(50G) full digit nth root of any real number/huge alg. expression, |n| an integer>1 +- HP Forums (https://www.hpmuseum.org/forum) +-- Forum: HP Software Libraries (https://www.hpmuseum.org/forum/forum-10.html) +--- Forum: General Software Library (https://www.hpmuseum.org/forum/forum-13.html) +--- Thread: (50G) full digit nth root of any real number/huge alg. expression, |n| an integer>1 (/thread-23796.html)

(50G) full digit nth root of any real number/huge alg. expression, |n| an integer>1 - Gil - 2025-07-13 To calculate full digit nth root (n beeing an integer, with abs(n)> 1, or a real number, with abs(n) >1, whose fraction part = 0) of any real number or (huge) algebraic expression containing +-*/ operations (but no functions like LN, SIN, EXP, etc.),  launch the following program rootN  placed in the same directory as the other program manyD. rootN "\\<< \"1 Arg:      Version3  {A n d}  or  {A   n }     . A :   real/integer/alg.exp . n th Root:   |n|\\>=2 & FP(n)=0 . d : 22\\<= d igits\\<=352 .If  {A   n }  \\->d=44 dig .X1=X0 - f(X0)/f'(XO) \" DROP DUP SIZE 2 == { 44 + } IFT OBJ\\-> DROP UNROT OVER TYPE UNROT DUP2 DUP 0 ==   IF   THEN 1 +   END 1 CF -3 CF -105 SF SWAP \\->NUM DUP 0 \\>= 1 -1 IFTE R\\->I DUP 8 ROLLD * SWAP INV ^ \\->NUM -105 CF DUP FP 0 == { R\\->I } IFT 0 0 0 0 0 0 0 \\-> s d t A n X l ii A2 ls ds A0 inp   \\<< A 'A0' STO s 'A' STO*     IF n ABS 2 < n FP 0 \\=/ OR     THEN { A0 n d } \"You must have: |n|\\>=2 & FP(n)=0\" DOERR     END A manyD PICK3 EVAL A0 SWAP DUP2 DTAG SWAP ABS SAME     IF     THEN DROP \\->STR     ELSE DUP TYPE 't' STO s * EVAL \\->STR SWAP \\->STR \" =\" + SWAP +     END t 9 ==     IF     THEN \" =\" + A0 \\->NUM +     END \" ^(1/\" + n R\\->I DUP 'n' STO + \"), \" + d 352 MIN 22 MAX 11 / LN 2 LN / \\->NUM CEIL 'd' STO 11 2 d ^ * R\\->I + \"d\" + 'inp' STO \"X\" 1 d     START \"X\" \"((n-1)/n*X+(A)/n/X^(n-1))\" SREPL DROP     NEXT \"X\" X \\->STR SREPL DROP \"A\" A \\->STR SREPL DROP \"'\" \"\" SREPL DROP \"n\" n \\->STR SREPL DROP \"'\" + \"'\" SWAP + OBJ\\-> manyD DEPTH DUP 'ds' STO \\->LIST 'l' STO 1 ds     FOR i l i GET DUP TYPE 12 ==       IF       THEN OBJ\\-> SWAP DROP \"Input\" SAME         IF         THEN i 'ii' STO ds 'i' STO         END       ELSE DROP       END     NEXT l OBJ\\-> ii 1 - - DROPN inp l REVLIST DUP 2 GET SWAP 1 GET DUP2 ==     IF     THEN SWAP DROP s * \"=\" \\->TAG     ELSE \\->STR 1 11 2 d ^ * SUB OBJ\\-> s * SWAP s * SWAP DUP2 SWAP DTAG \\->STR \".\" \"\" SREPL DROP DUP SIZE 1 + PICK3 SIZE       START \"0\" +       NEXT OBJ\\-> SAME       IF       THEN DROP \"=\" \\->TAG       ELSE SWAP \"~\" \\->TAG SWAP       END     END   \\>> \\>>" " manyD \\<< \" 1 Arg   Version 8  \\|>ex '7+3. ! * ( 2*10. ^ 600.  ) /(4-3.27 E-15 )* 4 . ^-7 .' or '73179/17^2/13' or .275 or 3.275 \\|>mix 4 operat.  + - * /  & any real/integ/frac   \\|>& x ^ y:   x real/integer/fract   y integer or FP(y)=0   \\-> '(2./3.)^3.' : ok     '(2./3.)^-3.': ok     '3^( 1/2 )':  not  ok \\|>& x ! : x\\>=0 &   x integer or FP(x)=0 \\|>xE-499xE0 xE499 OK:  | xEy |  <1E500, x real, -499\\<=y_integer\\<=499 \\|>for  | #| \\>= 1E500,   say 2E600,   write  ( 2.*10.^600. )    or  ( 2*10^600 )     always with ()  \\|> If intermed. | result |    \\<=1E-500 \\->impossible:   7 * (2^600) \\-> ok   7  /  (2^600) \\-> error   1.E-450*1.E-99\\-> err \\|>in '': no var-name & no \\v/ SIN LN EXP etc. \\|>local var  n  (min di- gits) by default= 360 . 6 lines below, you can change it. \" DROP STD RAD -3 CF -20 SF -21 CF -105 CF DTAG DUP DUP TYPE 0 0 0 0 0 0 { } { \"E-\" \"E\" } 360 \" default :  360 \" DROP \\-> x num t ls ds p p1 p2 o lx lE n   \\<<     \\<< DUP2 \\->STR SWAP DTAG \\->STR \"E\" POS 1 - OVER SIZE DUP2 \"\" UNROT       START \"0\" +       NEXT 4 ROLLD SUB \\=/       IF       THEN SWAP DTAG \"~\" \\->TAG SWAP       END DEPTH DUP 'ds' STO \\->LIST REVLIST 'ls' STO 2 ds       FOR i ls i GET DUP TYPE 12 ==         IF         THEN OBJ\\-> \"num\" ==           IF           THEN \"\\->wrong!\" \\->TAG i SWAP ls UNROT PUT ds 'i' STO           ELSE DROP           END         ELSE DROP         END       NEXT REVLIST OBJ\\-> DROP     \\>> \\-> msg     \\<< x \"Input\" \\->TAG t 0 \\=/ t 28 \\=/ AND       IF       THEN DUP         IFERR \\->NUM         THEN \"Try, instead input 'a', '1/(a)'. Res='1/Res'\" DOERR         END DUP DUP XPON ABS R\\->I 'n' STO+ 'num' STO \"num\" \\->TAG       END x \\->STR \".!\" \"!\" SREPL DROP \"'\" \"\" SREPL DROP 'x' STO 1 2       FOR i         WHILE x lE i GET POS DUP 'p' STO 0 \\=/         REPEAT 0 'p1' STO 0 'p2' STO p 1 - 1           FOR i x i i SUB 'o' STO             IF o \"+\" SAME o \"-\" SAME OR o \"*\" SAME OR o \"/\" SAME OR o \"^\" SAME OR o \"(\" SAME OR             THEN i 'p1' STO 0 'i' STO             END -1           STEP x 1 p1 SUB \"(\" + x p1 1 + p 1 - SUB + i 1 == \"/\" \"*\" IFTE + p i NEG 4 + + x SIZE           FOR i x i i SUB 'o' STO             IF o \"+\" SAME o \"-\" SAME OR o \"*\" SAME OR o \"/\" SAME OR o \"^\" SAME OR o \")\" SAME OR             THEN i 'p2' STO 0 'i' STO             END           NEXT p2 0 ==           IF           THEN x SIZE 1 + 'p2' STO           END x p i NEG 3 + + p2 1 - SUB OBJ\\-> ALOG + \")\" + x p2 x SIZE SUB + 'x' STO         END       NEXT       WHILE x \".\" POS DUP 'p' STO 0 \\=/       REPEAT 0 'p1' STO 0 'p2' STO 0 'o' STO p 1 - 1         FOR i x i i SUB 'o' STO           IF o \"+\" SAME o \"-\" SAME OR o \"*\" SAME OR o \"/\" SAME OR o \"^\" SAME OR o \"(\" SAME OR           THEN i 'p1' STO 0 'i' STO           END -1         STEP         IF p1 0 \\=/         THEN x 1 p1 SUB 'lx' STO+         END \"(\" x p1 1 + p 1 - SUB + 'lx' STO+ p 1 + x SIZE         FOR i x i i SUB 'o' STO           IF o \"+\" SAME o \"-\" SAME OR o \"*\" SAME OR o \"/\" SAME OR o \"^\" SAME OR o \")\" SAME OR           THEN i 'p2' STO 0 'i' STO           END         NEXT         IF p2 0 \\=/         THEN x p 1 + p2 1 - SUB \"/\" + p2 p - 1 - ALOG R\\->I + \")\" + 'lx' STO+ x p2 DUP SUB 'lx' STO+ x p2 1 + x SIZE SUB 'x' STO         ELSE x p 1 + x SIZE SUB \"/\" + x SIZE p - ALOG R\\->I + \")\" + 'lx' STO+ \"\" 'x' STO         END       END x 'lx' STO+ lx DUP SIZE 0 \\=/       IF       THEN REVLIST 'lx' STO \"\" 1 lx SIZE         FOR i lx i GET +         NEXT \"'\" + \"'\" SWAP + OBJ\\-> DUP2 SAME { DROP } IFT DUP EVAL DUP2 SAME { DROP } IFT EVAL DUP TYPE 9 ==         IF         THEN DUP OBJ\\-> DROP2 ABS SWAP ABS <           IF           THEN DUP PROPFRAC           END t 0 \\=/           IF           THEN DUP n ALOG R\\->I * PROPFRAC PROPFRAC DUP TYPE 9 ==             IF             THEN OBJ\\-> 3 DROPN             END DUP \\->STR DUP DUP SIZE DUP n 1 + - R\\->I ROT 1 1 SUB \".\" + 4 ROLL 4 ROLL 2 SWAP SUB + OBJ\\-> \"E\" + OVER + SWAP ABS 500 <             IF             THEN OBJ\\->             END SWAP DUP2 \\->STR SWAP DUP TYPE 2 \\=/ { \\->STR } IFT DUP \"E\" POS DUP 0 \\=/             IF             THEN 1 - 1 SWAP SUB             ELSE DROP             END \".\" \"\" SREPL DROP SIZE 1 + OVER SIZE DUP2 \"\" UNROT             START \"0\" +             NEXT 4 ROLLD SUB SAME NOT             IF             THEN SWAP \"~\" \\->TAG SWAP             END OVER DUP DTAG TYPE 2 \\=/             IF             THEN num - PICK3 / ABS .0000000001 >               IF               THEN msg EVAL               END             ELSE DROP msg EVAL             END           END         ELSE DUP DUP \\->STR DUP DUP 1 1 SUB \".\" + UNROT SIZE DUP 4 ROLLD 2 SWAP SUB + OBJ\\-> SWAP 1 - R\\->I DUP UNROT \"E\" SWAP + + SWAP ABS 499 \\<= { OBJ\\-> } IFT SWAP OVER DUP DTAG TYPE 2 \\=/           IF           THEN num - PICK3 / ABS .0000000001 >             IF             THEN msg EVAL             END           ELSE TYPE 2 ==             IF             THEN msg EVAL             ELSE DUP ABS LOG num ABS LOG -105 SF \\=/ { msg EVAL } IFT             END           END         END       END     \\>>   \\>> \\>> Example input  Cubic root of 29.1/1.3 with 60 digits accuracy {'29.1/1.3'   3  60}  rootN Example result  "'29.1/1.3' ='291/13' =22.3846153846 ^(1/3), 88d"  :~: 2.818273988 2818273987996238408541866938109913983599721872486686698084448911457926268410135830391437 Note The number relative to the digits accuracy output is fixed between 22 and 352 and is always a multiple of 22. If you don't specify in the input list the third element, ie the requested digit-precision, then the accuracy will be calculated with 44 digits.