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All functions are defined for \( x \in [0, 1] \).
\( \mathit{GS}_1\ (1.50):\quad y(x) = \dfrac{e^{-\frac{1}{2} \left( \frac{x-1}{0.436} \right)^2} - e^{-\frac{1}{2} \left( \frac{1}{0.436} \right)^2}}{1 - e^{-\frac{1}{2} \left( \frac{1}{0.436} \right)^2}} \)
(1)
\( \mathit{RC}\ (1.57):\quad y(x) = 0.5 - 0.5 \cos(\pi x) \)
(2)
\( \mathit{CF}\ (1.58):\quad y(x) = 0.46620824 - 0.49597453 \cos(\pi x) + 0.03379176 \cos(2\pi x) - 0.00402547 \cos(3\pi x) \)
(3)
\( \mathit{AP}_1\ (1.60):\quad y(x) = 1.0 - 3.05937423(x-1)^2 + 3.93982779(x-1)^4 - 2.83450610(x-1)^6 + 1.18652517(x-1)^8 - 0.23247263(x-1)^{10} \)
(4)
\( \mathit{AP}_2\ (1.61):\quad y(x) = 1.0 - 3.08953423(x-1)^2 + 4.02519021(x-1)^4 - 2.93077271(x-1)^6 + 1.24004322(x-1)^8 - 0.24492649(x-1)^{10} \)
(5)
\( \mathit{AP}_3\ (1.62):\quad y(x) = 1.0 - 3.13487054(x-1)^2 + 4.15502848(x-1)^4 - 3.07919687(x-1)^6 + 1.32362225(x-1)^8 - 0.26458332(x-1)^{10} \)
(6)
\( \mathit{CK}\ (1.64):\quad y(x) = 0.46096 - 0.5 \cos(\pi x) + 0.03904 \cos(2\pi x) \)
(7)
\( \mathit{GS}_2\ (1.65):\quad y(x) = \dfrac{e^{-\frac{1}{2} \left( \frac{x-1}{0.379} \right)^2} - e^{-\frac{1}{2} \left( \frac{1}{0.379} \right)^2}}{1 - e^{-\frac{1}{2} \left( \frac{1}{0.379} \right)^2}} \)
(8)
\( \mathit{BM}\ (1.82):\quad y(x) = (1 - \left|x - 1\right|) \cos(\pi \left|x - 1\right|) + \dfrac{1}{\pi} \sin(\pi \left|x - 1\right|) \)
(9)
\( \mathit{CP}\ (2.09):\quad y(x) = 0.36334998 - 0.48935168 \cos(\pi x) + 0.13665002 \cos(2\pi x) - 0.01064832 \cos(3\pi x) \)
(10)
Note 1: You may right-click on any function to export or copy it in various formats (such as MathML, LaTeX, or as an image), depending on your browser and MathJax options.
Note 2: For its application in SynthEdit Waveshaper2B module,
equation (9) of function \( \mathit{BM}\ (1.82)\) has been simplified to:
\( \mathit y(x) = x \cos(\pi (1-x)) + \frac{1}{\pi} \sin(\pi (1-x)) \).
All other functions are implemented exactly as shown above.
Summary of Function Properties on 0 ≤ x ≤ 1
|
Function
Name
|
Maximum slope
mmax
|
y(x) strictly increasing on (0,1)
Always goes up
|
0 ≤ y(x) ≤ 1
Stays between 0 and 1
|
y(0)=0
Starts at 0
|
y(1)=1
Ends at 1
|
y′(0)=0
Smooth start
|
y′(1)=0
Smooth finish
|
| GS1 |
≈ 1.50 |
✔ |
✔ |
✔ |
✔ |
y′(0) ≈ 0.41 |
✔ |
| RC |
≈ 1.57 |
✔ |
✔ |
✔ |
✔ |
✔ |
✔ |
| CF |
≈ 1.58 |
✔ |
✔ |
✔ |
✔ |
✔ |
✔ |
| AP1 |
≈ 1.60 |
✔ |
✔ |
✔ |
✔ |
y′(0) ≈ 0.20 |
✔ |
| AP2 |
≈ 1.61 |
✔ |
✔ |
✔ |
✔ |
y′(0) ≈ 0.19 |
✔ |
| AP3 |
≈ 1.62 |
✔ |
✔ |
✔ |
✔ |
y′(0) ≈ 0.18 |
✔ |
| CK |
≈ 1.64 |
✔ |
✔ |
✔ |
✔ |
✔ |
✔ |
| GS2 |
≈ 1.65 |
✔ |
✔ |
✔ |
✔ |
y′(0) ≈ 0.22 |
✔ |
| BM |
≈ 1.82 |
✔ |
✔ |
✔ |
✔ |
✔ |
✔ |
| CP |
≈ 2.09 |
✔ |
✔ |
✔ |
✔ |
✔ |
✔ |
