{"version":8,"graph":{"viewport":{"xmin":-2.957116334745052,"ymin":-4.304148094363356,"xmax":64.27995215648295,"ymax":42.28600291088193}},"randomSeed":"c861f8a1b55200259bf96a542957c4fd","expressions":{"list":[{"type":"text","id":"47","text":"Taken from https://en.wikipedia.org/wiki/Lambert_W#Numerical_evaluation"},{"type":"text","id":"49","text":"Given a W(x) input and an approximation, get a better approximation (Newton-Raphson method)"},{"type":"expression","id":"45","color":"#000000","latex":"w\\left(z,p\\right)=p-\\frac{pe^{p}-z}{e^{p}+pe^{p}}"},{"type":"folder","id":"69","title":"Demonstration (c^c ~= 6)","collapsed":true},{"type":"expression","id":"50","folderId":"69","color":"#388c46","latex":"a=\\ln6"},{"type":"expression","id":"52","folderId":"69","color":"#000000","latex":"b_{0}=0"},{"type":"expression","id":"58","folderId":"69","color":"#c74440","latex":"b_{1}=w\\left(a,b_{0}\\right)"},{"type":"expression","id":"53","folderId":"69","color":"#c74440","latex":"b_{2}=w\\left(a,b_{1}\\right)"},{"type":"expression","id":"59","folderId":"69","color":"#2d70b3","latex":"b_{3}=w\\left(a,b_{2}\\right)"},{"type":"expression","id":"60","folderId":"69","color":"#388c46","latex":"b_{4}=w\\left(a,b_{3}\\right)"},{"type":"expression","id":"51","folderId":"69","color":"#6042a6","latex":"c=e^{b_{4}}"},{"type":"expression","id":"65","folderId":"69","color":"#2d70b3","latex":"c^{c}"},{"type":"text","id":"71","text":"Apply several steps of w for a good approximation"},{"type":"expression","id":"72","color":"#c74440","latex":"W\\left(x\\right)=w\\left(x,w\\left(x,w\\left(x,w\\left(x,w\\left(x,w\\left(x,\\ln x\\right)\\right)\\right)\\right)\\right)\\right)"},{"type":"expression","id":"73","color":"#2d70b3","latex":"W\\left(9274\\right)e^{W\\left(9274\\right)}"},{"type":"folder","id":"92","title":"Graphable variants"},{"type":"text","id":"81","folderId":"92","text":"Graphable variant (lower precision and less recursion) (very laggy)"},{"type":"expression","id":"83","folderId":"92","color":"#c74440","latex":"W_{g}\\left(x\\right)=w\\left(x,w\\left(x,w\\left(x,w\\left(x,\\ln x\\right)\\right)\\right)\\right)","hidden":true},{"type":"expression","id":"87","folderId":"92","color":"#6042a6","latex":"G_{0}=\\left(1,100\\right)","hidden":true},{"type":"expression","id":"88","folderId":"92","color":"#000000","latex":"G_{1}=\\left[0,G_{0}.x,...G_{0}.y\\right]"},{"type":"expression","id":"94","folderId":"92","color":"#6042a6","latex":"G_{2}=W\\left(G_{1}\\right)"},{"type":"text","id":"97","folderId":"92","text":"Lower-frequency samples of W with linear interpolation (accelerate graphing while mostly maintaining precision)"},{"type":"expression","id":"95","folderId":"92","color":"#000000","latex":"G_{3}\\left(x\\right)=\\operatorname{floor}\\left(\\frac{x}{G_{0}.x}+1\\right)","hidden":true},{"type":"expression","id":"99","folderId":"92","color":"#388c46","latex":"G_{4}\\left(x\\right)=G_{2}\\left[G_{3}\\left(x\\right)\\right]+\\frac{\\operatorname{mod}\\left(x,G_{0}.x\\right)}{G_{0}.x}\\left(G_{2}\\left[G_{3}\\left(x\\right)+1\\right]-G_{2}\\left[G_{3}\\left(x\\right)\\right]\\right)"},{"type":"text","id":"86","folderId":"92","text":"Even lower precision, but faster graphing"},{"type":"expression","id":"84","folderId":"92","color":"#c74440","latex":"W_{h}\\left(x\\right)=w\\left(x,w\\left(x,\\ln x\\right)\\right)"},{"type":"folder","id":"105","title":"Fast method"},{"type":"expression","id":"106","folderId":"105","color":"#000000","latex":"W_{f}\\left(x\\right)=\\ln\\left(\\frac{x}{\\ln\\left(\\frac{x}{\\ln\\left(\\frac{x}{\\ln\\left(\\frac{x}{\\ln\\left(x\\right)}\\right)}\\right)}\\right)}\\right)"}]}}