812280720910422130024688

$\frac{4}{\pi}$

◆ fourpi

number math::fourpi = 12.56637061435917295385057353311801153678867759750042328389977836923126562514483599451213930136846827

$4\pi$

◆ gauss

number math::gauss = 0.8346268416740731862814297327990468089939930134903470024498273701036819927095264118696911603512753241

$G$ Gauss constant. $$G = \frac{\varpi}{\pi}$$

◆ golden

number math::golden = phi

Golden ratio $\varphi$. 1° metallic mean. $\frac{1 + \sqrt{5}}{2}$. Continued fraction $[1;\overline{1}]$ and positive solution to the equation $x^2 = x + 1$.

◆ golden_angle

number math::golden_angle = 2.399963229728653322231555506633613853124999011058115042935112750731307338239438790779962060660583964

Golden angle in radians $g$. The smaller angle that sections a circles's circumference according to the golden ratio $\varphi$.

◆ halfpi

number math::halfpi = 1.570796326794896619231321691639751442098584699687552910487472296153908203143104499314017412671058534

$\frac{\pi}{2}$

◆ inv2pi

number math::inv2pi = 0.1591549430918953357688837633725143620344596457404564487476673440588967976342265350901138027662530860

$\frac{1}{2\pi}$

◆ inv4pi

number math::inv4pi = 0.07957747154594766788444188168625718101722982287022822437383367202944839881711326754505690138312654298

$\frac{1}{4\pi}$

◆ invphi

number math::invphi = -psi

$\varphi^{-1} = \frac{-1 + \sqrt{5}}{2} = -\psi$

◆ invpi

number math::invpi = 0.3183098861837906715377675267450287240689192914809128974953346881177935952684530701802276055325061719

$\frac{1}{\pi}$

◆ invpsi

number math::invpsi = -phi

$\psi^{-1} = \frac{-1 - \sqrt{5}}{2} = -\varphi$

◆ invsqrt2

number math::invsqrt2 = 0.7071067811865475244008443621048490392848359376884740365883398689953662392310535194251937671638207864

$\frac{1}{\sqrt{2}}$

◆ invsqrt2pi

number math::invsqrt2pi = 0.3989422804014326779399460599343818684758586311649346576659258296706579258993018385012523339073069364

$\frac{1}{\sqrt{2\pi}}$

◆ invsqrtpi

number math::invsqrtpi = 0.5641895835477562869480794515607725858440506293289988568440857217106424684414934144867436602021073634

$\frac{1}{\sqrt{\pi}}$

◆ lemniscate

number math::lemniscate = 2.622057554292119810464839589891119413682754951431623162816821703800790587070414250230295532961429093

$\varpi$ Lemniscate constant. The ratio of a Bernoulli's lemniscate's perimeter to its diameter. $$\varpi = \frac{p}{d} = 2 \prod _{n=1}^{\infty}{ \left(1 + \frac{1}{2n}\right)^{\left(-1\right)^{n+1}} }$$

◆ ln10

number math::ln10 = 2.302585092994045684017991454684364207601101488628772976033327900967572609677352480235997205089598298

$\ln(10)$. Natural logarithm of 10

◆ ln2

number math::ln2 = 0.6931471805599453094172321214581765680755001343602552541206800094933936219696947156058633269964186875

$\ln(2)$. Natural logarithm of 2

◆ ln2pi

number math::ln2pi = 1.837877066409345483560659472811235279722794947275566825634303080965531391854520795389486597271908395

$\ln{(2\pi)}$. Natural logarithm of $2\pi$

◆ ln3

number math::ln3 = 1.0986122886681096913952452369225257046474905578227494517346943336374942932186089668736157548137320887

$\ln(3)$. Natural logarithm of 3

◆ ln4

number math::ln4 = 1.3862943611198906188344642429163531361510002687205105082413600189867872439393894312117266539928373750

$\ln(4)$. Natural logarithm of 4

◆ ln4pi

number math::ln4pi = 2.531024246969290792977891594269411847798295081635822079754983090458925013824215510995349924268327083

$\ln{(4\pi)}$. Natural logarithm of $4\pi$

◆ ln5

number math::ln5 = 1.6094379124341003746007593332261876395256013542685177219126478914741789877076577646301338780931796107

$\ln(5)$. Natural logarithm of 5

◆ lnhalf

number math::lnhalf = -0.6931471805599453094172321214581765680755001343602552541206800094933936219696947156058633269964186875

$\ln\left(\frac{1}{2}\right)$. Natural logarithm of $\frac{1}{2}$

◆ lnpi

number math::lnpi = 1.144729885849400174143427351353058711647294812915311571513623071472137769884826079783623270275489708

$\ln{(\pi)}$. Natural logarithm of $\pi$

◆ nickel

number math::nickel = 5.192582403567252015625355245770164778147560080822394418840194335008322981413829346438316890839917742

Nickel ratio. 5° metallic mean. $\frac{5 + \sqrt{29}}{2}$. Continued fraction $[5;\overline{5}]$ and positive solution to the equation $x^2 = 5x + 1$.

◆ parabolic

number math::parabolic = 2.295587149392638074034298049189490387597832203638583483929975346644109662684133126684094426237897616

$P$ Universal parabolic constant. The ratio of a parabola's latus rectum arc to its semi latus rectum. $$P = \sqrt{2} + \ln{\left(1 + \sqrt{2}\right)}$$

◆ phi

number math::phi = 1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374

$\varphi = \frac{1 + \sqrt{5}}{2}$

◆ pi

number math::pi = 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679

$\pi$ number. The ratio of a circle's circumference to its diameter. $\pi$ radians represent a half rotation ( $180°$). $$\pi = \frac{C}{d} = 2 \prod _{n=1}^{\infty}{ \left(1 + \frac{1}{n}\right)^{\left(-1\right)^{n+1}} }$$

◆ plastic

number math::plastic = 1.324717957244746025960908854478097340734404056901733364534015050302827851245547594054699347981787280

Plastic ratio. Real solution to the equation $x^3 = x + 1$.

◆ psi

number math::psi = -0.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374

$\psi = \frac{1 - \sqrt{5}}{2} = - \varphi^{-1}$

◆ quartpi

number math::quartpi = 0.7853981633974483096156608458198757210492923498437764552437361480769541015715522496570087063355292670

$\frac{\pi}{4}$

◆ silver

number math::silver = 2.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727

Silver ratio. 2° metallic mean. $1 + \sqrt{2}$. Continued fraction $[2;\overline{2}]$ and positive solution to the equation $x^2 = 2x + 1$.

◆ sqrt10

number math::sqrt10 = 3.1622776601683793319988935444327185337195551393252168268575048527925944386392382213442481083793002951

$\sqrt{10}$. Square root of 10

◆ sqrt11

number math::sqrt11 = 3.3166247903553998491149327366706866839270885455893535970586821461164846426090438467088433991282906509

$\sqrt{11}$. Square root of 11

◆ sqrt12

number math::sqrt12 = 3.4641016151377545870548926830117447338856105076207612561116139589038660338176000741622923735144971513

$\sqrt{12}$. Square root of 12

◆ sqrt13

number math::sqrt13 = 3.6055512754639892931192212674704959462512965738452462127104530562271669482930104452046190820184907176

$\sqrt{13}$. Square root of 13

◆ sqrt2

number math::sqrt2 = 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727

$\sqrt{2}$. Square root of 2

◆ sqrt2pi

number math::sqrt2pi = 2.506628274631000502415765284811045253006986740609938316629923576342293654607841974946595838378057266

$\sqrt{2\pi}$

◆ sqrt3

number math::sqrt3 = 1.7320508075688772935274463415058723669428052538103806280558069794519330169088000370811461867572485756

$\sqrt{3}$. Square root of 3

◆ sqrt4pi

number math::sqrt4pi = 3.544907701811032054596334966682290365595098912244774256427615579705822569182064362749901313477089331

$\sqrt{4\pi}$

◆ sqrt5

number math::sqrt5 = 2.2360679774997896964091736687312762354406183596115257242708972454105209256378048994144144083787822749

$\sqrt{5}$. Square root of 5

◆ sqrt6

number math::sqrt6 = 2.4494897427831780981972840747058913919659474806566701284326925672509603774573150265398594331046402348

$\sqrt{6}$. Square root of 6

◆ sqrt7

number math::sqrt7 = 2.6457513110645905905016157536392604257102591830824501803683344592010688232302836277603928864745436106

$\sqrt{7}$. Square root of 7

◆ sqrt8

number math::sqrt8 = 2.8284271247461900976033774484193961571393437507538961463533594759814649569242140777007750686552831454

$\sqrt{8}$. Square root of 8

◆ sqrthalfpi

number math::sqrthalfpi = 1.253314137315500251207882642405522626503493370304969158314961788171146827303920987473297919189028633

$\sqrt{\frac{\pi}{2}}$

◆ sqrtpi

number math::sqrtpi = 1.772453850905516027298167483341145182797549456122387128213807789852911284591032181374950656738544665

$\sqrt{\pi}$

◆ super_golden

number math::super_golden = 1.4655712318767680266567312252199391080255775684722857016431831112492629966850178404781258011949092701

Supergolden ratio. Real solution to the equation $x^3 = x^2 + 1$.

◆ tau

number math::tau = 6.283185307179586476925286766559005768394338798750211641949889184615632812572417997256069650684234136

$\tau$ number. The ratio of a circle's circumference to its radius. $\tau$ radians represent a full rotation ( $360°$). $$\tau = \frac{C}{r} = 2\pi$$

◆ tribonacci

number math::tribonacci = 1.839286755214161132551852564653286600424178746097592246778758639404203222081966425738435419428307014

Tribonacci constant. Real solution to the equation $x^3 = x^2 + x + 1$.

◆ twoinvpi

number math::twoinvpi = 0.6366197723675813430755350534900574481378385829618257949906693762355871905369061403604552110650123438

$\frac{2}{\pi}$

◆ twopi

number math::twopi = tau

$2\pi = \tau$

Functions

Variables

Function Documentation

◆ abs()

◆ abs2()

◆ acos()

◆ acosh()

◆ acot()

◆ acoth()

◆ acsc()

◆ acsch()

◆ add()

◆ approx()

◆ asec()

◆ asech()

◆ asin()

◆ asinh()

◆ atan()

◆ atanh()

◆ cbrt()

◆ ceil()

◆ cexpexp()

◆ cloglog()

◆ cmp()

◆ copysign()

◆ cosh()

◆ cosine()

◆ cot()

◆ coth()

◆ csc()

◆ csch()

◆ div()

◆ eq()

◆ eqBool()

◆ eqCollate()

◆ expm1()

◆ factorial()

◆ flipsign()

◆ floor()

◆ floorDiv()

◆ frac()

◆ gcd()

◆ ge()

◆ geBool()

◆ gt()

◆ gtBool()

◆ hypot()

◆ is()

◆ isBetween()

◆ lcm()

◆ le()

◆ leBool()

◆ ln()

◆ log10()

◆ log1mexp()

◆ log1p()

◆ log1pexp()

◆ log1pmx()

◆ log1psq()

◆ log2()

◆ log2mexp()

◆ logb()

◆ logcosh()

◆ logistic()

◆ logit()

◆ logmxp1()

◆ lt()

◆ ltBool()

◆ max()

◆ min()

◆ minmax()

◆ mod()

◆ modf()

◆ mul()

◆ muladd()

◆ ne()

◆ neBool()

◆ ni()

◆ pow()

◆ prodSeq()