ford_gradient2:
  Python version: 3.6.8
  Given mileage x and price y for used Fords,
  seek (m,b) so that y=b+mx approximates the data.
  Use gradient descent to estimate best b and m.

   it      x               f(x)         f'(x)

it: 0 x: [0.5 0. ] f(x) 1.7156939184673647 df(x) [2.40298507 3.9880597 ]
it: 1 x: [ 0.47597015 -0.0398806 ] f(x) 1.5676726347605254 df(x) [-0.0742806   2.03201433]
it: 2 x: [ 0.47671296 -0.06020074] f(x) 1.531666876341145 df(x) [-0.73912445  1.4821004 ]
it: 3 x: [ 0.4841042  -0.07502174] f(x) 1.5048361576684801 df(x) [-0.90896974  1.31662838]
it: 4 x: [ 0.4931939  -0.08818803] f(x) 1.479476858123571 df(x) [-0.94376382  1.25644481]
it: 5 x: [ 0.50263154 -0.10075248] f(x) 1.4549882740066096 df(x) [-0.94184948  1.22527439]
it: 6 x: [ 0.51205003 -0.11300522] f(x) 1.431300775534029 df(x) [-0.93009311  1.20227291]
it: 7 x: [ 0.52135096 -0.12502795] f(x) 1.4083851855451774 df(x) [-0.91583216  1.18174125]
it: 8 x: [ 0.53050928 -0.13684536] f(x) 1.3862161278685092 df(x) [-0.90106836  1.16211839]
it: 9 x: [ 0.53951997 -0.14846655] f(x) 1.3647692657958095 df(x) [-0.88634564  1.14297398]
it: 10 x: [ 0.54838342 -0.15989629] f(x) 1.3440210713130525 df(x) [-0.87180969  1.12418665]
it: 11 x: [ 0.55710152 -0.17113815] f(x) 1.32394878405655 df(x) [-0.85749744  1.10571952]
it: 12 x: [ 0.56567649 -0.18219535] f(x) 1.3045303852081431 df(x) [-0.84341613  1.08755887]
it: 13 x: [ 0.57411066 -0.19307094] f(x) 1.2857445732561952 df(x) [-0.82956496  1.06969735]
it: 14 x: [ 0.58240631 -0.20376791] f(x) 1.2675707406220147 df(x) [-0.81594097  1.05212941]
it: 15 x: [ 0.59056571 -0.2142892 ] f(x) 1.2499889510533717 df(x) [-0.80254064  1.03485005]
it: 16 x: [ 0.59859112 -0.2246377 ] f(x) 1.232979917754877 df(x) [-0.78936036  1.0178545 ]
it: 17 x: [ 0.60648472 -0.23481625] f(x) 1.2165249822308342 df(x) [-0.77639654  1.00113807]
it: 18 x: [ 0.61424869 -0.24482763] f(x) 1.2006060938173246 df(x) [-0.76364563  0.98469618]
it: 19 x: [ 0.62188515 -0.25467459] f(x) 1.1852057898810682 df(x) [-0.75110413  0.96852432]
it: 20 x: [ 0.62939619 -0.26435983] f(x) 1.1703071766633384 df(x) [-0.73876859  0.95261805]
it: 21 x: [ 0.63678387 -0.27388601] f(x) 1.155893910747921 df(x) [-0.72663565  0.93697301]
it: 22 x: [ 0.64405023 -0.28325574] f(x) 1.141950181132784 df(x) [-0.71470197  0.92158492]
it: 23 x: [ 0.65119725 -0.29247159] f(x) 1.1284606918857922 df(x) [-0.70296427  0.90644955]
it: 24 x: [ 0.65822689 -0.30153609] f(x) 1.115410645365444 df(x) [-0.69141935  0.89156275]
it: 25 x: [ 0.66514109 -0.31045172] f(x) 1.1027857259882154 df(x) [-0.68006403  0.87692043]
it: 26 x: [ 0.67194173 -0.31922092] f(x) 1.0905720845247162 df(x) [-0.66889521  0.86251859]
it: 27 x: [ 0.67863068 -0.32784611] f(x) 1.078756322907421 df(x) [-0.65790981  0.84835328]
it: 28 x: [ 0.68520978 -0.33632964] f(x) 1.067325479533315 df(x) [-0.64710483  0.83442061]
it: 29 x: [ 0.69168082 -0.34467385] f(x) 1.0562670150453306 df(x) [-0.63647729  0.82071675]
it: 30 x: [ 0.6980456  -0.35288101] f(x) 1.0455687985769788 df(x) [-0.6260243   0.80723796]
it: 31 x: [ 0.70430584 -0.36095339] f(x) 1.0352190944450834 df(x) [-0.61574298  0.79398053]
it: 32 x: [ 0.71046327 -0.3688932 ] f(x) 1.0252065492760214 df(x) [-0.60563051  0.78094083]
it: 33 x: [ 0.71651958 -0.37670261] f(x) 1.0155201795513493 df(x) [-0.59568412  0.76811528]
it: 34 x: [ 0.72247642 -0.38438376] f(x) 1.0061493595591502 df(x) [-0.58590108  0.75550037]
it: 35 x: [ 0.72833543 -0.39193876] f(x) 0.9970838097378858 df(x) [-0.57627871  0.74309264]
it: 36 x: [ 0.73409821 -0.39936969] f(x) 0.9883135853999687 df(x) [-0.56681437  0.73088868]
it: 37 x: [ 0.73976636 -0.40667858] f(x) 0.9798290658226809 df(x) [-0.55750547  0.71888515]
it: 38 x: [ 0.74534141 -0.41386743] f(x) 0.971620943694476 df(x) [-0.54834944  0.70707875]
it: 39 x: [ 0.75082491 -0.42093822] f(x) 0.9636802149050848 df(x) [-0.53934379  0.69546626]
it: 40 x: [ 0.75621835 -0.42789288] f(x) 0.9559981686682235 df(x) [-0.53048604  0.68404448]
it: 41 x: [ 0.76152321 -0.43473332] f(x) 0.9485663779660766 df(x) [-0.52177376  0.67281028]
it: 42 x: [ 0.76674094 -0.44146143] f(x) 0.9413766903050595 df(x) [-0.51320457  0.66176058]
it: 43 x: [ 0.77187299 -0.44807903] f(x) 0.9344212187727353 df(x) [-0.50477611  0.65089236]
it: 44 x: [ 0.77692075 -0.45458796] f(x) 0.9276923333860602 df(x) [-0.49648607  0.64020262]
it: 45 x: [ 0.78188561 -0.46098998] f(x) 0.9211826527214803 df(x) [-0.48833218  0.62968845]
it: 46 x: [ 0.78676893 -0.46728687] f(x) 0.9148850358176875 df(x) [-0.4803122   0.61934695]
it: 47 x: [ 0.79157205 -0.47348034] f(x) 0.908792574342161 df(x) [-0.47242394  0.60917529]
it: 48 x: [ 0.79629629 -0.47957209] f(x) 0.9028985850128922 df(x) [-0.46466523  0.59917068]
it: 49 x: [ 0.80094295 -0.48556379] f(x) 0.8971966022669888 df(x) [-0.45703394  0.58933038]
it: 50 x: [ 0.80551329 -0.4914571 ] f(x) 0.8916803711681088 df(x) [-0.44952798  0.57965169]
it: 51 x: [ 0.81000857 -0.49725362] f(x) 0.8863438405449473 df(x) [-0.44214529  0.57013196]
it: 52 x: [ 0.81443002 -0.50295494] f(x) 0.8811811563532495 df(x) [-0.43488385  0.56076857]
it: 53 x: [ 0.81877886 -0.50856262] f(x) 0.8761866552540661 df(x) [-0.42774167  0.55155895]
it: 54 x: [ 0.82305627 -0.51407821] f(x) 0.8713548584012079 df(x) [-0.42071678  0.54250059]
it: 55 x: [ 0.82726344 -0.51950322] f(x) 0.8666804654310859 df(x) [-0.41380727  0.533591  ]
it: 56 x: [ 0.83140151 -0.52483913] f(x) 0.8621583486483406 df(x) [-0.40701123  0.52482773]
it: 57 x: [ 0.83547163 -0.5300874 ] f(x) 0.8577835474008856 df(x) [-0.4003268   0.51620838]
it: 58 x: [ 0.83947489 -0.53524949] f(x) 0.8535512626381914 df(x) [-0.39375215  0.50773058]
it: 59 x: [ 0.84341242 -0.54032679] f(x) 0.8494568516468458 df(x) [-0.38728548  0.49939202]
it: 60 x: [ 0.84728527 -0.54532071] f(x) 0.8454958229576087 df(x) [-0.38092502  0.49119041]
it: 61 x: [ 0.85109452 -0.55023262] f(x) 0.841663831418381 df(x) [-0.37466901  0.48312349]
it: 62 x: [ 0.85484121 -0.55506385] f(x) 0.8379566734276764 df(x) [-0.36851575  0.47518906]
it: 63 x: [ 0.85852637 -0.55981574] f(x) 0.8343702823233756 df(x) [-0.36246354  0.46738493]
it: 64 x: [ 0.862151   -0.56448959] f(x) 0.8309007239216942 df(x) [-0.35651073  0.45970898]
it: 65 x: [ 0.86571611 -0.56908668] f(x) 0.8275441922014807 df(x) [-0.35065568  0.45215908]
it: 66 x: [ 0.86922267 -0.57360827] f(x) 0.8242970051291023 df(x) [-0.34489679  0.44473319]
it: 67 x: [ 0.87267164 -0.5780556 ] f(x) 0.8211556006193462 df(x) [-0.33923248  0.43742924]
it: 68 x: [ 0.87606396 -0.5824299 ] f(x) 0.8181165326278955 df(x) [-0.3336612   0.43024526]
it: 69 x: [ 0.87940057 -0.58673235] f(x) 0.8151764673711044 df(x) [-0.32818142  0.42317925]
it: 70 x: [ 0.88268239 -0.59096414] f(x) 0.812332179668919 df(x) [-0.32279163  0.4162293 ]
it: 71 x: [ 0.8859103  -0.59512644] f(x) 0.8095805494069315 df(x) [-0.31749036  0.40939348]
it: 72 x: [ 0.88908521 -0.59922037] f(x) 0.8069185581136922 df(x) [-0.31227615  0.40266993]
it: 73 x: [ 0.89220797 -0.60324707] f(x) 0.8043432856495194 df(x) [-0.30714758  0.3960568 ]
it: 74 x: [ 0.89527944 -0.60720764] f(x) 0.8018519070031762 df(x) [-0.30210323  0.38955228]
it: 75 x: [ 0.89830048 -0.61110316] f(x) 0.7994416891929038 df(x) [-0.29714173  0.38315459]
it: 76 x: [ 0.90127189 -0.61493471] f(x) 0.7971099882684048 df(x) [-0.29226171  0.37686196]
it: 77 x: [ 0.90419451 -0.61870333] f(x) 0.7948542464104957 df(x) [-0.28746184  0.37067268]
it: 78 x: [ 0.90706913 -0.62241005] f(x) 0.7926719891252413 df(x) [-0.2827408   0.36458505]
it: 79 x: [ 0.90989654 -0.6260559 ] f(x) 0.7905608225294941 df(x) [-0.27809729  0.3585974 ]
it: 80 x: [ 0.91267751 -0.62964188] f(x) 0.7885184307248635 df(x) [-0.27353004  0.35270808]
it: 81 x: [ 0.91541281 -0.63316896] f(x) 0.7865425732572318 df(x) [-0.2690378   0.34691549]
it: 82 x: [ 0.91810319 -0.63663811] f(x) 0.7846310826590295 df(x) [-0.26461934  0.34121803]
it: 83 x: [ 0.92074938 -0.64005029] f(x) 0.7827818620715778 df(x) [-0.26027345  0.33561413]
it: 84 x: [ 0.92335212 -0.64340643] f(x) 0.7809928829448838 df(x) [-0.25599892  0.33010228]
it: 85 x: [ 0.92591211 -0.64670746] f(x) 0.7792621828123717 df(x) [-0.2517946   0.32468094]
it: 86 x: [ 0.92843005 -0.64995427] f(x) 0.7775878631381041 df(x) [-0.24765933  0.31934864]
it: 87 x: [ 0.93090664 -0.65314775] f(x) 0.7759680872341352 df(x) [-0.24359197  0.31410392]
it: 88 x: [ 0.93334256 -0.65628879] f(x) 0.7744010782457075 df(x) [-0.23959141  0.30894532]
it: 89 x: [ 0.93573848 -0.65937825] f(x) 0.7728851172020864 df(x) [-0.23565655  0.30387145]
it: 90 x: [ 0.93809504 -0.66241696] f(x) 0.7714185411308905 df(x) [-0.23178632  0.29888091]
it: 91 x: [ 0.94041291 -0.66540577] f(x) 0.769999741233852 df(x) [-0.22797965  0.29397233]
it: 92 x: [ 0.9426927  -0.66834549] f(x) 0.7686271611220031 df(x) [-0.22423549  0.28914437]
it: 93 x: [ 0.94493506 -0.67123694] f(x) 0.7672992951083563 df(x) [-0.22055283  0.28439569]
it: 94 x: [ 0.94714059 -0.67408089] f(x) 0.7660146865562019 df(x) [-0.21693064  0.279725  ]
it: 95 x: [ 0.94930989 -0.67687814] f(x) 0.7647719262812129 df(x) [-0.21336795  0.27513102]
it: 96 x: [ 0.95144357 -0.67962945] f(x) 0.7635696510056044 df(x) [-0.20986376  0.27061249]
it: 97 x: [ 0.95354221 -0.68233558] f(x) 0.7624065418626499 df(x) [-0.20641713  0.26616817]
it: 98 x: [ 0.95560638 -0.68499726] f(x) 0.761281322949919 df(x) [-0.2030271   0.26179684]
it: 99 x: [ 0.95763665 -0.68761523] f(x) 0.7601927599296412 df(x) [-0.19969275  0.2574973 ]
it: 100 x: [ 0.95963358 -0.6901902 ] f(x) 0.7591396586746674 df(x) [-0.19641315  0.25326837]
it: 101 x: [ 0.96159771 -0.69272288] f(x) 0.7581208639585436 df(x) [-0.19318742  0.24910889]
it: 102 x: [ 0.96352959 -0.69521397] f(x) 0.7571352581882537 df(x) [-0.19001466  0.24501772]
it: 103 x: [ 0.96542973 -0.69766415] f(x) 0.7561817601782472 df(x) [-0.18689402  0.24099375]
it: 104 x: [ 0.96729867 -0.70007409] f(x) 0.7552593239644063 df(x) [-0.18382462  0.23703586]
it: 105 x: [ 0.96913692 -0.70244445] f(x) 0.7543669376566473 df(x) [-0.18080563  0.23314297]
it: 106 x: [ 0.97094498 -0.70477588] f(x) 0.7535036223289023 df(x) [-0.17783622  0.22931402]
it: 107 x: [ 0.97272334 -0.70706902] f(x) 0.7526684309452616 df(x) [-0.17491558  0.22554795]
it: 108 x: [ 0.97447249 -0.7093245 ] f(x) 0.7518604473210977 df(x) [-0.17204291  0.22184373]
it: 109 x: [ 0.97619292 -0.71154293] f(x) 0.7510787851180333 df(x) [-0.16921742  0.21820035]
it: 110 x: [ 0.9778851  -0.71372494] f(x) 0.7503225868716491 df(x) [-0.16643832  0.2146168 ]
it: 111 x: [ 0.97954948 -0.71587111] f(x) 0.7495910230508678 df(x) [-0.16370487  0.21109211]
it: 112 x: [ 0.98118653 -0.71798203] f(x) 0.748883291147977 df(x) [-0.16101632  0.2076253 ]
it: 113 x: [ 0.98279669 -0.72005828] f(x) 0.7481986147982984 df(x) [-0.15837191  0.20421543]
it: 114 x: [ 0.98438041 -0.72210043] f(x) 0.7475362429285343 df(x) [-0.15577094  0.20086156]
it: 115 x: [ 0.98593812 -0.72410905] f(x) 0.7468954489328598 df(x) [-0.15321268  0.19756277]
it: 116 x: [ 0.98747025 -0.72608468] f(x) 0.7462755298758519 df(x) [-0.15069644  0.19431815]
it: 117 x: [ 0.98897721 -0.72802786] f(x) 0.7456758057213878 df(x) [-0.14822152  0.19112683]
it: 118 x: [ 0.99045943 -0.72993913] f(x) 0.7450956185866616 df(x) [-0.14578725  0.18798791]
it: 119 x: [ 0.9919173  -0.73181901] f(x) 0.7445343320205025 df(x) [-0.14339296  0.18490055]
it: 120 x: [ 0.99335123 -0.73366801] f(x) 0.7439913303052033 df(x) [-0.14103799  0.18186389]
it: 121 x: [ 0.99476161 -0.73548665] f(x) 0.7434660177810942 df(x) [-0.13872169  0.17887711]
it: 122 x: [ 0.99614883 -0.73727542] f(x) 0.7429578181931167 df(x) [-0.13644344  0.17593937]
it: 123 x: [ 0.99751326 -0.73903482] f(x) 0.7424661740586878 df(x) [-0.1342026   0.17304988]
it: 124 x: [ 0.99885529 -0.74076531] f(x) 0.7419905460561534 df(x) [-0.13199856  0.17020785]
it: 125 x: [ 1.00017527 -0.74246739] f(x) 0.741530412433167 df(x) [-0.12983073  0.16741249]
it: 126 x: [ 1.00147358 -0.74414152] f(x) 0.7410852684343393 df(x) [-0.12769849  0.16466305]
it: 127 x: [ 1.00275056 -0.74578815] f(x) 0.740654625747534 df(x) [-0.12560127  0.16195875]
it: 128 x: [ 1.00400658 -0.74740774] f(x) 0.7402380119682015 df(x) [-0.1235385   0.15929887]
it: 129 x: [ 1.00524196 -0.74900072] f(x) 0.7398349700811615 df(x) [-0.1215096   0.15668267]
it: 130 x: [ 1.00645706 -0.75056755] f(x) 0.7394450579592708 df(x) [-0.11951402  0.15410944]
it: 131 x: [ 1.0076522  -0.75210865] f(x) 0.739067847878418 df(x) [-0.11755122  0.15157847]
it: 132 x: [ 1.00882771 -0.75362443] f(x) 0.7387029260483218 df(x) [-0.11562065  0.14908907]
it: 133 x: [ 1.00998392 -0.75511532] f(x) 0.7383498921586132 df(x) [-0.11372179  0.14664055]
it: 134 x: [ 1.01112113 -0.75658173] f(x) 0.7380083589397048 df(x) [-0.11185412  0.14423224]
it: 135 x: [ 1.01223968 -0.75802405] f(x) 0.737677951737965 df(x) [-0.11001712  0.14186349]
it: 136 x: [ 1.01333985 -0.75944268] f(x) 0.7373583081047321 df(x) [-0.10821028  0.13953364]
it: 137 x: [ 1.01442195 -0.76083802] f(x) 0.7370490773987175 df(x) [-0.10643312  0.13724205]
it: 138 x: [ 1.01548628 -0.76221044] f(x) 0.7367499204013606 df(x) [-0.10468515  0.13498809]
it: 139 x: [ 1.01653313 -0.76356032] f(x) 0.7364605089447136 df(x) [-0.10296589  0.13277116]
it: 140 x: [ 1.01756279 -0.76488803] f(x) 0.7361805255514501 df(x) [-0.10127486  0.13059063]
it: 141 x: [ 1.01857554 -0.76619394] f(x) 0.7359096630865997 df(x) [-0.0996116   0.12844591]
it: 142 x: [ 1.01957166 -0.7674784 ] f(x) 0.7356476244206281 df(x) [-0.09797566  0.12633642]
it: 143 x: [ 1.02055141 -0.76874176] f(x) 0.735394122103495 df(x) [-0.09636658  0.12426157]
it: 144 x: [ 1.02151508 -0.76998438] f(x) 0.7351488780493258 df(x) [-0.09478394  0.1222208 ]
it: 145 x: [ 1.02246292 -0.77120659] f(x) 0.73491162323136 df(x) [-0.09322728  0.12021354]
it: 146 x: [ 1.02339519 -0.77240872] f(x) 0.7346820973868347 df(x) [-0.09169619  0.11823925]
it: 147 x: [ 1.02431215 -0.77359111] f(x) 0.734460048731483 df(x) [-0.09019025  0.11629739]
it: 148 x: [ 1.02521405 -0.77475409] f(x) 0.7342452336833318 df(x) [-0.08870903  0.11438741]
it: 149 x: [ 1.02610115 -0.77589796] f(x) 0.7340374165955008 df(x) [-0.08725215  0.1125088 ]
it: 150 x: [ 1.02697367 -0.77702305] f(x) 0.7338363694977003 df(x) [-0.08581919  0.11066105]
it: 151 x: [ 1.02783186 -0.77812966] f(x) 0.7336418718461554 df(x) [-0.08440976  0.10884364]
it: 152 x: [ 1.02867596 -0.7792181 ] f(x) 0.7334537102816749 df(x) [-0.08302348  0.10705608]
it: 153 x: [ 1.02950619 -0.78028866] f(x) 0.7332716783956001 df(x) [-0.08165997  0.10529788]
it: 154 x: [ 1.03032279 -0.78134164] f(x) 0.7330955765033814 df(x) [-0.08031886  0.10356855]
it: 155 x: [ 1.03112598 -0.78237732] f(x) 0.7329252114255272 df(x) [-0.07899976  0.10186762]
it: 156 x: [ 1.03191598 -0.783396  ] f(x) 0.7327603962756922 df(x) [-0.07770233  0.10019463]
it: 157 x: [ 1.032693   -0.78439794] f(x) 0.7326009502556672 df(x) [-0.07642621  0.09854911]
it: 158 x: [ 1.03345726 -0.78538344] f(x) 0.7324466984570472 df(x) [-0.07517105  0.09693062]
it: 159 x: [ 1.03420897 -0.78635274] f(x) 0.7322974716693624 df(x) [-0.0739365   0.09533871]
it: 160 x: [ 1.03494834 -0.78730613] f(x) 0.7321531061944574 df(x) [-0.07272223  0.09377294]
it: 161 x: [ 1.03567556 -0.78824386] f(x) 0.7320134436669185 df(x) [-0.0715279   0.09223289]
it: 162 x: [ 1.03639084 -0.78916619] f(x) 0.7318783308803513 df(x) [-0.07035318  0.09071813]
it: 163 x: [ 1.03709437 -0.79007337] f(x) 0.7317476196193158 df(x) [-0.06919775  0.08922825]
it: 164 x: [ 1.03778635 -0.79096565] f(x) 0.7316211664967391 df(x) [-0.06806131  0.08776284]
it: 165 x: [ 1.03846696 -0.79184328] f(x) 0.7314988327966226 df(x) [-0.06694352  0.08632149]
it: 166 x: [ 1.0391364  -0.79270649] f(x) 0.7313804843218733 df(x) [-0.0658441   0.08490382]
it: 167 x: [ 1.03979484 -0.79355553] f(x) 0.7312659912470938 df(x) [-0.06476272  0.08350942]
it: 168 x: [ 1.04044247 -0.79439063] f(x) 0.7311552279761672 df(x) [-0.06369911  0.08213793]
it: 169 x: [ 1.04107946 -0.79521201] f(x) 0.7310480730044796 df(x) [-0.06265297  0.08078897]
it: 170 x: [ 1.04170599 -0.7960199 ] f(x) 0.7309444087856346 df(x) [-0.06162401  0.07946215]
it: 171 x: [ 1.04232223 -0.79681452] f(x) 0.7308441216025062 df(x) [-0.06061194  0.07815713]

  Step size ||dx|| less than dxtol =  0.001

  Estimating model parameters:
  x = normalized mileage
  y = normalized price
  Seek relationship y = b + m * x
  b = 1.04232
  m = -0.796815
  Norm of initial error is 1.71569
  Norm of total error is 0.730844

  Graphics saved in "ford_formula.png"

ford_gradient2:
  Normal end of execution.