/* cbrtl.c * * Cube root, long double precision * * * * SYNOPSIS: * * long double x, y, cbrtl(); * * y = cbrtl( x ); * * * * DESCRIPTION: * * Returns the cube root of the argument, which may be negative. * * Range reduction involves determining the power of 2 of * the argument. A polynomial of degree 2 applied to the * mantissa, and multiplication by the cube root of 1, 2, or 4 * approximates the root to within about 0.1%. Then Newton's * iteration is used three times to converge to an accurate * result. * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE .125,8 80000 7.0e-20 2.2e-20 * IEEE exp(+-707) 100000 7.0e-20 2.4e-20 * */ /* Cephes Math Library Release 2.2: January, 1991 Copyright 1984, 1991 by Stephen L. Moshier Direct inquiries to 30 Frost Street, Cambridge, MA 02140 */ #include <math.h> static long double CBRT2 = 1.2599210498948731647672L; static long double CBRT4 = 1.5874010519681994747517L; static long double CBRT2I = 0.79370052598409973737585L; static long double CBRT4I = 0.62996052494743658238361L; #ifdef ANSIPROT extern long double frexpl ( long double, int * ); extern long double ldexpl ( long double, int ); extern int isnanl ( long double ); #else long double frexpl(), ldexpl(); extern int isnanl(); #endif #ifdef INFINITIES extern long double INFINITYL; #endif long double cbrtl(x) long double x; { int e, rem, sign; long double z; #ifdef NANS if(isnanl(x)) return(x); #endif #ifdef INFINITIES if( x == INFINITYL) return(x); if( x == -INFINITYL) return(x); #endif if( x == 0 ) return( x ); if( x > 0 ) sign = 1; else { sign = -1; x = -x; } z = x; /* extract power of 2, leaving * mantissa between 0.5 and 1 */ x = frexpl( x, &e ); /* Approximate cube root of number between .5 and 1, * peak relative error = 1.2e-6 */ x = (((( 1.3584464340920900529734e-1L * x - 6.3986917220457538402318e-1L) * x + 1.2875551670318751538055e0L) * x - 1.4897083391357284957891e0L) * x + 1.3304961236013647092521e0L) * x + 3.7568280825958912391243e-1L; /* exponent divided by 3 */ if( e >= 0 ) { rem = e; e /= 3; rem -= 3*e; if( rem == 1 ) x *= CBRT2; else if( rem == 2 ) x *= CBRT4; } else { /* argument less than 1 */ e = -e; rem = e; e /= 3; rem -= 3*e; if( rem == 1 ) x *= CBRT2I; else if( rem == 2 ) x *= CBRT4I; e = -e; } /* multiply by power of 2 */ x = ldexpl( x, e ); /* Newton iteration */ x -= ( x - (z/(x*x)) )*0.3333333333333333333333L; x -= ( x - (z/(x*x)) )*0.3333333333333333333333L; if( sign < 0 ) x = -x; return(x); }