From f055ff24974a8fe8418bb90384e22f2adeb77e32 Mon Sep 17 00:00:00 2001
From: Connor <cloningdonor@gmail.com>
Date: Thu, 10 Aug 2017 12:15:39 -0700
Subject: [PATCH]

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+## stuff i've plotted in desmos
+
+in some order idk
+
+### [circumcircle](https://www.desmos.com/calculator/zaf29i2xrk)
+
+drag around the three points to see how their circumcircle is affected.
+sorry, i never finished simplifying the math here.
+
+### [parabola arc length](https://www.desmos.com/calculator/hrokhmphrf)
+
+not closed form, sorry. drag the points around like it's a jumprope.
+
+### ["diode"](https://www.desmos.com/calculator/vappug98ky)
+
+i didn't name this function; i have no idea if or how it relates to a diode.
+the red line (the line of interest) is actually just the 12th order smoothstep.
+
+btw, you can find smoothstep coefficients for any order
+by the triangular table of this integer sequence:
+[OEIS A091811](http://oeis.org/A091811)
+
+[punching it into wolfram][smoothie] (try changing n)
+
+[smoothie]: (https://www.wolframalpha.com/input/?i=ReplaceAll%5BBinomial%5Bn+%2B+k+-+2,+k+-+1%5D+Binomial%5B2+n+-+1,+n+-+k%5D,+%7Bk+-%3E+Range%5B1,n%5D%7D%5D+where+n+%3D+12)
+
+### [smoothstep alternatives](https://www.desmos.com/calculator/ibigyiumzb)
+
+refer to shadertoy:
+
+* [Inverse Smoothstep](https://www.shadertoy.com/view/MsSBRh)
+* [gain()](https://www.shadertoy.com/view/ldBfR1)
+* [Alt smoothstep](https://www.shadertoy.com/view/lsBfz1)
+
+### [tanh approximation](https://www.desmos.com/calculator/k3jzkhmoid)
+
+approximating tanh by taking its continued fraction form and (externally) optimizing the constants.
+probably not useful.
+
+### [TanhTest lolremez approx](https://www.desmos.com/calculator/carcltkgem)
+
+a polynomial approximation of a scaled tanh function
+made with [the excellent lolremez](https://github.com/samhocevar/lolremez).
+
+### [e^(-2/x) approximations](https://www.desmos.com/calculator/q8jtxgnmlh)
+
+i forget where i read about this but basically you can compute
+`e^(-2/x)` using `(x-1)/(x+1)` when x is large.
+
+the black line shows the absolute error.
+
+### [compressor gain](https://www.desmos.com/calculator/rizrdxsrbg)
+
+trying to design a cheap-to-compute gain curve for a dynamic range compressor.
+this thing uses just one division.
+iirc it didn't wind up being all that useful.
+
+both axes are in power decibels. note that it acts as a gate under -70 dB.
+
+### [compressor approx stuff](https://www.desmos.com/calculator/5wbup88lev)
+
+trying to compute abs(x) by polynomials, or something. weird stuff.
+
+### [doom turnspeeds](https://www.desmos.com/calculator/zfscpmsdbk)
+
+i used this to help design turnspeed smoothing in [my zdoom patch.](https://eaguru.guru/t/zdoom_smoothstep.patch)
+
+*note to self: i think i have a more recent version of that patch lying around somewhere.*
+
+### [Biweight Funtimes](https://www.desmos.com/calculator/mlnqmf6ogp)
+
+messing around with a variations of Tukey's Biweight?
+i think the only reason i kept this around was 'cause it's fun to watch animated.
+
+### [sincmod](https://www.desmos.com/calculator/lqgl4o0naz)
+
+something something windowing functions i dunno.
+
+### [SGDR](https://www.desmos.com/calculator/hlgqmyswy2)
+
+simple case of a [SGDR](https://arxiv.org/abs/1608.03983) curve.
+i just kept this around 'cause the logarithmic plot complements
+the linear plot in a kinda cool way.
+
+### [sqr/abs generalization](https://www.desmos.com/calculator/fagjg9vuz7)
+
+self-explanitory.
+
+### [SELU](https://www.desmos.com/calculator/bqu264oprw)
+
+[it's SELU!](https://arxiv.org/abs/1706.02515)
+
+### [GELU](https://www.desmos.com/calculator/ydzgtccsld)
+
+[GELU activation](https://arxiv.org/abs/1606.08415)
+alongside an approximation i lifted off reddit or someplace.
+
+### [erf cdf approximations](https://www.desmos.com/calculator/22r29cude2)
+
+approximate gaussian stuff, with inversions and integrals.
+
+### [squish](https://www.desmos.com/calculator/1qdx4ovpb6)
+
+possibly useful functions for squishing values.
+mostly based on sigmoidal functions.
+
+### [sin LUT approx 2](https://www.desmos.com/calculator/zdbgkxadns)
+
+approximating a quadrant of sin(x) with a polynomial and a LUT.
+
+### [circle calculus](https://www.desmos.com/calculator/yd5hyagaoj)
+
+some fun curves relating to unit circles.
+
+### [archimedes spiral](https://www.desmos.com/calculator/63vi3wnqug)
+
+it's just Archimedes' Spiral, plus some related functions.
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