{"version":9,"randomSeed":"391f502d38a3d6965ed8f2de9e778dfc","graph":{"viewport":{"xmin":-0.10233971539048944,"ymin":-0.006193638440850385,"xmax":1.4755708077447285,"ymax":0.1568223051210369},"squareAxes":false},"expressions":{"list":[{"type":"folder","id":"30","title":"abs error plots"},{"type":"expression","id":"68","folderId":"30","color":"#000000","latex":"x=\\frac{\\pi}{2}"},{"type":"text","id":"55","folderId":"30","text":"slight variant on Taylor series (two terms)"},{"type":"expression","id":"7","folderId":"30","color":"#c74440","latex":"\\left|\\sin\\left(x\\right)-\\left(x-\\frac{x^{3}}{8}\\right)\\right|"},{"type":"text","id":"57","folderId":"30","text":"proper Taylor series (two terms)"},{"type":"expression","id":"12","folderId":"30","color":"#c74440","latex":"\\left|\\sin\\left(x\\right)-\\left(x-\\frac{x^{3}}{6}\\right)\\right|"},{"type":"text","id":"59","folderId":"30","text":"x * sqrt(cos(x)), where cos(x) = two-term Taylor series"},{"type":"expression","id":"14","folderId":"30","color":"#c74440","latex":"\\left|\\sin\\left(x\\right)-\\left(x\\sqrt{1-\\frac{x^{2}}{4}}\\right)\\right|"},{"type":"text","id":"61","folderId":"30","text":"sin(x) * sqrt(1 - (sin x)^2 / 8), where sin(x) = improper two-term Taylor series"},{"type":"expression","id":"16","folderId":"30","color":"#fa7e19","latex":"\\left|\\sin\\left(x\\right)-\\left(x-\\frac{x^{3}}{8}\\right)\\sqrt{1-\\frac{\\left(x-\\frac{x^{3}}{8}\\right)^{2}}{8}}\\right|"},{"type":"text","id":"63","folderId":"30","text":"2 * sin(x/2) * cos(x/2), where 2 * sin(x/2) = augmented improper two-term Taylor series, cos(x / 2) = sqrt(1 - ((1/2) * (2 * sin(x / 2)))^2 )"},{"type":"expression","id":"43","folderId":"30","color":"#fa7e19","latex":"\\left|\\sin\\left(x\\right)-\\left(x-\\frac{x^{3}}{4\\cdot8}\\right)\\sqrt{1-\\left(\\frac{x}{2}-\\frac{x^{3}}{8\\cdot8}\\right)^{2}}\\right|"},{"type":"text","id":"65","folderId":"30","text":"same as above, but with proper Taylor series"},{"type":"expression","id":"41","folderId":"30","color":"#388c46","latex":"\\left|\\sin\\left(x\\right)-\\left(x-\\frac{x^{3}}{4\\cdot6}\\right)\\sqrt{1-\\left(\\frac{x}{2}-\\frac{x^{3}}{8\\cdot6}\\right)^{2}}\\right|"},{"type":"text","id":"67","folderId":"30","text":"2 * sin(x/2) * cos(x/2), where sin(x/2) = (x/2) * cos(x/4), cos(x/2) = sqrt(1 - sin(x / 2)^2), sin(x/4) = x^2 / 16, cos(x/4) = sqrt(1 - sin(x / 4)^2)"},{"type":"expression","id":"49","folderId":"30","color":"#388c46","latex":"\\left|\\sin\\left(x\\right)-\\left(x\\sqrt{1-\\frac{x^{2}}{16}}\\right)\\sqrt{1-\\left(\\frac{x}{2}\\sqrt{1-\\frac{x^{2}}{16}}\\right)^{2}}\\right|"},{"type":"text","id":"70","folderId":"30","text":"(a |-> sin(3 * a))(x / 3), where sin(3 * a) = -4 * a^3 + 3 * a"},{"type":"expression","id":"71","folderId":"30","color":"#fa7e19","latex":"\\left|\\sin\\left(x\\right)-\\left(-\\frac{4}{27}x^{3}+x\\right)\\right|"},{"type":"text","id":"73","folderId":"30","text":"(a |-> sin(9 * a))(x / 9), where sin(9 * a) comes from two iterations of the above formula"},{"type":"expression","id":"74","folderId":"30","color":"#388c46","latex":"\\left|\\sin\\left(x\\right)-\\left(\\frac{256}{387420489}x^{9}-\\frac{64}{531441}x^{7}+\\frac{16}{2187}x^{5}-\\frac{40}{243}x^{3}+x\\right)\\right|"}]}}