Short & Sweet Math Challenge #21: Powers that be

[Gerson W. Barbosa]

HP 50g, 181.5 bytes, 14 seconds. Only 6 constants, though.

Best regards,

Gerson.

%%HP: T(3)A(R)F(.);
\<< 1. 6.
  FOR i i X SWAP ^ [ 1. -1. -1. ] X PEVAL * [ 1. 0. -1. ] X PEVAL + EVAL DUP 'X' ZEROS DUP TYPE
    IF
    THEN OBJ\-> DROP NIP
    END
  NEXT
\>>

TEVAL

'X^3.+0.*X^2.+-1.*X-1.' 
                   1.32471795724 
'X^4.+-1.*X^3.+0.*X^2.-1.' 
                    1.3802775691 
'X^5.+-1.*X^4.+-1.*X^3.+X^2.-1.' 
                   1.44326879127 
'X^6.+-1.*X^5.+-1.*X^4.+X^2.-1.' 
                   1.50159480354 
'X^7.+-1.*X^6.+-1.*X^5.+X^2.-1.' 
                   1.54521564973 
'X^8.+-1.*X^7.+-1.*X^6.+X^2.-1.' 
                   1.57367896839
                       s:14.0446