{"version":9,"randomSeed":"6917a499b0f005631feca49ac731398a","graph":{"viewport":{"xmin":1.2534378633547054,"ymin":-1.950283256383137,"xmax":5.145221163553508,"ymax":1.7472278108810924}},"expressions":{"list":[{"type":"folder","id":"19","title":"functions to get roots from points","collapsed":true},{"type":"text","id":"21","folderId":"19","text":"see https://www.desmos.com/calculator/chnfh0nbqt"},{"type":"text","id":"34","folderId":"19","text":"coefficient helpers"},{"type":"expression","id":"22","folderId":"19","color":"#388c46","latex":"R_{k}\\left(A,B,C\\right)=\\frac{\\left(B.x-A.x\\right)\\left(C.y-A.y\\right)-\\left(B.y-A.y\\right)\\left(C.x-A.x\\right)}{\\left(B.x-A.x\\right)\\left(C.x-A.x\\right)\\left(C.x-B.x\\right)}"},{"type":"expression","id":"23","folderId":"19","color":"#6042a6","latex":"R_{m}\\left(A,B,C\\right)=\\frac{\\left(C.y-A.y\\right)-R_{k}\\left(A,B,C\\right)\\left(C.x-A.x\\right)^{2}}{C.x-A.x}"},{"type":"text","id":"36","folderId":"19","text":"coefficients"},{"type":"expression","id":"25","folderId":"19","color":"#c74440","latex":"R_{a}\\left(A,B,C\\right)=R_{k}\\left(A,B,C\\right)"},{"type":"expression","id":"26","folderId":"19","color":"#2d70b3","latex":"R_{b}\\left(A,B,C\\right)=-2R_{k}\\left(A,B,C\\right)A.x+R_{m}\\left(A,B,C\\right)"},{"type":"expression","id":"27","folderId":"19","color":"#388c46","latex":"R_{c}\\left(A,B,C\\right)=R_{k}\\left(A,B,C\\right)A.x^{2}-R_{m}\\left(A,B,C\\right)A.x+A.y"},{"type":"text","id":"38","folderId":"19","text":"discriminant"},{"type":"expression","id":"29","folderId":"19","color":"#000000","latex":"R_{d}\\left(A,B,C\\right)=R_{b}\\left(A,B,C\\right)^{2}-4R_{a}\\left(A,B,C\\right)R_{c}\\left(A,B,C\\right)"},{"type":"text","id":"40","folderId":"19","text":"two roots"},{"type":"expression","id":"28","folderId":"19","color":"#6042a6","latex":"R_{r1}\\left(A,B,C\\right)=\\frac{-R_{b}\\left(A,B,C\\right)-\\sqrt{R_{d}\\left(A,B,C\\right)}}{2R_{a}\\left(A,B,C\\right)}"},{"type":"expression","id":"31","folderId":"19","color":"#2d70b3","latex":"R_{r2}\\left(A,B,C\\right)=\\frac{-R_{b}\\left(A,B,C\\right)+\\sqrt{R_{d}\\left(A,B,C\\right)}}{2R_{a}\\left(A,B,C\\right)}"},{"type":"text","id":"42","folderId":"19","text":"select root between A.x and C.x (if it exists)"},{"type":"expression","id":"44","folderId":"19","color":"#000000","latex":"R_{q}\\left(a,b,c,d\\right)=\\left\\{a\\le b\\le c:b,d\\right\\}"},{"type":"expression","id":"43","folderId":"19","color":"#6042a6","latex":"R_{s}\\left(A,B,C\\right)=R_{q}\\left(A.x,R_{r1}\\left(A,B,C\\right),C.x,R_{q}\\left(A.x,R_{r2}\\left(A,B,C\\right),C.x,\\frac{0}{0}\\right)\\right)"},{"type":"expression","id":"8","color":"#2d70b3","latex":"f\\left(x\\right)=\\sin x"},{"type":"folder","id":"47","title":"step 0","collapsed":true},{"type":"expression","id":"9","folderId":"47","color":"#388c46","latex":"a_{0}=2"},{"type":"expression","id":"10","folderId":"47","color":"#6042a6","latex":"b_{0}=3"},{"type":"expression","id":"14","folderId":"47","color":"#388c46","latex":"c_{0}=4"},{"type":"expression","id":"12","folderId":"47","color":"#c74440","latex":"A_{0}=\\left(a_{0},f\\left(a_{0}\\right)\\right)"},{"type":"expression","id":"13","folderId":"47","color":"#2d70b3","latex":"B_{0}=\\left(b_{0},f\\left(b_{0}\\right)\\right)"},{"type":"expression","id":"15","folderId":"47","color":"#6042a6","latex":"C_{0}=\\left(c_{0},f\\left(c_{0}\\right)\\right)"},{"type":"folder","id":"49","title":"step 1","collapsed":true},{"type":"expression","id":"32","folderId":"49","color":"#388c46","latex":"b_{1}=R_{s}\\left(A_{0},B_{0},C_{0}\\right)"},{"type":"expression","id":"50","folderId":"49","color":"#6042a6","latex":"A_{1}=\\left\\{b_{1}>b_{0}:B_{0},A_{0}\\right\\}","hidden":true},{"type":"expression","id":"45","folderId":"49","color":"#c74440","latex":"B_{1}=\\left(b_{1},f\\left(b_{1}\\right)\\right)"},{"type":"expression","id":"52","folderId":"49","color":"#c74440","latex":"C_{1}=\\left\\{b_{1}>b_{0}:C_{0},B_{0}\\right\\}","hidden":true},{"type":"folder","id":"54","title":"step 2","collapsed":true},{"type":"expression","id":"55","folderId":"54","color":"#388c46","latex":"b_{2}=R_{s}\\left(A_{1},B_{1},C_{1}\\right)"},{"type":"expression","id":"56","folderId":"54","color":"#6042a6","latex":"A_{2}=\\left\\{b_{2}>b_{1}:B_{1},A_{1}\\right\\}"},{"type":"expression","id":"57","folderId":"54","color":"#000000","latex":"B_{2}=\\left(b_{2},f\\left(b_{2}\\right)\\right)"},{"type":"expression","id":"58","folderId":"54","color":"#c74440","latex":"C_{2}=\\left\\{b_{2}>b_{1}:C_{1},B_{1}\\right\\}"}]}}