Mapping-Formula Research Thread

⏣⌬⏣ ᎷᏗᎮᎮᎥᏁᎶ FᎧᏒᎷᏬᏝค Research Thread ⏣⌬⏣

Explicitly useful formulas should go here.
viewtopic.php?f=100&t=107015&hilit=useful+formulas
This is a Formula building/concept development/formula development/research thread

subject to revision. It may be somewhat common of me to revise this thread as posts are made periodically (with editorial notes made). This is a Work in Progress

⏣⌬⏣ The Syntax ⏣⌬⏣ - ( complete list to my knowledge as of 06/24/19 (Sorry for the errors here previously) )

Listed in Online/Offline help
+ addition ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀a+b
* multiplication ⠀⠀⠀⠀a*b
- subtraction ⠀⠀⠀⠀⠀a-b
/ division ⠀⠀⠀⠀⠀⠀⠀⠀⠀a/b
^ power ⠀⠀⠀⠀⠀⠀⠀⠀a^b
ATan2 arctangent angle/quadrant of a given number ⠀⠀⠀⠀⠀⠀Atan2(a,b)
ArcSin antisine ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ArcSin(a)
ArcCos anticosine ⠀⠀⠀⠀⠀⠀ArcCos(a)
ArcTg antitangent ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ArcTg(a)
Case returns b if a=1, else returns c ⠀⠀⠀⠀⠀Case(a,b,c)
Cos cosine ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀Cos(a)
CoSec cosecant ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀CoSec(a)
Ctg cotangent ⠀⠀⠀⠀⠀⠀⠀Ctg(a)
Exp exponent ⠀⠀⠀⠀⠀⠀⠀Exp(a)
Frac fractional part ⠀⠀⠀⠀⠀⠀Frac(a)
Fold Wavefolding operation Fold(a)
IfE returns 1 if a=b IfE(a,b)
IfG returns 1 if a>b ⠀⠀IfG(a,b)
IfGE returns 1 if a>=b ⠀⠀IfGE(a,b)
IfL returns 1 if a<b ⠀⠀⠀IfL(a,b)
IfLE returns 1 if a<=b ⠀⠀⠀IfLE(a,b)
Int integral part ⠀⠀⠀⠀⠀Int(a)
Intpower integer power, Value^Power Intpower(Value, power)
Ln logarithm base e Ln(a)
Log10 logarithm base 10 ⠀⠀⠀Log10(a)
Log2 logarithm base 2 ⠀⠀⠀⠀Log2(a)
LogN a base b calc LogN(a,b)
Max maximum ⠀⠀⠀⠀Max(a,b)
Min minimum ⠀⠀⠀⠀Min(a,b)
Neg negative ⠀⠀⠀⠀⠀Neg(a)
Pi Pi ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀Pi()
Round round to nearest integer ⠀⠀⠀⠀⠀Round(a)
POWER floating point power Power(a,b)
Pow floating point power Pow(a,b)
RadToDeg Radians to Degree RadToDeg(a)
DegToRad Degree to Radians DegToRad(a)
Sec secant ⠀⠀⠀⠀⠀Sec(a)
Sin sine ⠀⠀⠀⠀⠀⠀⠀⠀Sin(a)
Sqrt square root ⠀⠀⠀⠀Sqrt(a)
Sum sum of arguments ⠀⠀⠀⠀Sum(a,b)
Tg tangent ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀Tg(a)
Tension Tension ⠀⠀⠀⠀⠀ Tension(a,b)
Input value sent by the controller (MIDI or internal) ⠀⠀⠀⠀⠀Input
Warp warp signal around to warp point b Warp(a,b)

Unlisted
Tan (same as Tg) ⠀⠀⠀⠀ Tan(a)

(from Formula-Controller Syntax)

Inter returns 1 if a > b and a < c. If c < b, b & c values will be swapped ⠀⠀ Inter(a,b,c)
Internoswap as Inter but doesn't swap a & b ⠀⠀ Internoswap(a,b,c)
Rand Rand random value (0..1) ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ Rand
SeededRand controlled random value (0..1) ⠀⠀⠀⠀⠀ SeededRand(Seed)

Unlisted. Hyperbolic Functions

Sinh ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ Sinh(a)
Cosh ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ Cosh(a)
ArcCosh ⠀⠀⠀⠀⠀⠀⠀⠀⠀ ArcCosh(a)
ArcSinh ⠀⠀⠀⠀⠀⠀⠀⠀⠀ArcSinh(a)
Tanh ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ Tanh(a)

Other Functions
! ⠀⠀⠀⠀ Factorial
Floor ⠀⠀⠀⠀ Floor function
Ceil ⠀⠀⠀⠀ Ceiling function https://en.wikipedia.org/wiki/Floor_and_ceiling_functions
Trunc ⠀ In mathematics and computer science, truncation is limiting the number of digits right of the decimal point. Trunc(a)

Logical Operators

XOR Exclusive or or exclusive disjunction
NOT Not (a) Logical complement, negation unary single argument
AND Logical conjunction AND true if and only if all of its operands are true
DIV Modulus operation. remainder of euclidean division of one number by another
MOD Modulo finds the remainder of division of one number by another
OR OR is the truth functional operator of (inclusive) disjunction. true if only ONE or MORE operand is true
NAND inverse of and
NOR inverse of or
XNOR inverse of xor
shr integer shift bits right (divide) value shr amount
shl integer shift bits left (multiply) value shl amount
Fmod Floating point Modulo a fmod b
In Identifies elements in a set. (not sure) http://wiki.freepascal.org/In

misc. unlisted operators
> Greater than
< Less than
= Equal to
>= greater than or equal to
<= Less than or equal to

Unlisted supported Unicode functions.
≤ inequality less than or equal
≥ inequality greater than or equal to
∞ infinity
≡ identity, congruence modulo n or equivalence https://en.wikipedia.org/wiki/Modular_arithmetic

Unlisted non-expression-supported functions (there are no expressions for these)

(1 / ((Input^2)+1)) ARCCOTANGENT

(1 / (Input)*(Sqrt((Input^2)-1))) ARCSECANT

((1/(tanh(Input)))*0.5-0.5) Hyperbolic CoTangent (scaled)

((1/(Sinh(Input)))*0.5-0.5) Hyperbolic CoSecant (scaled)

((1/(cosh(Input)))*0.5+0.5) Hyperbolic Secant (scaled)

??? . I don't know. Comments would help.

Int (a,b,c) Not sure how this would be processed.

Tan(a,b) Suspected Format Function: I imagine the answer is here: http://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf
( ArcCos(a,b), Sin(a,b) , Cos(a,b) )... ect also acceptable

⌠ a ⌡ Labeled Top, Bottom half integral. No clue how these would be used in practice tbh

Exp(a,b)

⏣⌬⏣ Constants ⏣⌬⏣
( Most, if not all of these came from here https://en.wikipedia.org/wiki/List_of_mathematical_constants )
(listed from whole numbers to hundredth variation)
This is not everything on that page

Golden Ratio
1.6180339887498948482
0.16180339887498948482
0.016180339887498948482
0.0016180339887498948482

Golden Spiral
((1+sqrt(5)/2)^(2/))

Nested radical
((sqrt(21)+1)/2)

time constant
0.63212055882855767840
0.063212055882855767840
0.0063212055882855767840

connective constant
(sqrt(2+sqrt(2)))

bronze ratio
((3+sqrt 13)/2)

Volume of Reuleaux tetrahedron
0.42215773311582662702
0.042215773311582662702
0.0042215773311582662702

Hermite Constant Sphere packing 3D Kepler conjecture
(pi/(3 sqrt(2)))

Hermite Constant
(2/sqrt(3))

Delian constant cube root of 2
(2^(1/3))

Khinchin-Lévy constant
(2.71828182845904523536)^(\pi^2/(12 ln(2))
or
(0.0271828182845904523536)^(\pi^2/(12 ln(2))
or
(0.00271828182845904523536)^(\pi^2/(12 ln(2))

john constant
(2.71828182845904523536)^(pi/2)
or
(0.0271828182845904523536)^(pi/2)
or
(0.00271828182845904523536)^(pi/2)

levy constant
Pi^(2)/(6*ln(2))

Median of the Gumbel distribution
(-ln(ln(2)))
or
0.364621596072079117709
or
0.0364621596072079117709
or
0.00364621596072079117709

Pi^pi
(Pi^Pi)

triangular root of 2
(sqrt(17)-1)/2

Lebesgue constant
(1/3 + 2*sqrt(3)/pi)
1.43599112417691743235
or
0.0143599112417691743235
or
0.00143599112417691743235

Silver root
3.24697 96037 17467 06105
0.0324697960371746706105
0.0324697960371746706105

Tutte–Beraha constant
(2+2*cos(2Pi/7))

Plouffe's Gamma constant
ArcTg(1/2)/pi
0.14758361765043327417
or
0.014758361765043327417
or
0.0014758361765043327417

Grothendieck constant
pi/(log2(1+sqrt(2)))

Hausdorff dimension, Sierpinski triangle
1.58496250072115618145
or
0.0158496250072115618145
or
0.00158496250072115618145

Fractal dimension of the boundary of the dragon curve
1.52362 70862 02492 10627
or
0.0152362708620249210627
or
0.00152362708620249210627

Mills Constant
1.30637788386308069046
or
0.0130637788386308069046
or
0.00130637788386308069046861449260260571

EULERS NUMBER
Number e, Euler's number
2.71828182845904523536
or
0.0271828182845904523536
or
0.00271828182845904523536

MINIMUM FUNCTION VALUE
(2.71828182845904523536)^(-1/(2.71828182845904523536))
or
(0.0271828182845904523536)^(-1/(0.0271828182845904523536))
or
(0.00271828182845904523536)^(-1/(0.00271828182845904523536))

BUFFON CONSTANT
(2/Pi)
0.63661977236758134307
or
0.063661977236758134307
or
0.00063661977236758134307

IMAGINARY NUMBER i
sqrt(-1)

Ramanujan nested radical
((2+sqrt(5)+sqrt(15-6 sqrt(5)))/2)
2.74723827493230433305
or
0.0274723827493230433305
or
0.00274723827493230433305

⏣⌬⏣ Mapping Formula Designing / Fruity Formula Controller Patch Creation Resources ⏣⌬⏣

Derivative Calculator
https://www.derivative-calculator.net/

Integral Calculator
https://www.integral-calculator.com/

Currently researching "The Best" all around text editor for mapping formulas and FCC
https://en.wikipedia.org/wiki/List_of_text_editors

Many aspects of formula building can involve having to document multiple calculator entries. This will save you time.
https://www.dcode.fr/multiples-list-number

Online tools
http://www.unit-conversion.info/texttools/category/Line_tools#data

Other resources.

https://unaspectedstudios.wordpress.com/2016/03/09/fl-studio-understanding-mapping-formulas/2

⏣⌬⏣ closing thoughts (for now) ⏣⌬⏣
Going to keep up with this casually.
If anything I personally post ends up being very useful, I'll transfer the content to Nucleon's Useful Formula Thread

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