Thinking Functionally

I've been reading SICP and watching UC Berkeley's videos on programming paradigms. It uses Scheme and starts with functional programming, something I've been finding very enlightening compared to the C++ stuff I've done (and fun!)

Just now I wrote an Nth root solver:

; find nth power(define (nth x power)  (if (= power 1)      x      (* x (nth x (- power 1))))); find roots(define (rootsolve actual guess prec power count)  (define newguess (* (/ 1 power) (+ (* (- power 1) guess) (/ actual (nth guess (- power 1))))))  (if (> prec (abs (- actual (nth guess power))))      guess      (rootsolve actual newguess prec power (+ count 1))))    ; less precise root(define (root x power)  (rootsolve x (/ x 2.0) 0.001 power 0)); more precise root(define (root-prec x power)  (rootsolve x (/ x 2.0) 0.00000001 power 0))

And (root 25 2) reasonably solves as such:

> (root 25 2)0: 12.51: 7.252: 5.3491379310344823: 5.0113941065325524: 5.0000129530486845.000012953048684>

I just wiki'd an algorithm for nth roots, but I believe its a derivative of newton's method. The problem I'm having with it is for rather large values, say, (root 12844244 21), I'm starting at very poor guesses:

> (root 12844244 21)0: 6422122.01: 6116306.6666666662: 5825053.9682539673: 5547670.44595615854: 5283495.6628153895: 5031900.6312527516: 4792286.315478817: 4564082.2052179148: 4346744.9573503949: 4139757.1022384710: 3942625.81165568611: 3754881.725386367612: 3576077.833701302313: 3405788.413048858814: 3243608.012427484615: 3089150.488026175616: 2942048.08383445317: 2801950.55603281218: 2668524.339078868719: 2541451.751503684620: 2420430.23952731821: 2305171.656692683722: 2195401.577802555623: 2090858.645526243424: 1991293.948120231725: 1896470.42678117326: 1806162.311220164627: 1720154.582114442528: 1638242.459156611729: 1560230.913482487230: 1485934.203316654331: 1415175.431730146832: 1347786.125457282433: 1283605.833768840234: 1222481.746446514335: 1164268.329949061236: 1108826.980903867637: 1056025.696098921638: 1005738.758189449139: 957846.436370903940: 912234.701305622641: 868794.953624402442: 827423.765356573643: 788022.633672927344: 750497.746355168945: 714759.758433494146: 680723.579460470647: 648308.170914733848: 617436.353252127449: 588034.622144883250: 560032.973471317251: 533364.736639349752: 507966.4158469997353: 483777.538901904554: 460740.51323990955: 438800.488799913456: 417905.2274284889657: 398004.978503322858: 379052.3604793550359: 361002.248075576260: 343811.6648338820761: 327439.680794173462: 311847.315042069963: 296997.442897209464: 282854.7075211517565: 269385.435734430266: 256557.557842314567: 244340.5312783947768: 232705.2678841854869: 221624.064651605270: 211070.5377634335371: 201019.559774698672: 191447.1997854272273: 182330.666462311674: 173648.253773630175: 165379.2893082191376: 157504.0850554467877: 150003.890528996978: 142860.848122854279: 136057.950593194580: 129579.0005649471181: 123408.5719666162982: 117531.9733015393383: 111935.2126681326984: 106604.9644458406585: 101528.5375674672886: 96693.845302349887: 92089.3764784283788: 87704.1680746936889: 83527.7791187558890: 79550.2658273865591: 75762.1579308443392: 72154.4361246136593: 68718.5105948701494: 65446.2005665429895: 62329.7148252790396: 59361.63316693240697: 56534.8887304118198: 53842.75117182077599: 51278.8106398293100: 48836.962514123145101: 46511.39287059347102: 44296.56463866044103: 42187.20441777185104: 40178.28992168747105: 38265.038020654734106: 36442.8933530045107: 34707.51747905191108: 33054.778551478004109: 31480.741477598098110: 29981.658550093427111: 28553.960523898502112: 27194.248117998573113: 25899.283921903403114: 24665.98468752705115: 23491.413988120996116: 22372.7752267819117: 21307.404977887523118: 20292.766645607164119: 19326.444424387773120: 18406.13754703597121: 17529.65480670092122: 16694.90933971516123: 15899.913656871578124: 15142.774911306265125: 14421.690391720253126: 13734.943230209763127: 13080.898314485486128: 12457.998394748081129: 11864.760375950553130: 11299.771786619573131: 10761.687415828164132: 10249.226110312537133: 9761.167724107178134: 9296.350213435406135: 8853.66686993848136: 8432.063685655696137: 8030.536843481615138: 7648.130327125347139: 7283.933644881283140: 6937.079661791698141: 6606.742535039712142: 6292.135747656867143: 5992.510235863682144: 5707.152605584459145: 5435.3834338899605146: 5176.555651323772147: 4930.053001260734148: 4695.288572629271149: 4471.703402504067150: 4258.765145241968151: 4055.96680499235152: 3862.8255285641426153: 3678.8814557753735154: 3503.696624547975155: 3336.8539281409285156: 3177.956122038979157: 3026.6248781323607158: 2882.4998839355817159: 2745.2379847005536160: 2614.5123663814798161: 2490.011777506171162: 2371.4397881011146163: 2258.514083905823164: 2150.9657941960218165: 2048.5388516152584166: 1950.9893824907222167: 1858.0851261816401168: 1769.6048820777526169: 1685.337982931193170: 1605.0837932678028171: 1528.6512316836215172: 1455.8583158891634173: 1386.5317294182507174: 1320.5064089697626175: 1257.6251513997738176: 1197.7382394283559177: 1140.7030851698628178: 1086.3838906379644179: 1034.651324417109180: 985.38221373058181: 938.4592511719809182: 893.7707154018866183: 851.2102051446539184: 810.6763858520512185: 772.0727484305249186: 735.3073794576427187: 700.292742340612188: 666.9454688958209189: 635.1861608531627190: 604.9392008125359191: 576.1325722024151192: 548.6976878118238193: 522.5692264874513194: 497.68497760709647195: 473.9856929591395196: 451.4149456753709197: 429.91899588130565198: 409.4466627441006199: 389.9492026134291200: 371.3801929651706201: 353.695421871591202: 336.8527827348486203: 320.8121740331891204: 305.5354038411324205: 290.98609889631655206: 277.1296179964919207: 263.9329695204685208: 251.36473287663665209: 239.3949836920349210: 227.9952225638428211: 217.1383072036598212: 206.7983878130093213: 196.95084553619935214: 187.57223384399938215: 178.64022270857083216: 170.13354543673412217: 162.03194803498488218: 154.3161409856999219: 146.96775331971418220: 139.96928887591824221: 133.30408464373164222: 126.95627108926821223: 120.91073437073163224: 115.15308035307775225: 109.66960033626452226: 104.44723841549227: 99.47356039570477228: 94.73672418638549229: 90.2254516060814230: 85.92900152960134231: 81.83714431390604232: 77.94013744181527233: 74.22870232553835234: 70.69400221479842235: 67.32762115695087236: 64.12154395900082237: 61.068137103810294238: 58.16013057505742239: 55.390600547673735240: 52.75295290254642241: 50.24090752623468242: 47.84848335831874243: 45.569984150779746244: 43.39998490550452245: 41.33331895762335246: 39.365065673927247: 37.49053873707333248: 35.70527498768888249: 34.005023797798934250: 32.3857369502847251: 30.843559000271142252: 29.374818095496323253: 27.976017233806022254: 26.643825936958113255: 25.375072320912487256: 24.16673554372618257: 23.01593861307255258: 21.919941536259568259: 20.876134796437682260: 19.882033139464458261: 18.935269656632816262: 18.03359014917411263: 17.174847761118198264: 16.356997867731618265: 15.578093207363445266: 14.836279245108043267: 14.129789757245755268: 13.456942625948336269: 12.816135834236508270: 12.205843651653817271: 11.624613001575062272: 11.071060001500058273: 10.543866668095294274: 10.041777779138377275: 9.563597884893698276: 9.108188461803536277: 8.674465201717693278: 8.261395430207433279: 7.86799564781688280: 7.493329188397769281: 7.136503988952219282: 6.796670465673989283: 6.473019491131907284: 6.1647804677813465285: 5.871219493222397286: 5.591637612850939287: 5.325369155781161288: 5.0717801501805955289: 4.8302668145184290: 4.6002541218609245291: 4.381194435723326292: 4.1725662193424515293: 3.9738728288645904294: 3.784641423571073295: 3.6044220860537064296: 3.4327874036355905297: 3.269333182232988298: 3.113682076939584299: 2.965494870717574300: 2.8245019073472775301: 2.6905874849393436302: 2.5640114729390593303: 2.4459716140163272304: 2.339907021423322305: 2.2537477575579974306: 2.199930047543687307: 2.1819299313374882308: 2.18028084576654309: 2.1802682525677963310: 2.18026825184036182.1802682518403618>

is there a better function to find a decent place to start checking than dividing the input in half? Or some other fun things to do in Scheme?

p.s. <3 parenthesis h8 college apps kkthx bbai

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